Measurement invariance in the Vietnam Experience Study

Init

options(
  digits = 3
  )

library(pacman)
p_install(
  pwr
)

## Installing package into '/media/luks8tb2/Projects/VES item spearman/renv/library/R-4.0/x86_64-pc-linux-gnu'
## (as 'lib' is unspecified)

## 
## pwr installed

p_load(
  kirkegaard,
  haven,
  dplyr,
  readr,
  polycor,
  rms,
  mirt,
  lavaan,
  future,
  furrr,
  renv)

#parallel
cl = mirtCluster(7)
plan(multiprocess)

## Warning: [ONE-TIME WARNING] Forked processing ('multicore') is disabled
## in future (>= 1.13.0) when running R from RStudio, because it is
## considered unstable. Because of this, plan("multicore") will fall
## back to plan("sequential"), and plan("multiprocess") will fall back to
## plan("multisession") - not plan("multicore") as in the past. For more details,
## how to control forked processing or not, and how to silence this warning in
## future R sessions, see ?future::supportsMulticore

#check broom version
#https://github.com/harrelfe/rms/issues/93
assertthat::assert_that(version("broom") %>% str_match("[\\d.]+") %>% equals("0.5.6"))

## [1] TRUE

Version control

if (F) {
  #test if already init
  if (!file.exists("renv.lock")) {
    #initiatelize
    renv::init()
  } else {
    #update packages in renv
    renv::snapshot()
  }
}

Ad hoc functions

#compute z score gap using inverse normal
z_to_d = function(x, y) qnorm(x) - qnorm(y)

#rename items with counts
number_in_test = function(x) {
  # browser()
  #unique vals
  uniq_vals = unique(x)
  
  #make vector with count for each
  y = rep(NA, length(x))
  
  #replace each set with 1:n
  for (val in uniq_vals) y[x==val] = 1:(sum(x==val))
    
  y
}

#test
number_in_test(1:3)

## [1] 1 1 1

number_in_test(c(rep("a", 3), rep("b", 3), rep("c", 2), "a"))

## [1] 1 2 3 1 2 3 1 2 4

#redefine describe
describe = function(...) {
 y = psych::describe(...)
 class(y) = "data.frame"
 y
}

#mean of subgroup
mean_of_group = function(x, group) {
  assert_that(length(x) == length(group))
  wtd_mean(x[group])
}

DIF testing function

Since we are doing this a couple of times.

#define a convenience function
DIF_test = function(items, model, group, fscores_pars = list(full.scores = T, full.scores.SE = T), messages = T, method = "EM", technical = list()) {
# browser()
  #regular fit joint group
  if (messages) message("There are 8 steps")
  if (messages) message("Step 1: Initial joint fit\n")
  mirt_fit = mirt(items, model = model, method = method, technical = technical)
  
  #step 3
  if (!is.character(group) && !is.factor(group)) group = factor(group)
  if (messages) message("\nStep 2: Initial MI fit")
  mirt_fit_MI = multipleGroup(items, model = model, group = group, invariance = c('intercepts','slopes', 'free_means', 'free_var'), method = method, technical = technical)
  
  #DIFs
  if (messages) message("\nStep 3: Leave one out MI testing")
  DIFs = DIF(mirt_fit_MI, c('a1', 'd'), scheme = 'drop', method = method, technical = technical)
  DIFs = DIFs %>% rownames_to_column("item")
  DIFs$number = 1:nrow(DIFs)
  
  #adjust p values
  DIFs$p_adj = DIFs$p * nrow(DIFs)
  
  #with significant DIF
  DIFs_detected_liberal = DIFs %>% filter(p < .05)
  DIFs_detected_conservative = DIFs %>% filter(p_adj < .05)
  
  #subset itmes
  items_noDIF_liberal = items %>% select(!!setdiff(DIFs$item, DIFs_detected_liberal$item))
  items_noDIF_conservative = items %>% select(!!setdiff(DIFs$item, DIFs_detected_conservative$item))
  
  #subset models
  #tricky!
  #if its a g only model, we dont have to do anything
  #but if its complex we need name format or Q matrix format
  #extract loadings matrix
  #convert to Q matrix
  # browser()
  mirt_fit_loadings = mirt_fit@Fit$`F`
  model_noDIF_liberal_Q = mirt_fit_loadings %>% apply(MARGIN = 2, as.logical) %>% set_rownames(rownames(mirt_fit_loadings))
  model_noDIF_conservative_Q = model_noDIF_liberal_Q
  
  #set unused items' rows to FALSE
  model_noDIF_liberal_Q[DIFs_detected_liberal$item, ] = F
  model_noDIF_conservative_Q[DIFs_detected_conservative$item, ] = F

  #fit together without DIF
  if (messages) message("\nStep 4: Fit without DIF items, liberal threshold")
  mirt_fit_noDIF_liberal = mirt(items, model = mirt.model(model_noDIF_liberal_Q), method = method, technical = technical)
  if (messages) message("\nStep 5: Fit without DIF items, conservative threshold")
  mirt_fit_noDIF_conservative = mirt(items, model = mirt.model(model_noDIF_conservative_Q), method = method, technical = technical)
  
  #with anchors
  if (messages) message("\nStep 6: Fit with anchor items, liberal threshold")
  mirt_fit_anchors_liberal = multipleGroup(items, model = model, group = group, invariance = c(items_noDIF_liberal %>% names(), 'free_means', 'free_var'), method = method, technical = technical)
  if (messages) message("\nStep 7: Fit with anchor items, conservative threshold")
  mirt_fit_anchors_conservative = multipleGroup(items, model = model, group = group, invariance = c(items_noDIF_conservative %>% names(), 'free_means', 'free_var'), method = method, technical = technical)

  #get scores
  if (messages) message("\nStep 8: Get scores")
  orig_scores = do.call(what = mirt::fscores, args = c(list(object = mirt_fit), fscores_pars))
  noDIF_scores_liberal = do.call(what = mirt::fscores, args = c(list(object = mirt_fit_noDIF_liberal), fscores_pars))
  noDIF_scores_conservative = do.call(what = mirt::fscores, args = c(list(object = mirt_fit_noDIF_conservative), fscores_pars))
  anchor_scores_liberal = do.call(what = mirt::fscores, args = c(list(object = mirt_fit_anchors_liberal), fscores_pars))
  anchor_scores_conservative = do.call(what = mirt::fscores, args = c(list(object = mirt_fit_anchors_conservative), fscores_pars))

  #in a data frame
  scores = list(
    #original scores
    original = orig_scores,
    
    #after DIF removal
    noDIF_liberal = noDIF_scores_liberal,
    noDIF_conservative = noDIF_scores_conservative,
    
    #anchor scores
    anchor_liberal = anchor_scores_liberal,
    anchor_conservative = anchor_scores_conservative
  )
  
  #effect sizes
  #this only works with 1 dimensional models
  #https://groups.google.com/forum/#!topic/mirt-package/hAj7jfdzsxY
  if (ncol(mirt_fit_loadings) == 1) {
    #item level
    effect_size_items = list(
      liberal = empirical_ES(mirt_fit_anchors_liberal, DIF = T, plot = F),
      conservative = empirical_ES(mirt_fit_anchors_conservative, DIF = T, plot = F)
    )
    
    #test level
    effect_size_test = list(
      liberal = empirical_ES(mirt_fit_anchors_liberal, DIF = F, plot = F),
      conservative = empirical_ES(mirt_fit_anchors_conservative, DIF = F, plot = F)
    )
  } else {
    #we fill in NULLS to keep structure
    #item
    effect_size_items = list(
      liberal = NULL,
      conservative = NULL
    )
    
    #test
    effect_size_test = list(
      liberal = NULL,
      conservative = NULL
    )
  }
  
  #out
  list(
    scores = scores,
    fits = list(
      original = mirt_fit,
      noDIF_liberal = mirt_fit_noDIF_liberal,
      noDIF_conservative = mirt_fit_noDIF_conservative,
      anchor_liberal = mirt_fit_anchors_liberal,
      anchor_conservative = mirt_fit_anchors_conservative
    ),
    DIF_stats = DIFs,
    effect_size_items = effect_size_items,
    effect_size_test = effect_size_test
  )
  
}

#test that it works
if (F) {
  LSAT = ltm::LSAT %>% df_legalize_names()
  
  #DIF test it
  #this should show no difference because we split groups at random
  LSAT_diff = DIF_test(LSAT, model = mirt.model("g = 1-5"), group = sample(1:2, size = nrow(LSAT), replace = T))
  
  #results
  LSAT_diff$effect_size_test
  
  #reliability test
  empirical_rxx(LSAT_diff$scores$original)
  marginal_rxx(LSAT_diff$fits$original)
}

VES dataset

Data

#load data files
ves = read_rds("data/VES_dataset.rds")
var_table = read_csv("data/VES_dataset_variables.csv")

## 
## ── Column specification ────────────────────────────────────────────────────────
## cols(
##   number = col_double(),
##   code = col_character(),
##   name = col_character()
## )

ves_items = read_tsv("data/VES_items_binary.tsv")

## 
## ── Column specification ────────────────────────────────────────────────────────
## cols(
##   .default = col_double()
## )
## ℹ Use `spec()` for the full column specifications.

ves_MGCFA = read_csv("data/MGCFA_vesd4.csv")

## Warning: Missing column names filled in: 'X1' [1]

## 
## ── Column specification ────────────────────────────────────────────────────────
## cols(
##   .default = col_double()
## )
## ℹ Use `spec()` for the full column specifications.

#join in g from MGCFA/Lasker
ves = left_join(ves, ves_MGCFA %>% select(g, AOP_ID) %>% rename(g_MGCFA = g))

## Joining, by = "AOP_ID"

Recode

#rename items
item_prefix = names(ves_items) %>% str_match("[^\\d]+") %>% .[, 1]

#nicer names
item_prefix = item_prefix %>% plyr::mapvalues(c("WAIS_R_INF_ITEM_", "WAIS_R_BD_ITEM_", "CV", "RE"),
                                              c("WAIS_Inf", "WAIS_BD", "CVLT", "CFD"))
names(ves_items) = item_prefix + "_" + number_in_test(item_prefix)

#white black only
ves$race2 = case_when(ves$race %in% c("White", "Black") ~ as.character(ves$race),
                    T ~ NA_character_) %>% factor()

#impute zeros
#keep orig just in case
ves_items_no_imp = ves_items
ves_items = ves_items %>% map_df(~plyr::mapvalues(., NA, 0, warn_missing = F))

Descriptives

#descriptives
ves$race %>% table2()

ves$AGE %>% describe() %>% as.matrix()

##    vars    n mean   sd median trimmed  mad min max range skew kurtosis     se
## X1    1 4462 38.3 2.53     38    38.3 2.97  31  49    18 0.14 -0.00517 0.0378

Items

#items without extreme pass rates
pass_rates = ves_items %>% colMeans()

#subset
ves_items = ves_items[pass_rates > .05 & pass_rates < .95]

#item stats
#ne = no extreme items
ves_items_stats = tibble(
  item = colnames(ves_items),
  test = item %>% str_replace("_\\d+", ""),
  pass_rate = pass_rates[item]
)

IRT

#simple g model
irt_fit_1 = mirt::mirt(ves_items, model = 1)

## 
Iteration: 1, Log-Lik: -481404.135, Max-Change: 1.02920
Iteration: 2, Log-Lik: -475583.579, Max-Change: 0.24532
Iteration: 3, Log-Lik: -475064.093, Max-Change: 0.09392
Iteration: 4, Log-Lik: -474931.300, Max-Change: 0.04390
Iteration: 5, Log-Lik: -474887.136, Max-Change: 0.03900
Iteration: 6, Log-Lik: -474860.673, Max-Change: 0.03649
Iteration: 7, Log-Lik: -474840.527, Max-Change: 0.03153
Iteration: 8, Log-Lik: -474824.466, Max-Change: 0.02613
Iteration: 9, Log-Lik: -474811.903, Max-Change: 0.02364
Iteration: 10, Log-Lik: -474801.595, Max-Change: 0.02058
Iteration: 11, Log-Lik: -474793.348, Max-Change: 0.01810
Iteration: 12, Log-Lik: -474786.775, Max-Change: 0.01589
Iteration: 13, Log-Lik: -474781.555, Max-Change: 0.01399
Iteration: 14, Log-Lik: -474777.431, Max-Change: 0.01199
Iteration: 15, Log-Lik: -474774.322, Max-Change: 0.01035
Iteration: 16, Log-Lik: -474763.906, Max-Change: 0.00305
Iteration: 17, Log-Lik: -474763.646, Max-Change: 0.00268
Iteration: 18, Log-Lik: -474763.465, Max-Change: 0.00234
Iteration: 19, Log-Lik: -474763.091, Max-Change: 0.00289
Iteration: 20, Log-Lik: -474763.016, Max-Change: 0.00146
Iteration: 21, Log-Lik: -474762.963, Max-Change: 0.00132
Iteration: 22, Log-Lik: -474762.835, Max-Change: 0.00045
Iteration: 23, Log-Lik: -474762.830, Max-Change: 0.00042
Iteration: 24, Log-Lik: -474762.826, Max-Change: 0.00036
Iteration: 25, Log-Lik: -474762.819, Max-Change: 0.00028
Iteration: 26, Log-Lik: -474762.817, Max-Change: 0.00023
Iteration: 27, Log-Lik: -474762.816, Max-Change: 0.00022
Iteration: 28, Log-Lik: -474762.815, Max-Change: 0.00019
Iteration: 29, Log-Lik: -474762.814, Max-Change: 0.00017
Iteration: 30, Log-Lik: -474762.813, Max-Change: 0.00015
Iteration: 31, Log-Lik: -474762.813, Max-Change: 0.00013
Iteration: 32, Log-Lik: -474762.812, Max-Change: 0.00011
Iteration: 33, Log-Lik: -474762.812, Max-Change: 0.00010
Iteration: 34, Log-Lik: -474762.812, Max-Change: 0.00010
Iteration: 35, Log-Lik: -474762.812, Max-Change: 0.00008

irt_fit_1

## 
## Call:
## mirt::mirt(data = ves_items, model = 1)
## 
## Full-information item factor analysis with 1 factor(s).
## Converged within 1e-04 tolerance after 35 EM iterations.
## mirt version: 1.32.1 
## M-step optimizer: BFGS 
## EM acceleration: Ramsay 
## Number of rectangular quadrature: 61
## Latent density type: Gaussian 
## 
## Log-likelihood = -474763
## Estimated parameters: 384 
## AIC = 950294; AICc = 950366
## BIC = 952753; SABIC = 951532
## G2 (1e+10) = 874534, p = 1
## RMSEA = 0, CFI = NaN, TLI = NaN

irt_fit_1@Fit

## $G2
## [1] 874534
## 
## $p
## [1] 1
## 
## $TLI
## [1] NaN
## 
## $CFI
## [1] NaN
## 
## $RMSEA
## [1] 0
## 
## $df
## [1] 1e+10
## 
## $AIC
## [1] 950294
## 
## $AICc
## [1] 950366
## 
## $BIC
## [1] 952753
## 
## $SABIC
## [1] 951532
## 
## $DIC
## [1] 950294
## 
## $HQ
## [1] 951160
## 
## $logLik
## [1] -474763
## 
## $logPrior
## [1] 0
## 
## $SElogLik
## [1] 0
## 
## $F
##                  F1
## WAIS_Inf_7  0.54286
## WAIS_Inf_8  0.36039
## WAIS_Inf_9  0.65540
## WAIS_Inf_10 0.25401
## WAIS_Inf_11 0.66550
## WAIS_Inf_12 0.59109
## WAIS_Inf_13 0.50127
## WAIS_Inf_14 0.57232
## WAIS_Inf_15 0.50689
## WAIS_Inf_16 0.45829
## WAIS_Inf_17 0.60705
## WAIS_Inf_18 0.46761
## WAIS_Inf_19 0.55864
## WAIS_Inf_20 0.52376
## WAIS_Inf_21 0.39907
## WAIS_Inf_22 0.51821
## WAIS_Inf_23 0.34292
## WAIS_Inf_24 0.59824
## WAIS_Inf_25 0.57308
## WAIS_Inf_26 0.53374
## WAIS_Inf_27 0.48046
## WAIS_Inf_28 0.55987
## WAIS_BD_2   0.49310
## WAIS_BD_3   0.57812
## WAIS_BD_4   0.44171
## WAIS_BD_5   0.53470
## WAIS_BD_6   0.53594
## WAIS_BD_7   0.54340
## CVLT_1      0.25305
## CVLT_2      0.17550
## CVLT_3      0.26126
## CVLT_4      0.26377
## CVLT_5      0.01387
## CVLT_6      0.20756
## CVLT_7      0.03748
## CVLT_8      0.11595
## CVLT_9      0.16817
## CVLT_10     0.08296
## CVLT_11     0.17429
## CVLT_12     0.00475
## CVLT_13     0.09010
## CVLT_14     0.00226
## CVLT_15     0.04303
## CVLT_16     0.10085
## CVLT_17     0.22659
## CVLT_18     0.18158
## CVLT_19     0.30845
## CVLT_20     0.24335
## CVLT_21     0.13575
## CVLT_22     0.21288
## CVLT_23     0.15337
## CVLT_24     0.19216
## CVLT_25     0.16418
## CVLT_26     0.16653
## CVLT_27     0.24293
## CVLT_28     0.13450
## CVLT_29     0.11454
## CVLT_30     0.05120
## CVLT_31     0.08866
## CVLT_32     0.17317
## CVLT_33     0.29271
## CVLT_34     0.21059
## CVLT_35     0.30624
## CVLT_36     0.25349
## CVLT_37     0.17469
## CVLT_38     0.23897
## CVLT_39     0.21765
## CVLT_40     0.21872
## CVLT_41     0.17932
## CVLT_42     0.22478
## CVLT_43     0.25236
## CVLT_44     0.14725
## CVLT_45     0.20865
## CVLT_46     0.07120
## CVLT_47     0.11422
## CVLT_48     0.22791
## CVLT_49     0.30623
## CVLT_50     0.22480
## CVLT_51     0.32705
## CVLT_52     0.28174
## CVLT_53     0.22436
## CVLT_54     0.20295
## CVLT_55     0.26356
## CVLT_56     0.23693
## CVLT_57     0.22723
## CVLT_58     0.26333
## CVLT_59     0.29733
## CVLT_60     0.22375
## CVLT_61     0.20238
## CVLT_62     0.13251
## CVLT_63     0.14808
## CVLT_64     0.25851
## CVLT_65     0.31760
## CVLT_66     0.23779
## CVLT_67     0.36502
## CVLT_68     0.30952
## CVLT_69     0.21251
## CVLT_70     0.19552
## CVLT_71     0.27501
## CVLT_72     0.27496
## CVLT_73     0.27447
## CVLT_74     0.27896
## CVLT_75     0.28340
## CVLT_76     0.25607
## CVLT_77     0.21856
## CVLT_78     0.12624
## CVLT_79     0.16546
## CVLT_80     0.26548
## CVLT_81     0.25255
## CVLT_82     0.27509
## CVLT_83     0.37330
## CVLT_84     0.31864
## CVLT_85     0.20343
## CVLT_86     0.20890
## CVLT_87     0.27712
## CVLT_88     0.28673
## CVLT_89     0.30924
## CVLT_90     0.30499
## CVLT_91     0.29384
## CVLT_92     0.27131
## CVLT_93     0.24051
## CVLT_94     0.19866
## CVLT_95     0.19326
## CVLT_96     0.27613
## CVLT_97     0.29878
## CVLT_98     0.28267
## CVLT_99     0.40807
## CVLT_100    0.31666
## CVLT_101    0.25094
## CVLT_102    0.25214
## CVLT_103    0.28436
## CVLT_104    0.29063
## CVLT_105    0.35088
## CVLT_106    0.30689
## CVLT_107    0.31927
## CVLT_108    0.25626
## CVLT_109    0.27618
## CVLT_110    0.20901
## CVLT_111    0.19595
## CVLT_112    0.32552
## CFD_1       0.37874
## CFD_2       0.47768
## CFD_3       0.43567
## CFD_4       0.41815
## CFD_5       0.35205
## CFD_6       0.40161
## CFD_7       0.49450
## CFD_8       0.48978
## CFD_9       0.32861
## CFD_10      0.48093
## CFD_11      0.37205
## CFD_12      0.47527
## CFD_13      0.41257
## CFD_14      0.43523
## CFD_15      0.39084
## CFD_16      0.31632
## CFD_17      0.41041
## CFD_18      0.29801
## CFD_19      0.44765
## CFD_20      0.53151
## CFD_21      0.57590
## CFD_22      0.58355
## CFD_23      0.62160
## CFD_24      0.51409
## CFD_25      0.63335
## CFD_26      0.66144
## CFD_27      0.54595
## CFD_29      0.61760
## CFD_30      0.59107
## CFD_31      0.45162
## CFD_32      0.45642
## CFD_33      0.21252
## CFD_34      0.37211
## CFD_35      0.47240
## CFD_36      0.36597
## CFD_37      0.43856
## CFD_38      0.49822
## CFD_39      0.55177
## CFD_40      0.56117
## CFD_41      0.59500
## CFD_42      0.52469
## CFD_43      0.64427
## CFD_44      0.66356
## CFD_45      0.55380
## CFD_47      0.59665
## CFD_48      0.62010
## CFD_49      0.41968
## CFD_50      0.44562
## CFD_51      0.23706
## CFD_52      0.31844
## CFD_53      0.48949
## CFD_54      0.41620
## 
## $h2
##  WAIS_Inf_7  WAIS_Inf_8  WAIS_Inf_9 WAIS_Inf_10 WAIS_Inf_11 WAIS_Inf_12 
##    2.95e-01    1.30e-01    4.30e-01    6.45e-02    4.43e-01    3.49e-01 
## WAIS_Inf_13 WAIS_Inf_14 WAIS_Inf_15 WAIS_Inf_16 WAIS_Inf_17 WAIS_Inf_18 
##    2.51e-01    3.28e-01    2.57e-01    2.10e-01    3.69e-01    2.19e-01 
## WAIS_Inf_19 WAIS_Inf_20 WAIS_Inf_21 WAIS_Inf_22 WAIS_Inf_23 WAIS_Inf_24 
##    3.12e-01    2.74e-01    1.59e-01    2.69e-01    1.18e-01    3.58e-01 
## WAIS_Inf_25 WAIS_Inf_26 WAIS_Inf_27 WAIS_Inf_28   WAIS_BD_2   WAIS_BD_3 
##    3.28e-01    2.85e-01    2.31e-01    3.13e-01    2.43e-01    3.34e-01 
##   WAIS_BD_4   WAIS_BD_5   WAIS_BD_6   WAIS_BD_7      CVLT_1      CVLT_2 
##    1.95e-01    2.86e-01    2.87e-01    2.95e-01    6.40e-02    3.08e-02 
##      CVLT_3      CVLT_4      CVLT_5      CVLT_6      CVLT_7      CVLT_8 
##    6.83e-02    6.96e-02    1.92e-04    4.31e-02    1.40e-03    1.34e-02 
##      CVLT_9     CVLT_10     CVLT_11     CVLT_12     CVLT_13     CVLT_14 
##    2.83e-02    6.88e-03    3.04e-02    2.26e-05    8.12e-03    5.09e-06 
##     CVLT_15     CVLT_16     CVLT_17     CVLT_18     CVLT_19     CVLT_20 
##    1.85e-03    1.02e-02    5.13e-02    3.30e-02    9.51e-02    5.92e-02 
##     CVLT_21     CVLT_22     CVLT_23     CVLT_24     CVLT_25     CVLT_26 
##    1.84e-02    4.53e-02    2.35e-02    3.69e-02    2.70e-02    2.77e-02 
##     CVLT_27     CVLT_28     CVLT_29     CVLT_30     CVLT_31     CVLT_32 
##    5.90e-02    1.81e-02    1.31e-02    2.62e-03    7.86e-03    3.00e-02 
##     CVLT_33     CVLT_34     CVLT_35     CVLT_36     CVLT_37     CVLT_38 
##    8.57e-02    4.43e-02    9.38e-02    6.43e-02    3.05e-02    5.71e-02 
##     CVLT_39     CVLT_40     CVLT_41     CVLT_42     CVLT_43     CVLT_44 
##    4.74e-02    4.78e-02    3.22e-02    5.05e-02    6.37e-02    2.17e-02 
##     CVLT_45     CVLT_46     CVLT_47     CVLT_48     CVLT_49     CVLT_50 
##    4.35e-02    5.07e-03    1.30e-02    5.19e-02    9.38e-02    5.05e-02 
##     CVLT_51     CVLT_52     CVLT_53     CVLT_54     CVLT_55     CVLT_56 
##    1.07e-01    7.94e-02    5.03e-02    4.12e-02    6.95e-02    5.61e-02 
##     CVLT_57     CVLT_58     CVLT_59     CVLT_60     CVLT_61     CVLT_62 
##    5.16e-02    6.93e-02    8.84e-02    5.01e-02    4.10e-02    1.76e-02 
##     CVLT_63     CVLT_64     CVLT_65     CVLT_66     CVLT_67     CVLT_68 
##    2.19e-02    6.68e-02    1.01e-01    5.65e-02    1.33e-01    9.58e-02 
##     CVLT_69     CVLT_70     CVLT_71     CVLT_72     CVLT_73     CVLT_74 
##    4.52e-02    3.82e-02    7.56e-02    7.56e-02    7.53e-02    7.78e-02 
##     CVLT_75     CVLT_76     CVLT_77     CVLT_78     CVLT_79     CVLT_80 
##    8.03e-02    6.56e-02    4.78e-02    1.59e-02    2.74e-02    7.05e-02 
##     CVLT_81     CVLT_82     CVLT_83     CVLT_84     CVLT_85     CVLT_86 
##    6.38e-02    7.57e-02    1.39e-01    1.02e-01    4.14e-02    4.36e-02 
##     CVLT_87     CVLT_88     CVLT_89     CVLT_90     CVLT_91     CVLT_92 
##    7.68e-02    8.22e-02    9.56e-02    9.30e-02    8.63e-02    7.36e-02 
##     CVLT_93     CVLT_94     CVLT_95     CVLT_96     CVLT_97     CVLT_98 
##    5.78e-02    3.95e-02    3.74e-02    7.62e-02    8.93e-02    7.99e-02 
##     CVLT_99    CVLT_100    CVLT_101    CVLT_102    CVLT_103    CVLT_104 
##    1.67e-01    1.00e-01    6.30e-02    6.36e-02    8.09e-02    8.45e-02 
##    CVLT_105    CVLT_106    CVLT_107    CVLT_108    CVLT_109    CVLT_110 
##    1.23e-01    9.42e-02    1.02e-01    6.57e-02    7.63e-02    4.37e-02 
##    CVLT_111    CVLT_112       CFD_1       CFD_2       CFD_3       CFD_4 
##    3.84e-02    1.06e-01    1.43e-01    2.28e-01    1.90e-01    1.75e-01 
##       CFD_5       CFD_6       CFD_7       CFD_8       CFD_9      CFD_10 
##    1.24e-01    1.61e-01    2.45e-01    2.40e-01    1.08e-01    2.31e-01 
##      CFD_11      CFD_12      CFD_13      CFD_14      CFD_15      CFD_16 
##    1.38e-01    2.26e-01    1.70e-01    1.89e-01    1.53e-01    1.00e-01 
##      CFD_17      CFD_18      CFD_19      CFD_20      CFD_21      CFD_22 
##    1.68e-01    8.88e-02    2.00e-01    2.83e-01    3.32e-01    3.41e-01 
##      CFD_23      CFD_24      CFD_25      CFD_26      CFD_27      CFD_29 
##    3.86e-01    2.64e-01    4.01e-01    4.37e-01    2.98e-01    3.81e-01 
##      CFD_30      CFD_31      CFD_32      CFD_33      CFD_34      CFD_35 
##    3.49e-01    2.04e-01    2.08e-01    4.52e-02    1.38e-01    2.23e-01 
##      CFD_36      CFD_37      CFD_38      CFD_39      CFD_40      CFD_41 
##    1.34e-01    1.92e-01    2.48e-01    3.04e-01    3.15e-01    3.54e-01 
##      CFD_42      CFD_43      CFD_44      CFD_45      CFD_47      CFD_48 
##    2.75e-01    4.15e-01    4.40e-01    3.07e-01    3.56e-01    3.85e-01 
##      CFD_49      CFD_50      CFD_51      CFD_52      CFD_53      CFD_54 
##    1.76e-01    1.99e-01    5.62e-02    1.01e-01    2.40e-01    1.73e-01

Item stats

#extract and compute additional item stats
ves_items_stats = ves_items_stats %>% mutate(
  item_number = test %>% number_in_test(),
  difficulty = irt_fit_1 %>% coef() %>% map_dbl(~.[2]) %>% .[-193] %>% multiply_by(-1),
  BW_gap = map_dbl(ves_items, ~z_to_d(wtd_mean(.[ves$race == "White"]), wtd_mean(.[ves$race == "Black"]))),
  
  #get loadings manually
  g_loading = irt_fit_1@Fit$`F` %>% as.vector()
)

Main results

#correlations
ves_items_stats %>% .[-c(1:2)] %>% wtd.cors()

##             pass_rate item_number difficulty  BW_gap g_loading
## pass_rate      1.0000      0.0229    -0.9867  0.0196    0.0788
## item_number    0.0229      1.0000     0.0427 -0.3148   -0.2757
## difficulty    -0.9867      0.0427     1.0000 -0.0538   -0.1154
## BW_gap         0.0196     -0.3148    -0.0538  1.0000    0.8045
## g_loading      0.0788     -0.2757    -0.1154  0.8045    1.0000

#scatterplots
GG_scatter(ves_items_stats, "g_loading", "BW_gap", color = "test") +
  scale_color_discrete("Test") + 
  xlab("g-loading") + 
  ylab("black-white gap")

## `geom_smooth()` using formula 'y ~ x'

GG_save("figs/ves/SH_scatter.png")

## `geom_smooth()` using formula 'y ~ x'

#regression models
ves_sh_models = list(
  ols(BW_gap ~ g_loading, data = ves_items_stats),
  ols(BW_gap ~ g_loading * difficulty, data = ves_items_stats),
  ols(BW_gap ~ g_loading * difficulty + test, data = ves_items_stats),
  ols(BW_gap ~ g_loading * difficulty + test + item_number, data = ves_items_stats)
)

#summary table
summarize_models(ves_sh_models)

#full results
ves_sh_models

## [[1]]
## Linear Regression Model
##  
##  ols(formula = BW_gap ~ g_loading, data = ves_items_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     192    LR chi2    200.04    R2       0.647    
##  sigma0.1465    d.f.            1    R2 adj   0.645    
##  d.f.    190    Pr(> chi2) 0.0000    g        0.225    
##  
##  Residuals
##  
##       Min       1Q   Median       3Q      Max 
##  -0.75688 -0.08457 -0.01111  0.09043  0.42075 
##  
##  
##            Coef    S.E.   t     Pr(>|t|)
##  Intercept -0.1210 0.0245 -4.94 <0.0001 
##  g_loading  1.2379 0.0663 18.67 <0.0001 
##  
## 
## [[2]]
## Linear Regression Model
##  
##  ols(formula = BW_gap ~ g_loading * difficulty, data = ves_items_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     192    LR chi2    202.33    R2       0.651    
##  sigma0.1464    d.f.            3    R2 adj   0.646    
##  d.f.    188    Pr(> chi2) 0.0000    g        0.225    
##  
##  Residuals
##  
##       Min       1Q   Median       3Q      Max 
##  -0.73528 -0.07965 -0.01322  0.08953  0.41897 
##  
##  
##                         Coef    S.E.   t     Pr(>|t|)
##  Intercept              -0.1140 0.0249 -4.57 <0.0001 
##  g_loading               1.2317 0.0676 18.22 <0.0001 
##  difficulty              0.0384 0.0266  1.44 0.1504  
##  g_loading * difficulty -0.0742 0.0621 -1.20 0.2332  
##  
## 
## [[3]]
## Linear Regression Model
##  
##  ols(formula = BW_gap ~ g_loading * difficulty + test, data = ves_items_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     192    LR chi2    257.13    R2       0.738    
##  sigma0.1279    d.f.            6    R2 adj   0.729    
##  d.f.    185    Pr(> chi2) 0.0000    g        0.233    
##  
##  Residuals
##  
##        Min        1Q    Median        3Q       Max 
##  -0.711065 -0.073318  0.005434  0.082211  0.345845 
##  
##  
##                         Coef    S.E.   t     Pr(>|t|)
##  Intercept              -0.2524 0.0516 -4.89 <0.0001 
##  g_loading               1.3874 0.1030 13.47 <0.0001 
##  difficulty              0.0358 0.0239  1.50 0.1349  
##  test=CVLT               0.1175 0.0338  3.47 0.0006  
##  test=WAIS_BD            0.4069 0.0559  7.28 <0.0001 
##  test=WAIS_Inf           0.0703 0.0331  2.12 0.0351  
##  g_loading * difficulty -0.0522 0.0559 -0.93 0.3513  
##  
## 
## [[4]]
## Linear Regression Model
##  
##  ols(formula = BW_gap ~ g_loading * difficulty + test + item_number, 
##      data = ves_items_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     192    LR chi2    270.18    R2       0.755    
##  sigma0.1240    d.f.            7    R2 adj   0.746    
##  d.f.    184    Pr(> chi2) 0.0000    g        0.235    
##  
##  Residuals
##  
##        Min        1Q    Median        3Q       Max 
##  -0.705923 -0.077980 -0.001489  0.074368  0.339100 
##  
##  
##                         Coef    S.E.   t     Pr(>|t|)
##  Intercept              -0.2912 0.0512 -5.69 <0.0001 
##  g_loading               1.5503 0.1096 14.15 <0.0001 
##  difficulty              0.0177 0.0237  0.75 0.4551  
##  test=CVLT               0.1978 0.0397  4.99 <0.0001 
##  test=WAIS_BD            0.3725 0.0550  6.77 <0.0001 
##  test=WAIS_Inf           0.0469 0.0328  1.43 0.1541  
##  item_number            -0.0014 0.0004 -3.60 0.0004  
##  g_loading * difficulty -0.0034 0.0558 -0.06 0.9514  
## 

#effect size metrics
#model 4
lm(BW_gap ~ g_loading * difficulty + test + item_number, data = ves_items_stats) %>% car::Anova() %>% sjstats::anova_stats()

## Registered S3 methods overwritten by 'lme4':
##   method                          from
##   cooks.distance.influence.merMod car 
##   influence.merMod                car 
##   dfbeta.influence.merMod         car 
##   dfbetas.influence.merMod        car

#extract model estimates
model_ests = ves_sh_models %>% 
  map(function(x) {
  # browser()
  y = x$coefficients %>% as.data.frame()
  y = rownames_to_column(y)
  names(y) = c("predictor", "beta")
  y
}) %>% ldf_to_df(by_name = "model")

#sum stats for g-loadings
model_ests %>% filter(str_detect(predictor, "g_loading"), !str_detect(predictor, "\\*")) %>% .$beta %>% describe()

#intercepts
model_ests %>% filter(str_detect(predictor, "Intercept")) %>% .$beta %>% describe()

#gap size for perfect item
predict(ves_sh_models[[4]], 
        newdata = data.frame(
          g_loading = 1,
          difficulty = wtd_mean(ves_items_stats$difficulty),
          test = unique(ves_items_stats$test),
          item_number = wtd_mean(ves_items_stats$item_number))) %>% 
  describe()

Test gap size

#1 factor model
irt_fit_1_scores = fscores(irt_fit_1)
ves$irt_1_g = irt_fit_1_scores[, 1] %>% standardize(focal_group = (ves$race=="White"))
describeBy(ves$irt_1_g, ves$race2)

## 
##  Descriptive statistics by group 
## group: Black
##    vars   n  mean   sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 525 -0.95 0.95  -1.01   -0.96 0.92 -4.08 1.95  6.03 0.07    -0.04 0.04
## ------------------------------------------------------------ 
## group: White
##    vars    n mean sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 3654    0  1  -0.02       0 1.01 -3.29 3.62   6.9 0.04    -0.01 0.02

#g from 19 subtests
ves$g = ves$g %>% standardize(focal_group = (ves$race=="White"))
describeBy(ves$g, ves$race2)

## 
##  Descriptive statistics by group 
## group: Black
##    vars   n  mean   sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 525 -1.27 0.86  -1.32   -1.31 0.84 -3.77 1.68  5.45 0.39     0.04 0.04
## ------------------------------------------------------------ 
## group: White
##    vars    n mean sd median trimmed  mad   min  max range  skew kurtosis   se
## X1    1 3654    0  1   0.05    0.03 1.06 -3.16 2.37  5.53 -0.28    -0.44 0.02

#correlations
wtd.cors(ves[c("g", "g_MGCFA", "irt_1_g", "income", "education", "unemployment_3yrs")])

##                        g g_MGCFA irt_1_g income education unemployment_3yrs
## g                  1.000   0.976   0.831  0.397     0.545            -0.219
## g_MGCFA            0.976   1.000   0.741  0.386     0.589            -0.185
## irt_1_g            0.831   0.741   1.000  0.293     0.413            -0.163
## income             0.397   0.386   0.293  1.000     0.349            -0.484
## education          0.545   0.589   0.413  0.349     1.000            -0.152
## unemployment_3yrs -0.219  -0.185  -0.163 -0.484    -0.152             1.000

#black white gap
ves_bw = ves %>% 
  filter(!is.na(race2))

#1 factor model
ves_bw %>% 
  GG_denhist(var = "irt_1_g", group = "race2") +
  scale_fill_discrete("Race")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

GG_save("figs/ves/EFA_item_g.png")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

#gap sizes
SMD_matrix(ves_bw$irt_1_g, ves_bw$race2)

##        Black  White
## Black     NA -0.961
## White -0.961     NA

describeBy(ves_bw$irt_1_g, ves_bw$race2)

## 
##  Descriptive statistics by group 
## group: Black
##    vars   n  mean   sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 525 -0.95 0.95  -1.01   -0.96 0.92 -4.08 1.95  6.03 0.07    -0.04 0.04
## ------------------------------------------------------------ 
## group: White
##    vars    n mean sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 3654    0  1  -0.02       0 1.01 -3.29 3.62   6.9 0.04    -0.01 0.02

#subtest based g factor
ves_bw %>% 
  GG_denhist(var = "g", group = "race2") +
  scale_fill_discrete("Race")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

GG_save("figs/ves/EFA_g.png")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

SMD_matrix(ves_bw$g, ves_bw$race2)

##       Black White
## Black    NA -1.29
## White -1.29    NA

describeBy(ves_bw$g, ves_bw$race2)

## 
##  Descriptive statistics by group 
## group: Black
##    vars   n  mean   sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 525 -1.27 0.86  -1.32   -1.31 0.84 -3.77 1.68  5.45 0.39     0.04 0.04
## ------------------------------------------------------------ 
## group: White
##    vars    n mean sd median trimmed  mad   min  max range  skew kurtosis   se
## X1    1 3654    0  1   0.05    0.03 1.06 -3.16 2.37  5.53 -0.28    -0.44 0.02

DIF

Apply DIF tesing to the items.

All items together

Ignoring the unclear factor structure. Do it step by step, and plot problem items for inspection.

#items black-white only
ves_items_bw = ves_items[ves$race %in% c("Black", "White"), ]

#joint fit for bw
mirt_fit_g = cache_object({
  DIF_test(
  items = ves_items_bw,
  model = 1,
  group = ves_bw$race2
)
}, filename = "cache/mirt_fit_g.rds", renew = F)

## Cache found, reading object from disk

#plot the anchor fits
plot(mirt_fit_g$fits$anchor_liberal)

plot(mirt_fit_g$fits$anchor_conservative)

#effect sizes at test level
mirt_fit_g$effect_size_test

## $liberal
##           Effect Size   Value
## 1                STDS  0.3621
## 2                UTDS  5.1915
## 3              UETSDS  0.5378
## 4               ETSSD  0.0189
## 5         Starks.DTFR  0.3577
## 6               UDTFR  5.1883
## 7              UETSDN  0.5437
## 8 theta.of.max.test.D -2.2993
## 9           Test.Dmax -2.2124
## 
## $conservative
##           Effect Size    Value
## 1                STDS  0.17283
## 2                UTDS  2.64073
## 3              UETSDS  0.39186
## 4               ETSSD  0.00903
## 5         Starks.DTFR  0.17006
## 6               UDTFR  2.63697
## 7              UETSDN  0.39754
## 8 theta.of.max.test.D -2.28515
## 9           Test.Dmax -2.12613

#gap sizes by scoring method
(mirt_fit_g_gaps = describeBy(map_df(mirt_fit_g$scores, ~.[, 1] %>% standardize(focal_group = ves_bw$race2 == "White")), group = ves_bw$race2))

## 
##  Descriptive statistics by group 
## group: Black
##                     vars   n  mean   sd median trimmed  mad   min  max range
## original               1 525 -0.96 0.95  -1.00   -0.96 0.94 -4.11 1.96  6.07
## noDIF_liberal          2 525 -0.85 0.97  -0.89   -0.85 0.98 -4.05 2.03  6.08
## noDIF_conservative     3 525 -0.90 0.97  -0.89   -0.90 0.91 -4.10 2.13  6.23
## anchor_liberal         4 525 -0.95 0.97  -0.95   -0.96 0.99 -4.33 1.96  6.29
## anchor_conservative    5 525 -0.97 0.97  -0.96   -0.97 0.98 -4.29 1.88  6.17
##                      skew kurtosis   se
## original             0.06    -0.03 0.04
## noDIF_liberal        0.00    -0.21 0.04
## noDIF_conservative   0.04    -0.13 0.04
## anchor_liberal      -0.01    -0.06 0.04
## anchor_conservative  0.00    -0.06 0.04
## ------------------------------------------------------------ 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range skew
## original               1 3654    0  1  -0.01       0 1.00 -3.29 3.62  6.91 0.05
## noDIF_liberal          2 3654    0  1  -0.01       0 1.02 -3.57 3.64  7.22 0.05
## noDIF_conservative     3 3654    0  1  -0.01       0 1.01 -3.30 3.55  6.85 0.05
## anchor_liberal         4 3654    0  1  -0.01       0 1.00 -3.28 3.60  6.87 0.05
## anchor_conservative    5 3654    0  1  -0.01       0 1.00 -3.27 3.60  6.87 0.05
##                     kurtosis   se
## original               -0.01 0.02
## noDIF_liberal           0.08 0.02
## noDIF_conservative      0.04 0.02
## anchor_liberal         -0.01 0.02
## anchor_conservative    -0.02 0.02

#number of DIF items
mirt_fit_g$DIF_stats %>% select(p, p_adj) %>% {colSums(. < .05)}

##     p p_adj 
##    69    27

#reliabilities
map_dbl(mirt_fit_g$fits[1:3], marginal_rxx)

##           original      noDIF_liberal noDIF_conservative 
##              0.941              0.909              0.930

map_dbl(mirt_fit_g$scores, empirical_rxx)

##            original       noDIF_liberal  noDIF_conservative      anchor_liberal 
##               0.941               0.910               0.930               0.941 
## anchor_conservative 
##               0.941

Group difference and test length

Removing DIF items shrink group gaps because of item bias, but it will also lead to shrinking gaps by reducing reliability. To estimate this effect, we subset to random sets of items and score the group difference.

set.seed(1)

#pars to loop over
resampling_pars = tibble(
  n_items = sample(5:ncol(ves_items_bw), replace = T, size = 1000)
)

#function
gap_sampler = function(n_items) {
  #items
  i_items = (1:ncol(ves_items_bw)) %>% sample(size = n_items)
    
  #fit
  i_fit = mirt(ves_items_bw[i_items], 1, verbose = F)
  
  #score
  i_scores = mirt::fscores(i_fit) %>% as.vector() %>% standardize(focal_group = (ves_bw$race2 == "White"))
  
  y = tibble(
    n_items = n_items,
    #gap
    d = (i_scores[ves_bw$race2 == "Black"] %>% mean()) * -1
  )
}

#loop
resampled_tests = cache_object({
  future_pmap_dfr(resampling_pars, gap_sampler, .progress = T) %>% 
  mutate(DIF = F)
}, filename = "cache/resampled_tests_ves.rds", renew = F)

## Cache found, reading object from disk

#add our no DIF datapoints
resampled_tests = bind_rows(
  resampled_tests,
  tibble(
    n_items = c(sum(mirt_fit_g$DIF_stats$p > .05),
                sum(mirt_fit_g$DIF_stats$p_adj > .05)),
    d = -c(mirt_fit_g_gaps$Black$mean[2], mirt_fit_g_gaps$Black$mean[3]),
    DIF = T
    )
  )

#plot
resampled_tests %>% 
  filter(DIF == F) %>% 
  ggplot(aes(n_items, d)) +
  # geom_boxplot() +
  geom_point() +
  geom_smooth(se = F) +
  #specific points from analyses
  geom_point(data = resampled_tests %>% filter(DIF == T), color = "red") +
  scale_x_continuous("Number of items") +
  scale_y_continuous("Black-White gap size (d)", breaks = seq(-1, 2, .2)) +
  theme_classic()

## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'

GG_save("figs/ves/test_length_gap.png")

## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'

Subscales separately

This avoids the question of unknown factor structure.

#read from cache if available
#Loop over each scale, and do DIF inside it.
mirt_fit_g_by_test = cache_object(
  {
    #score results
    mirt_fit_g_by_test = list()
    
    #loop
    for (test_i in unique(ves_items_stats$test)) {
      message(test_i)
      test_i_items = ves_items_bw[ves_items_stats %>% filter(test == test_i) %>% pull(item)]
      
      #fit
      mirt_fit_g_by_test[[test_i]] = DIF_test(test_i_items, model = 1, group = (ves_bw$race2 == "White"))
  }
  
  mirt_fit_g_by_test
},
filename = "cache/mirt_fit_g_by_test.rds", renew = F
)

## Cache found, reading object from disk

#effect sizes by subtest
map(mirt_fit_g_by_test, ~.$effect_size_test)

## $WAIS_Inf
## $WAIS_Inf$liberal
##           Effect Size    Value
## 1                STDS -0.01709
## 2                UTDS  1.17270
## 3              UETSDS  0.17281
## 4               ETSSD -0.00382
## 5         Starks.DTFR -0.04901
## 6               UDTFR  1.12764
## 7              UETSDN  0.19411
## 8 theta.of.max.test.D -1.74167
## 9           Test.Dmax -0.93222
## 
## $WAIS_Inf$conservative
##           Effect Size  Value
## 1                STDS -0.170
## 2                UTDS  1.081
## 3              UETSDS  0.181
## 4               ETSSD -0.038
## 5         Starks.DTFR -0.185
## 6               UDTFR  1.047
## 7              UETSDN  0.197
## 8 theta.of.max.test.D -1.752
## 9           Test.Dmax -0.831
## 
## 
## $WAIS_BD
## $WAIS_BD$liberal
##           Effect Size Value
## 1                STDS 0.156
## 2                UTDS 0.174
## 3              UETSDS 0.158
## 4               ETSSD 0.147
## 5         Starks.DTFR 0.132
## 6               UDTFR 0.169
## 7              UETSDN 0.138
## 8 theta.of.max.test.D 1.558
## 9           Test.Dmax 0.196
## 
## $WAIS_BD$conservative
##           Effect Size Value
## 1                STDS 0.156
## 2                UTDS 0.174
## 3              UETSDS 0.158
## 4               ETSSD 0.147
## 5         Starks.DTFR 0.132
## 6               UDTFR 0.169
## 7              UETSDN 0.138
## 8 theta.of.max.test.D 1.558
## 9           Test.Dmax 0.196
## 
## 
## $CVLT
## $CVLT$liberal
##           Effect Size    Value
## 1                STDS  0.04893
## 2                UTDS  2.41124
## 3              UETSDS  0.08162
## 4               ETSSD  0.00435
## 5         Starks.DTFR  0.03954
## 6               UDTFR  2.41237
## 7              UETSDN  0.08334
## 8 theta.of.max.test.D -3.02806
## 9           Test.Dmax -1.11089
## 
## $CVLT$conservative
##           Effect Size    Value
## 1                STDS -0.05957
## 2                UTDS  0.71085
## 3              UETSDS  0.13603
## 4               ETSSD -0.00534
## 5         Starks.DTFR -0.06425
## 6               UDTFR  0.71014
## 7              UETSDN  0.14691
## 8 theta.of.max.test.D -2.96744
## 9           Test.Dmax -0.82089
## 
## 
## $CFD
## $CFD$liberal
##           Effect Size  Value
## 1                STDS -0.372
## 2                UTDS  1.191
## 3              UETSDS  0.409
## 4               ETSSD -0.045
## 5         Starks.DTFR -0.354
## 6               UDTFR  1.166
## 7              UETSDN  0.406
## 8 theta.of.max.test.D -2.588
## 9           Test.Dmax  0.809
## 
## $CFD$conservative
##           Effect Size   Value
## 1                STDS -0.1076
## 2                UTDS  0.7729
## 3              UETSDS  0.1337
## 4               ETSSD -0.0132
## 5         Starks.DTFR -0.0972
## 6               UDTFR  0.7498
## 7              UETSDN  0.1349
## 8 theta.of.max.test.D -2.3460
## 9           Test.Dmax  0.4875

#gap sizes
mirt_fit_g_gaps_by_test = map(mirt_fit_g_by_test, function(x) {
  (describeBy(map_df(x$scores, ~.[, 1] %>% standardize(focal_group = ves_bw$race2 == "White")), group = ves_bw$race2))
})
mirt_fit_g_gaps_by_test

## $WAIS_Inf
## 
##  Descriptive statistics by group 
## group: Black
##                     vars   n  mean   sd median trimmed  mad   min  max range
## original               1 525 -0.77 1.00  -0.73   -0.78 0.93 -2.90 2.05  4.95
## noDIF_liberal          2 525 -0.71 0.94  -0.72   -0.77 0.94 -1.98 1.96  3.94
## noDIF_conservative     3 525 -0.77 0.98  -0.72   -0.82 1.09 -2.47 1.94  4.41
## anchor_liberal         4 525 -0.94 1.13  -0.84   -0.90 1.01 -3.46 1.95  5.41
## anchor_conservative    5 525 -0.96 1.11  -0.85   -0.93 0.98 -3.48 1.90  5.38
##                      skew kurtosis   se
## original             0.09    -0.32 0.04
## noDIF_liberal        0.37    -0.40 0.04
## noDIF_conservative   0.26    -0.45 0.04
## anchor_liberal      -0.24    -0.33 0.05
## anchor_conservative -0.19    -0.32 0.05
## ------------------------------------------------------------ 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 3654    0  1   0.07    0.03 1.01 -3.08 2.05  5.13
## noDIF_liberal          2 3654    0  1   0.06    0.01 1.09 -1.98 1.96  3.94
## noDIF_conservative     3 3654    0  1   0.05    0.03 1.08 -2.47 1.94  4.41
## anchor_liberal         4 3654    0  1   0.07    0.03 1.03 -3.02 2.04  5.06
## anchor_conservative    5 3654    0  1   0.07    0.03 1.03 -3.01 2.04  5.05
##                      skew kurtosis   se
## original            -0.36    -0.02 0.02
## noDIF_liberal       -0.12    -0.65 0.02
## noDIF_conservative  -0.20    -0.50 0.02
## anchor_liberal      -0.33    -0.12 0.02
## anchor_conservative -0.33    -0.11 0.02
## 
## $WAIS_BD
## 
##  Descriptive statistics by group 
## group: Black
##                     vars   n  mean   sd median trimmed  mad   min  max range
## original               1 525 -1.03 1.01  -0.96   -1.06 0.98 -2.68 1.29  3.97
## noDIF_liberal          2 525 -0.88 0.97  -1.28   -0.98 1.04 -1.98 0.99  2.97
## noDIF_conservative     3 525 -0.88 0.97  -1.28   -0.98 1.04 -1.98 0.99  2.97
## anchor_liberal         4 525 -1.52 1.22  -1.42   -1.53 1.24 -3.59 1.10  4.68
## anchor_conservative    5 525 -1.52 1.22  -1.42   -1.53 1.24 -3.59 1.10  4.68
##                     skew kurtosis   se
## original            0.34    -0.42 0.04
## noDIF_liberal       0.79    -0.55 0.04
## noDIF_conservative  0.79    -0.55 0.04
## anchor_liberal      0.19    -0.59 0.05
## anchor_conservative 0.19    -0.59 0.05
## ------------------------------------------------------------ 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 3654    0  1   0.19    0.07 1.25 -2.68 1.29  3.97
## noDIF_liberal          2 3654    0  1  -0.13    0.07 1.65 -1.98 0.99  2.97
## noDIF_conservative     3 3654    0  1  -0.13    0.07 1.65 -1.98 0.99  2.97
## anchor_liberal         4 3654    0  1   0.19    0.07 1.21 -2.62 1.29  3.91
## anchor_conservative    5 3654    0  1   0.19    0.07 1.21 -2.62 1.29  3.91
##                      skew kurtosis   se
## original            -0.42    -0.56 0.02
## noDIF_liberal       -0.39    -1.30 0.02
## noDIF_conservative  -0.39    -1.30 0.02
## anchor_liberal      -0.42    -0.63 0.02
## anchor_conservative -0.42    -0.63 0.02
## 
## $CVLT
## 
##  Descriptive statistics by group 
## group: Black
##                     vars   n  mean   sd median trimmed  mad   min  max range
## original               1 525 -0.52 0.98  -0.60   -0.55 0.96 -2.81 2.52  5.33
## noDIF_liberal          2 525 -0.45 0.99  -0.47   -0.47 1.02 -2.71 2.76  5.47
## noDIF_conservative     3 525 -0.51 1.00  -0.59   -0.55 0.98 -2.76 2.72  5.48
## anchor_liberal         4 525 -0.55 0.98  -0.62   -0.57 0.94 -2.90 2.41  5.30
## anchor_conservative    5 525 -0.59 0.99  -0.65   -0.61 0.97 -2.88 2.41  5.29
##                     skew kurtosis   se
## original            0.33     0.13 0.04
## noDIF_liberal       0.28    -0.16 0.04
## noDIF_conservative  0.34     0.13 0.04
## anchor_liberal      0.27     0.08 0.04
## anchor_conservative 0.29     0.06 0.04
## ------------------------------------------------------------ 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range skew
## original               1 3654    0  1  -0.04   -0.02 0.95 -3.74 3.57  7.31 0.20
## noDIF_liberal          2 3654    0  1  -0.03   -0.01 0.98 -3.51 3.65  7.16 0.10
## noDIF_conservative     3 3654    0  1  -0.04   -0.02 0.96 -3.67 3.65  7.32 0.19
## anchor_liberal         4 3654    0  1  -0.04   -0.02 0.95 -3.70 3.59  7.28 0.20
## anchor_conservative    5 3654    0  1  -0.04   -0.02 0.95 -3.72 3.59  7.30 0.20
##                     kurtosis   se
## original                0.27 0.02
## noDIF_liberal           0.05 0.02
## noDIF_conservative      0.22 0.02
## anchor_liberal          0.27 0.02
## anchor_conservative     0.27 0.02
## 
## $CFD
## 
##  Descriptive statistics by group 
## group: Black
##                     vars   n  mean   sd median trimmed  mad   min  max range
## original               1 525 -0.75 1.06  -0.74   -0.73 1.11 -4.35 2.15  6.50
## noDIF_liberal          2 525 -0.73 1.02  -0.79   -0.73 1.06 -3.82 2.38  6.20
## noDIF_conservative     3 525 -0.73 1.04  -0.75   -0.74 1.05 -4.11 2.31  6.42
## anchor_liberal         4 525 -0.83 1.07  -0.83   -0.82 1.14 -4.58 1.98  6.56
## anchor_conservative    5 525 -0.83 1.09  -0.80   -0.81 1.16 -4.66 2.07  6.73
##                      skew kurtosis   se
## original            -0.11     0.01 0.05
## noDIF_liberal        0.09    -0.10 0.04
## noDIF_conservative   0.03    -0.05 0.05
## anchor_liberal      -0.13    -0.01 0.05
## anchor_conservative -0.13     0.01 0.05
## ------------------------------------------------------------ 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 3654    0  1   0.03    0.02 1.02 -4.35 3.20  7.55
## noDIF_liberal          2 3654    0  1   0.04    0.01 1.01 -3.82 3.03  6.84
## noDIF_conservative     3 3654    0  1   0.03    0.01 1.02 -4.11 3.17  7.28
## anchor_liberal         4 3654    0  1   0.03    0.02 1.02 -4.28 3.17  7.45
## anchor_conservative    5 3654    0  1   0.03    0.02 1.03 -4.28 3.17  7.45
##                      skew kurtosis   se
## original            -0.18     0.08 0.02
## noDIF_liberal       -0.08    -0.02 0.02
## noDIF_conservative  -0.12     0.07 0.02
## anchor_liberal      -0.17     0.06 0.02
## anchor_conservative -0.18     0.06 0.02

#number of DIF items by test
map(mirt_fit_g_by_test, function(x) {
  x$DIF_stats %>% select(p, p_adj) %>% {colSums(. < .05)}
})

## $WAIS_Inf
##     p p_adj 
##    13    11 
## 
## $WAIS_BD
##     p p_adj 
##     3     3 
## 
## $CVLT
##     p p_adj 
##    36     8 
## 
## $CFD
##     p p_adj 
##    18    10

#use non-DIF items from each of the 4 testings
mirt_fit_g_by_test_noDIF_item_names = map(mirt_fit_g_by_test, ~.$DIF_stats %>% filter(p > .05) %>% pull(item)) %>% unlist()
mirt_fit_g_by_test_noDIF_items = ves_items_bw[mirt_fit_g_by_test_noDIF_item_names]
ncol(mirt_fit_g_by_test_noDIF_items)

## [1] 122

#fit again
mirt_fit_g_by_test_noDIF_fit = mirt(mirt_fit_g_by_test_noDIF_items, model = 1)

## 
Iteration: 1, Log-Lik: -288487.777, Max-Change: 0.67220
Iteration: 2, Log-Lik: -285654.251, Max-Change: 0.19860
Iteration: 3, Log-Lik: -285240.201, Max-Change: 0.11396
Iteration: 4, Log-Lik: -285107.161, Max-Change: 0.06622
Iteration: 5, Log-Lik: -285052.347, Max-Change: 0.03952
Iteration: 6, Log-Lik: -285024.707, Max-Change: 0.02631
Iteration: 7, Log-Lik: -285008.378, Max-Change: 0.02370
Iteration: 8, Log-Lik: -284997.680, Max-Change: 0.02180
Iteration: 9, Log-Lik: -284990.546, Max-Change: 0.01897
Iteration: 10, Log-Lik: -284985.756, Max-Change: 0.01605
Iteration: 11, Log-Lik: -284982.542, Max-Change: 0.01342
Iteration: 12, Log-Lik: -284980.470, Max-Change: 0.01072
Iteration: 13, Log-Lik: -284976.465, Max-Change: 0.00247
Iteration: 14, Log-Lik: -284976.377, Max-Change: 0.00220
Iteration: 15, Log-Lik: -284976.322, Max-Change: 0.00180
Iteration: 16, Log-Lik: -284976.232, Max-Change: 0.00029
Iteration: 17, Log-Lik: -284976.230, Max-Change: 0.00023
Iteration: 18, Log-Lik: -284976.229, Max-Change: 0.00020
Iteration: 19, Log-Lik: -284976.227, Max-Change: 0.00014
Iteration: 20, Log-Lik: -284976.226, Max-Change: 0.00013
Iteration: 21, Log-Lik: -284976.226, Max-Change: 0.00012
Iteration: 22, Log-Lik: -284976.226, Max-Change: 0.00010
Iteration: 23, Log-Lik: -284976.226, Max-Change: 0.00009

ves_bw$g_noDIF2 = mirt::fscores(mirt_fit_g_by_test_noDIF_fit) %>% as.vector() %>% standardize(focal_group = (ves_bw$race2 == "White"))

#gap
GG_denhist(ves_bw, "g_noDIF2", "race2") +
  scale_fill_discrete("Race")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

describeBy(ves_bw$g_noDIF2, ves_bw$race2)

## 
##  Descriptive statistics by group 
## group: Black
##    vars   n  mean   sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 525 -0.88 0.96  -0.89   -0.89 0.97 -4.37 2.01  6.38 0.02    -0.05 0.04
## ------------------------------------------------------------ 
## group: White
##    vars    n mean sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 3654    0  1   0.01       0 1.01 -3.29 3.39  6.68 0.06     0.04 0.02

NLSY 79 dataset

NLSY script

#need to switch the wd briefly
setwd("data/NLSY79items/")
source("NLSY79items.R")
setwd("../..")

Recode

#rename object
nlsy = vallabels(new_data)

#fix names
names(nlsy) = varlabels %>% str_legalize() %>% str_uniquify()

#SIRE
nlsy$SIRE = nlsy$RACL_ETHNIC_COHORT_SCRNR_79 %>% 
  mapvalues(
    nlsy$RACL_ETHNIC_COHORT_SCRNR_79 %>% levels(),                                                        c("Hispanic", "Black", "White")
    ) %>% 
  fct_relevel("White", "Hispanic", "Black")

#numbers
nlsy$SIRE %>% table2()

#all data
nlsy_all = nlsy

Item data

#nlsy_items
nlsy_items = nlsy_all %>% 
  select(
    PROFILES_ARITHMETIC_REASONING_ITEM_01_XRND:PROFILES_ARITHMETIC_REASONING_ITEM_30_XRND,
    PROFILES_WORD_KNOWLEDGE_ITEM_01_XRND:PROFILES_WORD_KNOWLEDGE_ITEM_35_XRND,
    PROFILES_PARAGRAPH_COMPREHENSION_ITEM_01_XRND:PROFILES_PARAGRAPH_COMPREHENSION_ITEM_15_XRND,
    PROFILES_MATHEMATICS_KNOWLEDGE_ITEM_01_XRND:PROFILES_MATHEMATICS_KNOWLEDGE_ITEM_25_XRND
    )

#fix item names
names(nlsy_items) = names(nlsy_items) %>% str_replace("PROFILES_", "") %>% str_replace("_XRND", "")

#how many nlsy_items
nlsy_items %>% ncol()

## [1] 105

#count miss again
nlsy_items %>% miss_amount()

## cases with missing data  vars with missing data cells with missing data 
##                   0.440                   1.000                   0.146

#miss counts
items_miss = nlsy_items %>% miss_by_case()

#filter out cases with too many missing item data
#this is to avoid empty responses in the item data
#not doing this causes some errors in some subanalyses, but does not affect results much
items_cases_keep = items_miss < 22
nlsy_items = nlsy_items[items_cases_keep, ]
nlsy = nlsy_all[items_cases_keep, ]

#count miss again
nlsy_items %>% miss_amount()

## cases with missing data  vars with missing data cells with missing data 
##                  0.3400                  1.0000                  0.0201

#recode to integers
nlsy_items %<>% map_df(~as.factor(.) %>% as.integer() %>% subtract(1))

#data subsets for pairwise comparisons
nlsy_items_bw = nlsy_items[nlsy$SIRE %in% c("White", "Black"), ]
nlsy_items_hw = nlsy_items[nlsy$SIRE %in% c("White", "Hispanic"), ]
nlsy_bw = nlsy[nlsy$SIRE %in% c("White", "Black"), ] %>% mutate(SIRE = SIRE %>% fct_drop())
nlsy_hw = nlsy[nlsy$SIRE %in% c("White", "Hispanic"), ] %>% mutate(SIRE = SIRE %>% fct_drop())

IRT

g-only model

#fit simple g model
irt_1g = mirt(nlsy_items, model = 1)

## 
Iteration: 1, Log-Lik: -574897.494, Max-Change: 1.97577
Iteration: 2, Log-Lik: -563437.782, Max-Change: 1.43840
Iteration: 3, Log-Lik: -562691.186, Max-Change: 0.17093
Iteration: 4, Log-Lik: -562338.844, Max-Change: 0.08478
Iteration: 5, Log-Lik: -562171.218, Max-Change: 0.10526
Iteration: 6, Log-Lik: -562038.760, Max-Change: 0.07871
Iteration: 7, Log-Lik: -561924.305, Max-Change: 0.06469
Iteration: 8, Log-Lik: -561805.978, Max-Change: 0.05805
Iteration: 9, Log-Lik: -561711.330, Max-Change: 0.06977
Iteration: 10, Log-Lik: -561660.487, Max-Change: 0.04987
Iteration: 11, Log-Lik: -561615.300, Max-Change: 0.09198
Iteration: 12, Log-Lik: -561570.274, Max-Change: 0.06749
Iteration: 13, Log-Lik: -561534.949, Max-Change: 0.06835
Iteration: 14, Log-Lik: -561507.476, Max-Change: 0.07873
Iteration: 15, Log-Lik: -561482.967, Max-Change: 0.08394
Iteration: 16, Log-Lik: -561461.827, Max-Change: 0.07006
Iteration: 17, Log-Lik: -561442.719, Max-Change: 0.07994
Iteration: 18, Log-Lik: -561426.298, Max-Change: 0.06885
Iteration: 19, Log-Lik: -561411.544, Max-Change: 0.07154
Iteration: 20, Log-Lik: -561398.615, Max-Change: 0.06289
Iteration: 21, Log-Lik: -561386.946, Max-Change: 0.06330
Iteration: 22, Log-Lik: -561376.617, Max-Change: 0.05671
Iteration: 23, Log-Lik: -561367.253, Max-Change: 0.05560
Iteration: 24, Log-Lik: -561358.886, Max-Change: 0.02168
Iteration: 25, Log-Lik: -561347.869, Max-Change: 0.01774
Iteration: 26, Log-Lik: -561338.363, Max-Change: 0.01443
Iteration: 27, Log-Lik: -561330.266, Max-Change: 0.01587
Iteration: 28, Log-Lik: -561323.072, Max-Change: 0.01337
Iteration: 29, Log-Lik: -561316.758, Max-Change: 0.01128
Iteration: 30, Log-Lik: -561311.177, Max-Change: 0.01285
Iteration: 31, Log-Lik: -561306.130, Max-Change: 0.00993
Iteration: 32, Log-Lik: -561301.641, Max-Change: 0.00898
Iteration: 33, Log-Lik: -561297.503, Max-Change: 0.00955
Iteration: 34, Log-Lik: -561282.104, Max-Change: 0.00850
Iteration: 35, Log-Lik: -561279.491, Max-Change: 0.00538
Iteration: 36, Log-Lik: -561277.249, Max-Change: 0.00483
Iteration: 37, Log-Lik: -561265.613, Max-Change: 0.00356
Iteration: 38, Log-Lik: -561264.216, Max-Change: 0.00340
Iteration: 39, Log-Lik: -561262.891, Max-Change: 0.00339
Iteration: 40, Log-Lik: -561255.857, Max-Change: 0.00391
Iteration: 41, Log-Lik: -561254.912, Max-Change: 0.00374
Iteration: 42, Log-Lik: -561254.010, Max-Change: 0.00368
Iteration: 43, Log-Lik: -561249.195, Max-Change: 0.00261
Iteration: 44, Log-Lik: -561248.586, Max-Change: 0.00269
Iteration: 45, Log-Lik: -561248.009, Max-Change: 0.00269
Iteration: 46, Log-Lik: -561244.907, Max-Change: 0.00256
Iteration: 47, Log-Lik: -561244.500, Max-Change: 0.00252
Iteration: 48, Log-Lik: -561244.111, Max-Change: 0.00248
Iteration: 49, Log-Lik: -561242.013, Max-Change: 0.00372
Iteration: 50, Log-Lik: -561241.646, Max-Change: 0.00213
Iteration: 51, Log-Lik: -561241.389, Max-Change: 0.00349
Iteration: 52, Log-Lik: -561240.758, Max-Change: 0.00196
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Iteration: 55, Log-Lik: -561239.858, Max-Change: 0.00165
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Iteration: 127, Log-Lik: -561235.585, Max-Change: 0.00010
Iteration: 128, Log-Lik: -561235.584, Max-Change: 0.00010

irt_1g

## 
## Call:
## mirt(data = nlsy_items, model = 1)
## 
## Full-information item factor analysis with 1 factor(s).
## Converged within 1e-04 tolerance after 128 EM iterations.
## mirt version: 1.32.1 
## M-step optimizer: BFGS 
## EM acceleration: Ramsay 
## Number of rectangular quadrature: 61
## Latent density type: Gaussian 
## 
## Log-likelihood = -561236
## Estimated parameters: 210 
## AIC = 1122891; AICc = 1122900
## BIC = 1124422; SABIC = 1123754

irt_1g %>% summary()

##                                    F1     h2
## ARITHMETIC_REASONING_ITEM_01    0.545 0.2968
## ARITHMETIC_REASONING_ITEM_02    0.676 0.4575
## ARITHMETIC_REASONING_ITEM_03    0.725 0.5249
## ARITHMETIC_REASONING_ITEM_04    0.798 0.6372
## ARITHMETIC_REASONING_ITEM_05    0.626 0.3923
## ARITHMETIC_REASONING_ITEM_06    0.678 0.4597
## ARITHMETIC_REASONING_ITEM_07    0.597 0.3563
## ARITHMETIC_REASONING_ITEM_08    0.662 0.4388
## ARITHMETIC_REASONING_ITEM_09    0.702 0.4929
## ARITHMETIC_REASONING_ITEM_10    0.796 0.6329
## ARITHMETIC_REASONING_ITEM_11    0.817 0.6679
## ARITHMETIC_REASONING_ITEM_12    0.654 0.4278
## ARITHMETIC_REASONING_ITEM_13    0.616 0.3796
## ARITHMETIC_REASONING_ITEM_14    0.691 0.4782
## ARITHMETIC_REASONING_ITEM_15    0.682 0.4645
## ARITHMETIC_REASONING_ITEM_16    0.541 0.2929
## ARITHMETIC_REASONING_ITEM_17    0.453 0.2056
## ARITHMETIC_REASONING_ITEM_18    0.537 0.2882
## ARITHMETIC_REASONING_ITEM_19    0.581 0.3380
## ARITHMETIC_REASONING_ITEM_20    0.607 0.3680
## ARITHMETIC_REASONING_ITEM_21    0.499 0.2488
## ARITHMETIC_REASONING_ITEM_22    0.579 0.3358
## ARITHMETIC_REASONING_ITEM_23    0.495 0.2451
## ARITHMETIC_REASONING_ITEM_24    0.556 0.3087
## ARITHMETIC_REASONING_ITEM_25    0.623 0.3878
## ARITHMETIC_REASONING_ITEM_26    0.558 0.3109
## ARITHMETIC_REASONING_ITEM_27    0.669 0.4482
## ARITHMETIC_REASONING_ITEM_28    0.519 0.2695
## ARITHMETIC_REASONING_ITEM_29    0.500 0.2505
## ARITHMETIC_REASONING_ITEM_30    0.523 0.2732
## WORD_KNOWLEDGE_ITEM_01          0.875 0.7650
## WORD_KNOWLEDGE_ITEM_02          0.852 0.7256
## WORD_KNOWLEDGE_ITEM_03          0.863 0.7444
## WORD_KNOWLEDGE_ITEM_04          0.671 0.4505
## WORD_KNOWLEDGE_ITEM_05          0.793 0.6290
## WORD_KNOWLEDGE_ITEM_06          0.876 0.7669
## WORD_KNOWLEDGE_ITEM_07          0.720 0.5190
## WORD_KNOWLEDGE_ITEM_08          0.847 0.7169
## WORD_KNOWLEDGE_ITEM_09          0.649 0.4209
## WORD_KNOWLEDGE_ITEM_10          0.886 0.7855
## WORD_KNOWLEDGE_ITEM_11          0.820 0.6717
## WORD_KNOWLEDGE_ITEM_12          0.734 0.5383
## WORD_KNOWLEDGE_ITEM_13          0.837 0.7013
## WORD_KNOWLEDGE_ITEM_14          0.720 0.5189
## WORD_KNOWLEDGE_ITEM_15          0.849 0.7204
## WORD_KNOWLEDGE_ITEM_16          0.737 0.5428
## WORD_KNOWLEDGE_ITEM_17          0.803 0.6444
## WORD_KNOWLEDGE_ITEM_18          0.738 0.5452
## WORD_KNOWLEDGE_ITEM_19          0.652 0.4252
## WORD_KNOWLEDGE_ITEM_20          0.800 0.6395
## WORD_KNOWLEDGE_ITEM_21          0.640 0.4100
## WORD_KNOWLEDGE_ITEM_22          0.725 0.5257
## WORD_KNOWLEDGE_ITEM_23          0.688 0.4736
## WORD_KNOWLEDGE_ITEM_24          0.477 0.2279
## WORD_KNOWLEDGE_ITEM_25          0.549 0.3013
## WORD_KNOWLEDGE_ITEM_26          0.710 0.5035
## WORD_KNOWLEDGE_ITEM_27          0.536 0.2876
## WORD_KNOWLEDGE_ITEM_28          0.597 0.3563
## WORD_KNOWLEDGE_ITEM_29          0.398 0.1582
## WORD_KNOWLEDGE_ITEM_30          0.456 0.2082
## WORD_KNOWLEDGE_ITEM_31          0.835 0.6977
## WORD_KNOWLEDGE_ITEM_32          0.408 0.1665
## WORD_KNOWLEDGE_ITEM_33          0.422 0.1777
## WORD_KNOWLEDGE_ITEM_34          0.603 0.3639
## WORD_KNOWLEDGE_ITEM_35          0.699 0.4893
## PARAGRAPH_COMPREHENSION_ITEM_01 0.725 0.5253
## PARAGRAPH_COMPREHENSION_ITEM_02 0.764 0.5841
## PARAGRAPH_COMPREHENSION_ITEM_03 0.874 0.7644
## PARAGRAPH_COMPREHENSION_ITEM_04 0.551 0.3034
## PARAGRAPH_COMPREHENSION_ITEM_05 0.739 0.5466
## PARAGRAPH_COMPREHENSION_ITEM_06 0.578 0.3337
## PARAGRAPH_COMPREHENSION_ITEM_07 0.718 0.5148
## PARAGRAPH_COMPREHENSION_ITEM_08 0.458 0.2094
## PARAGRAPH_COMPREHENSION_ITEM_09 0.680 0.4618
## PARAGRAPH_COMPREHENSION_ITEM_10 0.601 0.3612
## PARAGRAPH_COMPREHENSION_ITEM_11 0.457 0.2089
## PARAGRAPH_COMPREHENSION_ITEM_12 0.556 0.3086
## PARAGRAPH_COMPREHENSION_ITEM_13 0.754 0.5692
## PARAGRAPH_COMPREHENSION_ITEM_14 0.592 0.3508
## PARAGRAPH_COMPREHENSION_ITEM_15 0.111 0.0123
## MATHEMATICS_KNOWLEDGE_ITEM_01   0.822 0.6752
## MATHEMATICS_KNOWLEDGE_ITEM_02   0.576 0.3318
## MATHEMATICS_KNOWLEDGE_ITEM_03   0.752 0.5659
## MATHEMATICS_KNOWLEDGE_ITEM_04   0.575 0.3303
## MATHEMATICS_KNOWLEDGE_ITEM_05   0.597 0.3563
## MATHEMATICS_KNOWLEDGE_ITEM_06   0.669 0.4477
## MATHEMATICS_KNOWLEDGE_ITEM_07   0.619 0.3832
## MATHEMATICS_KNOWLEDGE_ITEM_08   0.620 0.3847
## MATHEMATICS_KNOWLEDGE_ITEM_09   0.625 0.3909
## MATHEMATICS_KNOWLEDGE_ITEM_10   0.587 0.3449
## MATHEMATICS_KNOWLEDGE_ITEM_11   0.511 0.2616
## MATHEMATICS_KNOWLEDGE_ITEM_12   0.663 0.4390
## MATHEMATICS_KNOWLEDGE_ITEM_13   0.664 0.4407
## MATHEMATICS_KNOWLEDGE_ITEM_14   0.724 0.5249
## MATHEMATICS_KNOWLEDGE_ITEM_15   0.260 0.0674
## MATHEMATICS_KNOWLEDGE_ITEM_16   0.576 0.3316
## MATHEMATICS_KNOWLEDGE_ITEM_17   0.666 0.4438
## MATHEMATICS_KNOWLEDGE_ITEM_18   0.587 0.3449
## MATHEMATICS_KNOWLEDGE_ITEM_19   0.492 0.2420
## MATHEMATICS_KNOWLEDGE_ITEM_20   0.460 0.2121
## MATHEMATICS_KNOWLEDGE_ITEM_21   0.410 0.1681
## MATHEMATICS_KNOWLEDGE_ITEM_22   0.484 0.2344
## MATHEMATICS_KNOWLEDGE_ITEM_23   0.579 0.3352
## MATHEMATICS_KNOWLEDGE_ITEM_24   0.534 0.2849
## MATHEMATICS_KNOWLEDGE_ITEM_25   0.547 0.2993
## 
## SS loadings:  44.4 
## Proportion Var:  0.423 
## 
## Factor correlations: 
## 
##    F1
## F1  1

plot(irt_1g)

#scores for fit
irt_1g_scores = mirt::fscores(irt_1g, full.scores = T, full.scores.SE = T)
nlsy$g = irt_1g_scores[, 1] %>% standardize(focal_group = (nlsy$SIRE == "White"))

#plot gap
GG_denhist(nlsy, "g", "SIRE") +
  scale_fill_discrete("Race")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

GG_save("figs/nlsy/EFA_item_g.png")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

describeBy(nlsy$g, nlsy$SIRE)

## 
##  Descriptive statistics by group 
## group: White
##    vars    n mean sd median trimmed mad   min  max range skew kurtosis   se
## X1    1 6352    0  1  -0.12   -0.05   1 -2.16 2.74   4.9 0.44    -0.33 0.01
## ------------------------------------------------------------ 
## group: Hispanic
##    vars    n  mean   sd median trimmed  mad  min  max range skew kurtosis   se
## X1    1 1682 -0.72 0.83  -0.86   -0.81 0.73 -2.2 2.74  4.95 1.12     1.25 0.02
## ------------------------------------------------------------ 
## group: Black
##    vars    n  mean   sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 2774 -0.97 0.71   -1.1   -1.05 0.63 -2.43 2.74  5.17 1.21     1.88 0.01

Jensen’s method

Item stats

#extract and compute additional item stats
nlsy_item_stats = tibble(
  item = names(nlsy_items),
  test = str_match(item, "(\\w+)_ITEM")[, 2] %>% str_to_sentence() %>% str_clean(),
  item_number = test %>% number_in_test(),
  pass_rate = nlsy_items %>% map_dbl(wtd_mean),
  pass_rate_white = nlsy_items[nlsy$SIRE == "White", ] %>% map_dbl(wtd_mean),
  pass_rate_black = nlsy_items[nlsy$SIRE == "Black", ] %>% map_dbl(wtd_mean),
  pass_rate_hispanic = nlsy_items[nlsy$SIRE == "Hispanic", ] %>% map_dbl(wtd_mean),
  difficulty = irt_1g %>% coef() %>% map_dbl(~.[2]) %>% {.[-length(.)]} %>% multiply_by(-1),
  BW_gap = map_dbl(nlsy_items, ~z_to_d(wtd_mean(.[nlsy$SIRE == "White"]), wtd_mean(.[nlsy$SIRE == "Black"]))),
  HW_gap = map_dbl(nlsy_items, ~z_to_d(wtd_mean(.[nlsy$SIRE == "White"]), wtd_mean(.[nlsy$SIRE == "Hispanic"]))),
  
  #get loadings manually
  g_loading = irt_1g@Fit$`F` %>% as.vector()
)

Results

#correlations
nlsy_item_stats %>% .[-c(1:2)] %>% wtd.cors()

##                    item_number pass_rate pass_rate_white pass_rate_black
## item_number              1.000    -0.639          -0.639         -0.6247
## pass_rate               -0.639     1.000           0.987          0.9633
## pass_rate_white         -0.639     0.987           1.000          0.9094
## pass_rate_black         -0.625     0.963           0.909          1.0000
## pass_rate_hispanic      -0.614     0.973           0.931          0.9727
## difficulty               0.619    -0.961          -0.945         -0.9294
## BW_gap                  -0.192     0.311           0.447          0.0648
## HW_gap                  -0.253     0.337           0.465          0.1481
## g_loading               -0.419     0.604           0.679          0.4434
##                    pass_rate_hispanic difficulty  BW_gap HW_gap g_loading
## item_number                    -0.614      0.619 -0.1918 -0.253    -0.419
## pass_rate                       0.973     -0.961  0.3113  0.337     0.604
## pass_rate_white                 0.931     -0.945  0.4470  0.465     0.679
## pass_rate_black                 0.973     -0.929  0.0648  0.148     0.443
## pass_rate_hispanic              1.000     -0.945  0.1733  0.129     0.511
## difficulty                     -0.945      1.000 -0.3624 -0.356    -0.691
## BW_gap                          0.173     -0.362  1.0000  0.836     0.785
## HW_gap                          0.129     -0.356  0.8361  1.000     0.689
## g_loading                       0.511     -0.691  0.7850  0.689     1.000

#scatterplots
GG_scatter(nlsy_item_stats, "g_loading", "BW_gap", color = "test") +
  scale_color_discrete("Test") + 
  xlab("g-loading") + 
  ylab("Black-White gap")

## `geom_smooth()` using formula 'y ~ x'

GG_save("figs/nlsy/BW_item_scatter.png")

## `geom_smooth()` using formula 'y ~ x'

GG_scatter(nlsy_item_stats, "g_loading", "HW_gap", color = "test") +
  scale_color_discrete("Test") + 
  xlab("g-loading") + 
  ylab("Hispanic-White gap")

## `geom_smooth()` using formula 'y ~ x'

GG_save("figs/nlsy/HW_item_scatter.png")

## `geom_smooth()` using formula 'y ~ x'

#regression models for blacks
g_models = list(
  ols(BW_gap ~ g_loading, data = nlsy_item_stats),
  ols(BW_gap ~ g_loading + difficulty, data = nlsy_item_stats),
  ols(BW_gap ~ g_loading + difficulty + test, data = nlsy_item_stats),
  ols(BW_gap ~ g_loading + difficulty + test + item_number, data = nlsy_item_stats)
)

#summary output
g_models %>% summarize_models()

#full output
g_models

## [[1]]
## Linear Regression Model
##  
##  ols(formula = BW_gap ~ g_loading, data = nlsy_item_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     105    LR chi2    100.58    R2       0.616    
##  sigma0.1190    d.f.            1    R2 adj   0.613    
##  d.f.    103    Pr(> chi2) 0.0000    g        0.168    
##  
##  Residuals
##  
##        Min        1Q    Median        3Q       Max 
##  -0.271008 -0.080651 -0.002254  0.095168  0.317354 
##  
##  
##            Coef    S.E.   t     Pr(>|t|)
##  Intercept -0.0755 0.0549 -1.37 0.1724  
##  g_loading  1.0865 0.0845 12.86 <0.0001 
##  
## 
## [[2]]
## Linear Regression Model
##  
##  ols(formula = BW_gap ~ g_loading + difficulty, data = nlsy_item_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     105    LR chi2    118.98    R2       0.678    
##  sigma0.1096    d.f.            2    R2 adj   0.672    
##  d.f.    102    Pr(> chi2) 0.0000    g        0.171    
##  
##  Residuals
##  
##        Min        1Q    Median        3Q       Max 
##  -0.235825 -0.069087  0.007028  0.080069  0.280716 
##  
##  
##             Coef    S.E.   t     Pr(>|t|)
##  Intercept  -0.2391 0.0627 -3.82 0.0002  
##  g_loading   1.4146 0.1075 13.16 <0.0001 
##  difficulty  0.0531 0.0120  4.42 <0.0001 
##  
## 
## [[3]]
## Linear Regression Model
##  
##  ols(formula = BW_gap ~ g_loading + difficulty + test, data = nlsy_item_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     105    LR chi2    141.15    R2       0.739    
##  sigma0.1001    d.f.            5    R2 adj   0.726    
##  d.f.     99    Pr(> chi2) 0.0000    g        0.180    
##  
##  Residuals
##  
##        Min        1Q    Median        3Q       Max 
##  -0.208424 -0.060846 -0.006877  0.064008  0.246914 
##  
##  
##                               Coef    S.E.   t     Pr(>|t|)
##  Intercept                    -0.2237 0.0618 -3.62 0.0005  
##  g_loading                     1.4460 0.1008 14.34 <0.0001 
##  difficulty                    0.0713 0.0124  5.74 <0.0001 
##  test=Mathematics knowledge   -0.1106 0.0273 -4.05 0.0001  
##  test=Paragraph comprehension  0.0185 0.0333  0.55 0.5810  
##  test=Word knowledge           0.0113 0.0274  0.41 0.6818  
##  
## 
## [[4]]
## Linear Regression Model
##  
##  ols(formula = BW_gap ~ g_loading + difficulty + test + item_number, 
##      data = nlsy_item_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     105    LR chi2    148.26    R2       0.756    
##  sigma0.0972    d.f.            6    R2 adj   0.741    
##  d.f.     98    Pr(> chi2) 0.0000    g        0.183    
##  
##  Residuals
##  
##       Min       1Q   Median       3Q      Max 
##  -0.20660 -0.06652 -0.01335  0.06597  0.22694 
##  
##  
##                               Coef    S.E.   t     Pr(>|t|)
##  Intercept                    -0.1140 0.0732 -1.56 0.1224  
##  g_loading                     1.4392 0.0980 14.69 <0.0001 
##  difficulty                    0.1104 0.0192  5.75 <0.0001 
##  test=Mathematics knowledge   -0.1376 0.0284 -4.84 <0.0001 
##  test=Paragraph comprehension  0.0066 0.0327  0.20 0.8413  
##  test=Word knowledge           0.0738 0.0358  2.07 0.0415  
##  item_number                  -0.0058 0.0022 -2.62 0.0102  
## 

#effect size metrics
#model 4
lm(BW_gap ~ g_loading + difficulty + test + item_number, data = nlsy_item_stats) %>% car::Anova() %>% sjstats::anova_stats()

#extract model estimates
model_ests = g_models %>% 
  map(function(x) {
    # browser()
    y = x$coefficients %>% as.data.frame()
    y = rownames_to_column(y)
    names(y) = c("predictor", "beta")
    y
  }) %>% ldf_to_df(by_name = "model")

#sum stats for g-loadings
model_ests %>% 
  filter(str_detect(predictor, "g_loading"), !str_detect(predictor, "\\*")) %>% 
  .$beta %>% 
  describe()

#intercepts
model_ests %>% 
  filter(str_detect(predictor, "Intercept")) %>% 
  .$beta %>% 
  describe()

#gap size for perfect item
predict(g_models[[4]], data.frame(
  g_loading = 1,
  difficulty = 0,
  test = unique(nlsy_item_stats$test),
  item_number = wtd_mean(nlsy_item_stats$item_number))) %>% 
  describe()

Hispanics

#regression models for hispanics
g_models_HW = list(
  ols(HW_gap ~ g_loading, data = nlsy_item_stats),
  ols(HW_gap ~ g_loading + difficulty, data = nlsy_item_stats),
  ols(HW_gap ~ g_loading + difficulty + test, data = nlsy_item_stats),
  ols(HW_gap ~ g_loading + difficulty + test + item_number, data = nlsy_item_stats)
)

#summary output
g_models_HW %>% summarize_models()

#full output
g_models_HW

## [[1]]
## Linear Regression Model
##  
##  ols(formula = HW_gap ~ g_loading, data = nlsy_item_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     105    LR chi2     67.65    R2       0.475    
##  sigma0.1233    d.f.            1    R2 adj   0.470    
##  d.f.    103    Pr(> chi2) 0.0000    g        0.130    
##  
##  Residuals
##  
##         Min         1Q     Median         3Q        Max 
##  -0.2988626 -0.0728836 -0.0006697  0.0626271  0.4737261 
##  
##  
##            Coef    S.E.   t     Pr(>|t|)
##  Intercept -0.0871 0.0569 -1.53 0.1290  
##  g_loading  0.8447 0.0875  9.65 <0.0001 
##  
## 
## [[2]]
## Linear Regression Model
##  
##  ols(formula = HW_gap ~ g_loading + difficulty, data = nlsy_item_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     105    LR chi2     73.25    R2       0.502    
##  sigma0.1207    d.f.            2    R2 adj   0.492    
##  d.f.    102    Pr(> chi2) 0.0000    g        0.131    
##  
##  Residuals
##  
##        Min        1Q    Median        3Q       Max 
##  -0.300896 -0.074127 -0.008877  0.061828  0.460414 
##  
##  
##             Coef    S.E.   t     Pr(>|t|)
##  Intercept  -0.1835 0.0690 -2.66 0.0091  
##  g_loading   1.0380 0.1184  8.77 <0.0001 
##  difficulty  0.0313 0.0132  2.37 0.0199  
##  
## 
## [[3]]
## Linear Regression Model
##  
##  ols(formula = HW_gap ~ g_loading + difficulty + test, data = nlsy_item_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     105    LR chi2     78.09    R2       0.525    
##  sigma0.1197    d.f.            5    R2 adj   0.501    
##  d.f.     99    Pr(> chi2) 0.0000    g        0.135    
##  
##  Residuals
##  
##        Min        1Q    Median        3Q       Max 
##  -0.313793 -0.064017 -0.002207  0.045920  0.448669 
##  
##  
##                               Coef    S.E.   t     Pr(>|t|)
##  Intercept                    -0.1942 0.0739 -2.63 0.0099  
##  g_loading                     1.0736 0.1206  8.90 <0.0001 
##  difficulty                    0.0427 0.0148  2.88 0.0049  
##  test=Mathematics knowledge   -0.0470 0.0326 -1.44 0.1528  
##  test=Paragraph comprehension  0.0380 0.0399  0.95 0.3423  
##  test=Word knowledge           0.0105 0.0328  0.32 0.7499  
##  
## 
## [[4]]
## Linear Regression Model
##  
##  ols(formula = HW_gap ~ g_loading + difficulty + test + item_number, 
##      data = nlsy_item_stats)
##  
##                  Model Likelihood    Discrimination    
##                        Ratio Test           Indexes    
##  Obs     105    LR chi2     83.74    R2       0.550    
##  sigma0.1171    d.f.            6    R2 adj   0.522    
##  d.f.     98    Pr(> chi2) 0.0000    g        0.139    
##  
##  Residuals
##  
##        Min        1Q    Median        3Q       Max 
##  -0.338591 -0.077149  0.002304  0.056938  0.436485 
##  
##  
##                               Coef    S.E.   t     Pr(>|t|)
##  Intercept                    -0.0768 0.0881 -0.87 0.3853  
##  g_loading                     1.0663 0.1180  9.04 <0.0001 
##  difficulty                    0.0846 0.0231  3.66 0.0004  
##  test=Mathematics knowledge   -0.0759 0.0342 -2.22 0.0290  
##  test=Paragraph comprehension  0.0253 0.0394  0.64 0.5218  
##  test=Word knowledge           0.0774 0.0431  1.80 0.0753  
##  item_number                  -0.0062 0.0027 -2.33 0.0220  
## 

#effect size metrics
#model 4
lm(HW_gap ~ g_loading + difficulty + test + item_number, data = nlsy_item_stats) %>% car::Anova() %>% sjstats::anova_stats()

#extract model estimates
model_ests = g_models_HW %>% 
  map(function(x) {
    # browser()
    y = x$coefficients %>% as.data.frame()
    y = rownames_to_column(y)
    names(y) = c("predictor", "beta")
    y
  }) %>% ldf_to_df(by_name = "model")

#sum stats for g-loadings
model_ests %>% 
  filter(str_detect(predictor, "g_loading"), !str_detect(predictor, "\\*")) %>% 
  .$beta %>% 
  describe()

#intercepts
model_ests %>% 
  filter(str_detect(predictor, "Intercept")) %>% 
  .$beta %>% 
  describe()

#gap size for perfect item
predict(g_models_HW[[4]], data.frame(
  g_loading = 1,
  difficulty = 0,
  test = unique(nlsy_item_stats$test),
  item_number = wtd_mean(nlsy_item_stats$item_number))) %>% 
  describe()

DIF

All items together

#fit DIF testing all items together
#all 3 groups
#DIF function does not work with >2 groups
#because the effect size function only works for pairwise comparisons
# nlsy_1g_dif = cache_object({
#   DIF_test(
#   items = nlsy_items,
#   model = 1,
#   group = nlsy$SIRE
# )
# }, filename = "cache/nlsy_mirt_fit_g.rds")

#BW
nlsy_1g_dif_bw = cache_object({
  DIF_test(
  items = nlsy_items_bw,
  model = 1,
  group = nlsy_bw$SIRE
)
}, filename = "cache/nlsy_mirt_fit_g_bw.rds", renew = F)

## Cache found, reading object from disk

#plot the anchor fits
plot(nlsy_1g_dif_bw$fits$anchor_liberal)

plot(nlsy_1g_dif_bw$fits$anchor_conservative)

#effect sizes at test level
nlsy_1g_dif_bw$effect_size_test

## $liberal
##           Effect Size   Value
## 1                STDS  0.6051
## 2                UTDS  5.2638
## 3              UETSDS  0.9686
## 4               ETSSD  0.0337
## 5         Starks.DTFR  0.1291
## 6               UDTFR  5.2203
## 7              UETSDN  1.0130
## 8 theta.of.max.test.D  1.5804
## 9           Test.Dmax -3.0612
## 
## $conservative
##           Effect Size   Value
## 1                STDS  0.6392
## 2                UTDS  4.8520
## 3              UETSDS  0.9711
## 4               ETSSD  0.0359
## 5         Starks.DTFR  0.2114
## 6               UDTFR  4.7884
## 7              UETSDN  1.0197
## 8 theta.of.max.test.D  1.6083
## 9           Test.Dmax -2.7180

#gap sizes by scoring method
(nlsy_fit_bw_g_gaps = describeBy(map_df(nlsy_1g_dif_bw$scores, ~.[, 1] %>% standardize(focal_group = nlsy_bw$SIRE == "White")), group = nlsy_bw$SIRE))

## 
##  Descriptive statistics by group 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 6352    0  1  -0.12   -0.05 0.99 -2.15 2.77  4.92
## noDIF_liberal          2 6352    0  1   0.10    0.06 1.09 -3.18 1.39  4.57
## noDIF_conservative     3 6352    0  1   0.03    0.02 1.11 -2.72 1.80  4.52
## anchor_liberal         4 6352    0  1  -0.11   -0.05 1.00 -2.18 2.76  4.94
## anchor_conservative    5 6352    0  1  -0.11   -0.05 1.00 -2.17 2.76  4.93
##                      skew kurtosis   se
## original             0.46    -0.31 0.01
## noDIF_liberal       -0.40    -0.63 0.01
## noDIF_conservative  -0.13    -0.75 0.01
## anchor_liberal       0.44    -0.33 0.01
## anchor_conservative  0.44    -0.33 0.01
## ------------------------------------------------------------ 
## group: Black
##                     vars    n  mean   sd median trimmed  mad   min  max range
## original               1 2774 -0.96 0.70  -1.10   -1.05 0.63 -2.43 2.77  5.19
## noDIF_liberal          2 2774 -1.01 0.93  -1.18   -1.08 0.90 -3.18 1.39  4.57
## noDIF_conservative     3 2774 -1.01 0.84  -1.17   -1.08 0.79 -3.01 1.80  4.81
## anchor_liberal         4 2774 -0.99 0.68  -1.12   -1.07 0.63 -2.32 2.04  4.36
## anchor_conservative    5 2774 -1.00 0.66  -1.12   -1.07 0.61 -2.30 1.96  4.25
##                     skew kurtosis   se
## original            1.22     1.93 0.01
## noDIF_liberal       0.60    -0.16 0.02
## noDIF_conservative  0.82     0.40 0.02
## anchor_liberal      1.03     1.08 0.01
## anchor_conservative 1.03     1.08 0.01

#number of DIF items
nlsy_1g_dif_bw$DIF_stats %>% select(p, p_adj) %>% {colSums(. < .05)}

##     p p_adj 
##    90    76

#reliabilities
map_dbl(nlsy_1g_dif_bw$fits[1:3], marginal_rxx)

##           original      noDIF_liberal noDIF_conservative 
##              0.963              0.788              0.872

map_dbl(nlsy_1g_dif_bw$scores, empirical_rxx)

##            original       noDIF_liberal  noDIF_conservative      anchor_liberal 
##               0.962               0.801               0.877               0.964 
## anchor_conservative 
##               0.964

Hispanics

#HW
nlsy_1g_dif_hw = cache_object({
  DIF_test(
  items = nlsy_items_hw,
  model = 1,
  group = nlsy_hw$SIRE
)
}, filename = "cache/nlsy_mirt_fit_g_hw.rds", renew = F)

## Cache found, reading object from disk

#plot the anchor fits
plot(nlsy_1g_dif_hw$fits$anchor_liberal)

plot(nlsy_1g_dif_hw$fits$anchor_conservative)

#effect sizes at test level
nlsy_1g_dif_hw$effect_size_test

## $liberal
##           Effect Size   Value
## 1                STDS  0.3816
## 2                UTDS  4.3220
## 3              UETSDS  0.5794
## 4               ETSSD  0.0194
## 5         Starks.DTFR  0.1873
## 6               UDTFR  4.1228
## 7              UETSDN  0.6056
## 8 theta.of.max.test.D  1.6322
## 9           Test.Dmax -0.9710
## 
## $conservative
##           Effect Size   Value
## 1                STDS -0.3197
## 2                UTDS  3.1108
## 3              UETSDS  0.3197
## 4               ETSSD -0.0163
## 5         Starks.DTFR -0.3350
## 6               UDTFR  2.9119
## 7              UETSDN  0.3350
## 8 theta.of.max.test.D  1.5652
## 9           Test.Dmax -0.8020

#gap sizes by scoring method
(nlsy_fit_hw_g_gaps = describeBy(map_df(nlsy_1g_dif_hw$scores, ~.[, 1] %>% standardize(focal_group = nlsy_hw$SIRE == "White")), group = nlsy_hw$SIRE))

## 
##  Descriptive statistics by group 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 6352    0  1  -0.11   -0.05 0.99 -2.16 2.76  4.92
## noDIF_liberal          2 6352    0  1   0.02    0.01 1.12 -2.65 1.80  4.45
## noDIF_conservative     3 6352    0  1  -0.05   -0.02 1.08 -2.60 2.32  4.92
## anchor_liberal         4 6352    0  1  -0.11   -0.05 1.00 -2.17 2.76  4.93
## anchor_conservative    5 6352    0  1  -0.11   -0.05 1.00 -2.16 2.76  4.92
##                      skew kurtosis   se
## original             0.45    -0.32 0.01
## noDIF_liberal       -0.08    -0.75 0.01
## noDIF_conservative   0.20    -0.59 0.01
## anchor_liberal       0.45    -0.33 0.01
## anchor_conservative  0.45    -0.32 0.01
## ------------------------------------------------------------ 
## group: Hispanic
##                     vars    n  mean   sd median trimmed  mad   min  max range
## original               1 1682 -0.72 0.82  -0.87   -0.81 0.74 -2.23 2.76  4.98
## noDIF_liberal          2 1682 -0.73 0.92  -0.87   -0.80 0.92 -2.90 1.80  4.70
## noDIF_conservative     3 1682 -0.70 0.86  -0.85   -0.78 0.80 -2.35 2.32  4.67
## anchor_liberal         4 1682 -0.74 0.79  -0.87   -0.82 0.72 -2.15 2.27  4.42
## anchor_conservative    5 1682 -0.72 0.79  -0.85   -0.80 0.73 -2.17 2.27  4.44
##                     skew kurtosis   se
## original            1.12     1.27 0.02
## noDIF_liberal       0.61    -0.13 0.02
## noDIF_conservative  0.83     0.42 0.02
## anchor_liberal      0.98     0.75 0.02
## anchor_conservative 0.97     0.72 0.02

#number of DIF items
nlsy_1g_dif_hw$DIF_stats %>% select(p, p_adj) %>% {colSums(. < .05)}

##     p p_adj 
##    78    42

#reliabilities
map_dbl(nlsy_1g_dif_hw$fits[1:3], marginal_rxx)

##           original      noDIF_liberal noDIF_conservative 
##              0.959              0.866              0.929

map_dbl(nlsy_1g_dif_hw$scores, empirical_rxx)

##            original       noDIF_liberal  noDIF_conservative      anchor_liberal 
##               0.959               0.871               0.929               0.960 
## anchor_conservative 
##               0.959

Group difference and test length

Blacks only

set.seed(1)

#pars to loop over
resampling_pars = tibble(
  n_items = sample(5:ncol(nlsy_items_bw), replace = T, size = 1000)
)

#function
gap_sampler = function(n_items) {
  #items
  i_items = (1:ncol(nlsy_items_bw)) %>% sample(size = n_items)
    
  #fit
  i_fit = mirt(nlsy_items_bw[i_items], 1, verbose = F, technical = list(removeEmptyRows=T))
  
  #score
  i_scores = mirt::fscores(i_fit) %>% as.vector() %>% standardize(focal_group = (nlsy_bw$SIRE == "White"))
  
  y = tibble(
    n_items = n_items,
    #gap
    d = (i_scores[nlsy_bw$SIRE == "Black"] %>% wtd_mean()) * -1
  )
}

#loop
nlsy_resampled_tests = cache_object({
  future_pmap_dfr(resampling_pars, gap_sampler, .progress = T) %>% 
  mutate(DIF = F)
}, filename = "cache/nlsy_resampled_tests.rds", renew = F)

## Cache not found, evaluating expression

## Warning: UNRELIABLE VALUE: Future ('<none>') unexpectedly generated random
## numbers without specifying argument '[future.]seed'. There is a risk that those
## random numbers are not statistically sound and the overall results might be
## invalid. To fix this, specify argument '[future.]seed', e.g. 'seed=TRUE'. This
## ensures that proper, parallel-safe random numbers are produced via the L'Ecuyer-
## CMRG method. To disable this check, use [future].seed=NULL, or set option
## 'future.rng.onMisuse' to "ignore".

## Warning: UNRELIABLE VALUE: Future ('<none>') unexpectedly generated random
## numbers without specifying argument '[future.]seed'. There is a risk that those
## random numbers are not statistically sound and the overall results might be
## invalid. To fix this, specify argument '[future.]seed', e.g. 'seed=TRUE'. This
## ensures that proper, parallel-safe random numbers are produced via the L'Ecuyer-
## CMRG method. To disable this check, use [future].seed=NULL, or set option
## 'future.rng.onMisuse' to "ignore".

## Warning: UNRELIABLE VALUE: Future ('<none>') unexpectedly generated random
## numbers without specifying argument '[future.]seed'. There is a risk that those
## random numbers are not statistically sound and the overall results might be
## invalid. To fix this, specify argument '[future.]seed', e.g. 'seed=TRUE'. This
## ensures that proper, parallel-safe random numbers are produced via the L'Ecuyer-
## CMRG method. To disable this check, use [future].seed=NULL, or set option
## 'future.rng.onMisuse' to "ignore".

## Warning: UNRELIABLE VALUE: Future ('<none>') unexpectedly generated random
## numbers without specifying argument '[future.]seed'. There is a risk that those
## random numbers are not statistically sound and the overall results might be
## invalid. To fix this, specify argument '[future.]seed', e.g. 'seed=TRUE'. This
## ensures that proper, parallel-safe random numbers are produced via the L'Ecuyer-
## CMRG method. To disable this check, use [future].seed=NULL, or set option
## 'future.rng.onMisuse' to "ignore".

## Warning: UNRELIABLE VALUE: Future ('<none>') unexpectedly generated random
## numbers without specifying argument '[future.]seed'. There is a risk that those
## random numbers are not statistically sound and the overall results might be
## invalid. To fix this, specify argument '[future.]seed', e.g. 'seed=TRUE'. This
## ensures that proper, parallel-safe random numbers are produced via the L'Ecuyer-
## CMRG method. To disable this check, use [future].seed=NULL, or set option
## 'future.rng.onMisuse' to "ignore".

## Warning: UNRELIABLE VALUE: Future ('<none>') unexpectedly generated random
## numbers without specifying argument '[future.]seed'. There is a risk that those
## random numbers are not statistically sound and the overall results might be
## invalid. To fix this, specify argument '[future.]seed', e.g. 'seed=TRUE'. This
## ensures that proper, parallel-safe random numbers are produced via the L'Ecuyer-
## CMRG method. To disable this check, use [future].seed=NULL, or set option
## 'future.rng.onMisuse' to "ignore".

## Warning: UNRELIABLE VALUE: Future ('<none>') unexpectedly generated random
## numbers without specifying argument '[future.]seed'. There is a risk that those
## random numbers are not statistically sound and the overall results might be
## invalid. To fix this, specify argument '[future.]seed', e.g. 'seed=TRUE'. This
## ensures that proper, parallel-safe random numbers are produced via the L'Ecuyer-
## CMRG method. To disable this check, use [future].seed=NULL, or set option
## 'future.rng.onMisuse' to "ignore".

## Warning: UNRELIABLE VALUE: Future ('<none>') unexpectedly generated random
## numbers without specifying argument '[future.]seed'. There is a risk that those
## random numbers are not statistically sound and the overall results might be
## invalid. To fix this, specify argument '[future.]seed', e.g. 'seed=TRUE'. This
## ensures that proper, parallel-safe random numbers are produced via the L'Ecuyer-
## CMRG method. To disable this check, use [future].seed=NULL, or set option
## 'future.rng.onMisuse' to "ignore".

#add our no DIF datapoints
nlsy_resampled_tests = bind_rows(
  nlsy_resampled_tests,
  tibble(
    n_items = c(sum(nlsy_1g_dif_bw$DIF_stats$p > .05),
                sum(nlsy_1g_dif_bw$DIF_stats$p_adj > .05)),
    d = -c(nlsy_fit_bw_g_gaps$Black$mean[2], nlsy_fit_bw_g_gaps$Black$mean[3]),
    DIF = T
    )
  )

#plot
nlsy_resampled_tests %>% 
  filter(DIF == F) %>% 
  ggplot(aes(n_items, d)) +
  # geom_boxplot() +
  geom_point() +
  geom_smooth(se = F) +
  #specific points from analyses
  geom_point(data = nlsy_resampled_tests %>% filter(DIF == T), color = "red") +
  scale_x_continuous("Number of items") +
  scale_y_continuous("Black-White gap size (d)", breaks = seq(-1, 2, .2)) +
  theme_classic()

## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'

GG_save("figs/nlsy/test_length_gap.png")

## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'

Subscales separately

This avoids the question of unknown factor structure.

#read from cache if available
#Loop over each scale, and do DIF inside it.
nlsy_1g_dif_by_test_bw = cache_object(
  {
    nlsy_1g_dif_by_test_bw = list()
    for (test_i in unique(nlsy_item_stats$test)) {
      message(test_i)
      test_i_items = nlsy_items_bw[nlsy_item_stats %>% filter(test == test_i) %>% pull(item)]
      
      #fit
      nlsy_1g_dif_by_test_bw[[test_i]] = DIF_test(
        test_i_items,
        model = 1,
        group = (nlsy_bw$SIRE == "White")
        )
      
    }
    
    nlsy_1g_dif_by_test_bw
  },
  filename = "cache/nlsy_1g_dif_by_test_bw.rds", renew = F
)

## Cache not found, evaluating expression

## Arithmetic reasoning

## There are 8 steps

## Step 1: Initial joint fit

## 
Iteration: 1, Log-Lik: -145090.152, Max-Change: 0.83834
Iteration: 2, Log-Lik: -142498.036, Max-Change: 0.42539
Iteration: 3, Log-Lik: -141976.022, Max-Change: 0.20627
Iteration: 4, Log-Lik: -141773.002, Max-Change: 0.14375
Iteration: 5, Log-Lik: -141668.458, Max-Change: 0.10836
Iteration: 6, Log-Lik: -141608.219, Max-Change: 0.07899
Iteration: 7, Log-Lik: -141573.627, Max-Change: 0.06589
Iteration: 8, Log-Lik: -141552.774, Max-Change: 0.05733
Iteration: 9, Log-Lik: -141539.336, Max-Change: 0.03877
Iteration: 10, Log-Lik: -141530.975, Max-Change: 0.02697
Iteration: 11, Log-Lik: -141525.165, Max-Change: 0.02353
Iteration: 12, Log-Lik: -141521.530, Max-Change: 0.02094
Iteration: 13, Log-Lik: -141516.124, Max-Change: 0.01214
Iteration: 14, Log-Lik: -141514.463, Max-Change: 0.00727
Iteration: 15, Log-Lik: -141513.192, Max-Change: 0.00634
Iteration: 16, Log-Lik: -141508.560, Max-Change: 0.00466
Iteration: 17, Log-Lik: -141508.273, Max-Change: 0.00434
Iteration: 18, Log-Lik: -141508.022, Max-Change: 0.00416
Iteration: 19, Log-Lik: -141507.014, Max-Change: 0.00261
Iteration: 20, Log-Lik: -141506.952, Max-Change: 0.00213
Iteration: 21, Log-Lik: -141506.900, Max-Change: 0.00141
Iteration: 22, Log-Lik: -141506.809, Max-Change: 0.00168
Iteration: 23, Log-Lik: -141506.777, Max-Change: 0.00139
Iteration: 24, Log-Lik: -141506.751, Max-Change: 0.00130
Iteration: 25, Log-Lik: -141506.714, Max-Change: 0.00087
Iteration: 26, Log-Lik: -141506.697, Max-Change: 0.00097
Iteration: 27, Log-Lik: -141506.685, Max-Change: 0.00080
Iteration: 28, Log-Lik: -141506.632, Max-Change: 0.00021
Iteration: 29, Log-Lik: -141506.630, Max-Change: 0.00022
Iteration: 30, Log-Lik: -141506.628, Max-Change: 0.00023
Iteration: 31, Log-Lik: -141506.617, Max-Change: 0.00022
Iteration: 32, Log-Lik: -141506.616, Max-Change: 0.00020
Iteration: 33, Log-Lik: -141506.615, Max-Change: 0.00018
Iteration: 34, Log-Lik: -141506.610, Max-Change: 0.00012
Iteration: 35, Log-Lik: -141506.609, Max-Change: 0.00013
Iteration: 36, Log-Lik: -141506.609, Max-Change: 0.00013
Iteration: 37, Log-Lik: -141506.606, Max-Change: 0.00011
Iteration: 38, Log-Lik: -141506.605, Max-Change: 0.00010
Iteration: 39, Log-Lik: -141506.605, Max-Change: 0.00010

## 
## Step 2: Initial MI fit

## 
Iteration: 1, Log-Lik: -145090.152, Max-Change: 0.80462
Iteration: 2, Log-Lik: -141531.534, Max-Change: 0.27781
Iteration: 3, Log-Lik: -141276.293, Max-Change: 0.10350
Iteration: 4, Log-Lik: -141188.735, Max-Change: 0.04985
Iteration: 5, Log-Lik: -141122.726, Max-Change: 0.05835
Iteration: 6, Log-Lik: -141064.263, Max-Change: 0.05947
Iteration: 7, Log-Lik: -141011.142, Max-Change: 0.05826
Iteration: 8, Log-Lik: -140962.585, Max-Change: 0.05636
Iteration: 9, Log-Lik: -140918.065, Max-Change: 0.05430
Iteration: 10, Log-Lik: -140877.150, Max-Change: 0.05223
Iteration: 11, Log-Lik: -140839.469, Max-Change: 0.05022
Iteration: 12, Log-Lik: -140804.701, Max-Change: 0.04827
Iteration: 13, Log-Lik: -140772.564, Max-Change: 0.04639
Iteration: 14, Log-Lik: -140742.810, Max-Change: 0.04458
Iteration: 15, Log-Lik: -140715.220, Max-Change: 0.04284
Iteration: 16, Log-Lik: -140689.602, Max-Change: 0.04115
Iteration: 17, Log-Lik: -140665.784, Max-Change: 0.03953
Iteration: 18, Log-Lik: -140643.612, Max-Change: 0.03797
Iteration: 19, Log-Lik: -140622.952, Max-Change: 0.03647
Iteration: 20, Log-Lik: -140603.680, Max-Change: 0.03502
Iteration: 21, Log-Lik: -140585.687, Max-Change: 0.03363
Iteration: 22, Log-Lik: -140568.875, Max-Change: 0.03229
Iteration: 23, Log-Lik: -140553.155, Max-Change: 0.03100
Iteration: 24, Log-Lik: -140538.446, Max-Change: 0.02976
Iteration: 25, Log-Lik: -140524.676, Max-Change: 0.02857
Iteration: 26, Log-Lik: -140511.778, Max-Change: 0.02742
Iteration: 27, Log-Lik: -140499.692, Max-Change: 0.02632
Iteration: 28, Log-Lik: -140488.362, Max-Change: 0.02525
Iteration: 29, Log-Lik: -140477.738, Max-Change: 0.02423
Iteration: 30, Log-Lik: -140467.774, Max-Change: 0.02325
Iteration: 31, Log-Lik: -140458.426, Max-Change: 0.02231
Iteration: 32, Log-Lik: -140449.657, Max-Change: 0.02141
Iteration: 33, Log-Lik: -140441.428, Max-Change: 0.02054
Iteration: 34, Log-Lik: -140433.707, Max-Change: 0.01970
Iteration: 35, Log-Lik: -140426.462, Max-Change: 0.01890
Iteration: 36, Log-Lik: -140419.663, Max-Change: 0.01813
Iteration: 37, Log-Lik: -140413.286, Max-Change: 0.01739
Iteration: 38, Log-Lik: -140407.302, Max-Change: 0.01668
Iteration: 39, Log-Lik: -140401.690, Max-Change: 0.01600
Iteration: 40, Log-Lik: -140396.427, Max-Change: 0.01535
Iteration: 41, Log-Lik: -140391.492, Max-Change: 0.01472
Iteration: 42, Log-Lik: -140386.865, Max-Change: 0.01412
Iteration: 43, Log-Lik: -140382.529, Max-Change: 0.01354
Iteration: 44, Log-Lik: -140378.467, Max-Change: 0.01299
Iteration: 45, Log-Lik: -140374.661, Max-Change: 0.01246
Iteration: 46, Log-Lik: -140360.769, Max-Change: 0.04975
Iteration: 47, Log-Lik: -140354.226, Max-Change: 0.01692
Iteration: 48, Log-Lik: -140351.896, Max-Change: 0.01125
Iteration: 49, Log-Lik: -140344.406, Max-Change: 0.03728
Iteration: 50, Log-Lik: -140340.254, Max-Change: 0.01384
Iteration: 51, Log-Lik: -140338.855, Max-Change: 0.00914
Iteration: 52, Log-Lik: -140334.812, Max-Change: 0.02723
Iteration: 53, Log-Lik: -140332.261, Max-Change: 0.01085
Iteration: 54, Log-Lik: -140331.401, Max-Change: 0.00712
Iteration: 55, Log-Lik: -140329.232, Max-Change: 0.01994
Iteration: 56, Log-Lik: -140327.703, Max-Change: 0.00795
Iteration: 57, Log-Lik: -140327.165, Max-Change: 0.00532
Iteration: 58, Log-Lik: -140325.894, Max-Change: 0.01593
Iteration: 59, Log-Lik: -140324.801, Max-Change: 0.00605
Iteration: 60, Log-Lik: -140324.461, Max-Change: 0.00393
Iteration: 61, Log-Lik: -140323.787, Max-Change: 0.01055
Iteration: 62, Log-Lik: -140323.143, Max-Change: 0.00419
Iteration: 63, Log-Lik: -140322.926, Max-Change: 0.00283
Iteration: 64, Log-Lik: -140322.509, Max-Change: 0.00884
Iteration: 65, Log-Lik: -140321.991, Max-Change: 0.00324
Iteration: 66, Log-Lik: -140321.847, Max-Change: 0.00206
Iteration: 67, Log-Lik: -140321.632, Max-Change: 0.00521
Iteration: 68, Log-Lik: -140321.352, Max-Change: 0.00211
Iteration: 69, Log-Lik: -140321.259, Max-Change: 0.00146
Iteration: 70, Log-Lik: -140321.117, Max-Change: 0.00472
Iteration: 71, Log-Lik: -140320.863, Max-Change: 0.00170
Iteration: 72, Log-Lik: -140320.798, Max-Change: 0.00106
Iteration: 73, Log-Lik: -140320.723, Max-Change: 0.00256
Iteration: 74, Log-Lik: -140320.594, Max-Change: 0.00106
Iteration: 75, Log-Lik: -140320.552, Max-Change: 0.00074
Iteration: 76, Log-Lik: -140320.500, Max-Change: 0.00242
Iteration: 77, Log-Lik: -140320.376, Max-Change: 0.00088
Iteration: 78, Log-Lik: -140320.345, Max-Change: 0.00054
Iteration: 79, Log-Lik: -140320.315, Max-Change: 0.00129
Iteration: 80, Log-Lik: -140320.252, Max-Change: 0.00055
Iteration: 81, Log-Lik: -140320.232, Max-Change: 0.00038
Iteration: 82, Log-Lik: -140320.211, Max-Change: 0.00124
Iteration: 83, Log-Lik: -140320.149, Max-Change: 0.00045
Iteration: 84, Log-Lik: -140320.134, Max-Change: 0.00028
Iteration: 85, Log-Lik: -140320.121, Max-Change: 0.00068
Iteration: 86, Log-Lik: -140320.088, Max-Change: 0.00028
Iteration: 87, Log-Lik: -140320.078, Max-Change: 0.00020
Iteration: 88, Log-Lik: -140320.069, Max-Change: 0.00062
Iteration: 89, Log-Lik: -140320.038, Max-Change: 0.00023
Iteration: 90, Log-Lik: -140320.030, Max-Change: 0.00015
Iteration: 91, Log-Lik: -140320.024, Max-Change: 0.00034
Iteration: 92, Log-Lik: -140320.007, Max-Change: 0.00015
Iteration: 93, Log-Lik: -140320.002, Max-Change: 0.00010
Iteration: 94, Log-Lik: -140319.998, Max-Change: 0.00031
Iteration: 95, Log-Lik: -140319.982, Max-Change: 0.00012
Iteration: 96, Log-Lik: -140319.979, Max-Change: 0.00008

## 
## Step 3: Leave one out MI testing

## 
## Step 4: Fit without DIF items, liberal threshold

## 
Iteration: 1, Log-Lik: -166983.833, Max-Change: 0.48491
Iteration: 2, Log-Lik: -166038.542, Max-Change: 0.41768
Iteration: 3, Log-Lik: -165801.936, Max-Change: 0.23988
Iteration: 4, Log-Lik: -165746.195, Max-Change: 0.12418
Iteration: 5, Log-Lik: -165728.152, Max-Change: 0.05158
Iteration: 6, Log-Lik: -165724.101, Max-Change: 0.04031
Iteration: 7, Log-Lik: -165720.979, Max-Change: 0.01011
Iteration: 8, Log-Lik: -165720.654, Max-Change: 0.00532
Iteration: 9, Log-Lik: -165720.580, Max-Change: 0.00330
Iteration: 10, Log-Lik: -165720.525, Max-Change: 0.00120
Iteration: 11, Log-Lik: -165720.517, Max-Change: 0.00090
Iteration: 12, Log-Lik: -165720.511, Max-Change: 0.00047
Iteration: 13, Log-Lik: -165720.509, Max-Change: 0.00036
Iteration: 14, Log-Lik: -165720.507, Max-Change: 0.00027
Iteration: 15, Log-Lik: -165720.506, Max-Change: 0.00025
Iteration: 16, Log-Lik: -165720.502, Max-Change: 0.00011
Iteration: 17, Log-Lik: -165720.502, Max-Change: 0.00009

## 
## Step 5: Fit without DIF items, conservative threshold

## 
Iteration: 1, Log-Lik: -161765.368, Max-Change: 0.56630
Iteration: 2, Log-Lik: -160317.924, Max-Change: 0.28133
Iteration: 3, Log-Lik: -160039.142, Max-Change: 0.21292
Iteration: 4, Log-Lik: -159937.032, Max-Change: 0.14449
Iteration: 5, Log-Lik: -159890.757, Max-Change: 0.09728
Iteration: 6, Log-Lik: -159874.957, Max-Change: 0.04213
Iteration: 7, Log-Lik: -159868.175, Max-Change: 0.02971
Iteration: 8, Log-Lik: -159864.490, Max-Change: 0.02074
Iteration: 9, Log-Lik: -159863.054, Max-Change: 0.01272
Iteration: 10, Log-Lik: -159862.018, Max-Change: 0.00484
Iteration: 11, Log-Lik: -159861.696, Max-Change: 0.00372
Iteration: 12, Log-Lik: -159861.492, Max-Change: 0.00312
Iteration: 13, Log-Lik: -159861.161, Max-Change: 0.00166
Iteration: 14, Log-Lik: -159861.127, Max-Change: 0.00153
Iteration: 15, Log-Lik: -159861.102, Max-Change: 0.00138
Iteration: 16, Log-Lik: -159861.075, Max-Change: 0.00094
Iteration: 17, Log-Lik: -159861.065, Max-Change: 0.00078
Iteration: 18, Log-Lik: -159861.059, Max-Change: 0.00066
Iteration: 19, Log-Lik: -159861.041, Max-Change: 0.00007

## 
## Step 6: Fit with anchor items, liberal threshold

## 
Iteration: 1, Log-Lik: -145090.152, Max-Change: 0.75888
Iteration: 2, Log-Lik: -141019.432, Max-Change: 0.30392
Iteration: 3, Log-Lik: -140760.720, Max-Change: 0.13489
Iteration: 4, Log-Lik: -140678.023, Max-Change: 0.06564
Iteration: 5, Log-Lik: -140624.570, Max-Change: 0.05276
Iteration: 6, Log-Lik: -140580.857, Max-Change: 0.04741
Iteration: 7, Log-Lik: -140542.224, Max-Change: 0.04738
Iteration: 8, Log-Lik: -140507.126, Max-Change: 0.04697
Iteration: 9, Log-Lik: -140474.885, Max-Change: 0.04606
Iteration: 10, Log-Lik: -140445.130, Max-Change: 0.04495
Iteration: 11, Log-Lik: -140417.599, Max-Change: 0.04371
Iteration: 12, Log-Lik: -140392.088, Max-Change: 0.04241
Iteration: 13, Log-Lik: -140368.425, Max-Change: 0.04111
Iteration: 14, Log-Lik: -140346.449, Max-Change: 0.03980
Iteration: 15, Log-Lik: -140326.015, Max-Change: 0.03856
Iteration: 16, Log-Lik: -140307.002, Max-Change: 0.03731
Iteration: 17, Log-Lik: -140289.294, Max-Change: 0.03610
Iteration: 18, Log-Lik: -140272.780, Max-Change: 0.03494
Iteration: 19, Log-Lik: -140257.368, Max-Change: 0.03378
Iteration: 20, Log-Lik: -140242.967, Max-Change: 0.03269
Iteration: 21, Log-Lik: -140229.503, Max-Change: 0.03162
Iteration: 22, Log-Lik: -140216.896, Max-Change: 0.03059
Iteration: 23, Log-Lik: -140205.083, Max-Change: 0.02959
Iteration: 24, Log-Lik: -140194.005, Max-Change: 0.02862
Iteration: 25, Log-Lik: -140183.603, Max-Change: 0.02769
Iteration: 26, Log-Lik: -140173.830, Max-Change: 0.02679
Iteration: 27, Log-Lik: -140164.637, Max-Change: 0.02590
Iteration: 28, Log-Lik: -140155.984, Max-Change: 0.02505
Iteration: 29, Log-Lik: -140147.831, Max-Change: 0.02422
Iteration: 30, Log-Lik: -140140.144, Max-Change: 0.02343
Iteration: 31, Log-Lik: -140132.889, Max-Change: 0.02265
Iteration: 32, Log-Lik: -140126.034, Max-Change: 0.02191
Iteration: 33, Log-Lik: -140119.553, Max-Change: 0.02119
Iteration: 34, Log-Lik: -140113.422, Max-Change: 0.02049
Iteration: 35, Log-Lik: -140107.613, Max-Change: 0.01983
Iteration: 36, Log-Lik: -140102.112, Max-Change: 0.01916
Iteration: 37, Log-Lik: -140096.891, Max-Change: 0.01854
Iteration: 38, Log-Lik: -140091.937, Max-Change: 0.01793
Iteration: 39, Log-Lik: -140087.231, Max-Change: 0.01734
Iteration: 40, Log-Lik: -140082.758, Max-Change: 0.01677
Iteration: 41, Log-Lik: -140078.503, Max-Change: 0.01622
Iteration: 42, Log-Lik: -140074.453, Max-Change: 0.01569
Iteration: 43, Log-Lik: -140058.025, Max-Change: 0.04846
Iteration: 44, Log-Lik: -140051.278, Max-Change: 0.01253
Iteration: 45, Log-Lik: -140048.505, Max-Change: 0.01204
Iteration: 46, Log-Lik: -140037.709, Max-Change: 0.04121
Iteration: 47, Log-Lik: -140032.685, Max-Change: 0.01164
Iteration: 48, Log-Lik: -140030.708, Max-Change: 0.00896
Iteration: 49, Log-Lik: -140023.292, Max-Change: 0.03515
Iteration: 50, Log-Lik: -140019.448, Max-Change: 0.01228
Iteration: 51, Log-Lik: -140017.986, Max-Change: 0.00865
Iteration: 52, Log-Lik: -140012.768, Max-Change: 0.03032
Iteration: 53, Log-Lik: -140009.751, Max-Change: 0.01220
Iteration: 54, Log-Lik: -140008.651, Max-Change: 0.00856
Iteration: 55, Log-Lik: -140005.039, Max-Change: 0.02706
Iteration: 56, Log-Lik: -140002.901, Max-Change: 0.01122
Iteration: 57, Log-Lik: -140002.053, Max-Change: 0.00804
Iteration: 58, Log-Lik: -139999.377, Max-Change: 0.02598
Iteration: 59, Log-Lik: -139997.660, Max-Change: 0.01039
Iteration: 60, Log-Lik: -139997.003, Max-Change: 0.00740
Iteration: 61, Log-Lik: -139995.088, Max-Change: 0.02250
Iteration: 62, Log-Lik: -139993.840, Max-Change: 0.00911
Iteration: 63, Log-Lik: -139993.328, Max-Change: 0.00662
Iteration: 64, Log-Lik: -139991.855, Max-Change: 0.02135
Iteration: 65, Log-Lik: -139990.782, Max-Change: 0.00825
Iteration: 66, Log-Lik: -139990.384, Max-Change: 0.00587
Iteration: 67, Log-Lik: -139989.346, Max-Change: 0.01716
Iteration: 68, Log-Lik: -139988.592, Max-Change: 0.00691
Iteration: 69, Log-Lik: -139988.281, Max-Change: 0.00508
Iteration: 70, Log-Lik: -139987.458, Max-Change: 0.01680
Iteration: 71, Log-Lik: -139986.753, Max-Change: 0.00625
Iteration: 72, Log-Lik: -139986.510, Max-Change: 0.00440
Iteration: 73, Log-Lik: -139985.956, Max-Change: 0.01220
Iteration: 74, Log-Lik: -139985.501, Max-Change: 0.00499
Iteration: 75, Log-Lik: -139985.312, Max-Change: 0.00372
Iteration: 76, Log-Lik: -139984.852, Max-Change: 0.01302
Iteration: 77, Log-Lik: -139984.365, Max-Change: 0.00462
Iteration: 78, Log-Lik: -139984.216, Max-Change: 0.00317
Iteration: 79, Log-Lik: -139983.929, Max-Change: 0.00809
Iteration: 80, Log-Lik: -139983.661, Max-Change: 0.00344
Iteration: 81, Log-Lik: -139983.546, Max-Change: 0.00263
Iteration: 82, Log-Lik: -139983.294, Max-Change: 0.00951
Iteration: 83, Log-Lik: -139982.969, Max-Change: 0.00330
Iteration: 84, Log-Lik: -139982.876, Max-Change: 0.00223
Iteration: 85, Log-Lik: -139982.722, Max-Change: 0.00547
Iteration: 86, Log-Lik: -139982.554, Max-Change: 0.00237
Iteration: 87, Log-Lik: -139982.483, Max-Change: 0.00183
Iteration: 88, Log-Lik: -139982.346, Max-Change: 0.00664
Iteration: 89, Log-Lik: -139982.134, Max-Change: 0.00230
Iteration: 90, Log-Lik: -139982.075, Max-Change: 0.00155
Iteration: 91, Log-Lik: -139981.990, Max-Change: 0.00379
Iteration: 92, Log-Lik: -139981.878, Max-Change: 0.00164
Iteration: 93, Log-Lik: -139981.834, Max-Change: 0.00127
Iteration: 94, Log-Lik: -139981.758, Max-Change: 0.00460
Iteration: 95, Log-Lik: -139981.619, Max-Change: 0.00158
Iteration: 96, Log-Lik: -139981.581, Max-Change: 0.00107
Iteration: 97, Log-Lik: -139981.532, Max-Change: 0.00264
Iteration: 98, Log-Lik: -139981.457, Max-Change: 0.00113
Iteration: 99, Log-Lik: -139981.429, Max-Change: 0.00087
Iteration: 100, Log-Lik: -139981.387, Max-Change: 0.00315
Iteration: 101, Log-Lik: -139981.294, Max-Change: 0.00109
Iteration: 102, Log-Lik: -139981.270, Max-Change: 0.00073
Iteration: 103, Log-Lik: -139981.241, Max-Change: 0.00182
Iteration: 104, Log-Lik: -139981.191, Max-Change: 0.00078
Iteration: 105, Log-Lik: -139981.172, Max-Change: 0.00060
Iteration: 106, Log-Lik: -139981.148, Max-Change: 0.00215
Iteration: 107, Log-Lik: -139981.086, Max-Change: 0.00074
Iteration: 108, Log-Lik: -139981.070, Max-Change: 0.00050
Iteration: 109, Log-Lik: -139981.054, Max-Change: 0.00123
Iteration: 110, Log-Lik: -139981.020, Max-Change: 0.00053
Iteration: 111, Log-Lik: -139981.008, Max-Change: 0.00041
Iteration: 112, Log-Lik: -139980.994, Max-Change: 0.00147
Iteration: 113, Log-Lik: -139980.952, Max-Change: 0.00051
Iteration: 114, Log-Lik: -139980.941, Max-Change: 0.00034
Iteration: 115, Log-Lik: -139980.931, Max-Change: 0.00084
Iteration: 116, Log-Lik: -139980.908, Max-Change: 0.00036
Iteration: 117, Log-Lik: -139980.900, Max-Change: 0.00028
Iteration: 118, Log-Lik: -139980.891, Max-Change: 0.00101
Iteration: 119, Log-Lik: -139980.863, Max-Change: 0.00035
Iteration: 120, Log-Lik: -139980.856, Max-Change: 0.00023
Iteration: 121, Log-Lik: -139980.850, Max-Change: 0.00059
Iteration: 122, Log-Lik: -139980.834, Max-Change: 0.00025
Iteration: 123, Log-Lik: -139980.828, Max-Change: 0.00019
Iteration: 124, Log-Lik: -139980.823, Max-Change: 0.00069
Iteration: 125, Log-Lik: -139980.804, Max-Change: 0.00023
Iteration: 126, Log-Lik: -139980.799, Max-Change: 0.00016
Iteration: 127, Log-Lik: -139980.795, Max-Change: 0.00043
Iteration: 128, Log-Lik: -139980.783, Max-Change: 0.00018
Iteration: 129, Log-Lik: -139980.780, Max-Change: 0.00013
Iteration: 130, Log-Lik: -139980.776, Max-Change: 0.00046
Iteration: 131, Log-Lik: -139980.763, Max-Change: 0.00016
Iteration: 132, Log-Lik: -139980.760, Max-Change: 0.00011
Iteration: 133, Log-Lik: -139980.758, Max-Change: 0.00031
Iteration: 134, Log-Lik: -139980.749, Max-Change: 0.00012
Iteration: 135, Log-Lik: -139980.747, Max-Change: 0.00009

## 
## Step 7: Fit with anchor items, conservative threshold

## 
Iteration: 1, Log-Lik: -145090.152, Max-Change: 0.75894
Iteration: 2, Log-Lik: -141117.322, Max-Change: 0.33279
Iteration: 3, Log-Lik: -140879.005, Max-Change: 0.15389
Iteration: 4, Log-Lik: -140797.208, Max-Change: 0.07685
Iteration: 5, Log-Lik: -140737.518, Max-Change: 0.05354
Iteration: 6, Log-Lik: -140685.694, Max-Change: 0.05425
Iteration: 7, Log-Lik: -140639.023, Max-Change: 0.05315
Iteration: 8, Log-Lik: -140596.555, Max-Change: 0.05153
Iteration: 9, Log-Lik: -140557.736, Max-Change: 0.04979
Iteration: 10, Log-Lik: -140522.171, Max-Change: 0.04800
Iteration: 11, Log-Lik: -140489.507, Max-Change: 0.04629
Iteration: 12, Log-Lik: -140459.453, Max-Change: 0.04462
Iteration: 13, Log-Lik: -140431.754, Max-Change: 0.04300
Iteration: 14, Log-Lik: -140406.170, Max-Change: 0.04147
Iteration: 15, Log-Lik: -140382.507, Max-Change: 0.03992
Iteration: 16, Log-Lik: -140360.583, Max-Change: 0.03850
Iteration: 17, Log-Lik: -140340.241, Max-Change: 0.03711
Iteration: 18, Log-Lik: -140321.336, Max-Change: 0.03577
Iteration: 19, Log-Lik: -140303.746, Max-Change: 0.03446
Iteration: 20, Log-Lik: -140287.354, Max-Change: 0.03321
Iteration: 21, Log-Lik: -140272.053, Max-Change: 0.03201
Iteration: 22, Log-Lik: -140257.761, Max-Change: 0.03083
Iteration: 23, Log-Lik: -140244.386, Max-Change: 0.02971
Iteration: 24, Log-Lik: -140231.862, Max-Change: 0.02865
Iteration: 25, Log-Lik: -140220.115, Max-Change: 0.02758
Iteration: 26, Log-Lik: -140209.089, Max-Change: 0.02660
Iteration: 27, Log-Lik: -140198.724, Max-Change: 0.02563
Iteration: 28, Log-Lik: -140188.974, Max-Change: 0.02470
Iteration: 29, Log-Lik: -140179.797, Max-Change: 0.02378
Iteration: 30, Log-Lik: -140171.147, Max-Change: 0.02292
Iteration: 31, Log-Lik: -140162.986, Max-Change: 0.02209
Iteration: 32, Log-Lik: -140155.283, Max-Change: 0.02128
Iteration: 33, Log-Lik: -140148.006, Max-Change: 0.02051
Iteration: 34, Log-Lik: -140141.125, Max-Change: 0.01976
Iteration: 35, Log-Lik: -140134.616, Max-Change: 0.01904
Iteration: 36, Log-Lik: -140128.454, Max-Change: 0.01837
Iteration: 37, Log-Lik: -140122.615, Max-Change: 0.01769
Iteration: 38, Log-Lik: -140117.082, Max-Change: 0.01706
Iteration: 39, Log-Lik: -140111.835, Max-Change: 0.01644
Iteration: 40, Log-Lik: -140106.855, Max-Change: 0.01585
Iteration: 41, Log-Lik: -140102.129, Max-Change: 0.01528
Iteration: 42, Log-Lik: -140097.641, Max-Change: 0.01473
Iteration: 43, Log-Lik: -140079.993, Max-Change: 0.05216
Iteration: 44, Log-Lik: -140072.357, Max-Change: 0.01520
Iteration: 45, Log-Lik: -140069.340, Max-Change: 0.01089
Iteration: 46, Log-Lik: -140058.249, Max-Change: 0.04341
Iteration: 47, Log-Lik: -140052.672, Max-Change: 0.01543
Iteration: 48, Log-Lik: -140050.600, Max-Change: 0.01078
Iteration: 49, Log-Lik: -140043.367, Max-Change: 0.03602
Iteration: 50, Log-Lik: -140039.283, Max-Change: 0.01467
Iteration: 51, Log-Lik: -140037.831, Max-Change: 0.01013
Iteration: 52, Log-Lik: -140033.300, Max-Change: 0.03086
Iteration: 53, Log-Lik: -140030.679, Max-Change: 0.01260
Iteration: 54, Log-Lik: -140029.639, Max-Change: 0.00893
Iteration: 55, Log-Lik: -140026.455, Max-Change: 0.02926
Iteration: 56, Log-Lik: -140024.344, Max-Change: 0.01120
Iteration: 57, Log-Lik: -140023.602, Max-Change: 0.00773
Iteration: 58, Log-Lik: -140021.618, Max-Change: 0.02206
Iteration: 59, Log-Lik: -140020.320, Max-Change: 0.00890
Iteration: 60, Log-Lik: -140019.784, Max-Change: 0.00643
Iteration: 61, Log-Lik: -140018.307, Max-Change: 0.02173
Iteration: 62, Log-Lik: -140017.108, Max-Change: 0.00788
Iteration: 63, Log-Lik: -140016.724, Max-Change: 0.00535
Iteration: 64, Log-Lik: -140015.860, Max-Change: 0.01414
Iteration: 65, Log-Lik: -140015.203, Max-Change: 0.00581
Iteration: 66, Log-Lik: -140014.925, Max-Change: 0.00429
Iteration: 67, Log-Lik: -140014.242, Max-Change: 0.01510
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Iteration: 71, Log-Lik: -140012.614, Max-Change: 0.00366
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Iteration: 76, Log-Lik: -140011.476, Max-Change: 0.00548
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Iteration: 78, Log-Lik: -140011.197, Max-Change: 0.00173
Iteration: 79, Log-Lik: -140011.054, Max-Change: 0.00615
Iteration: 80, Log-Lik: -140010.824, Max-Change: 0.00211
Iteration: 81, Log-Lik: -140010.762, Max-Change: 0.00140
Iteration: 82, Log-Lik: -140010.679, Max-Change: 0.00346
Iteration: 83, Log-Lik: -140010.559, Max-Change: 0.00144
Iteration: 84, Log-Lik: -140010.515, Max-Change: 0.00108
Iteration: 85, Log-Lik: -140010.448, Max-Change: 0.00384
Iteration: 86, Log-Lik: -140010.312, Max-Change: 0.00131
Iteration: 87, Log-Lik: -140010.277, Max-Change: 0.00087
Iteration: 88, Log-Lik: -140010.236, Max-Change: 0.00216
Iteration: 89, Log-Lik: -140010.164, Max-Change: 0.00090
Iteration: 90, Log-Lik: -140010.138, Max-Change: 0.00067
Iteration: 91, Log-Lik: -140010.106, Max-Change: 0.00238
Iteration: 92, Log-Lik: -140010.024, Max-Change: 0.00082
Iteration: 93, Log-Lik: -140010.003, Max-Change: 0.00054
Iteration: 94, Log-Lik: -140009.982, Max-Change: 0.00135
Iteration: 95, Log-Lik: -140009.938, Max-Change: 0.00056
Iteration: 96, Log-Lik: -140009.923, Max-Change: 0.00042
Iteration: 97, Log-Lik: -140009.906, Max-Change: 0.00147
Iteration: 98, Log-Lik: -140009.857, Max-Change: 0.00050
Iteration: 99, Log-Lik: -140009.845, Max-Change: 0.00034
Iteration: 100, Log-Lik: -140009.833, Max-Change: 0.00086
Iteration: 101, Log-Lik: -140009.806, Max-Change: 0.00035
Iteration: 102, Log-Lik: -140009.797, Max-Change: 0.00026
Iteration: 103, Log-Lik: -140009.788, Max-Change: 0.00091
Iteration: 104, Log-Lik: -140009.758, Max-Change: 0.00031
Iteration: 105, Log-Lik: -140009.751, Max-Change: 0.00021
Iteration: 106, Log-Lik: -140009.744, Max-Change: 0.00054
Iteration: 107, Log-Lik: -140009.727, Max-Change: 0.00022
Iteration: 108, Log-Lik: -140009.721, Max-Change: 0.00016
Iteration: 109, Log-Lik: -140009.716, Max-Change: 0.00055
Iteration: 110, Log-Lik: -140009.698, Max-Change: 0.00019
Iteration: 111, Log-Lik: -140009.694, Max-Change: 0.00013
Iteration: 112, Log-Lik: -140009.690, Max-Change: 0.00032
Iteration: 113, Log-Lik: -140009.680, Max-Change: 0.00013
Iteration: 114, Log-Lik: -140009.677, Max-Change: 0.00010

## 
## Step 8: Get scores

## Word knowledge

## There are 8 steps

## Step 1: Initial joint fit

## 
Iteration: 1, Log-Lik: -144411.779, Max-Change: 2.96970
Iteration: 2, Log-Lik: -140381.484, Max-Change: 1.51690
Iteration: 3, Log-Lik: -139436.487, Max-Change: 0.46455
Iteration: 4, Log-Lik: -139067.146, Max-Change: 0.25781
Iteration: 5, Log-Lik: -138877.738, Max-Change: 0.15204
Iteration: 6, Log-Lik: -138758.635, Max-Change: 0.09989
Iteration: 7, Log-Lik: -138680.751, Max-Change: 0.08206
Iteration: 8, Log-Lik: -138620.310, Max-Change: 0.09756
Iteration: 9, Log-Lik: -138573.022, Max-Change: 0.09183
Iteration: 10, Log-Lik: -138534.442, Max-Change: 0.09339
Iteration: 11, Log-Lik: -138501.088, Max-Change: 0.08423
Iteration: 12, Log-Lik: -138473.040, Max-Change: 0.10269
Iteration: 13, Log-Lik: -138447.934, Max-Change: 0.08340
Iteration: 14, Log-Lik: -138426.704, Max-Change: 0.10310
Iteration: 15, Log-Lik: -138407.248, Max-Change: 0.08031
Iteration: 16, Log-Lik: -138389.425, Max-Change: 0.10051
Iteration: 17, Log-Lik: -138373.567, Max-Change: 0.07743
Iteration: 18, Log-Lik: -138359.948, Max-Change: 0.08923
Iteration: 19, Log-Lik: -138347.306, Max-Change: 0.06787
Iteration: 20, Log-Lik: -138336.141, Max-Change: 0.08082
Iteration: 21, Log-Lik: -138325.622, Max-Change: 0.06646
Iteration: 22, Log-Lik: -138316.271, Max-Change: 0.07875
Iteration: 23, Log-Lik: -138307.940, Max-Change: 0.06075
Iteration: 24, Log-Lik: -138300.063, Max-Change: 0.07341
Iteration: 25, Log-Lik: -138293.166, Max-Change: 0.05655
Iteration: 26, Log-Lik: -138286.396, Max-Change: 0.06921
Iteration: 27, Log-Lik: -138280.698, Max-Change: 0.04898
Iteration: 28, Log-Lik: -138274.791, Max-Change: 0.06279
Iteration: 29, Log-Lik: -138269.998, Max-Change: 0.04307
Iteration: 30, Log-Lik: -138264.863, Max-Change: 0.05777
Iteration: 31, Log-Lik: -138260.854, Max-Change: 0.03734
Iteration: 32, Log-Lik: -138256.449, Max-Change: 0.05156
Iteration: 33, Log-Lik: -138253.052, Max-Change: 0.03413
Iteration: 34, Log-Lik: -138250.811, Max-Change: 0.05243
Iteration: 35, Log-Lik: -138247.480, Max-Change: 0.03601
Iteration: 36, Log-Lik: -138244.697, Max-Change: 0.04470
Iteration: 37, Log-Lik: -138243.003, Max-Change: 0.04178
Iteration: 38, Log-Lik: -138240.071, Max-Change: 0.04631
Iteration: 39, Log-Lik: -138237.798, Max-Change: 0.01497
Iteration: 40, Log-Lik: -138235.265, Max-Change: 0.02130
Iteration: 41, Log-Lik: -138232.752, Max-Change: 0.03132
Iteration: 42, Log-Lik: -138230.980, Max-Change: 0.02411
Iteration: 43, Log-Lik: -138229.700, Max-Change: 0.02961
Iteration: 44, Log-Lik: -138228.050, Max-Change: 0.02074
Iteration: 45, Log-Lik: -138226.569, Max-Change: 0.01712
Iteration: 46, Log-Lik: -138224.937, Max-Change: 0.02711
Iteration: 47, Log-Lik: -138223.436, Max-Change: 0.02238
Iteration: 48, Log-Lik: -138222.303, Max-Change: 0.01097
Iteration: 49, Log-Lik: -138220.940, Max-Change: 0.00662
Iteration: 50, Log-Lik: -138219.446, Max-Change: 0.00686
Iteration: 51, Log-Lik: -138218.105, Max-Change: 0.00677
Iteration: 52, Log-Lik: -138211.798, Max-Change: 0.00428
Iteration: 53, Log-Lik: -138211.254, Max-Change: 0.00412
Iteration: 54, Log-Lik: -138210.769, Max-Change: 0.00400
Iteration: 55, Log-Lik: -138208.515, Max-Change: 0.00303
Iteration: 56, Log-Lik: -138208.307, Max-Change: 0.00307
Iteration: 57, Log-Lik: -138208.119, Max-Change: 0.00287
Iteration: 58, Log-Lik: -138207.474, Max-Change: 0.00276
Iteration: 59, Log-Lik: -138207.356, Max-Change: 0.00139
Iteration: 60, Log-Lik: -138207.278, Max-Change: 0.00236
Iteration: 61, Log-Lik: -138207.111, Max-Change: 0.00178
Iteration: 62, Log-Lik: -138207.051, Max-Change: 0.00181
Iteration: 63, Log-Lik: -138206.995, Max-Change: 0.00108
Iteration: 64, Log-Lik: -138206.913, Max-Change: 0.00170
Iteration: 65, Log-Lik: -138206.869, Max-Change: 0.00154
Iteration: 66, Log-Lik: -138206.832, Max-Change: 0.00129
Iteration: 67, Log-Lik: -138206.683, Max-Change: 0.00091
Iteration: 68, Log-Lik: -138206.664, Max-Change: 0.00078
Iteration: 69, Log-Lik: -138206.651, Max-Change: 0.00067
Iteration: 70, Log-Lik: -138206.609, Max-Change: 0.00060
Iteration: 71, Log-Lik: -138206.597, Max-Change: 0.00058
Iteration: 72, Log-Lik: -138206.588, Max-Change: 0.00055
Iteration: 73, Log-Lik: -138206.543, Max-Change: 0.00044
Iteration: 74, Log-Lik: -138206.538, Max-Change: 0.00042
Iteration: 75, Log-Lik: -138206.534, Max-Change: 0.00022
Iteration: 76, Log-Lik: -138206.532, Max-Change: 0.00014
Iteration: 77, Log-Lik: -138206.530, Max-Change: 0.00015
Iteration: 78, Log-Lik: -138206.528, Max-Change: 0.00015
Iteration: 79, Log-Lik: -138206.518, Max-Change: 0.00015
Iteration: 80, Log-Lik: -138206.517, Max-Change: 0.00014
Iteration: 81, Log-Lik: -138206.516, Max-Change: 0.00014
Iteration: 82, Log-Lik: -138206.509, Max-Change: 0.00011
Iteration: 83, Log-Lik: -138206.509, Max-Change: 0.00011
Iteration: 84, Log-Lik: -138206.508, Max-Change: 0.00011
Iteration: 85, Log-Lik: -138206.504, Max-Change: 0.00010
Iteration: 86, Log-Lik: -138206.503, Max-Change: 0.00010

## 
## Step 2: Initial MI fit

## 
Iteration: 1, Log-Lik: -144411.779, Max-Change: 1.11668
Iteration: 2, Log-Lik: -138802.052, Max-Change: 0.29070
Iteration: 3, Log-Lik: -138366.185, Max-Change: 0.12397
Iteration: 4, Log-Lik: -138221.773, Max-Change: 0.06214
Iteration: 5, Log-Lik: -138126.160, Max-Change: 0.04536
Iteration: 6, Log-Lik: -138046.154, Max-Change: 0.05005
Iteration: 7, Log-Lik: -137974.972, Max-Change: 0.05256
Iteration: 8, Log-Lik: -137910.576, Max-Change: 0.05388
Iteration: 9, Log-Lik: -137851.910, Max-Change: 0.05287
Iteration: 10, Log-Lik: -137798.322, Max-Change: 0.05084
Iteration: 11, Log-Lik: -137749.283, Max-Change: 0.04994
Iteration: 12, Log-Lik: -137704.246, Max-Change: 0.04857
Iteration: 13, Log-Lik: -137662.855, Max-Change: 0.04729
Iteration: 14, Log-Lik: -137624.725, Max-Change: 0.04543
Iteration: 15, Log-Lik: -137589.573, Max-Change: 0.04482
Iteration: 16, Log-Lik: -137557.122, Max-Change: 0.04251
Iteration: 17, Log-Lik: -137527.122, Max-Change: 0.04203
Iteration: 18, Log-Lik: -137499.349, Max-Change: 0.04035
Iteration: 19, Log-Lik: -137473.581, Max-Change: 0.03926
Iteration: 20, Log-Lik: -137449.694, Max-Change: 0.03785
Iteration: 21, Log-Lik: -137427.511, Max-Change: 0.03677
Iteration: 22, Log-Lik: -137406.892, Max-Change: 0.03573
Iteration: 23, Log-Lik: -137387.698, Max-Change: 0.03463
Iteration: 24, Log-Lik: -137369.841, Max-Change: 0.03371
Iteration: 25, Log-Lik: -137353.199, Max-Change: 0.03269
Iteration: 26, Log-Lik: -137337.677, Max-Change: 0.03159
Iteration: 27, Log-Lik: -137323.198, Max-Change: 0.03044
Iteration: 28, Log-Lik: -137309.693, Max-Change: 0.02969
Iteration: 29, Log-Lik: -137297.078, Max-Change: 0.02883
Iteration: 30, Log-Lik: -137285.285, Max-Change: 0.02788
Iteration: 31, Log-Lik: -137274.265, Max-Change: 0.02707
Iteration: 32, Log-Lik: -137263.955, Max-Change: 0.02623
Iteration: 33, Log-Lik: -137254.305, Max-Change: 0.02551
Iteration: 34, Log-Lik: -137245.269, Max-Change: 0.02467
Iteration: 35, Log-Lik: -137236.806, Max-Change: 0.02395
Iteration: 36, Log-Lik: -137228.883, Max-Change: 0.02311
Iteration: 37, Log-Lik: -137221.460, Max-Change: 0.02257
Iteration: 38, Log-Lik: -137214.507, Max-Change: 0.02177
Iteration: 39, Log-Lik: -137207.986, Max-Change: 0.02110
Iteration: 40, Log-Lik: -137201.868, Max-Change: 0.02044
Iteration: 41, Log-Lik: -137196.137, Max-Change: 0.01981
Iteration: 42, Log-Lik: -137190.762, Max-Change: 0.01922
Iteration: 43, Log-Lik: -137185.716, Max-Change: 0.01864
Iteration: 44, Log-Lik: -137180.981, Max-Change: 0.01808
Iteration: 45, Log-Lik: -137176.538, Max-Change: 0.01750
Iteration: 46, Log-Lik: -137155.272, Max-Change: 0.04412
Iteration: 47, Log-Lik: -137152.040, Max-Change: 0.01294
Iteration: 48, Log-Lik: -137149.386, Max-Change: 0.01309
Iteration: 49, Log-Lik: -137136.853, Max-Change: 0.03620
Iteration: 50, Log-Lik: -137135.077, Max-Change: 0.01012
Iteration: 51, Log-Lik: -137133.459, Max-Change: 0.01008
Iteration: 52, Log-Lik: -137125.951, Max-Change: 0.02984
Iteration: 53, Log-Lik: -137125.023, Max-Change: 0.00795
Iteration: 54, Log-Lik: -137124.018, Max-Change: 0.00768
Iteration: 55, Log-Lik: -137119.477, Max-Change: 0.02452
Iteration: 56, Log-Lik: -137119.049, Max-Change: 0.00755
Iteration: 57, Log-Lik: -137118.419, Max-Change: 0.00589
Iteration: 58, Log-Lik: -137115.657, Max-Change: 0.02002
Iteration: 59, Log-Lik: -137115.519, Max-Change: 0.00685
Iteration: 60, Log-Lik: -137115.125, Max-Change: 0.00457
Iteration: 61, Log-Lik: -137113.442, Max-Change: 0.01626
Iteration: 62, Log-Lik: -137113.468, Max-Change: 0.00602
Iteration: 63, Log-Lik: -137113.220, Max-Change: 0.00398
Iteration: 64, Log-Lik: -137112.194, Max-Change: 0.01312
Iteration: 65, Log-Lik: -137112.303, Max-Change: 0.00515
Iteration: 66, Log-Lik: -137112.147, Max-Change: 0.00339
Iteration: 67, Log-Lik: -137111.523, Max-Change: 0.01054
Iteration: 68, Log-Lik: -137111.668, Max-Change: 0.00432
Iteration: 69, Log-Lik: -137111.571, Max-Change: 0.00284
Iteration: 70, Log-Lik: -137111.192, Max-Change: 0.00842
Iteration: 71, Log-Lik: -137111.345, Max-Change: 0.00358
Iteration: 72, Log-Lik: -137111.284, Max-Change: 0.00234
Iteration: 73, Log-Lik: -137111.056, Max-Change: 0.00670
Iteration: 74, Log-Lik: -137111.201, Max-Change: 0.00292
Iteration: 75, Log-Lik: -137111.162, Max-Change: 0.00191
Iteration: 76, Log-Lik: -137111.026, Max-Change: 0.00531
Iteration: 77, Log-Lik: -137111.155, Max-Change: 0.00236
Iteration: 78, Log-Lik: -137111.131, Max-Change: 0.00154
Iteration: 79, Log-Lik: -137111.051, Max-Change: 0.00413
Iteration: 80, Log-Lik: -137111.161, Max-Change: 0.00188
Iteration: 81, Log-Lik: -137111.146, Max-Change: 0.00123
Iteration: 82, Log-Lik: -137111.099, Max-Change: 0.00327
Iteration: 83, Log-Lik: -137111.191, Max-Change: 0.00151
Iteration: 84, Log-Lik: -137111.182, Max-Change: 0.00098
Iteration: 85, Log-Lik: -137111.155, Max-Change: 0.00254
Iteration: 86, Log-Lik: -137111.230, Max-Change: 0.00119
Iteration: 87, Log-Lik: -137111.225, Max-Change: 0.00078
Iteration: 88, Log-Lik: -137111.209, Max-Change: 0.00203
Iteration: 89, Log-Lik: -137111.272, Max-Change: 0.00095
Iteration: 90, Log-Lik: -137111.268, Max-Change: 0.00062
Iteration: 91, Log-Lik: -137111.260, Max-Change: 0.00156
Iteration: 92, Log-Lik: -137111.309, Max-Change: 0.00074
Iteration: 93, Log-Lik: -137111.307, Max-Change: 0.00049
Iteration: 94, Log-Lik: -137111.302, Max-Change: 0.00127
Iteration: 95, Log-Lik: -137111.343, Max-Change: 0.00060
Iteration: 96, Log-Lik: -137111.342, Max-Change: 0.00039
Iteration: 97, Log-Lik: -137111.340, Max-Change: 0.00096
Iteration: 98, Log-Lik: -137111.371, Max-Change: 0.00046
Iteration: 99, Log-Lik: -137111.371, Max-Change: 0.00031
Iteration: 100, Log-Lik: -137111.370, Max-Change: 0.00079
Iteration: 101, Log-Lik: -137111.396, Max-Change: 0.00038
Iteration: 102, Log-Lik: -137111.396, Max-Change: 0.00024
Iteration: 103, Log-Lik: -137111.396, Max-Change: 0.00057
Iteration: 104, Log-Lik: -137111.415, Max-Change: 0.00028
Iteration: 105, Log-Lik: -137111.415, Max-Change: 0.00019
Iteration: 106, Log-Lik: -137111.415, Max-Change: 0.00050
Iteration: 107, Log-Lik: -137111.432, Max-Change: 0.00024
Iteration: 108, Log-Lik: -137111.432, Max-Change: 0.00015
Iteration: 109, Log-Lik: -137111.432, Max-Change: 0.00035
Iteration: 110, Log-Lik: -137111.444, Max-Change: 0.00017
Iteration: 111, Log-Lik: -137111.444, Max-Change: 0.00012
Iteration: 112, Log-Lik: -137111.444, Max-Change: 0.00031
Iteration: 113, Log-Lik: -137111.455, Max-Change: 0.00015
Iteration: 114, Log-Lik: -137111.455, Max-Change: 0.00010

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 3: Leave one out MI testing

## 
## Step 4: Fit without DIF items, liberal threshold

## 
Iteration: 1, Log-Lik: -171289.510, Max-Change: 0.35162
Iteration: 2, Log-Lik: -170639.768, Max-Change: 0.25103
Iteration: 3, Log-Lik: -170517.665, Max-Change: 0.18211
Iteration: 4, Log-Lik: -170472.526, Max-Change: 0.10846
Iteration: 5, Log-Lik: -170458.581, Max-Change: 0.06636
Iteration: 6, Log-Lik: -170454.304, Max-Change: 0.04422
Iteration: 7, Log-Lik: -170451.857, Max-Change: 0.01370
Iteration: 8, Log-Lik: -170451.680, Max-Change: 0.00776
Iteration: 9, Log-Lik: -170451.614, Max-Change: 0.00589
Iteration: 10, Log-Lik: -170451.547, Max-Change: 0.00023
Iteration: 11, Log-Lik: -170451.547, Max-Change: 0.00061
Iteration: 12, Log-Lik: -170451.546, Max-Change: 0.00017
Iteration: 13, Log-Lik: -170451.546, Max-Change: 0.00014
Iteration: 14, Log-Lik: -170451.546, Max-Change: 0.00041
Iteration: 15, Log-Lik: -170451.546, Max-Change: 0.00011
Iteration: 16, Log-Lik: -170451.546, Max-Change: 0.00009

## 
## Step 5: Fit without DIF items, conservative threshold

## 
Iteration: 1, Log-Lik: -167322.817, Max-Change: 0.64661
Iteration: 2, Log-Lik: -165561.651, Max-Change: 0.47070
Iteration: 3, Log-Lik: -165149.184, Max-Change: 0.24301
Iteration: 4, Log-Lik: -165054.810, Max-Change: 0.22008
Iteration: 5, Log-Lik: -165005.831, Max-Change: 0.13841
Iteration: 6, Log-Lik: -164981.174, Max-Change: 0.08730
Iteration: 7, Log-Lik: -164966.739, Max-Change: 0.06509
Iteration: 8, Log-Lik: -164960.283, Max-Change: 0.04529
Iteration: 9, Log-Lik: -164955.852, Max-Change: 0.03827
Iteration: 10, Log-Lik: -164954.400, Max-Change: 0.02572
Iteration: 11, Log-Lik: -164951.976, Max-Change: 0.01530
Iteration: 12, Log-Lik: -164950.437, Max-Change: 0.00962
Iteration: 13, Log-Lik: -164948.137, Max-Change: 0.00547
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Iteration: 42, Log-Lik: -164947.071, Max-Change: 0.00011
Iteration: 43, Log-Lik: -164947.071, Max-Change: 0.00009

## 
## Step 6: Fit with anchor items, liberal threshold

## 
Iteration: 1, Log-Lik: -144411.779, Max-Change: 1.35005
Iteration: 2, Log-Lik: -137975.911, Max-Change: 0.34050
Iteration: 3, Log-Lik: -137517.612, Max-Change: 0.23203
Iteration: 4, Log-Lik: -137341.260, Max-Change: 0.15019
Iteration: 5, Log-Lik: -137236.058, Max-Change: 0.11241
Iteration: 6, Log-Lik: -137162.196, Max-Change: 0.07983
Iteration: 7, Log-Lik: -137105.858, Max-Change: 0.05936
Iteration: 8, Log-Lik: -137060.513, Max-Change: 0.04577
Iteration: 9, Log-Lik: -137022.408, Max-Change: 0.03667
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Iteration: 11, Log-Lik: -136959.940, Max-Change: 0.03697
Iteration: 12, Log-Lik: -136933.344, Max-Change: 0.03736
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Iteration: 14, Log-Lik: -136886.438, Max-Change: 0.03656
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Iteration: 177, Log-Lik: -136500.458, Max-Change: 0.00009

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 7: Fit with anchor items, conservative threshold

## 
Iteration: 1, Log-Lik: -144411.779, Max-Change: 1.35485
Iteration: 2, Log-Lik: -138091.259, Max-Change: 0.40544
Iteration: 3, Log-Lik: -137681.397, Max-Change: 0.23826
Iteration: 4, Log-Lik: -137532.889, Max-Change: 0.17307
Iteration: 5, Log-Lik: -137439.373, Max-Change: 0.10791
Iteration: 6, Log-Lik: -137367.068, Max-Change: 0.07918
Iteration: 7, Log-Lik: -137305.729, Max-Change: 0.05991
Iteration: 8, Log-Lik: -137251.385, Max-Change: 0.04784
Iteration: 9, Log-Lik: -137202.345, Max-Change: 0.04836
Iteration: 10, Log-Lik: -137157.590, Max-Change: 0.04767
Iteration: 11, Log-Lik: -137116.491, Max-Change: 0.04559
Iteration: 12, Log-Lik: -137078.613, Max-Change: 0.04461
Iteration: 13, Log-Lik: -137043.622, Max-Change: 0.04362
Iteration: 14, Log-Lik: -137011.243, Max-Change: 0.04257
Iteration: 15, Log-Lik: -136981.190, Max-Change: 0.04187
Iteration: 16, Log-Lik: -136953.250, Max-Change: 0.03960
Iteration: 17, Log-Lik: -136927.331, Max-Change: 0.03947
Iteration: 18, Log-Lik: -136903.137, Max-Change: 0.03811
Iteration: 19, Log-Lik: -136880.697, Max-Change: 0.03667
Iteration: 20, Log-Lik: -136859.782, Max-Change: 0.03663
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Iteration: 137, Log-Lik: -136527.293, Max-Change: 0.00028
Iteration: 138, Log-Lik: -136527.293, Max-Change: 0.00022
Iteration: 139, Log-Lik: -136527.294, Max-Change: 0.00061
Iteration: 140, Log-Lik: -136527.309, Max-Change: 0.00031
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Iteration: 143, Log-Lik: -136527.316, Max-Change: 0.00021
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Iteration: 146, Log-Lik: -136527.329, Max-Change: 0.00023
Iteration: 147, Log-Lik: -136527.328, Max-Change: 0.00015
Iteration: 148, Log-Lik: -136527.328, Max-Change: 0.00028
Iteration: 149, Log-Lik: -136527.335, Max-Change: 0.00016
Iteration: 150, Log-Lik: -136527.335, Max-Change: 0.00012
Iteration: 151, Log-Lik: -136527.335, Max-Change: 0.00034
Iteration: 152, Log-Lik: -136527.344, Max-Change: 0.00017
Iteration: 153, Log-Lik: -136527.344, Max-Change: 0.00011
Iteration: 154, Log-Lik: -136527.344, Max-Change: 0.00020
Iteration: 155, Log-Lik: -136527.348, Max-Change: 0.00012
Iteration: 156, Log-Lik: -136527.348, Max-Change: 0.00009

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 8: Get scores

## Paragraph comprehension

## There are 8 steps

## Step 1: Initial joint fit

## 
Iteration: 1, Log-Lik: -72763.816, Max-Change: 0.83600
Iteration: 2, Log-Lik: -71150.120, Max-Change: 0.44535
Iteration: 3, Log-Lik: -70865.899, Max-Change: 0.26289
Iteration: 4, Log-Lik: -70779.817, Max-Change: 0.13437
Iteration: 5, Log-Lik: -70742.614, Max-Change: 0.09281
Iteration: 6, Log-Lik: -70725.284, Max-Change: 0.06719
Iteration: 7, Log-Lik: -70717.117, Max-Change: 0.03520
Iteration: 8, Log-Lik: -70712.199, Max-Change: 0.02701
Iteration: 9, Log-Lik: -70708.520, Max-Change: 0.01849
Iteration: 10, Log-Lik: -70706.855, Max-Change: 0.01294
Iteration: 11, Log-Lik: -70705.609, Max-Change: 0.00896
Iteration: 12, Log-Lik: -70704.695, Max-Change: 0.00698
Iteration: 13, Log-Lik: -70702.892, Max-Change: 0.00461
Iteration: 14, Log-Lik: -70702.805, Max-Change: 0.00246
Iteration: 15, Log-Lik: -70702.744, Max-Change: 0.00282
Iteration: 16, Log-Lik: -70702.616, Max-Change: 0.00099
Iteration: 17, Log-Lik: -70702.612, Max-Change: 0.00038
Iteration: 18, Log-Lik: -70702.609, Max-Change: 0.00075
Iteration: 19, Log-Lik: -70702.606, Max-Change: 0.00018
Iteration: 20, Log-Lik: -70702.605, Max-Change: 0.00012
Iteration: 21, Log-Lik: -70702.605, Max-Change: 0.00014
Iteration: 22, Log-Lik: -70702.603, Max-Change: 0.00017
Iteration: 23, Log-Lik: -70702.602, Max-Change: 0.00014
Iteration: 24, Log-Lik: -70702.602, Max-Change: 0.00012
Iteration: 25, Log-Lik: -70702.602, Max-Change: 0.00005

## 
## Step 2: Initial MI fit

## 
Iteration: 1, Log-Lik: -72763.816, Max-Change: 0.82214
Iteration: 2, Log-Lik: -70832.701, Max-Change: 0.39609
Iteration: 3, Log-Lik: -70543.198, Max-Change: 0.19339
Iteration: 4, Log-Lik: -70438.384, Max-Change: 0.09827
Iteration: 5, Log-Lik: -70365.914, Max-Change: 0.06651
Iteration: 6, Log-Lik: -70306.520, Max-Change: 0.07743
Iteration: 7, Log-Lik: -70255.935, Max-Change: 0.07936
Iteration: 8, Log-Lik: -70212.337, Max-Change: 0.07753
Iteration: 9, Log-Lik: -70174.531, Max-Change: 0.07381
Iteration: 10, Log-Lik: -70141.622, Max-Change: 0.07042
Iteration: 11, Log-Lik: -70112.853, Max-Change: 0.06607
Iteration: 12, Log-Lik: -70087.637, Max-Change: 0.06232
Iteration: 13, Log-Lik: -70065.462, Max-Change: 0.05859
Iteration: 14, Log-Lik: -70045.912, Max-Change: 0.05504
Iteration: 15, Log-Lik: -70028.630, Max-Change: 0.05171
Iteration: 16, Log-Lik: -70013.316, Max-Change: 0.04859
Iteration: 17, Log-Lik: -69999.717, Max-Change: 0.04562
Iteration: 18, Log-Lik: -69987.614, Max-Change: 0.04282
Iteration: 19, Log-Lik: -69976.820, Max-Change: 0.04019
Iteration: 20, Log-Lik: -69967.179, Max-Change: 0.03783
Iteration: 21, Log-Lik: -69958.549, Max-Change: 0.03549
Iteration: 22, Log-Lik: -69950.813, Max-Change: 0.03338
Iteration: 23, Log-Lik: -69943.865, Max-Change: 0.03130
Iteration: 24, Log-Lik: -69937.617, Max-Change: 0.02944
Iteration: 25, Log-Lik: -69931.991, Max-Change: 0.02769
Iteration: 26, Log-Lik: -69926.917, Max-Change: 0.02606
Iteration: 27, Log-Lik: -69922.335, Max-Change: 0.02451
Iteration: 28, Log-Lik: -69911.550, Max-Change: 0.05640
Iteration: 29, Log-Lik: -69902.475, Max-Change: 0.01344
Iteration: 30, Log-Lik: -69900.237, Max-Change: 0.01341
Iteration: 31, Log-Lik: -69895.597, Max-Change: 0.03309
Iteration: 32, Log-Lik: -69892.231, Max-Change: 0.01336
Iteration: 33, Log-Lik: -69890.881, Max-Change: 0.01087
Iteration: 34, Log-Lik: -69888.475, Max-Change: 0.02774
Iteration: 35, Log-Lik: -69886.043, Max-Change: 0.01359
Iteration: 36, Log-Lik: -69885.181, Max-Change: 0.01054
Iteration: 37, Log-Lik: -69883.748, Max-Change: 0.01552
Iteration: 38, Log-Lik: -69882.852, Max-Change: 0.01089
Iteration: 39, Log-Lik: -69882.285, Max-Change: 0.00892
Iteration: 40, Log-Lik: -69881.523, Max-Change: 0.01898
Iteration: 41, Log-Lik: -69880.325, Max-Change: 0.01138
Iteration: 42, Log-Lik: -69879.936, Max-Change: 0.00838
Iteration: 43, Log-Lik: -69879.416, Max-Change: 0.00948
Iteration: 44, Log-Lik: -69879.136, Max-Change: 0.00752
Iteration: 45, Log-Lik: -69878.892, Max-Change: 0.00641
Iteration: 46, Log-Lik: -69878.597, Max-Change: 0.01263
Iteration: 47, Log-Lik: -69878.091, Max-Change: 0.00836
Iteration: 48, Log-Lik: -69877.914, Max-Change: 0.00603
Iteration: 49, Log-Lik: -69877.700, Max-Change: 0.00651
Iteration: 50, Log-Lik: -69877.592, Max-Change: 0.00519
Iteration: 51, Log-Lik: -69877.487, Max-Change: 0.00446
Iteration: 52, Log-Lik: -69877.366, Max-Change: 0.00900
Iteration: 53, Log-Lik: -69877.169, Max-Change: 0.00581
Iteration: 54, Log-Lik: -69877.089, Max-Change: 0.00416
Iteration: 55, Log-Lik: -69876.998, Max-Change: 0.00443
Iteration: 56, Log-Lik: -69876.958, Max-Change: 0.00352
Iteration: 57, Log-Lik: -69876.914, Max-Change: 0.00303
Iteration: 58, Log-Lik: -69876.864, Max-Change: 0.00617
Iteration: 59, Log-Lik: -69876.797, Max-Change: 0.00394
Iteration: 60, Log-Lik: -69876.761, Max-Change: 0.00281
Iteration: 61, Log-Lik: -69876.722, Max-Change: 0.00297
Iteration: 62, Log-Lik: -69876.709, Max-Change: 0.00236
Iteration: 63, Log-Lik: -69876.691, Max-Change: 0.00203
Iteration: 64, Log-Lik: -69876.671, Max-Change: 0.00415
Iteration: 65, Log-Lik: -69876.653, Max-Change: 0.00263
Iteration: 66, Log-Lik: -69876.637, Max-Change: 0.00187
Iteration: 67, Log-Lik: -69876.621, Max-Change: 0.00198
Iteration: 68, Log-Lik: -69876.618, Max-Change: 0.00156
Iteration: 69, Log-Lik: -69876.611, Max-Change: 0.00135
Iteration: 70, Log-Lik: -69876.603, Max-Change: 0.00276
Iteration: 71, Log-Lik: -69876.603, Max-Change: 0.00174
Iteration: 72, Log-Lik: -69876.596, Max-Change: 0.00124
Iteration: 73, Log-Lik: -69876.589, Max-Change: 0.00130
Iteration: 74, Log-Lik: -69876.590, Max-Change: 0.00103
Iteration: 75, Log-Lik: -69876.587, Max-Change: 0.00089
Iteration: 76, Log-Lik: -69876.584, Max-Change: 0.00182
Iteration: 77, Log-Lik: -69876.590, Max-Change: 0.00114
Iteration: 78, Log-Lik: -69876.586, Max-Change: 0.00081
Iteration: 79, Log-Lik: -69876.583, Max-Change: 0.00085
Iteration: 80, Log-Lik: -69876.585, Max-Change: 0.00068
Iteration: 81, Log-Lik: -69876.584, Max-Change: 0.00058
Iteration: 82, Log-Lik: -69876.583, Max-Change: 0.00120
Iteration: 83, Log-Lik: -69876.588, Max-Change: 0.00075
Iteration: 84, Log-Lik: -69876.587, Max-Change: 0.00053
Iteration: 85, Log-Lik: -69876.586, Max-Change: 0.00056
Iteration: 86, Log-Lik: -69876.587, Max-Change: 0.00044
Iteration: 87, Log-Lik: -69876.587, Max-Change: 0.00038
Iteration: 88, Log-Lik: -69876.587, Max-Change: 0.00078
Iteration: 89, Log-Lik: -69876.591, Max-Change: 0.00049
Iteration: 90, Log-Lik: -69876.590, Max-Change: 0.00035
Iteration: 91, Log-Lik: -69876.590, Max-Change: 0.00037
Iteration: 92, Log-Lik: -69876.591, Max-Change: 0.00029
Iteration: 93, Log-Lik: -69876.591, Max-Change: 0.00025
Iteration: 94, Log-Lik: -69876.591, Max-Change: 0.00051
Iteration: 95, Log-Lik: -69876.595, Max-Change: 0.00032
Iteration: 96, Log-Lik: -69876.594, Max-Change: 0.00023
Iteration: 97, Log-Lik: -69876.594, Max-Change: 0.00024
Iteration: 98, Log-Lik: -69876.595, Max-Change: 0.00019
Iteration: 99, Log-Lik: -69876.595, Max-Change: 0.00016
Iteration: 100, Log-Lik: -69876.595, Max-Change: 0.00033
Iteration: 101, Log-Lik: -69876.597, Max-Change: 0.00021
Iteration: 102, Log-Lik: -69876.597, Max-Change: 0.00015
Iteration: 103, Log-Lik: -69876.597, Max-Change: 0.00016
Iteration: 104, Log-Lik: -69876.597, Max-Change: 0.00012
Iteration: 105, Log-Lik: -69876.598, Max-Change: 0.00011
Iteration: 106, Log-Lik: -69876.598, Max-Change: 0.00022
Iteration: 107, Log-Lik: -69876.599, Max-Change: 0.00014
Iteration: 108, Log-Lik: -69876.599, Max-Change: 0.00010

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 3: Leave one out MI testing

## 
## Step 4: Fit without DIF items, liberal threshold

## 
Iteration: 1, Log-Lik: -77381.066, Max-Change: 0.66616
Iteration: 2, Log-Lik: -76467.493, Max-Change: 0.37653
Iteration: 3, Log-Lik: -76227.874, Max-Change: 0.31099
Iteration: 4, Log-Lik: -76145.787, Max-Change: 0.19739
Iteration: 5, Log-Lik: -76116.741, Max-Change: 0.13652
Iteration: 6, Log-Lik: -76103.395, Max-Change: 0.08201
Iteration: 7, Log-Lik: -76098.080, Max-Change: 0.05426
Iteration: 8, Log-Lik: -76096.156, Max-Change: 0.03777
Iteration: 9, Log-Lik: -76095.110, Max-Change: 0.02329
Iteration: 10, Log-Lik: -76094.444, Max-Change: 0.01136
Iteration: 11, Log-Lik: -76094.233, Max-Change: 0.00558
Iteration: 12, Log-Lik: -76094.121, Max-Change: 0.00439
Iteration: 13, Log-Lik: -76093.957, Max-Change: 0.00070
Iteration: 14, Log-Lik: -76093.954, Max-Change: 0.00025
Iteration: 15, Log-Lik: -76093.953, Max-Change: 0.00023
Iteration: 16, Log-Lik: -76093.950, Max-Change: 0.00013
Iteration: 17, Log-Lik: -76093.950, Max-Change: 0.00012
Iteration: 18, Log-Lik: -76093.950, Max-Change: 0.00054
Iteration: 19, Log-Lik: -76093.949, Max-Change: 0.00025
Iteration: 20, Log-Lik: -76093.949, Max-Change: 0.00007

## 
## Step 5: Fit without DIF items, conservative threshold

## 
Iteration: 1, Log-Lik: -76936.031, Max-Change: 0.67793
Iteration: 2, Log-Lik: -75756.465, Max-Change: 0.40234
Iteration: 3, Log-Lik: -75480.251, Max-Change: 0.31125
Iteration: 4, Log-Lik: -75385.709, Max-Change: 0.20751
Iteration: 5, Log-Lik: -75359.553, Max-Change: 0.13247
Iteration: 6, Log-Lik: -75348.969, Max-Change: 0.08314
Iteration: 7, Log-Lik: -75344.572, Max-Change: 0.05410
Iteration: 8, Log-Lik: -75342.443, Max-Change: 0.03528
Iteration: 9, Log-Lik: -75341.332, Max-Change: 0.02299
Iteration: 10, Log-Lik: -75340.388, Max-Change: 0.00589
Iteration: 11, Log-Lik: -75340.208, Max-Change: 0.00473
Iteration: 12, Log-Lik: -75340.096, Max-Change: 0.00330
Iteration: 13, Log-Lik: -75339.953, Max-Change: 0.00140
Iteration: 14, Log-Lik: -75339.938, Max-Change: 0.00129
Iteration: 15, Log-Lik: -75339.932, Max-Change: 0.00090
Iteration: 16, Log-Lik: -75339.920, Max-Change: 0.00052
Iteration: 17, Log-Lik: -75339.919, Max-Change: 0.00017
Iteration: 18, Log-Lik: -75339.918, Max-Change: 0.00015
Iteration: 19, Log-Lik: -75339.916, Max-Change: 0.00011
Iteration: 20, Log-Lik: -75339.916, Max-Change: 0.00009

## 
## Step 6: Fit with anchor items, liberal threshold

## 
Iteration: 1, Log-Lik: -72763.816, Max-Change: 0.82213
Iteration: 2, Log-Lik: -70577.263, Max-Change: 0.38991
Iteration: 3, Log-Lik: -70302.964, Max-Change: 0.19302
Iteration: 4, Log-Lik: -70206.600, Max-Change: 0.10149
Iteration: 5, Log-Lik: -70141.847, Max-Change: 0.06157
Iteration: 6, Log-Lik: -70089.119, Max-Change: 0.06846
Iteration: 7, Log-Lik: -70044.089, Max-Change: 0.07179
Iteration: 8, Log-Lik: -70005.054, Max-Change: 0.07114
Iteration: 9, Log-Lik: -69970.989, Max-Change: 0.06844
Iteration: 10, Log-Lik: -69941.138, Max-Change: 0.06590
Iteration: 11, Log-Lik: -69914.901, Max-Change: 0.06239
Iteration: 12, Log-Lik: -69891.772, Max-Change: 0.05917
Iteration: 13, Log-Lik: -69871.330, Max-Change: 0.05587
Iteration: 14, Log-Lik: -69853.215, Max-Change: 0.05281
Iteration: 15, Log-Lik: -69837.127, Max-Change: 0.04986
Iteration: 16, Log-Lik: -69822.803, Max-Change: 0.04707
Iteration: 17, Log-Lik: -69810.024, Max-Change: 0.04444
Iteration: 18, Log-Lik: -69798.596, Max-Change: 0.04196
Iteration: 19, Log-Lik: -69788.354, Max-Change: 0.03963
Iteration: 20, Log-Lik: -69779.158, Max-Change: 0.03741
Iteration: 21, Log-Lik: -69770.883, Max-Change: 0.03529
Iteration: 22, Log-Lik: -69763.421, Max-Change: 0.03334
Iteration: 23, Log-Lik: -69756.680, Max-Change: 0.03151
Iteration: 24, Log-Lik: -69750.579, Max-Change: 0.02976
Iteration: 25, Log-Lik: -69745.046, Max-Change: 0.02809
Iteration: 26, Log-Lik: -69740.019, Max-Change: 0.02656
Iteration: 27, Log-Lik: -69735.444, Max-Change: 0.02509
Iteration: 28, Log-Lik: -69724.121, Max-Change: 0.05582
Iteration: 29, Log-Lik: -69714.880, Max-Change: 0.01425
Iteration: 30, Log-Lik: -69712.487, Max-Change: 0.01462
Iteration: 31, Log-Lik: -69707.458, Max-Change: 0.02884
Iteration: 32, Log-Lik: -69704.600, Max-Change: 0.01090
Iteration: 33, Log-Lik: -69703.007, Max-Change: 0.01086
Iteration: 34, Log-Lik: -69700.063, Max-Change: 0.03382
Iteration: 35, Log-Lik: -69696.392, Max-Change: 0.01309
Iteration: 36, Log-Lik: -69695.300, Max-Change: 0.01003
Iteration: 37, Log-Lik: -69693.503, Max-Change: 0.01504
Iteration: 38, Log-Lik: -69692.471, Max-Change: 0.01051
Iteration: 39, Log-Lik: -69691.670, Max-Change: 0.00908
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Iteration: 140, Log-Lik: -69679.488, Max-Change: 0.00011
Iteration: 141, Log-Lik: -69679.488, Max-Change: 0.00009

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 7: Fit with anchor items, conservative threshold

## 
Iteration: 1, Log-Lik: -72763.816, Max-Change: 0.82215
Iteration: 2, Log-Lik: -70615.201, Max-Change: 0.39170
Iteration: 3, Log-Lik: -70336.613, Max-Change: 0.19264
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## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 8: Get scores

## Mathematics knowledge

## There are 8 steps

## Step 1: Initial joint fit

## 
Iteration: 1, Log-Lik: -127518.405, Max-Change: 0.50923
Iteration: 2, Log-Lik: -125558.322, Max-Change: 0.28126
Iteration: 3, Log-Lik: -125188.541, Max-Change: 0.16871
Iteration: 4, Log-Lik: -125023.496, Max-Change: 0.11510
Iteration: 5, Log-Lik: -124947.932, Max-Change: 0.07884
Iteration: 6, Log-Lik: -124913.758, Max-Change: 0.06145
Iteration: 7, Log-Lik: -124896.724, Max-Change: 0.04340
Iteration: 8, Log-Lik: -124887.436, Max-Change: 0.02748
Iteration: 9, Log-Lik: -124882.874, Max-Change: 0.02121
Iteration: 10, Log-Lik: -124879.272, Max-Change: 0.01252
Iteration: 11, Log-Lik: -124878.161, Max-Change: 0.00733
Iteration: 12, Log-Lik: -124877.636, Max-Change: 0.00611
Iteration: 13, Log-Lik: -124876.909, Max-Change: 0.00275
Iteration: 14, Log-Lik: -124876.828, Max-Change: 0.00195
Iteration: 15, Log-Lik: -124876.768, Max-Change: 0.00108
Iteration: 16, Log-Lik: -124876.698, Max-Change: 0.00131
Iteration: 17, Log-Lik: -124876.668, Max-Change: 0.00081
Iteration: 18, Log-Lik: -124876.649, Max-Change: 0.00070
Iteration: 19, Log-Lik: -124876.610, Max-Change: 0.00073
Iteration: 20, Log-Lik: -124876.602, Max-Change: 0.00060
Iteration: 21, Log-Lik: -124876.594, Max-Change: 0.00042
Iteration: 22, Log-Lik: -124876.582, Max-Change: 0.00060
Iteration: 23, Log-Lik: -124876.579, Max-Change: 0.00022
Iteration: 24, Log-Lik: -124876.577, Max-Change: 0.00020
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Iteration: 26, Log-Lik: -124876.568, Max-Change: 0.00018
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Iteration: 28, Log-Lik: -124876.563, Max-Change: 0.00010
Iteration: 29, Log-Lik: -124876.563, Max-Change: 0.00010

## 
## Step 2: Initial MI fit

## 
Iteration: 1, Log-Lik: -127518.405, Max-Change: 0.64194
Iteration: 2, Log-Lik: -124828.619, Max-Change: 0.21298
Iteration: 3, Log-Lik: -124660.770, Max-Change: 0.08962
Iteration: 4, Log-Lik: -124605.564, Max-Change: 0.04191
Iteration: 5, Log-Lik: -124562.302, Max-Change: 0.04154
Iteration: 6, Log-Lik: -124523.980, Max-Change: 0.03996
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Iteration: 14, Log-Lik: -124322.540, Max-Change: 0.03056
Iteration: 15, Log-Lik: -124306.269, Max-Change: 0.02922
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Iteration: 17, Log-Lik: -124277.656, Max-Change: 0.02666
Iteration: 18, Log-Lik: -124265.071, Max-Change: 0.02545
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Iteration: 20, Log-Lik: -124242.834, Max-Change: 0.02321
Iteration: 21, Log-Lik: -124233.010, Max-Change: 0.02214
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Iteration: 23, Log-Lik: -124215.587, Max-Change: 0.02019
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Iteration: 90, Log-Lik: -124112.437, Max-Change: 0.00009

## 
## Step 3: Leave one out MI testing

## 
## Step 4: Fit without DIF items, liberal threshold

## 
Iteration: 1, Log-Lik: -146455.662, Max-Change: 0.09192
Iteration: 2, Log-Lik: -146402.961, Max-Change: 0.06168
Iteration: 3, Log-Lik: -146395.920, Max-Change: 0.05312
Iteration: 4, Log-Lik: -146391.085, Max-Change: 0.04224
Iteration: 5, Log-Lik: -146388.019, Max-Change: 0.03525
Iteration: 6, Log-Lik: -146385.988, Max-Change: 0.02858
Iteration: 7, Log-Lik: -146382.786, Max-Change: 0.00279
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Iteration: 9, Log-Lik: -146382.775, Max-Change: 0.00177
Iteration: 10, Log-Lik: -146382.773, Max-Change: 0.00039
Iteration: 11, Log-Lik: -146382.772, Max-Change: 0.00021
Iteration: 12, Log-Lik: -146382.771, Max-Change: 0.00080
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Iteration: 16, Log-Lik: -146382.770, Max-Change: 0.00012
Iteration: 17, Log-Lik: -146382.770, Max-Change: 0.00006

## 
## Step 5: Fit without DIF items, conservative threshold

## 
Iteration: 1, Log-Lik: -144809.528, Max-Change: 0.20117
Iteration: 2, Log-Lik: -144600.470, Max-Change: 0.16240
Iteration: 3, Log-Lik: -144526.711, Max-Change: 0.11073
Iteration: 4, Log-Lik: -144499.006, Max-Change: 0.05786
Iteration: 5, Log-Lik: -144491.401, Max-Change: 0.03702
Iteration: 6, Log-Lik: -144488.948, Max-Change: 0.02448
Iteration: 7, Log-Lik: -144487.648, Max-Change: 0.00813
Iteration: 8, Log-Lik: -144487.565, Max-Change: 0.00670
Iteration: 9, Log-Lik: -144487.528, Max-Change: 0.00262
Iteration: 10, Log-Lik: -144487.525, Max-Change: 0.00062
Iteration: 11, Log-Lik: -144487.522, Max-Change: 0.00052
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Iteration: 14, Log-Lik: -144487.517, Max-Change: 0.00014
Iteration: 15, Log-Lik: -144487.517, Max-Change: 0.00060
Iteration: 16, Log-Lik: -144487.517, Max-Change: 0.00009

## 
## Step 6: Fit with anchor items, liberal threshold

## 
Iteration: 1, Log-Lik: -127518.405, Max-Change: 0.80962
Iteration: 2, Log-Lik: -124137.855, Max-Change: 0.17789
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Iteration: 222, Log-Lik: -123612.666, Max-Change: 0.00037
Iteration: 223, Log-Lik: -123612.667, Max-Change: 0.00093
Iteration: 224, Log-Lik: -123612.675, Max-Change: 0.00043
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Iteration: 229, Log-Lik: -123612.692, Max-Change: 0.00085
Iteration: 230, Log-Lik: -123612.699, Max-Change: 0.00039
Iteration: 231, Log-Lik: -123612.701, Max-Change: 0.00032
Iteration: 232, Log-Lik: -123612.702, Max-Change: 0.00117
Iteration: 233, Log-Lik: -123612.712, Max-Change: 0.00043
Iteration: 234, Log-Lik: -123612.714, Max-Change: 0.00031
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Iteration: 300, Log-Lik: -123612.886, Max-Change: 0.00012
Iteration: 301, Log-Lik: -123612.887, Max-Change: 0.00029
Iteration: 302, Log-Lik: -123612.889, Max-Change: 0.00013
Iteration: 303, Log-Lik: -123612.890, Max-Change: 0.00011
Iteration: 304, Log-Lik: -123612.891, Max-Change: 0.00040
Iteration: 305, Log-Lik: -123612.895, Max-Change: 0.00015
Iteration: 306, Log-Lik: -123612.896, Max-Change: 0.00011
Iteration: 307, Log-Lik: -123612.896, Max-Change: 0.00025
Iteration: 308, Log-Lik: -123612.899, Max-Change: 0.00012
Iteration: 309, Log-Lik: -123612.899, Max-Change: 0.00010

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 7: Fit with anchor items, conservative threshold

## 
Iteration: 1, Log-Lik: -127518.405, Max-Change: 0.80937
Iteration: 2, Log-Lik: -124199.383, Max-Change: 0.17558
Iteration: 3, Log-Lik: -124016.117, Max-Change: 0.08221
Iteration: 4, Log-Lik: -123964.305, Max-Change: 0.05364
Iteration: 5, Log-Lik: -123934.679, Max-Change: 0.04097
Iteration: 6, Log-Lik: -123912.662, Max-Change: 0.03246
Iteration: 7, Log-Lik: -123894.165, Max-Change: 0.03057
Iteration: 8, Log-Lik: -123877.719, Max-Change: 0.03130
Iteration: 9, Log-Lik: -123862.709, Max-Change: 0.03092
Iteration: 10, Log-Lik: -123848.859, Max-Change: 0.03048
Iteration: 11, Log-Lik: -123836.010, Max-Change: 0.02958
Iteration: 12, Log-Lik: -123824.062, Max-Change: 0.02888
Iteration: 13, Log-Lik: -123812.938, Max-Change: 0.02801
Iteration: 14, Log-Lik: -123802.577, Max-Change: 0.02713
Iteration: 15, Log-Lik: -123792.916, Max-Change: 0.02641
Iteration: 16, Log-Lik: -123783.899, Max-Change: 0.02541
Iteration: 17, Log-Lik: -123775.486, Max-Change: 0.02455
Iteration: 18, Log-Lik: -123767.631, Max-Change: 0.02399
Iteration: 19, Log-Lik: -123760.290, Max-Change: 0.02305
Iteration: 20, Log-Lik: -123753.431, Max-Change: 0.02228
Iteration: 21, Log-Lik: -123747.014, Max-Change: 0.02153
Iteration: 22, Log-Lik: -123741.011, Max-Change: 0.02076
Iteration: 23, Log-Lik: -123735.390, Max-Change: 0.02020
Iteration: 24, Log-Lik: -123730.126, Max-Change: 0.01945
Iteration: 25, Log-Lik: -123725.192, Max-Change: 0.01878
Iteration: 26, Log-Lik: -123720.564, Max-Change: 0.01817
Iteration: 27, Log-Lik: -123716.221, Max-Change: 0.01760
Iteration: 28, Log-Lik: -123698.790, Max-Change: 0.05452
Iteration: 29, Log-Lik: -123692.430, Max-Change: 0.01433
Iteration: 30, Log-Lik: -123689.790, Max-Change: 0.01320
Iteration: 31, Log-Lik: -123679.200, Max-Change: 0.04405
Iteration: 32, Log-Lik: -123675.176, Max-Change: 0.01102
Iteration: 33, Log-Lik: -123673.497, Max-Change: 0.00992
Iteration: 34, Log-Lik: -123666.749, Max-Change: 0.03580
Iteration: 35, Log-Lik: -123664.105, Max-Change: 0.00872
Iteration: 36, Log-Lik: -123662.984, Max-Change: 0.00747
Iteration: 37, Log-Lik: -123658.549, Max-Change: 0.02973
Iteration: 38, Log-Lik: -123656.694, Max-Change: 0.00951
Iteration: 39, Log-Lik: -123655.914, Max-Change: 0.00686
Iteration: 40, Log-Lik: -123652.896, Max-Change: 0.02488
Iteration: 41, Log-Lik: -123651.545, Max-Change: 0.00976
Iteration: 42, Log-Lik: -123650.985, Max-Change: 0.00703
Iteration: 43, Log-Lik: -123648.919, Max-Change: 0.02313
Iteration: 44, Log-Lik: -123647.991, Max-Change: 0.00941
Iteration: 45, Log-Lik: -123647.574, Max-Change: 0.00689
Iteration: 46, Log-Lik: -123646.102, Max-Change: 0.02209
Iteration: 47, Log-Lik: -123645.442, Max-Change: 0.00887
Iteration: 48, Log-Lik: -123645.126, Max-Change: 0.00658
Iteration: 49, Log-Lik: -123644.042, Max-Change: 0.02082
Iteration: 50, Log-Lik: -123643.551, Max-Change: 0.00825
Iteration: 51, Log-Lik: -123643.308, Max-Change: 0.00615
Iteration: 52, Log-Lik: -123642.502, Max-Change: 0.01912
Iteration: 53, Log-Lik: -123642.137, Max-Change: 0.00754
Iteration: 54, Log-Lik: -123641.950, Max-Change: 0.00566
Iteration: 55, Log-Lik: -123641.340, Max-Change: 0.01759
Iteration: 56, Log-Lik: -123641.060, Max-Change: 0.00685
Iteration: 57, Log-Lik: -123640.916, Max-Change: 0.00515
Iteration: 58, Log-Lik: -123640.458, Max-Change: 0.01570
Iteration: 59, Log-Lik: -123640.249, Max-Change: 0.00613
Iteration: 60, Log-Lik: -123640.137, Max-Change: 0.00463
Iteration: 61, Log-Lik: -123639.788, Max-Change: 0.01425
Iteration: 62, Log-Lik: -123639.625, Max-Change: 0.00548
Iteration: 63, Log-Lik: -123639.539, Max-Change: 0.00413
Iteration: 64, Log-Lik: -123639.277, Max-Change: 0.01240
Iteration: 65, Log-Lik: -123639.158, Max-Change: 0.00481
Iteration: 66, Log-Lik: -123639.091, Max-Change: 0.00365
Iteration: 67, Log-Lik: -123638.891, Max-Change: 0.01122
Iteration: 68, Log-Lik: -123638.796, Max-Change: 0.00427
Iteration: 69, Log-Lik: -123638.745, Max-Change: 0.00321
Iteration: 70, Log-Lik: -123638.597, Max-Change: 0.00952
Iteration: 71, Log-Lik: -123638.529, Max-Change: 0.00369
Iteration: 72, Log-Lik: -123638.489, Max-Change: 0.00280
Iteration: 73, Log-Lik: -123638.375, Max-Change: 0.00868
Iteration: 74, Log-Lik: -123638.319, Max-Change: 0.00326
Iteration: 75, Log-Lik: -123638.289, Max-Change: 0.00244
Iteration: 76, Log-Lik: -123638.206, Max-Change: 0.00710
Iteration: 77, Log-Lik: -123638.168, Max-Change: 0.00277
Iteration: 78, Log-Lik: -123638.144, Max-Change: 0.00211
Iteration: 79, Log-Lik: -123638.079, Max-Change: 0.00661
Iteration: 80, Log-Lik: -123638.047, Max-Change: 0.00246
Iteration: 81, Log-Lik: -123638.029, Max-Change: 0.00183
Iteration: 82, Log-Lik: -123637.983, Max-Change: 0.00517
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Iteration: 85, Log-Lik: -123637.911, Max-Change: 0.00509
Iteration: 86, Log-Lik: -123637.891, Max-Change: 0.00184
Iteration: 87, Log-Lik: -123637.880, Max-Change: 0.00136
Iteration: 88, Log-Lik: -123637.855, Max-Change: 0.00369
Iteration: 89, Log-Lik: -123637.843, Max-Change: 0.00149
Iteration: 90, Log-Lik: -123637.835, Max-Change: 0.00116
Iteration: 91, Log-Lik: -123637.814, Max-Change: 0.00398
Iteration: 92, Log-Lik: -123637.801, Max-Change: 0.00139
Iteration: 93, Log-Lik: -123637.794, Max-Change: 0.00100
Iteration: 94, Log-Lik: -123637.781, Max-Change: 0.00251
Iteration: 95, Log-Lik: -123637.775, Max-Change: 0.00107
Iteration: 96, Log-Lik: -123637.770, Max-Change: 0.00085
Iteration: 97, Log-Lik: -123637.758, Max-Change: 0.00313
Iteration: 98, Log-Lik: -123637.750, Max-Change: 0.00104
Iteration: 99, Log-Lik: -123637.745, Max-Change: 0.00073
Iteration: 100, Log-Lik: -123637.738, Max-Change: 0.00169
Iteration: 101, Log-Lik: -123637.735, Max-Change: 0.00076
Iteration: 102, Log-Lik: -123637.732, Max-Change: 0.00062
Iteration: 103, Log-Lik: -123637.725, Max-Change: 0.00229
Iteration: 104, Log-Lik: -123637.720, Max-Change: 0.00076
Iteration: 105, Log-Lik: -123637.718, Max-Change: 0.00054
Iteration: 106, Log-Lik: -123637.714, Max-Change: 0.00124
Iteration: 107, Log-Lik: -123637.712, Max-Change: 0.00056
Iteration: 108, Log-Lik: -123637.710, Max-Change: 0.00045
Iteration: 109, Log-Lik: -123637.706, Max-Change: 0.00167
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Iteration: 112, Log-Lik: -123637.699, Max-Change: 0.00089
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Iteration: 114, Log-Lik: -123637.697, Max-Change: 0.00033
Iteration: 115, Log-Lik: -123637.694, Max-Change: 0.00122
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Iteration: 117, Log-Lik: -123637.692, Max-Change: 0.00028
Iteration: 118, Log-Lik: -123637.690, Max-Change: 0.00063
Iteration: 119, Log-Lik: -123637.689, Max-Change: 0.00029
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Iteration: 121, Log-Lik: -123637.687, Max-Change: 0.00089
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Iteration: 124, Log-Lik: -123637.685, Max-Change: 0.00049
Iteration: 125, Log-Lik: -123637.684, Max-Change: 0.00022
Iteration: 126, Log-Lik: -123637.684, Max-Change: 0.00017
Iteration: 127, Log-Lik: -123637.683, Max-Change: 0.00063
Iteration: 128, Log-Lik: -123637.682, Max-Change: 0.00022
Iteration: 129, Log-Lik: -123637.682, Max-Change: 0.00015
Iteration: 130, Log-Lik: -123637.681, Max-Change: 0.00032
Iteration: 131, Log-Lik: -123637.681, Max-Change: 0.00015
Iteration: 132, Log-Lik: -123637.680, Max-Change: 0.00013
Iteration: 133, Log-Lik: -123637.680, Max-Change: 0.00046
Iteration: 134, Log-Lik: -123637.680, Max-Change: 0.00016
Iteration: 135, Log-Lik: -123637.679, Max-Change: 0.00011
Iteration: 136, Log-Lik: -123637.679, Max-Change: 0.00022
Iteration: 137, Log-Lik: -123637.679, Max-Change: 0.00011
Iteration: 138, Log-Lik: -123637.678, Max-Change: 0.00009

## 
## Step 8: Get scores

#more and after test level
map(nlsy_1g_dif_by_test_bw, ~.$effect_size_test)

## $`Arithmetic reasoning`
## $`Arithmetic reasoning`$liberal
##           Effect Size  Value
## 1                STDS 0.5133
## 2                UTDS 1.2149
## 3              UETSDS 0.7984
## 4               ETSSD 0.0813
## 5         Starks.DTFR 0.5716
## 6               UDTFR 1.1410
## 7              UETSDN 0.7338
## 8 theta.of.max.test.D 3.3963
## 9           Test.Dmax 1.7068
## 
## $`Arithmetic reasoning`$conservative
##           Effect Size  Value
## 1                STDS 0.3559
## 2                UTDS 0.9713
## 3              UETSDS 0.6173
## 4               ETSSD 0.0557
## 5         Starks.DTFR 0.3893
## 6               UDTFR 0.8835
## 7              UETSDN 0.5458
## 8 theta.of.max.test.D 3.5194
## 9           Test.Dmax 1.3262
## 
## 
## $`Word knowledge`
## $`Word knowledge`$liberal
##           Effect Size   Value
## 1                STDS  0.1031
## 2                UTDS  1.2816
## 3              UETSDS  0.1973
## 4               ETSSD  0.0161
## 5         Starks.DTFR  0.1075
## 6               UDTFR  1.2863
## 7              UETSDN  0.1832
## 8 theta.of.max.test.D -1.6994
## 9           Test.Dmax  0.6248
## 
## $`Word knowledge`$conservative
##           Effect Size    Value
## 1                STDS -0.02132
## 2                UTDS  1.19450
## 3              UETSDS  0.28257
## 4               ETSSD -0.00338
## 5         Starks.DTFR  0.00588
## 6               UDTFR  1.20401
## 7              UETSDN  0.24747
## 8 theta.of.max.test.D -0.32500
## 9           Test.Dmax -0.69171
## 
## 
## $`Paragraph comprehension`
## $`Paragraph comprehension`$liberal
##           Effect Size   Value
## 1                STDS -0.0716
## 2                UTDS  0.5428
## 3              UETSDS  0.0777
## 4               ETSSD -0.0279
## 5         Starks.DTFR -0.0820
## 6               UDTFR  0.5651
## 7              UETSDN  0.0847
## 8 theta.of.max.test.D -2.1093
## 9           Test.Dmax  0.1903
## 
## $`Paragraph comprehension`$conservative
##           Effect Size   Value
## 1                STDS -0.0652
## 2                UTDS  0.4440
## 3              UETSDS  0.0713
## 4               ETSSD -0.0255
## 5         Starks.DTFR -0.0785
## 6               UDTFR  0.4671
## 7              UETSDN  0.0816
## 8 theta.of.max.test.D -2.1242
## 9           Test.Dmax  0.1782
## 
## 
## $`Mathematics knowledge`
## $`Mathematics knowledge`$liberal
##           Effect Size  Value
## 1                STDS 0.4088
## 2                UTDS 1.3997
## 3              UETSDS 0.7106
## 4               ETSSD 0.0755
## 5         Starks.DTFR 0.4978
## 6               UDTFR 1.3896
## 7              UETSDN 0.7032
## 8 theta.of.max.test.D 2.7713
## 9           Test.Dmax 1.4609
## 
## $`Mathematics knowledge`$conservative
##           Effect Size  Value
## 1                STDS 0.2175
## 2                UTDS 1.1266
## 3              UETSDS 0.2214
## 4               ETSSD 0.0385
## 5         Starks.DTFR 0.1769
## 6               UDTFR 1.1218
## 7              UETSDN 0.1814
## 8 theta.of.max.test.D 3.5506
## 9           Test.Dmax 0.7048

#gap sizes by test and scoring method
nlsy_fit_g_gaps_by_test_bw = map(nlsy_1g_dif_by_test_bw, function(x) {
  (describeBy(map_df(x$scores, ~.[, 1] %>% standardize(focal_group = nlsy_bw$SIRE == "White")), group = nlsy_bw$SIRE))
})
nlsy_fit_g_gaps_by_test_bw

## $`Arithmetic reasoning`
## 
##  Descriptive statistics by group 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 6352    0  1  -0.12   -0.04 1.11 -2.75 2.02  4.78
## noDIF_liberal          2 6352    0  1   0.06    0.08 1.39 -2.58 1.19  3.77
## noDIF_conservative     3 6352    0  1   0.01    0.03 1.20 -2.77 1.50  4.27
## anchor_liberal         4 6352    0  1  -0.13   -0.04 1.11 -2.73 2.02  4.74
## anchor_conservative    5 6352    0  1  -0.13   -0.04 1.11 -2.72 2.01  4.73
##                      skew kurtosis   se
## original             0.30    -0.77 0.01
## noDIF_liberal       -0.45    -0.83 0.01
## noDIF_conservative  -0.15    -0.93 0.01
## anchor_liberal       0.32    -0.79 0.01
## anchor_conservative  0.32    -0.79 0.01
## ------------------------------------------------------------ 
## group: Black
##                     vars    n  mean   sd median trimmed  mad   min  max range
## original               1 2774 -0.91 0.68  -1.06   -0.99 0.53 -3.03 2.02  5.05
## noDIF_liberal          2 2774 -0.90 0.88  -1.02   -0.96 0.83 -2.58 1.19  3.77
## noDIF_conservative     3 2774 -0.93 0.81  -1.06   -1.00 0.73 -2.77 1.50  4.27
## anchor_liberal         4 2774 -1.00 0.62  -1.13   -1.06 0.53 -2.70 1.21  3.92
## anchor_conservative    5 2774 -1.00 0.60  -1.12   -1.06 0.52 -2.67 1.17  3.85
##                     skew kurtosis   se
## original            1.24     2.27 0.01
## noDIF_liberal       0.51    -0.21 0.02
## noDIF_conservative  0.70     0.43 0.02
## anchor_liberal      0.92     0.79 0.01
## anchor_conservative 0.92     0.81 0.01
## 
## $`Word knowledge`
## 
##  Descriptive statistics by group 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 6352    0  1   0.01    0.01 1.10 -2.98 1.70  4.69
## noDIF_liberal          2 6352    0  1   0.27    0.12 1.14 -2.60 1.04  3.64
## noDIF_conservative     3 6352    0  1   0.21    0.10 1.20 -3.07 1.17  4.24
## anchor_liberal         4 6352    0  1   0.01    0.01 1.11 -2.90 1.70  4.60
## anchor_conservative    5 6352    0  1   0.01    0.01 1.11 -2.93 1.70  4.63
##                      skew kurtosis   se
## original            -0.10    -0.72 0.01
## noDIF_liberal       -0.68    -0.44 0.01
## noDIF_conservative  -0.66    -0.36 0.01
## anchor_liberal      -0.08    -0.76 0.01
## anchor_conservative -0.08    -0.76 0.01
## ------------------------------------------------------------ 
## group: Black
##                     vars    n  mean   sd median trimmed  mad   min  max range
## original               1 2774 -1.04 0.87  -1.18   -1.12 0.80 -3.27 1.70  4.98
## noDIF_liberal          2 2774 -0.88 1.04  -0.98   -0.91 1.10 -2.60 1.04  3.64
## noDIF_conservative     3 2774 -1.05 1.05  -1.21   -1.10 1.08 -3.07 1.17  4.24
## anchor_liberal         4 2774 -1.10 0.74  -1.20   -1.15 0.75 -2.92 0.86  3.78
## anchor_conservative    5 2774 -1.14 0.75  -1.24   -1.19 0.76 -2.98 0.86  3.84
##                     skew kurtosis   se
## original            0.84     0.53 0.02
## noDIF_liberal       0.23    -0.76 0.02
## noDIF_conservative  0.44    -0.58 0.02
## anchor_liberal      0.53    -0.26 0.01
## anchor_conservative 0.53    -0.26 0.01
## 
## $`Paragraph comprehension`
## 
##  Descriptive statistics by group 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 6352    0  1   0.15    0.07 0.96 -3.31 1.37  4.68
## noDIF_liberal          2 6352    0  1   0.28    0.13 1.10 -2.96 1.02  3.98
## noDIF_conservative     3 6352    0  1   0.21    0.10 1.21 -2.99 1.17  4.17
## anchor_liberal         4 6352    0  1   0.14    0.07 1.01 -3.16 1.37  4.53
## anchor_conservative    5 6352    0  1   0.14    0.07 1.01 -3.16 1.37  4.53
##                      skew kurtosis   se
## original            -0.56    -0.42 0.01
## noDIF_liberal       -0.78    -0.25 0.01
## noDIF_conservative  -0.68    -0.36 0.01
## anchor_liberal      -0.53    -0.50 0.01
## anchor_conservative -0.53    -0.51 0.01
## ------------------------------------------------------------ 
## group: Black
##                     vars    n  mean   sd median trimmed  mad   min  max range
## original               1 2774 -0.97 0.99  -1.05   -1.00 1.10 -3.31 1.37  4.68
## noDIF_liberal          2 2774 -0.94 1.08  -0.99   -0.98 1.27 -2.96 1.02  3.98
## noDIF_conservative     3 2774 -0.95 1.04  -1.05   -0.98 1.16 -2.99 1.17  4.17
## anchor_liberal         4 2774 -1.21 0.81  -1.22   -1.22 0.95 -3.17 0.48  3.65
## anchor_conservative    5 2774 -1.22 0.80  -1.23   -1.23 0.95 -3.14 0.44  3.58
##                     skew kurtosis   se
## original            0.30    -0.60 0.02
## noDIF_liberal       0.20    -0.86 0.02
## noDIF_conservative  0.23    -0.76 0.02
## anchor_liberal      0.08    -0.85 0.02
## anchor_conservative 0.08    -0.84 0.02
## 
## $`Mathematics knowledge`
## 
##  Descriptive statistics by group 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 6352    0  1  -0.18   -0.06 1.07 -2.46 2.10  4.55
## noDIF_liberal          2 6352    0  1   0.85    0.15 0.00 -2.05 0.85  2.90
## noDIF_conservative     3 6352    0  1   0.08    0.05 1.12 -2.28 1.36  3.64
## anchor_liberal         4 6352    0  1  -0.19   -0.06 1.05 -2.45 2.14  4.59
## anchor_conservative    5 6352    0  1  -0.19   -0.06 1.05 -2.45 2.12  4.57
##                      skew kurtosis   se
## original             0.42    -0.74 0.01
## noDIF_liberal       -0.75    -0.66 0.01
## noDIF_conservative  -0.19    -0.96 0.01
## anchor_liberal       0.47    -0.69 0.01
## anchor_conservative  0.45    -0.71 0.01
## ------------------------------------------------------------ 
## group: Black
##                     vars    n  mean   sd median trimmed  mad   min  max range
## original               1 2774 -0.72 0.74  -0.87   -0.81 0.57 -2.63 2.10  4.73
## noDIF_liberal          2 2774 -0.46 1.08  -0.61   -0.42 2.15 -2.05 0.85  2.90
## noDIF_conservative     3 2774 -0.63 0.91  -0.58   -0.67 0.85 -2.28 1.36  3.64
## anchor_liberal         4 2774 -0.82 0.72  -0.97   -0.89 0.59 -2.53 1.56  4.09
## anchor_conservative    5 2774 -0.77 0.63  -0.90   -0.84 0.51 -2.27 1.31  3.58
##                      skew kurtosis   se
## original             1.19     1.75 0.01
## noDIF_liberal       -0.15    -1.21 0.02
## noDIF_conservative   0.34    -0.41 0.02
## anchor_liberal       0.98     0.80 0.01
## anchor_conservative  0.99     0.85 0.01

#number of DIF items by test
map(nlsy_1g_dif_by_test_bw, function(x) {
  x$DIF_stats %>% select(p, p_adj) %>% {colSums(. < .05)}
})

## $`Arithmetic reasoning`
##     p p_adj 
##    23    17 
## 
## $`Word knowledge`
##     p p_adj 
##    30    25 
## 
## $`Paragraph comprehension`
##     p p_adj 
##     8     6 
## 
## $`Mathematics knowledge`
##     p p_adj 
##    23    19

#use non-DIF items from each of the 4 testings
nlsy_1g_dif_by_test_bw_noDIF_items = map(nlsy_1g_dif_by_test_bw, ~.$DIF_stats %>% filter(p_adj > .05) %>% pull(item)) %>% do.call(what = c)
length(nlsy_1g_dif_by_test_bw_noDIF_items)

## [1] 38

#fit again
nlsy_1g_dif_by_test_bw_noDIF_items_fit = mirt(nlsy_items_bw[nlsy_1g_dif_by_test_bw_noDIF_items], model = 1)

## 
Iteration: 1, Log-Lik: -180673.363, Max-Change: 0.91765
Iteration: 2, Log-Lik: -176823.629, Max-Change: 0.73741
Iteration: 3, Log-Lik: -176447.211, Max-Change: 0.25771
Iteration: 4, Log-Lik: -176234.660, Max-Change: 0.21884
Iteration: 5, Log-Lik: -176121.572, Max-Change: 0.15930
Iteration: 6, Log-Lik: -176046.838, Max-Change: 0.17543
Iteration: 7, Log-Lik: -175998.523, Max-Change: 0.08967
Iteration: 8, Log-Lik: -175963.972, Max-Change: 0.08703
Iteration: 9, Log-Lik: -175940.711, Max-Change: 0.07809
Iteration: 10, Log-Lik: -175923.995, Max-Change: 0.07592
Iteration: 11, Log-Lik: -175911.512, Max-Change: 0.03811
Iteration: 12, Log-Lik: -175901.148, Max-Change: 0.04854
Iteration: 13, Log-Lik: -175893.128, Max-Change: 0.05054
Iteration: 14, Log-Lik: -175886.111, Max-Change: 0.04358
Iteration: 15, Log-Lik: -175880.144, Max-Change: 0.04751
Iteration: 16, Log-Lik: -175874.960, Max-Change: 0.03824
Iteration: 17, Log-Lik: -175870.334, Max-Change: 0.04178
Iteration: 18, Log-Lik: -175866.151, Max-Change: 0.03369
Iteration: 19, Log-Lik: -175863.727, Max-Change: 0.01976
Iteration: 20, Log-Lik: -175859.008, Max-Change: 0.00991
Iteration: 21, Log-Lik: -175855.087, Max-Change: 0.00954
Iteration: 22, Log-Lik: -175838.467, Max-Change: 0.01261
Iteration: 23, Log-Lik: -175837.229, Max-Change: 0.00697
Iteration: 24, Log-Lik: -175836.157, Max-Change: 0.00669
Iteration: 25, Log-Lik: -175831.508, Max-Change: 0.00515
Iteration: 26, Log-Lik: -175831.118, Max-Change: 0.00444
Iteration: 27, Log-Lik: -175830.778, Max-Change: 0.00415
Iteration: 28, Log-Lik: -175829.285, Max-Change: 0.00168
Iteration: 29, Log-Lik: -175829.182, Max-Change: 0.00248
Iteration: 30, Log-Lik: -175829.080, Max-Change: 0.00234
Iteration: 31, Log-Lik: -175828.628, Max-Change: 0.00106
Iteration: 32, Log-Lik: -175828.597, Max-Change: 0.00134
Iteration: 33, Log-Lik: -175828.566, Max-Change: 0.00115
Iteration: 34, Log-Lik: -175828.505, Max-Change: 0.00076
Iteration: 35, Log-Lik: -175828.482, Max-Change: 0.00093
Iteration: 36, Log-Lik: -175828.467, Max-Change: 0.00077
Iteration: 37, Log-Lik: -175828.397, Max-Change: 0.00069
Iteration: 38, Log-Lik: -175828.391, Max-Change: 0.00055
Iteration: 39, Log-Lik: -175828.386, Max-Change: 0.00020
Iteration: 40, Log-Lik: -175828.383, Max-Change: 0.00019
Iteration: 41, Log-Lik: -175828.380, Max-Change: 0.00020
Iteration: 42, Log-Lik: -175828.377, Max-Change: 0.00020
Iteration: 43, Log-Lik: -175828.365, Max-Change: 0.00019
Iteration: 44, Log-Lik: -175828.363, Max-Change: 0.00018
Iteration: 45, Log-Lik: -175828.362, Max-Change: 0.00017
Iteration: 46, Log-Lik: -175828.354, Max-Change: 0.00013
Iteration: 47, Log-Lik: -175828.353, Max-Change: 0.00013
Iteration: 48, Log-Lik: -175828.353, Max-Change: 0.00013
Iteration: 49, Log-Lik: -175828.348, Max-Change: 0.00012
Iteration: 50, Log-Lik: -175828.348, Max-Change: 0.00012
Iteration: 51, Log-Lik: -175828.347, Max-Change: 0.00011
Iteration: 52, Log-Lik: -175828.345, Max-Change: 0.00009

nlsy_1g_dif_by_test_bw_noDIF_items_fit_scores = fscores(nlsy_1g_dif_by_test_bw_noDIF_items_fit, full.scores = T, full.scores.SE = T)

#gap
nlsy_bw$g_noDIF2 = nlsy_1g_dif_by_test_bw_noDIF_items_fit_scores[, 1] %>% as.vector() %>% standardize(focal_group = (nlsy_bw$SIRE == "White"))

#gap
GG_denhist(nlsy_bw, "g_noDIF2", "SIRE") +
  scale_fill_discrete("Race")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

describeBy(nlsy_bw$g_noDIF2, nlsy_bw$SIRE)

## 
##  Descriptive statistics by group 
## group: White
##    vars    n mean sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 6352    0  1  -0.04   -0.01 1.08 -2.72 2.01  4.74 0.05    -0.72 0.01
## ------------------------------------------------------------ 
## group: Black
##    vars    n  mean   sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 2774 -1.01 0.81  -1.18   -1.09 0.74 -2.73 2.01  4.75 0.93     0.71 0.02

#reliability
empirical_rxx(nlsy_1g_dif_by_test_bw_noDIF_items_fit_scores)

##    F1 
## 0.902

marginal_rxx(nlsy_1g_dif_by_test_bw_noDIF_items_fit)

## [1] 0.898

Hispanics

#read from cache if available
#Loop over each scale, and do DIF inside it.
nlsy_1g_dif_by_test_hw = cache_object(
  {
    nlsy_1g_dif_by_test_hw = list()
    for (test_i in unique(nlsy_item_stats$test)) {
      message(test_i)
      test_i_items = nlsy_items_hw[nlsy_item_stats %>% filter(test == test_i) %>% pull(item)]
      
      #fit
      nlsy_1g_dif_by_test_hw[[test_i]] = DIF_test(
        test_i_items,
        model = 1,
        group = (nlsy_hw$SIRE == "White")
        )
      
    }
    
    nlsy_1g_dif_by_test_hw
  },
  filename = "cache/nlsy_1g_dif_by_test_hw.rds", renew = F
)

## Cache not found, evaluating expression

## Arithmetic reasoning

## There are 8 steps

## Step 1: Initial joint fit

## 
Iteration: 1, Log-Lik: -125406.165, Max-Change: 0.80035
Iteration: 2, Log-Lik: -123467.365, Max-Change: 0.55100
Iteration: 3, Log-Lik: -122996.946, Max-Change: 0.22161
Iteration: 4, Log-Lik: -122760.695, Max-Change: 0.13689
Iteration: 5, Log-Lik: -122647.654, Max-Change: 0.11423
Iteration: 6, Log-Lik: -122576.143, Max-Change: 0.08093
Iteration: 7, Log-Lik: -122533.992, Max-Change: 0.05792
Iteration: 8, Log-Lik: -122509.399, Max-Change: 0.05069
Iteration: 9, Log-Lik: -122493.029, Max-Change: 0.03617
Iteration: 10, Log-Lik: -122482.785, Max-Change: 0.04292
Iteration: 11, Log-Lik: -122474.933, Max-Change: 0.02558
Iteration: 12, Log-Lik: -122469.423, Max-Change: 0.02306
Iteration: 13, Log-Lik: -122465.377, Max-Change: 0.02709
Iteration: 14, Log-Lik: -122462.190, Max-Change: 0.02244
Iteration: 15, Log-Lik: -122459.644, Max-Change: 0.01741
Iteration: 16, Log-Lik: -122457.517, Max-Change: 0.00863
Iteration: 17, Log-Lik: -122455.790, Max-Change: 0.00882
Iteration: 18, Log-Lik: -122454.422, Max-Change: 0.00698
Iteration: 19, Log-Lik: -122452.263, Max-Change: 0.00718
Iteration: 20, Log-Lik: -122451.370, Max-Change: 0.00610
Iteration: 21, Log-Lik: -122450.618, Max-Change: 0.00574
Iteration: 22, Log-Lik: -122447.572, Max-Change: 0.00430
Iteration: 23, Log-Lik: -122447.379, Max-Change: 0.00326
Iteration: 24, Log-Lik: -122447.218, Max-Change: 0.00315
Iteration: 25, Log-Lik: -122446.537, Max-Change: 0.00188
Iteration: 26, Log-Lik: -122446.491, Max-Change: 0.00169
Iteration: 27, Log-Lik: -122446.453, Max-Change: 0.00155
Iteration: 28, Log-Lik: -122446.290, Max-Change: 0.00081
Iteration: 29, Log-Lik: -122446.282, Max-Change: 0.00063
Iteration: 30, Log-Lik: -122446.274, Max-Change: 0.00060
Iteration: 31, Log-Lik: -122446.238, Max-Change: 0.00021
Iteration: 32, Log-Lik: -122446.236, Max-Change: 0.00017
Iteration: 33, Log-Lik: -122446.234, Max-Change: 0.00018
Iteration: 34, Log-Lik: -122446.226, Max-Change: 0.00019
Iteration: 35, Log-Lik: -122446.225, Max-Change: 0.00017
Iteration: 36, Log-Lik: -122446.224, Max-Change: 0.00016
Iteration: 37, Log-Lik: -122446.220, Max-Change: 0.00011
Iteration: 38, Log-Lik: -122446.219, Max-Change: 0.00011
Iteration: 39, Log-Lik: -122446.219, Max-Change: 0.00011
Iteration: 40, Log-Lik: -122446.216, Max-Change: 0.00052
Iteration: 41, Log-Lik: -122446.215, Max-Change: 0.00007

## 
## Step 2: Initial MI fit

## 
Iteration: 1, Log-Lik: -125406.165, Max-Change: 0.76708
Iteration: 2, Log-Lik: -122608.045, Max-Change: 0.24253
Iteration: 3, Log-Lik: -122457.296, Max-Change: 0.08671
Iteration: 4, Log-Lik: -122424.130, Max-Change: 0.03566
Iteration: 5, Log-Lik: -122402.091, Max-Change: 0.02833
Iteration: 6, Log-Lik: -122382.525, Max-Change: 0.03100
Iteration: 7, Log-Lik: -122364.451, Max-Change: 0.03124
Iteration: 8, Log-Lik: -122347.654, Max-Change: 0.03075
Iteration: 9, Log-Lik: -122332.016, Max-Change: 0.03003
Iteration: 10, Log-Lik: -122317.437, Max-Change: 0.02927
Iteration: 11, Log-Lik: -122303.831, Max-Change: 0.02850
Iteration: 12, Log-Lik: -122291.119, Max-Change: 0.02775
Iteration: 13, Log-Lik: -122279.231, Max-Change: 0.02702
Iteration: 14, Log-Lik: -122268.101, Max-Change: 0.02630
Iteration: 15, Log-Lik: -122257.673, Max-Change: 0.02561
Iteration: 16, Log-Lik: -122247.893, Max-Change: 0.02494
Iteration: 17, Log-Lik: -122238.711, Max-Change: 0.02429
Iteration: 18, Log-Lik: -122230.086, Max-Change: 0.02366
Iteration: 19, Log-Lik: -122221.975, Max-Change: 0.02304
Iteration: 20, Log-Lik: -122214.343, Max-Change: 0.02244
Iteration: 21, Log-Lik: -122207.157, Max-Change: 0.02185
Iteration: 22, Log-Lik: -122200.385, Max-Change: 0.02128
Iteration: 23, Log-Lik: -122193.998, Max-Change: 0.02074
Iteration: 24, Log-Lik: -122187.972, Max-Change: 0.02020
Iteration: 25, Log-Lik: -122182.282, Max-Change: 0.01968
Iteration: 26, Log-Lik: -122176.906, Max-Change: 0.01917
Iteration: 27, Log-Lik: -122171.824, Max-Change: 0.01867
Iteration: 28, Log-Lik: -122167.017, Max-Change: 0.01819
Iteration: 29, Log-Lik: -122162.468, Max-Change: 0.01772
Iteration: 30, Log-Lik: -122158.160, Max-Change: 0.01727
Iteration: 31, Log-Lik: -122139.594, Max-Change: 0.04380
Iteration: 32, Log-Lik: -122133.436, Max-Change: 0.01387
Iteration: 33, Log-Lik: -122130.626, Max-Change: 0.01389
Iteration: 34, Log-Lik: -122118.612, Max-Change: 0.03642
Iteration: 35, Log-Lik: -122114.418, Max-Change: 0.01118
Iteration: 36, Log-Lik: -122112.540, Max-Change: 0.01122
Iteration: 37, Log-Lik: -122104.586, Max-Change: 0.03056
Iteration: 38, Log-Lik: -122101.648, Max-Change: 0.00903
Iteration: 39, Log-Lik: -122100.364, Max-Change: 0.00909
Iteration: 40, Log-Lik: -122095.009, Max-Change: 0.02579
Iteration: 41, Log-Lik: -122092.906, Max-Change: 0.00731
Iteration: 42, Log-Lik: -122092.015, Max-Change: 0.00737
Iteration: 43, Log-Lik: -122088.363, Max-Change: 0.02184
Iteration: 44, Log-Lik: -122086.836, Max-Change: 0.00593
Iteration: 45, Log-Lik: -122086.209, Max-Change: 0.00599
Iteration: 46, Log-Lik: -122083.695, Max-Change: 0.01851
Iteration: 47, Log-Lik: -122082.575, Max-Change: 0.00483
Iteration: 48, Log-Lik: -122082.131, Max-Change: 0.00488
Iteration: 49, Log-Lik: -122080.386, Max-Change: 0.01570
Iteration: 50, Log-Lik: -122079.560, Max-Change: 0.00460
Iteration: 51, Log-Lik: -122079.244, Max-Change: 0.00398
Iteration: 52, Log-Lik: -122078.027, Max-Change: 0.01331
Iteration: 53, Log-Lik: -122077.415, Max-Change: 0.00433
Iteration: 54, Log-Lik: -122077.189, Max-Change: 0.00326
Iteration: 55, Log-Lik: -122076.338, Max-Change: 0.01128
Iteration: 56, Log-Lik: -122075.883, Max-Change: 0.00398
Iteration: 57, Log-Lik: -122075.721, Max-Change: 0.00267
Iteration: 58, Log-Lik: -122075.124, Max-Change: 0.00955
Iteration: 59, Log-Lik: -122074.785, Max-Change: 0.00360
Iteration: 60, Log-Lik: -122074.669, Max-Change: 0.00241
Iteration: 61, Log-Lik: -122074.249, Max-Change: 0.00807
Iteration: 62, Log-Lik: -122073.997, Max-Change: 0.00321
Iteration: 63, Log-Lik: -122073.914, Max-Change: 0.00215
Iteration: 64, Log-Lik: -122073.618, Max-Change: 0.00681
Iteration: 65, Log-Lik: -122073.430, Max-Change: 0.00282
Iteration: 66, Log-Lik: -122073.370, Max-Change: 0.00189
Iteration: 67, Log-Lik: -122073.161, Max-Change: 0.00574
Iteration: 68, Log-Lik: -122073.020, Max-Change: 0.00247
Iteration: 69, Log-Lik: -122072.977, Max-Change: 0.00165
Iteration: 70, Log-Lik: -122072.830, Max-Change: 0.00500
Iteration: 71, Log-Lik: -122072.724, Max-Change: 0.00214
Iteration: 72, Log-Lik: -122072.693, Max-Change: 0.00143
Iteration: 73, Log-Lik: -122072.589, Max-Change: 0.00435
Iteration: 74, Log-Lik: -122072.509, Max-Change: 0.00184
Iteration: 75, Log-Lik: -122072.487, Max-Change: 0.00123
Iteration: 76, Log-Lik: -122072.414, Max-Change: 0.00375
Iteration: 77, Log-Lik: -122072.353, Max-Change: 0.00158
Iteration: 78, Log-Lik: -122072.337, Max-Change: 0.00105
Iteration: 79, Log-Lik: -122072.285, Max-Change: 0.00316
Iteration: 80, Log-Lik: -122072.241, Max-Change: 0.00133
Iteration: 81, Log-Lik: -122072.229, Max-Change: 0.00089
Iteration: 82, Log-Lik: -122072.192, Max-Change: 0.00274
Iteration: 83, Log-Lik: -122072.156, Max-Change: 0.00114
Iteration: 84, Log-Lik: -122072.148, Max-Change: 0.00076
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Iteration: 114, Log-Lik: -122071.919, Max-Change: 0.00014
Iteration: 115, Log-Lik: -122071.918, Max-Change: 0.00041
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Iteration: 118, Log-Lik: -122071.914, Max-Change: 0.00037
Iteration: 119, Log-Lik: -122071.912, Max-Change: 0.00015
Iteration: 120, Log-Lik: -122071.911, Max-Change: 0.00010

## 
## Step 3: Leave one out MI testing

## 
## Step 4: Fit without DIF items, liberal threshold

## 
Iteration: 1, Log-Lik: -138480.931, Max-Change: 0.61531
Iteration: 2, Log-Lik: -137188.706, Max-Change: 0.35193
Iteration: 3, Log-Lik: -136925.519, Max-Change: 0.22765
Iteration: 4, Log-Lik: -136826.248, Max-Change: 0.14112
Iteration: 5, Log-Lik: -136783.050, Max-Change: 0.10017
Iteration: 6, Log-Lik: -136763.176, Max-Change: 0.05989
Iteration: 7, Log-Lik: -136755.783, Max-Change: 0.03937
Iteration: 8, Log-Lik: -136751.906, Max-Change: 0.02503
Iteration: 9, Log-Lik: -136749.630, Max-Change: 0.01669
Iteration: 10, Log-Lik: -136747.549, Max-Change: 0.00805
Iteration: 11, Log-Lik: -136746.882, Max-Change: 0.00605
Iteration: 12, Log-Lik: -136746.407, Max-Change: 0.00498
Iteration: 13, Log-Lik: -136745.393, Max-Change: 0.00321
Iteration: 14, Log-Lik: -136745.326, Max-Change: 0.00238
Iteration: 15, Log-Lik: -136745.284, Max-Change: 0.00188
Iteration: 16, Log-Lik: -136745.189, Max-Change: 0.00108
Iteration: 17, Log-Lik: -136745.183, Max-Change: 0.00081
Iteration: 18, Log-Lik: -136745.178, Max-Change: 0.00025
Iteration: 19, Log-Lik: -136745.176, Max-Change: 0.00024
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Iteration: 21, Log-Lik: -136745.173, Max-Change: 0.00025
Iteration: 22, Log-Lik: -136745.166, Max-Change: 0.00021
Iteration: 23, Log-Lik: -136745.165, Max-Change: 0.00018
Iteration: 24, Log-Lik: -136745.164, Max-Change: 0.00016
Iteration: 25, Log-Lik: -136745.162, Max-Change: 0.00046
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Iteration: 28, Log-Lik: -136745.161, Max-Change: 0.00039
Iteration: 29, Log-Lik: -136745.161, Max-Change: 0.00007

## 
## Step 5: Fit without DIF items, conservative threshold

## 
Iteration: 1, Log-Lik: -131248.704, Max-Change: 0.76232
Iteration: 2, Log-Lik: -129466.133, Max-Change: 0.42310
Iteration: 3, Log-Lik: -129000.441, Max-Change: 0.26848
Iteration: 4, Log-Lik: -128817.588, Max-Change: 0.14063
Iteration: 5, Log-Lik: -128720.755, Max-Change: 0.12654
Iteration: 6, Log-Lik: -128666.553, Max-Change: 0.08299
Iteration: 7, Log-Lik: -128639.023, Max-Change: 0.05370
Iteration: 8, Log-Lik: -128623.216, Max-Change: 0.04883
Iteration: 9, Log-Lik: -128612.837, Max-Change: 0.03274
Iteration: 10, Log-Lik: -128605.894, Max-Change: 0.02297
Iteration: 11, Log-Lik: -128601.139, Max-Change: 0.01850
Iteration: 12, Log-Lik: -128597.896, Max-Change: 0.01648
Iteration: 13, Log-Lik: -128594.986, Max-Change: 0.01378
Iteration: 14, Log-Lik: -128593.346, Max-Change: 0.01001
Iteration: 15, Log-Lik: -128591.880, Max-Change: 0.00840
Iteration: 16, Log-Lik: -128588.435, Max-Change: 0.00621
Iteration: 17, Log-Lik: -128587.989, Max-Change: 0.00518
Iteration: 18, Log-Lik: -128587.623, Max-Change: 0.00491
Iteration: 19, Log-Lik: -128586.263, Max-Change: 0.00296
Iteration: 20, Log-Lik: -128586.198, Max-Change: 0.00223
Iteration: 21, Log-Lik: -128586.150, Max-Change: 0.00197
Iteration: 22, Log-Lik: -128585.974, Max-Change: 0.00078
Iteration: 23, Log-Lik: -128585.968, Max-Change: 0.00057
Iteration: 24, Log-Lik: -128585.963, Max-Change: 0.00054
Iteration: 25, Log-Lik: -128585.940, Max-Change: 0.00015
Iteration: 26, Log-Lik: -128585.939, Max-Change: 0.00013
Iteration: 27, Log-Lik: -128585.938, Max-Change: 0.00014
Iteration: 28, Log-Lik: -128585.935, Max-Change: 0.00072
Iteration: 29, Log-Lik: -128585.933, Max-Change: 0.00011
Iteration: 30, Log-Lik: -128585.933, Max-Change: 0.00007

## 
## Step 6: Fit with anchor items, liberal threshold

## 
Iteration: 1, Log-Lik: -125406.165, Max-Change: 0.76692
Iteration: 2, Log-Lik: -122458.328, Max-Change: 0.32697
Iteration: 3, Log-Lik: -122311.209, Max-Change: 0.16171
Iteration: 4, Log-Lik: -122277.448, Max-Change: 0.07884
Iteration: 5, Log-Lik: -122256.848, Max-Change: 0.04643
Iteration: 6, Log-Lik: -122239.233, Max-Change: 0.02907
Iteration: 7, Log-Lik: -122223.107, Max-Change: 0.02914
Iteration: 8, Log-Lik: -122208.100, Max-Change: 0.02864
Iteration: 9, Log-Lik: -122194.077, Max-Change: 0.02801
Iteration: 10, Log-Lik: -122180.936, Max-Change: 0.02727
Iteration: 11, Log-Lik: -122168.617, Max-Change: 0.02663
Iteration: 12, Log-Lik: -122157.062, Max-Change: 0.02597
Iteration: 13, Log-Lik: -122146.203, Max-Change: 0.02533
Iteration: 14, Log-Lik: -122136.003, Max-Change: 0.02470
Iteration: 15, Log-Lik: -122126.408, Max-Change: 0.02411
Iteration: 16, Log-Lik: -122117.374, Max-Change: 0.02350
Iteration: 17, Log-Lik: -122108.876, Max-Change: 0.02298
Iteration: 18, Log-Lik: -122100.859, Max-Change: 0.02238
Iteration: 19, Log-Lik: -122093.304, Max-Change: 0.02187
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Iteration: 32, Log-Lik: -122009.450, Max-Change: 0.01452
Iteration: 33, Log-Lik: -122006.711, Max-Change: 0.01409
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Iteration: 46, Log-Lik: -121959.447, Max-Change: 0.01876
Iteration: 47, Log-Lik: -121958.245, Max-Change: 0.00546
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Iteration: 55, Log-Lik: -121950.998, Max-Change: 0.01235
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Iteration: 136, Log-Lik: -121944.299, Max-Change: 0.00054
Iteration: 137, Log-Lik: -121944.297, Max-Change: 0.00021
Iteration: 138, Log-Lik: -121944.296, Max-Change: 0.00014
Iteration: 139, Log-Lik: -121944.296, Max-Change: 0.00039
Iteration: 140, Log-Lik: -121944.294, Max-Change: 0.00017
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Iteration: 142, Log-Lik: -121944.293, Max-Change: 0.00041
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Iteration: 145, Log-Lik: -121944.291, Max-Change: 0.00031
Iteration: 146, Log-Lik: -121944.290, Max-Change: 0.00013
Iteration: 147, Log-Lik: -121944.290, Max-Change: 0.00010

## 
## Step 7: Fit with anchor items, conservative threshold

## 
Iteration: 1, Log-Lik: -125406.165, Max-Change: 0.76708
Iteration: 2, Log-Lik: -122501.799, Max-Change: 0.24211
Iteration: 3, Log-Lik: -122356.315, Max-Change: 0.08676
Iteration: 4, Log-Lik: -122323.429, Max-Change: 0.03678
Iteration: 5, Log-Lik: -122301.778, Max-Change: 0.02703
Iteration: 6, Log-Lik: -122282.683, Max-Change: 0.02978
Iteration: 7, Log-Lik: -122265.072, Max-Change: 0.03010
Iteration: 8, Log-Lik: -122248.712, Max-Change: 0.02970
Iteration: 9, Log-Lik: -122233.484, Max-Change: 0.02903
Iteration: 10, Log-Lik: -122219.287, Max-Change: 0.02830
Iteration: 11, Log-Lik: -122206.042, Max-Change: 0.02757
Iteration: 12, Log-Lik: -122193.671, Max-Change: 0.02687
Iteration: 13, Log-Lik: -122182.104, Max-Change: 0.02615
Iteration: 14, Log-Lik: -122171.278, Max-Change: 0.02548
Iteration: 15, Log-Lik: -122161.137, Max-Change: 0.02483
Iteration: 16, Log-Lik: -122151.629, Max-Change: 0.02419
Iteration: 17, Log-Lik: -122142.707, Max-Change: 0.02357
Iteration: 18, Log-Lik: -122134.328, Max-Change: 0.02296
Iteration: 19, Log-Lik: -122126.451, Max-Change: 0.02237
Iteration: 20, Log-Lik: -122119.042, Max-Change: 0.02180
Iteration: 21, Log-Lik: -122112.069, Max-Change: 0.02124
Iteration: 22, Log-Lik: -122105.498, Max-Change: 0.02073
Iteration: 23, Log-Lik: -122099.302, Max-Change: 0.02019
Iteration: 24, Log-Lik: -122093.458, Max-Change: 0.01968
Iteration: 25, Log-Lik: -122087.939, Max-Change: 0.01919
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Iteration: 31, Log-Lik: -122046.597, Max-Change: 0.04168
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Iteration: 59, Log-Lik: -121982.159, Max-Change: 0.00353
Iteration: 60, Log-Lik: -121982.028, Max-Change: 0.00240
Iteration: 61, Log-Lik: -121981.544, Max-Change: 0.00850
Iteration: 62, Log-Lik: -121981.262, Max-Change: 0.00324
Iteration: 63, Log-Lik: -121981.166, Max-Change: 0.00221
Iteration: 64, Log-Lik: -121980.815, Max-Change: 0.00729
Iteration: 65, Log-Lik: -121980.600, Max-Change: 0.00294
Iteration: 66, Log-Lik: -121980.529, Max-Change: 0.00200
Iteration: 67, Log-Lik: -121980.275, Max-Change: 0.00625
Iteration: 68, Log-Lik: -121980.110, Max-Change: 0.00263
Iteration: 69, Log-Lik: -121980.059, Max-Change: 0.00179
Iteration: 70, Log-Lik: -121979.873, Max-Change: 0.00561
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Iteration: 82, Log-Lik: -121979.064, Max-Change: 0.00343
Iteration: 83, Log-Lik: -121979.020, Max-Change: 0.00139
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Iteration: 116, Log-Lik: -121978.691, Max-Change: 0.00026
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Iteration: 121, Log-Lik: -121978.684, Max-Change: 0.00043
Iteration: 122, Log-Lik: -121978.682, Max-Change: 0.00019
Iteration: 123, Log-Lik: -121978.682, Max-Change: 0.00014
Iteration: 124, Log-Lik: -121978.681, Max-Change: 0.00045
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Iteration: 126, Log-Lik: -121978.678, Max-Change: 0.00012
Iteration: 127, Log-Lik: -121978.677, Max-Change: 0.00032
Iteration: 128, Log-Lik: -121978.676, Max-Change: 0.00014
Iteration: 129, Log-Lik: -121978.676, Max-Change: 0.00010
Iteration: 130, Log-Lik: -121978.675, Max-Change: 0.00034
Iteration: 131, Log-Lik: -121978.673, Max-Change: 0.00013
Iteration: 132, Log-Lik: -121978.673, Max-Change: 0.00009

## 
## Step 8: Get scores

## Word knowledge

## There are 8 steps

## Step 1: Initial joint fit

## 
Iteration: 1, Log-Lik: -121450.786, Max-Change: 2.13490
Iteration: 2, Log-Lik: -118451.648, Max-Change: 0.89758
Iteration: 3, Log-Lik: -117877.696, Max-Change: 0.33415
Iteration: 4, Log-Lik: -117622.108, Max-Change: 0.14321
Iteration: 5, Log-Lik: -117474.275, Max-Change: 0.18444
Iteration: 6, Log-Lik: -117379.802, Max-Change: 0.09635
Iteration: 7, Log-Lik: -117313.939, Max-Change: 0.13009
Iteration: 8, Log-Lik: -117265.222, Max-Change: 0.08623
Iteration: 9, Log-Lik: -117227.296, Max-Change: 0.12600
Iteration: 10, Log-Lik: -117199.373, Max-Change: 0.13565
Iteration: 11, Log-Lik: -117177.173, Max-Change: 0.11692
Iteration: 12, Log-Lik: -117158.232, Max-Change: 0.12818
Iteration: 13, Log-Lik: -117140.925, Max-Change: 0.11166
Iteration: 14, Log-Lik: -117126.591, Max-Change: 0.11865
Iteration: 15, Log-Lik: -117113.274, Max-Change: 0.09586
Iteration: 16, Log-Lik: -117101.742, Max-Change: 0.04493
Iteration: 17, Log-Lik: -117087.121, Max-Change: 0.03028
Iteration: 18, Log-Lik: -117074.471, Max-Change: 0.05431
Iteration: 19, Log-Lik: -117065.425, Max-Change: 0.09853
Iteration: 20, Log-Lik: -117057.691, Max-Change: 0.06887
Iteration: 21, Log-Lik: -117051.081, Max-Change: 0.08417
Iteration: 22, Log-Lik: -117045.121, Max-Change: 0.06616
Iteration: 23, Log-Lik: -117039.773, Max-Change: 0.07524
Iteration: 24, Log-Lik: -117034.850, Max-Change: 0.06521
Iteration: 25, Log-Lik: -117030.357, Max-Change: 0.08408
Iteration: 26, Log-Lik: -117026.272, Max-Change: 0.06765
Iteration: 27, Log-Lik: -117022.494, Max-Change: 0.07674
Iteration: 28, Log-Lik: -117020.556, Max-Change: 0.02098
Iteration: 29, Log-Lik: -117015.471, Max-Change: 0.02510
Iteration: 30, Log-Lik: -117012.081, Max-Change: 0.02166
Iteration: 31, Log-Lik: -117009.392, Max-Change: 0.03602
Iteration: 32, Log-Lik: -117006.328, Max-Change: 0.04490
Iteration: 33, Log-Lik: -117004.006, Max-Change: 0.04141
Iteration: 34, Log-Lik: -117003.050, Max-Change: 0.01766
Iteration: 35, Log-Lik: -116999.667, Max-Change: 0.02357
Iteration: 36, Log-Lik: -116997.652, Max-Change: 0.02924
Iteration: 37, Log-Lik: -116996.470, Max-Change: 0.03225
Iteration: 38, Log-Lik: -116994.744, Max-Change: 0.03267
Iteration: 39, Log-Lik: -116993.301, Max-Change: 0.03893
Iteration: 40, Log-Lik: -116992.617, Max-Change: 0.01235
Iteration: 41, Log-Lik: -116990.601, Max-Change: 0.01955
Iteration: 42, Log-Lik: -116989.376, Max-Change: 0.02627
Iteration: 43, Log-Lik: -116988.647, Max-Change: 0.01015
Iteration: 44, Log-Lik: -116987.144, Max-Change: 0.01970
Iteration: 45, Log-Lik: -116986.238, Max-Change: 0.00646
Iteration: 46, Log-Lik: -116985.052, Max-Change: 0.01616
Iteration: 47, Log-Lik: -116984.242, Max-Change: 0.00579
Iteration: 48, Log-Lik: -116983.207, Max-Change: 0.01976
Iteration: 49, Log-Lik: -116982.324, Max-Change: 0.01168
Iteration: 50, Log-Lik: -116981.606, Max-Change: 0.00549
Iteration: 51, Log-Lik: -116980.887, Max-Change: 0.00443
Iteration: 52, Log-Lik: -116977.672, Max-Change: 0.00831
Iteration: 53, Log-Lik: -116977.359, Max-Change: 0.00295
Iteration: 54, Log-Lik: -116977.121, Max-Change: 0.00299
Iteration: 55, Log-Lik: -116976.028, Max-Change: 0.00259
Iteration: 56, Log-Lik: -116975.934, Max-Change: 0.00291
Iteration: 57, Log-Lik: -116975.849, Max-Change: 0.00290
Iteration: 58, Log-Lik: -116975.500, Max-Change: 0.00089
Iteration: 59, Log-Lik: -116975.471, Max-Change: 0.00098
Iteration: 60, Log-Lik: -116975.441, Max-Change: 0.00091
Iteration: 61, Log-Lik: -116975.302, Max-Change: 0.00063
Iteration: 62, Log-Lik: -116975.294, Max-Change: 0.00054
Iteration: 63, Log-Lik: -116975.286, Max-Change: 0.00051
Iteration: 64, Log-Lik: -116975.244, Max-Change: 0.00039
Iteration: 65, Log-Lik: -116975.239, Max-Change: 0.00038
Iteration: 66, Log-Lik: -116975.235, Max-Change: 0.00036
Iteration: 67, Log-Lik: -116975.216, Max-Change: 0.00012
Iteration: 68, Log-Lik: -116975.215, Max-Change: 0.00010

## 
## Step 2: Initial MI fit

## 
Iteration: 1, Log-Lik: -121450.786, Max-Change: 1.12220
Iteration: 2, Log-Lik: -117429.773, Max-Change: 0.29009
Iteration: 3, Log-Lik: -117148.942, Max-Change: 0.11218
Iteration: 4, Log-Lik: -117081.940, Max-Change: 0.05607
Iteration: 5, Log-Lik: -117044.201, Max-Change: 0.03163
Iteration: 6, Log-Lik: -117013.312, Max-Change: 0.02855
Iteration: 7, Log-Lik: -116985.393, Max-Change: 0.03141
Iteration: 8, Log-Lik: -116959.591, Max-Change: 0.03285
Iteration: 9, Log-Lik: -116935.582, Max-Change: 0.03282
Iteration: 10, Log-Lik: -116913.232, Max-Change: 0.03276
Iteration: 11, Log-Lik: -116892.348, Max-Change: 0.03186
Iteration: 12, Log-Lik: -116872.849, Max-Change: 0.03120
Iteration: 13, Log-Lik: -116854.592, Max-Change: 0.03063
Iteration: 14, Log-Lik: -116837.499, Max-Change: 0.03021
Iteration: 15, Log-Lik: -116821.476, Max-Change: 0.02948
Iteration: 16, Log-Lik: -116806.454, Max-Change: 0.02877
Iteration: 17, Log-Lik: -116792.351, Max-Change: 0.02842
Iteration: 18, Log-Lik: -116779.101, Max-Change: 0.02732
Iteration: 19, Log-Lik: -116766.663, Max-Change: 0.02703
Iteration: 20, Log-Lik: -116754.941, Max-Change: 0.02668
Iteration: 21, Log-Lik: -116743.909, Max-Change: 0.02599
Iteration: 22, Log-Lik: -116733.518, Max-Change: 0.02543
Iteration: 23, Log-Lik: -116723.725, Max-Change: 0.02485
Iteration: 24, Log-Lik: -116714.490, Max-Change: 0.02427
Iteration: 25, Log-Lik: -116705.772, Max-Change: 0.02379
Iteration: 26, Log-Lik: -116697.563, Max-Change: 0.02352
Iteration: 27, Log-Lik: -116689.802, Max-Change: 0.02275
Iteration: 28, Log-Lik: -116682.475, Max-Change: 0.02221
Iteration: 29, Log-Lik: -116675.551, Max-Change: 0.02175
Iteration: 30, Log-Lik: -116669.006, Max-Change: 0.02171
Iteration: 31, Log-Lik: -116662.803, Max-Change: 0.02092
Iteration: 32, Log-Lik: -116656.952, Max-Change: 0.02064
Iteration: 33, Log-Lik: -116651.396, Max-Change: 0.02016
Iteration: 34, Log-Lik: -116646.135, Max-Change: 0.01974
Iteration: 35, Log-Lik: -116641.154, Max-Change: 0.01939
Iteration: 36, Log-Lik: -116636.434, Max-Change: 0.01901
Iteration: 37, Log-Lik: -116631.960, Max-Change: 0.01844
Iteration: 38, Log-Lik: -116627.723, Max-Change: 0.01813
Iteration: 39, Log-Lik: -116623.698, Max-Change: 0.01777
Iteration: 40, Log-Lik: -116602.976, Max-Change: 0.03753
Iteration: 41, Log-Lik: -116600.427, Max-Change: 0.01435
Iteration: 42, Log-Lik: -116597.856, Max-Change: 0.01475
Iteration: 43, Log-Lik: -116584.012, Max-Change: 0.03200
Iteration: 44, Log-Lik: -116582.649, Max-Change: 0.01211
Iteration: 45, Log-Lik: -116580.949, Max-Change: 0.01238
Iteration: 46, Log-Lik: -116571.545, Max-Change: 0.02743
Iteration: 47, Log-Lik: -116570.905, Max-Change: 0.01034
Iteration: 48, Log-Lik: -116569.768, Max-Change: 0.01043
Iteration: 49, Log-Lik: -116563.313, Max-Change: 0.02361
Iteration: 50, Log-Lik: -116563.111, Max-Change: 0.00866
Iteration: 51, Log-Lik: -116562.348, Max-Change: 0.00879
Iteration: 52, Log-Lik: -116557.885, Max-Change: 0.02036
Iteration: 53, Log-Lik: -116557.944, Max-Change: 0.00721
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Iteration: 55, Log-Lik: -116554.332, Max-Change: 0.01759
Iteration: 56, Log-Lik: -116554.541, Max-Change: 0.00608
Iteration: 57, Log-Lik: -116554.200, Max-Change: 0.00623
Iteration: 58, Log-Lik: -116552.038, Max-Change: 0.01523
Iteration: 59, Log-Lik: -116552.326, Max-Change: 0.00514
Iteration: 60, Log-Lik: -116552.099, Max-Change: 0.00524
Iteration: 61, Log-Lik: -116550.589, Max-Change: 0.01318
Iteration: 62, Log-Lik: -116550.913, Max-Change: 0.00432
Iteration: 63, Log-Lik: -116550.764, Max-Change: 0.00441
Iteration: 64, Log-Lik: -116549.708, Max-Change: 0.01141
Iteration: 65, Log-Lik: -116550.040, Max-Change: 0.00366
Iteration: 66, Log-Lik: -116549.945, Max-Change: 0.00369
Iteration: 67, Log-Lik: -116549.207, Max-Change: 0.00986
Iteration: 68, Log-Lik: -116549.530, Max-Change: 0.00309
Iteration: 69, Log-Lik: -116549.470, Max-Change: 0.00311
Iteration: 70, Log-Lik: -116548.955, Max-Change: 0.00853
Iteration: 71, Log-Lik: -116549.260, Max-Change: 0.00261
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Iteration: 73, Log-Lik: -116548.866, Max-Change: 0.00736
Iteration: 74, Log-Lik: -116549.148, Max-Change: 0.00220
Iteration: 75, Log-Lik: -116549.128, Max-Change: 0.00220
Iteration: 76, Log-Lik: -116548.880, Max-Change: 0.00635
Iteration: 77, Log-Lik: -116549.135, Max-Change: 0.00189
Iteration: 78, Log-Lik: -116549.126, Max-Change: 0.00186
Iteration: 79, Log-Lik: -116548.955, Max-Change: 0.00547
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Iteration: 85, Log-Lik: -116549.191, Max-Change: 0.00406
Iteration: 86, Log-Lik: -116549.370, Max-Change: 0.00140
Iteration: 87, Log-Lik: -116549.375, Max-Change: 0.00111
Iteration: 88, Log-Lik: -116549.322, Max-Change: 0.00348
Iteration: 89, Log-Lik: -116549.479, Max-Change: 0.00126
Iteration: 90, Log-Lik: -116549.485, Max-Change: 0.00094
Iteration: 91, Log-Lik: -116549.451, Max-Change: 0.00299
Iteration: 92, Log-Lik: -116549.587, Max-Change: 0.00112
Iteration: 93, Log-Lik: -116549.594, Max-Change: 0.00079
Iteration: 94, Log-Lik: -116549.572, Max-Change: 0.00257
Iteration: 95, Log-Lik: -116549.691, Max-Change: 0.00099
Iteration: 96, Log-Lik: -116549.698, Max-Change: 0.00067
Iteration: 97, Log-Lik: -116549.684, Max-Change: 0.00220
Iteration: 98, Log-Lik: -116549.787, Max-Change: 0.00087
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Iteration: 100, Log-Lik: -116549.786, Max-Change: 0.00189
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Iteration: 105, Log-Lik: -116549.960, Max-Change: 0.00045
Iteration: 106, Log-Lik: -116549.958, Max-Change: 0.00138
Iteration: 107, Log-Lik: -116550.024, Max-Change: 0.00058
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Iteration: 110, Log-Lik: -116550.086, Max-Change: 0.00051
Iteration: 111, Log-Lik: -116550.090, Max-Change: 0.00034
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Iteration: 114, Log-Lik: -116550.144, Max-Change: 0.00029
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Iteration: 116, Log-Lik: -116550.188, Max-Change: 0.00038
Iteration: 117, Log-Lik: -116550.191, Max-Change: 0.00026
Iteration: 118, Log-Lik: -116550.193, Max-Change: 0.00074
Iteration: 119, Log-Lik: -116550.229, Max-Change: 0.00033
Iteration: 120, Log-Lik: -116550.231, Max-Change: 0.00022
Iteration: 121, Log-Lik: -116550.234, Max-Change: 0.00063
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Iteration: 123, Log-Lik: -116550.267, Max-Change: 0.00019
Iteration: 124, Log-Lik: -116550.269, Max-Change: 0.00054
Iteration: 125, Log-Lik: -116550.295, Max-Change: 0.00025
Iteration: 126, Log-Lik: -116550.297, Max-Change: 0.00016
Iteration: 127, Log-Lik: -116550.299, Max-Change: 0.00046
Iteration: 128, Log-Lik: -116550.321, Max-Change: 0.00021
Iteration: 129, Log-Lik: -116550.322, Max-Change: 0.00014
Iteration: 130, Log-Lik: -116550.324, Max-Change: 0.00039
Iteration: 131, Log-Lik: -116550.343, Max-Change: 0.00018
Iteration: 132, Log-Lik: -116550.345, Max-Change: 0.00012
Iteration: 133, Log-Lik: -116550.347, Max-Change: 0.00034
Iteration: 134, Log-Lik: -116550.363, Max-Change: 0.00016
Iteration: 135, Log-Lik: -116550.364, Max-Change: 0.00011
Iteration: 136, Log-Lik: -116550.366, Max-Change: 0.00029
Iteration: 137, Log-Lik: -116550.379, Max-Change: 0.00014
Iteration: 138, Log-Lik: -116550.380, Max-Change: 0.00009

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 3: Leave one out MI testing

## 
## Step 4: Fit without DIF items, liberal threshold

## 
Iteration: 1, Log-Lik: -142116.133, Max-Change: 0.64324
Iteration: 2, Log-Lik: -141333.460, Max-Change: 0.41710
Iteration: 3, Log-Lik: -141133.651, Max-Change: 0.27561
Iteration: 4, Log-Lik: -141079.169, Max-Change: 0.14201
Iteration: 5, Log-Lik: -141064.769, Max-Change: 0.09616
Iteration: 6, Log-Lik: -141059.306, Max-Change: 0.04184
Iteration: 7, Log-Lik: -141057.794, Max-Change: 0.02714
Iteration: 8, Log-Lik: -141057.231, Max-Change: 0.01594
Iteration: 9, Log-Lik: -141056.994, Max-Change: 0.00965
Iteration: 10, Log-Lik: -141056.828, Max-Change: 0.00261
Iteration: 11, Log-Lik: -141056.802, Max-Change: 0.00304
Iteration: 12, Log-Lik: -141056.778, Max-Change: 0.00034
Iteration: 13, Log-Lik: -141056.778, Max-Change: 0.00034
Iteration: 14, Log-Lik: -141056.776, Max-Change: 0.00153
Iteration: 15, Log-Lik: -141056.771, Max-Change: 0.00025
Iteration: 16, Log-Lik: -141056.771, Max-Change: 0.00023
Iteration: 17, Log-Lik: -141056.770, Max-Change: 0.00079
Iteration: 18, Log-Lik: -141056.769, Max-Change: 0.00018
Iteration: 19, Log-Lik: -141056.769, Max-Change: 0.00080
Iteration: 20, Log-Lik: -141056.769, Max-Change: 0.00034
Iteration: 21, Log-Lik: -141056.768, Max-Change: 0.00064
Iteration: 22, Log-Lik: -141056.769, Max-Change: 0.00048
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Iteration: 32, Log-Lik: -141056.768, Max-Change: 0.00020
Iteration: 33, Log-Lik: -141056.768, Max-Change: 0.00008

## 
## Step 5: Fit without DIF items, conservative threshold

## 
Iteration: 1, Log-Lik: -138266.058, Max-Change: 0.76899
Iteration: 2, Log-Lik: -136895.161, Max-Change: 0.64515
Iteration: 3, Log-Lik: -136617.230, Max-Change: 0.30725
Iteration: 4, Log-Lik: -136485.831, Max-Change: 0.22820
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Iteration: 6, Log-Lik: -136394.072, Max-Change: 0.12491
Iteration: 7, Log-Lik: -136375.519, Max-Change: 0.08630
Iteration: 8, Log-Lik: -136359.584, Max-Change: 0.09338
Iteration: 9, Log-Lik: -136348.651, Max-Change: 0.05592
Iteration: 10, Log-Lik: -136341.027, Max-Change: 0.07776
Iteration: 11, Log-Lik: -136334.702, Max-Change: 0.05215
Iteration: 12, Log-Lik: -136330.091, Max-Change: 0.06103
Iteration: 13, Log-Lik: -136326.077, Max-Change: 0.05182
Iteration: 14, Log-Lik: -136323.168, Max-Change: 0.05171
Iteration: 15, Log-Lik: -136320.783, Max-Change: 0.03892
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Iteration: 17, Log-Lik: -136317.278, Max-Change: 0.01136
Iteration: 18, Log-Lik: -136315.610, Max-Change: 0.00867
Iteration: 19, Log-Lik: -136311.186, Max-Change: 0.00320
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Iteration: 23, Log-Lik: -136311.063, Max-Change: 0.00089
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Iteration: 32, Log-Lik: -136311.032, Max-Change: 0.00023
Iteration: 33, Log-Lik: -136311.032, Max-Change: 0.00010

## 
## Step 6: Fit with anchor items, liberal threshold

## 
Iteration: 1, Log-Lik: -121450.786, Max-Change: 1.26312
Iteration: 2, Log-Lik: -116671.154, Max-Change: 0.28708
Iteration: 3, Log-Lik: -116383.645, Max-Change: 0.16969
Iteration: 4, Log-Lik: -116306.237, Max-Change: 0.11100
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Iteration: 7, Log-Lik: -116223.396, Max-Change: 0.04347
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Iteration: 12, Log-Lik: -116152.083, Max-Change: 0.02474
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Iteration: 14, Log-Lik: -116129.323, Max-Change: 0.02420
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Iteration: 18, Log-Lik: -116090.212, Max-Change: 0.02228
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Iteration: 26, Log-Lik: -116031.567, Max-Change: 0.01969
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Iteration: 252, Log-Lik: -115898.568, Max-Change: 0.00009

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 7: Fit with anchor items, conservative threshold

## 
Iteration: 1, Log-Lik: -121450.786, Max-Change: 1.27718
Iteration: 2, Log-Lik: -116737.826, Max-Change: 0.27502
Iteration: 3, Log-Lik: -116475.539, Max-Change: 0.17861
Iteration: 4, Log-Lik: -116407.804, Max-Change: 0.11837
Iteration: 5, Log-Lik: -116372.892, Max-Change: 0.07564
Iteration: 6, Log-Lik: -116346.177, Max-Change: 0.05784
Iteration: 7, Log-Lik: -116322.891, Max-Change: 0.04146
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## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 8: Get scores

## Paragraph comprehension

## There are 8 steps

## Step 1: Initial joint fit

## 
Iteration: 1, Log-Lik: -61652.456, Max-Change: 0.87650
Iteration: 2, Log-Lik: -60402.586, Max-Change: 0.50201
Iteration: 3, Log-Lik: -60181.430, Max-Change: 0.24615
Iteration: 4, Log-Lik: -60115.427, Max-Change: 0.13536
Iteration: 5, Log-Lik: -60082.711, Max-Change: 0.08955
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Iteration: 7, Log-Lik: -60060.097, Max-Change: 0.04403
Iteration: 8, Log-Lik: -60054.928, Max-Change: 0.02723
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Iteration: 23, Log-Lik: -60043.457, Max-Change: 0.00039
Iteration: 24, Log-Lik: -60043.457, Max-Change: 0.00009

## 
## Step 2: Initial MI fit

## 
Iteration: 1, Log-Lik: -61652.456, Max-Change: 0.82317
Iteration: 2, Log-Lik: -60240.991, Max-Change: 0.38984
Iteration: 3, Log-Lik: -60050.448, Max-Change: 0.19018
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Iteration: 128, Log-Lik: -59710.638, Max-Change: 0.00011
Iteration: 129, Log-Lik: -59710.638, Max-Change: 0.00009

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 3: Leave one out MI testing

## 
## Step 4: Fit without DIF items, liberal threshold

## 
Iteration: 1, Log-Lik: -66348.738, Max-Change: 0.58323
Iteration: 2, Log-Lik: -65860.021, Max-Change: 0.50863
Iteration: 3, Log-Lik: -65670.210, Max-Change: 0.33640
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Iteration: 31, Log-Lik: -65582.527, Max-Change: 0.00010

## 
## Step 5: Fit without DIF items, conservative threshold

## 
Iteration: 1, Log-Lik: -63972.096, Max-Change: 0.75188
Iteration: 2, Log-Lik: -63204.029, Max-Change: 0.47628
Iteration: 3, Log-Lik: -63018.836, Max-Change: 0.33489
Iteration: 4, Log-Lik: -62948.638, Max-Change: 0.20558
Iteration: 5, Log-Lik: -62928.888, Max-Change: 0.14502
Iteration: 6, Log-Lik: -62919.546, Max-Change: 0.09265
Iteration: 7, Log-Lik: -62915.536, Max-Change: 0.06294
Iteration: 8, Log-Lik: -62913.390, Max-Change: 0.04236
Iteration: 9, Log-Lik: -62912.169, Max-Change: 0.02862
Iteration: 10, Log-Lik: -62911.047, Max-Change: 0.01471
Iteration: 11, Log-Lik: -62910.712, Max-Change: 0.00624
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Iteration: 15, Log-Lik: -62910.204, Max-Change: 0.00061
Iteration: 16, Log-Lik: -62910.197, Max-Change: 0.00009

## 
## Step 6: Fit with anchor items, liberal threshold

## 
Iteration: 1, Log-Lik: -61652.456, Max-Change: 0.82391
Iteration: 2, Log-Lik: -60103.024, Max-Change: 0.37992
Iteration: 3, Log-Lik: -59925.422, Max-Change: 0.18340
Iteration: 4, Log-Lik: -59883.942, Max-Change: 0.09784
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## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 7: Fit with anchor items, conservative threshold

## 
Iteration: 1, Log-Lik: -61652.456, Max-Change: 0.82410
Iteration: 2, Log-Lik: -60157.855, Max-Change: 0.38425
Iteration: 3, Log-Lik: -59973.191, Max-Change: 0.18927
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Iteration: 12, Log-Lik: -59778.678, Max-Change: 0.04617
Iteration: 13, Log-Lik: -59767.479, Max-Change: 0.04421
Iteration: 14, Log-Lik: -59757.290, Max-Change: 0.04236
Iteration: 15, Log-Lik: -59748.018, Max-Change: 0.04094
Iteration: 16, Log-Lik: -59739.565, Max-Change: 0.03935
Iteration: 17, Log-Lik: -59731.857, Max-Change: 0.03773
Iteration: 18, Log-Lik: -59724.820, Max-Change: 0.03634
Iteration: 19, Log-Lik: -59718.387, Max-Change: 0.03499
Iteration: 20, Log-Lik: -59712.503, Max-Change: 0.03360
Iteration: 21, Log-Lik: -59707.119, Max-Change: 0.03227
Iteration: 22, Log-Lik: -59702.188, Max-Change: 0.03107
Iteration: 23, Log-Lik: -59697.664, Max-Change: 0.02989
Iteration: 24, Log-Lik: -59693.516, Max-Change: 0.02859
Iteration: 25, Log-Lik: -59681.414, Max-Change: 0.05170
Iteration: 26, Log-Lik: -59673.499, Max-Change: 0.01872
Iteration: 27, Log-Lik: -59671.326, Max-Change: 0.02020
Iteration: 28, Log-Lik: -59665.257, Max-Change: 0.03854
Iteration: 29, Log-Lik: -59661.085, Max-Change: 0.01409
Iteration: 30, Log-Lik: -59659.861, Max-Change: 0.01499
Iteration: 31, Log-Lik: -59656.570, Max-Change: 0.03050
Iteration: 32, Log-Lik: -59654.114, Max-Change: 0.01008
Iteration: 33, Log-Lik: -59653.379, Max-Change: 0.01092
Iteration: 34, Log-Lik: -59651.513, Max-Change: 0.02164
Iteration: 35, Log-Lik: -59650.313, Max-Change: 0.00783
Iteration: 36, Log-Lik: -59649.827, Max-Change: 0.00833
Iteration: 37, Log-Lik: -59648.685, Max-Change: 0.01818
Iteration: 38, Log-Lik: -59647.850, Max-Change: 0.00581
Iteration: 39, Log-Lik: -59647.514, Max-Change: 0.00628
Iteration: 40, Log-Lik: -59646.775, Max-Change: 0.01264
Iteration: 41, Log-Lik: -59646.365, Max-Change: 0.00527
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Iteration: 43, Log-Lik: -59645.623, Max-Change: 0.01261
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Iteration: 48, Log-Lik: -59644.377, Max-Change: 0.00387
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Iteration: 78, Log-Lik: -59642.383, Max-Change: 0.00152
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Iteration: 101, Log-Lik: -59642.328, Max-Change: 0.00069
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Iteration: 128, Log-Lik: -59642.362, Max-Change: 0.00031
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Iteration: 130, Log-Lik: -59642.363, Max-Change: 0.00027
Iteration: 131, Log-Lik: -59642.364, Max-Change: 0.00021
Iteration: 132, Log-Lik: -59642.364, Max-Change: 0.00019
Iteration: 133, Log-Lik: -59642.365, Max-Change: 0.00042
Iteration: 134, Log-Lik: -59642.368, Max-Change: 0.00025
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Iteration: 136, Log-Lik: -59642.368, Max-Change: 0.00021
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Iteration: 138, Log-Lik: -59642.370, Max-Change: 0.00015
Iteration: 139, Log-Lik: -59642.370, Max-Change: 0.00033
Iteration: 140, Log-Lik: -59642.372, Max-Change: 0.00019
Iteration: 141, Log-Lik: -59642.373, Max-Change: 0.00014
Iteration: 142, Log-Lik: -59642.373, Max-Change: 0.00017
Iteration: 143, Log-Lik: -59642.374, Max-Change: 0.00013
Iteration: 144, Log-Lik: -59642.374, Max-Change: 0.00012
Iteration: 145, Log-Lik: -59642.374, Max-Change: 0.00026
Iteration: 146, Log-Lik: -59642.376, Max-Change: 0.00015
Iteration: 147, Log-Lik: -59642.376, Max-Change: 0.00011
Iteration: 148, Log-Lik: -59642.377, Max-Change: 0.00013
Iteration: 149, Log-Lik: -59642.377, Max-Change: 0.00010
Iteration: 150, Log-Lik: -59642.378, Max-Change: 0.00009

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 8: Get scores

## Mathematics knowledge

## There are 8 steps

## Step 1: Initial joint fit

## 
Iteration: 1, Log-Lik: -111250.918, Max-Change: 0.52069
Iteration: 2, Log-Lik: -109352.557, Max-Change: 0.25779
Iteration: 3, Log-Lik: -109037.705, Max-Change: 0.16192
Iteration: 4, Log-Lik: -108894.057, Max-Change: 0.11902
Iteration: 5, Log-Lik: -108829.258, Max-Change: 0.08124
Iteration: 6, Log-Lik: -108791.709, Max-Change: 0.03808
Iteration: 7, Log-Lik: -108783.502, Max-Change: 0.02507
Iteration: 8, Log-Lik: -108779.230, Max-Change: 0.04142
Iteration: 9, Log-Lik: -108768.803, Max-Change: 0.03342
Iteration: 10, Log-Lik: -108764.985, Max-Change: 0.01969
Iteration: 11, Log-Lik: -108762.375, Max-Change: 0.01410
Iteration: 12, Log-Lik: -108760.960, Max-Change: 0.01065
Iteration: 13, Log-Lik: -108759.559, Max-Change: 0.00593
Iteration: 14, Log-Lik: -108759.124, Max-Change: 0.00431
Iteration: 15, Log-Lik: -108758.844, Max-Change: 0.00360
Iteration: 16, Log-Lik: -108758.270, Max-Change: 0.00190
Iteration: 17, Log-Lik: -108758.196, Max-Change: 0.00165
Iteration: 18, Log-Lik: -108758.138, Max-Change: 0.00152
Iteration: 19, Log-Lik: -108757.925, Max-Change: 0.00075
Iteration: 20, Log-Lik: -108757.919, Max-Change: 0.00064
Iteration: 21, Log-Lik: -108757.912, Max-Change: 0.00060
Iteration: 22, Log-Lik: -108757.887, Max-Change: 0.00021
Iteration: 23, Log-Lik: -108757.886, Max-Change: 0.00013
Iteration: 24, Log-Lik: -108757.885, Max-Change: 0.00013
Iteration: 25, Log-Lik: -108757.881, Max-Change: 0.00012
Iteration: 26, Log-Lik: -108757.880, Max-Change: 0.00011
Iteration: 27, Log-Lik: -108757.880, Max-Change: 0.00010

## 
## Step 2: Initial MI fit

## 
Iteration: 1, Log-Lik: -111250.918, Max-Change: 0.65769
Iteration: 2, Log-Lik: -108866.425, Max-Change: 0.17850
Iteration: 3, Log-Lik: -108754.531, Max-Change: 0.07138
Iteration: 4, Log-Lik: -108731.376, Max-Change: 0.03368
Iteration: 5, Log-Lik: -108714.485, Max-Change: 0.02502
Iteration: 6, Log-Lik: -108699.306, Max-Change: 0.02471
Iteration: 7, Log-Lik: -108685.394, Max-Change: 0.02403
Iteration: 8, Log-Lik: -108672.602, Max-Change: 0.02329
Iteration: 9, Log-Lik: -108660.820, Max-Change: 0.02267
Iteration: 10, Log-Lik: -108649.954, Max-Change: 0.02217
Iteration: 11, Log-Lik: -108639.920, Max-Change: 0.02156
Iteration: 12, Log-Lik: -108630.643, Max-Change: 0.02100
Iteration: 13, Log-Lik: -108622.057, Max-Change: 0.02041
Iteration: 14, Log-Lik: -108614.100, Max-Change: 0.01984
Iteration: 15, Log-Lik: -108606.719, Max-Change: 0.01927
Iteration: 16, Log-Lik: -108599.865, Max-Change: 0.01873
Iteration: 17, Log-Lik: -108593.493, Max-Change: 0.01826
Iteration: 18, Log-Lik: -108587.565, Max-Change: 0.01773
Iteration: 19, Log-Lik: -108582.044, Max-Change: 0.01721
Iteration: 20, Log-Lik: -108576.897, Max-Change: 0.01671
Iteration: 21, Log-Lik: -108572.096, Max-Change: 0.01625
Iteration: 22, Log-Lik: -108567.612, Max-Change: 0.01578
Iteration: 23, Log-Lik: -108563.423, Max-Change: 0.01533
Iteration: 24, Log-Lik: -108559.504, Max-Change: 0.01489
Iteration: 25, Log-Lik: -108543.304, Max-Change: 0.04422
Iteration: 26, Log-Lik: -108538.048, Max-Change: 0.01152
Iteration: 27, Log-Lik: -108535.722, Max-Change: 0.01147
Iteration: 28, Log-Lik: -108525.879, Max-Change: 0.03544
Iteration: 29, Log-Lik: -108522.763, Max-Change: 0.00945
Iteration: 30, Log-Lik: -108521.334, Max-Change: 0.00916
Iteration: 31, Log-Lik: -108515.164, Max-Change: 0.02880
Iteration: 32, Log-Lik: -108513.253, Max-Change: 0.00742
Iteration: 33, Log-Lik: -108512.350, Max-Change: 0.00721
Iteration: 34, Log-Lik: -108508.394, Max-Change: 0.02365
Iteration: 35, Log-Lik: -108507.194, Max-Change: 0.00596
Iteration: 36, Log-Lik: -108506.613, Max-Change: 0.00573
Iteration: 37, Log-Lik: -108504.033, Max-Change: 0.01954
Iteration: 38, Log-Lik: -108503.271, Max-Change: 0.00477
Iteration: 39, Log-Lik: -108502.893, Max-Change: 0.00455
Iteration: 40, Log-Lik: -108501.187, Max-Change: 0.01621
Iteration: 41, Log-Lik: -108500.705, Max-Change: 0.00383
Iteration: 42, Log-Lik: -108500.458, Max-Change: 0.00365
Iteration: 43, Log-Lik: -108499.320, Max-Change: 0.01350
Iteration: 44, Log-Lik: -108499.020, Max-Change: 0.00370
Iteration: 45, Log-Lik: -108498.859, Max-Change: 0.00291
Iteration: 46, Log-Lik: -108498.095, Max-Change: 0.01125
Iteration: 47, Log-Lik: -108497.915, Max-Change: 0.00352
Iteration: 48, Log-Lik: -108497.812, Max-Change: 0.00242
Iteration: 49, Log-Lik: -108497.297, Max-Change: 0.00939
Iteration: 50, Log-Lik: -108497.195, Max-Change: 0.00326
Iteration: 51, Log-Lik: -108497.131, Max-Change: 0.00224
Iteration: 52, Log-Lik: -108496.783, Max-Change: 0.00784
Iteration: 53, Log-Lik: -108496.734, Max-Change: 0.00296
Iteration: 54, Log-Lik: -108496.695, Max-Change: 0.00203
Iteration: 55, Log-Lik: -108496.461, Max-Change: 0.00654
Iteration: 56, Log-Lik: -108496.445, Max-Change: 0.00264
Iteration: 57, Log-Lik: -108496.423, Max-Change: 0.00181
Iteration: 58, Log-Lik: -108496.265, Max-Change: 0.00559
Iteration: 59, Log-Lik: -108496.271, Max-Change: 0.00233
Iteration: 60, Log-Lik: -108496.260, Max-Change: 0.00159
Iteration: 61, Log-Lik: -108496.155, Max-Change: 0.00488
Iteration: 62, Log-Lik: -108496.173, Max-Change: 0.00202
Iteration: 63, Log-Lik: -108496.168, Max-Change: 0.00139
Iteration: 64, Log-Lik: -108496.099, Max-Change: 0.00422
Iteration: 65, Log-Lik: -108496.123, Max-Change: 0.00174
Iteration: 66, Log-Lik: -108496.123, Max-Change: 0.00120
Iteration: 67, Log-Lik: -108496.077, Max-Change: 0.00365
Iteration: 68, Log-Lik: -108496.104, Max-Change: 0.00150
Iteration: 69, Log-Lik: -108496.106, Max-Change: 0.00104
Iteration: 70, Log-Lik: -108496.076, Max-Change: 0.00310
Iteration: 71, Log-Lik: -108496.103, Max-Change: 0.00128
Iteration: 72, Log-Lik: -108496.107, Max-Change: 0.00089
Iteration: 73, Log-Lik: -108496.087, Max-Change: 0.00268
Iteration: 74, Log-Lik: -108496.113, Max-Change: 0.00110
Iteration: 75, Log-Lik: -108496.118, Max-Change: 0.00076
Iteration: 76, Log-Lik: -108496.105, Max-Change: 0.00228
Iteration: 77, Log-Lik: -108496.129, Max-Change: 0.00094
Iteration: 78, Log-Lik: -108496.134, Max-Change: 0.00065
Iteration: 79, Log-Lik: -108496.126, Max-Change: 0.00196
Iteration: 80, Log-Lik: -108496.148, Max-Change: 0.00080
Iteration: 81, Log-Lik: -108496.153, Max-Change: 0.00056
Iteration: 82, Log-Lik: -108496.148, Max-Change: 0.00164
Iteration: 83, Log-Lik: -108496.167, Max-Change: 0.00068
Iteration: 84, Log-Lik: -108496.172, Max-Change: 0.00047
Iteration: 85, Log-Lik: -108496.170, Max-Change: 0.00143
Iteration: 86, Log-Lik: -108496.187, Max-Change: 0.00058
Iteration: 87, Log-Lik: -108496.191, Max-Change: 0.00040
Iteration: 88, Log-Lik: -108496.190, Max-Change: 0.00115
Iteration: 89, Log-Lik: -108496.204, Max-Change: 0.00048
Iteration: 90, Log-Lik: -108496.208, Max-Change: 0.00034
Iteration: 91, Log-Lik: -108496.208, Max-Change: 0.00106
Iteration: 92, Log-Lik: -108496.221, Max-Change: 0.00042
Iteration: 93, Log-Lik: -108496.224, Max-Change: 0.00029
Iteration: 94, Log-Lik: -108496.225, Max-Change: 0.00084
Iteration: 95, Log-Lik: -108496.236, Max-Change: 0.00035
Iteration: 96, Log-Lik: -108496.239, Max-Change: 0.00024
Iteration: 97, Log-Lik: -108496.239, Max-Change: 0.00076
Iteration: 98, Log-Lik: -108496.249, Max-Change: 0.00030
Iteration: 99, Log-Lik: -108496.252, Max-Change: 0.00021
Iteration: 100, Log-Lik: -108496.253, Max-Change: 0.00057
Iteration: 101, Log-Lik: -108496.261, Max-Change: 0.00024
Iteration: 102, Log-Lik: -108496.263, Max-Change: 0.00017
Iteration: 103, Log-Lik: -108496.264, Max-Change: 0.00057
Iteration: 104, Log-Lik: -108496.271, Max-Change: 0.00022
Iteration: 105, Log-Lik: -108496.273, Max-Change: 0.00015
Iteration: 106, Log-Lik: -108496.274, Max-Change: 0.00038
Iteration: 107, Log-Lik: -108496.279, Max-Change: 0.00017
Iteration: 108, Log-Lik: -108496.281, Max-Change: 0.00012
Iteration: 109, Log-Lik: -108496.282, Max-Change: 0.00041
Iteration: 110, Log-Lik: -108496.287, Max-Change: 0.00016
Iteration: 111, Log-Lik: -108496.289, Max-Change: 0.00011
Iteration: 112, Log-Lik: -108496.290, Max-Change: 0.00027
Iteration: 113, Log-Lik: -108496.293, Max-Change: 0.00012
Iteration: 114, Log-Lik: -108496.294, Max-Change: 0.00009

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 3: Leave one out MI testing

## 
## Step 4: Fit without DIF items, liberal threshold

## 
Iteration: 1, Log-Lik: -124923.204, Max-Change: 0.44306
Iteration: 2, Log-Lik: -124261.810, Max-Change: 0.28035
Iteration: 3, Log-Lik: -124090.517, Max-Change: 0.16091
Iteration: 4, Log-Lik: -124044.408, Max-Change: 0.08682
Iteration: 5, Log-Lik: -124031.961, Max-Change: 0.05059
Iteration: 6, Log-Lik: -124027.799, Max-Change: 0.02832
Iteration: 7, Log-Lik: -124026.543, Max-Change: 0.01513
Iteration: 8, Log-Lik: -124026.108, Max-Change: 0.00840
Iteration: 9, Log-Lik: -124025.944, Max-Change: 0.00490
Iteration: 10, Log-Lik: -124025.846, Max-Change: 0.00174
Iteration: 11, Log-Lik: -124025.822, Max-Change: 0.00129
Iteration: 12, Log-Lik: -124025.813, Max-Change: 0.00075
Iteration: 13, Log-Lik: -124025.801, Max-Change: 0.00030
Iteration: 14, Log-Lik: -124025.799, Max-Change: 0.00022
Iteration: 15, Log-Lik: -124025.798, Max-Change: 0.00019
Iteration: 16, Log-Lik: -124025.795, Max-Change: 0.00016
Iteration: 17, Log-Lik: -124025.795, Max-Change: 0.00013
Iteration: 18, Log-Lik: -124025.795, Max-Change: 0.00011
Iteration: 19, Log-Lik: -124025.794, Max-Change: 0.00004

## 
## Step 5: Fit without DIF items, conservative threshold

## 
Iteration: 1, Log-Lik: -120604.700, Max-Change: 0.56289
Iteration: 2, Log-Lik: -119374.953, Max-Change: 0.27371
Iteration: 3, Log-Lik: -119146.152, Max-Change: 0.20439
Iteration: 4, Log-Lik: -119044.549, Max-Change: 0.10821
Iteration: 5, Log-Lik: -119011.441, Max-Change: 0.05422
Iteration: 6, Log-Lik: -119001.916, Max-Change: 0.03152
Iteration: 7, Log-Lik: -118998.136, Max-Change: 0.02445
Iteration: 8, Log-Lik: -118996.070, Max-Change: 0.01406
Iteration: 9, Log-Lik: -118995.166, Max-Change: 0.00854
Iteration: 10, Log-Lik: -118994.446, Max-Change: 0.00274
Iteration: 11, Log-Lik: -118994.322, Max-Change: 0.00236
Iteration: 12, Log-Lik: -118994.247, Max-Change: 0.00204
Iteration: 13, Log-Lik: -118994.104, Max-Change: 0.00102
Iteration: 14, Log-Lik: -118994.097, Max-Change: 0.00066
Iteration: 15, Log-Lik: -118994.092, Max-Change: 0.00058
Iteration: 16, Log-Lik: -118994.076, Max-Change: 0.00026
Iteration: 17, Log-Lik: -118994.076, Max-Change: 0.00012
Iteration: 18, Log-Lik: -118994.076, Max-Change: 0.00007

## 
## Step 6: Fit with anchor items, liberal threshold

## 
Iteration: 1, Log-Lik: -111250.918, Max-Change: 0.77817
Iteration: 2, Log-Lik: -108581.996, Max-Change: 0.16757
Iteration: 3, Log-Lik: -108461.122, Max-Change: 0.08865
Iteration: 4, Log-Lik: -108436.807, Max-Change: 0.04992
Iteration: 5, Log-Lik: -108423.086, Max-Change: 0.02974
Iteration: 6, Log-Lik: -108411.951, Max-Change: 0.02328
Iteration: 7, Log-Lik: -108401.942, Max-Change: 0.02340
Iteration: 8, Log-Lik: -108392.655, Max-Change: 0.02350
Iteration: 9, Log-Lik: -108383.952, Max-Change: 0.02305
Iteration: 10, Log-Lik: -108375.767, Max-Change: 0.02278
Iteration: 11, Log-Lik: -108368.058, Max-Change: 0.02216
Iteration: 12, Log-Lik: -108360.797, Max-Change: 0.02156
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Iteration: 155, Log-Lik: -108230.127, Max-Change: 0.00014
Iteration: 156, Log-Lik: -108230.127, Max-Change: 0.00010
Iteration: 157, Log-Lik: -108230.128, Max-Change: 0.00026
Iteration: 158, Log-Lik: -108230.130, Max-Change: 0.00011
Iteration: 159, Log-Lik: -108230.131, Max-Change: 0.00009

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 7: Fit with anchor items, conservative threshold

## 
Iteration: 1, Log-Lik: -111250.918, Max-Change: 0.77816
Iteration: 2, Log-Lik: -108608.553, Max-Change: 0.17227
Iteration: 3, Log-Lik: -108486.853, Max-Change: 0.09414
Iteration: 4, Log-Lik: -108463.279, Max-Change: 0.05239
Iteration: 5, Log-Lik: -108448.996, Max-Change: 0.03016
Iteration: 6, Log-Lik: -108436.820, Max-Change: 0.02513
Iteration: 7, Log-Lik: -108425.691, Max-Change: 0.02493
Iteration: 8, Log-Lik: -108415.345, Max-Change: 0.02456
Iteration: 9, Log-Lik: -108405.685, Max-Change: 0.02408
Iteration: 10, Log-Lik: -108396.652, Max-Change: 0.02353
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Iteration: 12, Log-Lik: -108380.272, Max-Change: 0.02220
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Iteration: 15, Log-Lik: -108359.359, Max-Change: 0.02022
Iteration: 16, Log-Lik: -108353.232, Max-Change: 0.01992
Iteration: 17, Log-Lik: -108347.479, Max-Change: 0.01940
Iteration: 18, Log-Lik: -108342.076, Max-Change: 0.01878
Iteration: 19, Log-Lik: -108336.998, Max-Change: 0.01826
Iteration: 20, Log-Lik: -108332.225, Max-Change: 0.01775
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Iteration: 22, Log-Lik: -108309.069, Max-Change: 0.05121
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Iteration: 128, Log-Lik: -108250.862, Max-Change: 0.00011
Iteration: 129, Log-Lik: -108250.862, Max-Change: 0.00009

## Warning: Log-likelihood was decreasing near the ML solution. EM method may be
## unstable

## 
## Step 8: Get scores

#more and after test level
map(nlsy_1g_dif_by_test_hw, ~.$effect_size_test)

## $`Arithmetic reasoning`
## $`Arithmetic reasoning`$liberal
##           Effect Size   Value
## 1                STDS 0.05650
## 2                UTDS 0.60493
## 3              UETSDS 0.26396
## 4               ETSSD 0.00857
## 5         Starks.DTFR 0.03857
## 6               UDTFR 0.56803
## 7              UETSDN 0.22768
## 8 theta.of.max.test.D 3.03773
## 9           Test.Dmax 0.57225
## 
## $`Arithmetic reasoning`$conservative
##           Effect Size  Value
## 1                STDS 0.0740
## 2                UTDS 0.3719
## 3              UETSDS 0.1777
## 4               ETSSD 0.0112
## 5         Starks.DTFR 0.0776
## 6               UDTFR 0.3386
## 7              UETSDN 0.1552
## 8 theta.of.max.test.D 2.9329
## 9           Test.Dmax 0.3942
## 
## 
## $`Word knowledge`
## $`Word knowledge`$liberal
##           Effect Size    Value
## 1                STDS -0.04505
## 2                UTDS  1.62044
## 3              UETSDS  0.14877
## 4               ETSSD -0.00705
## 5         Starks.DTFR -0.04269
## 6               UDTFR  1.66044
## 7              UETSDN  0.13532
## 8 theta.of.max.test.D -0.39953
## 9           Test.Dmax -0.34043
## 
## $`Word knowledge`$conservative
##           Effect Size   Value
## 1                STDS -0.1659
## 2                UTDS  1.4559
## 3              UETSDS  0.1906
## 4               ETSSD -0.0259
## 5         Starks.DTFR -0.1722
## 6               UDTFR  1.4949
## 7              UETSDN  0.1874
## 8 theta.of.max.test.D -0.2611
## 9           Test.Dmax -0.4141
## 
## 
## $`Paragraph comprehension`
## $`Paragraph comprehension`$liberal
##           Effect Size   Value
## 1                STDS -0.1386
## 2                UTDS  0.4808
## 3              UETSDS  0.1386
## 4               ETSSD -0.0541
## 5         Starks.DTFR -0.1427
## 6               UDTFR  0.4834
## 7              UETSDN  0.1428
## 8 theta.of.max.test.D  0.2487
## 9           Test.Dmax -0.1598
## 
## $`Paragraph comprehension`$conservative
##           Effect Size   Value
## 1                STDS -0.1030
## 2                UTDS  0.3330
## 3              UETSDS  0.1030
## 4               ETSSD -0.0402
## 5         Starks.DTFR -0.1036
## 6               UDTFR  0.3381
## 7              UETSDN  0.1036
## 8 theta.of.max.test.D  2.3337
## 9           Test.Dmax -0.1373
## 
## 
## $`Mathematics knowledge`
## $`Mathematics knowledge`$liberal
##           Effect Size   Value
## 1                STDS -0.2616
## 2                UTDS  0.9083
## 3              UETSDS  0.3133
## 4               ETSSD -0.0448
## 5         Starks.DTFR -0.3655
## 6               UDTFR  0.9218
## 7              UETSDN  0.3997
## 8 theta.of.max.test.D  1.1358
## 9           Test.Dmax -0.5792
## 
## $`Mathematics knowledge`$conservative
##           Effect Size   Value
## 1                STDS -0.0997
## 2                UTDS  0.7137
## 3              UETSDS  0.2549
## 4               ETSSD -0.0170
## 5         Starks.DTFR -0.2022
## 6               UDTFR  0.7029
## 7              UETSDN  0.2971
## 8 theta.of.max.test.D -1.4964
## 9           Test.Dmax  0.4686

#gap sizes by test and scoring method
nlsy_fit_g_gaps_by_test_hw = map(nlsy_1g_dif_by_test_hw, function(x) {
  (describeBy(map_df(x$scores, ~.[, 1] %>% standardize(focal_group = nlsy_hw$SIRE == "White")), group = nlsy_hw$SIRE))
})
nlsy_fit_g_gaps_by_test_hw

## $`Arithmetic reasoning`
## 
##  Descriptive statistics by group 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 6352    0  1  -0.13   -0.04 1.10 -2.72 2.04  4.76
## noDIF_liberal          2 6352    0  1  -0.03    0.01 1.19 -2.70 1.58  4.28
## noDIF_conservative     3 6352    0  1  -0.07   -0.02 1.15 -2.64 1.86  4.50
## anchor_liberal         4 6352    0  1  -0.13   -0.05 1.10 -2.71 2.06  4.76
## anchor_conservative    5 6352    0  1  -0.14   -0.05 1.10 -2.70 2.06  4.76
##                      skew kurtosis   se
## original             0.33    -0.76 0.01
## noDIF_liberal       -0.02    -1.03 0.01
## noDIF_conservative   0.14    -0.87 0.01
## anchor_liberal       0.35    -0.74 0.01
## anchor_conservative  0.35    -0.74 0.01
## ------------------------------------------------------------ 
## group: Hispanic
##                     vars    n  mean   sd median trimmed  mad   min  max range
## original               1 1682 -0.65 0.81  -0.85   -0.74 0.70 -2.46 2.04  4.50
## noDIF_liberal          2 1682 -0.65 0.88  -0.84   -0.72 0.82 -2.70 1.58  4.28
## noDIF_conservative     3 1682 -0.65 0.85  -0.83   -0.73 0.79 -2.64 1.86  4.50
## anchor_liberal         4 1682 -0.72 0.72  -0.88   -0.79 0.67 -2.34 1.40  3.75
## anchor_conservative    5 1682 -0.71 0.72  -0.88   -0.78 0.68 -2.36 1.41  3.77
##                     skew kurtosis   se
## original            1.02     0.79 0.02
## noDIF_liberal       0.66    -0.15 0.02
## noDIF_conservative  0.77     0.18 0.02
## anchor_liberal      0.82     0.15 0.02
## anchor_conservative 0.81     0.13 0.02
## 
## $`Word knowledge`
## 
##  Descriptive statistics by group 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 6352    0  1   0.01    0.01 1.10 -2.93 1.70  4.63
## noDIF_liberal          2 6352    0  1   0.71    0.18 0.00 -3.37 0.71  4.08
## noDIF_conservative     3 6352    0  1   0.23    0.14 1.07 -3.63 0.96  4.58
## anchor_liberal         4 6352    0  1   0.00    0.01 1.11 -2.88 1.71  4.60
## anchor_conservative    5 6352    0  1   0.00    0.01 1.10 -2.89 1.71  4.60
##                      skew kurtosis   se
## original            -0.10    -0.74 0.01
## noDIF_liberal       -1.19     0.37 0.01
## noDIF_conservative  -0.85    -0.20 0.01
## anchor_liberal      -0.07    -0.75 0.01
## anchor_conservative -0.07    -0.75 0.01
## ------------------------------------------------------------ 
## group: Hispanic
##                     vars    n  mean   sd median trimmed  mad   min  max range
## original               1 1682 -0.78 0.92  -0.93   -0.85 0.87 -2.73 1.70  4.43
## noDIF_liberal          2 1682 -0.68 1.16  -0.57   -0.61 1.77 -3.37 0.71  4.08
## noDIF_conservative     3 1682 -0.76 1.08  -0.82   -0.76 1.28 -3.37 0.96  4.32
## anchor_liberal         4 1682 -0.85 0.81  -0.94   -0.89 0.83 -2.69 1.05  3.74
## anchor_conservative    5 1682 -0.86 0.80  -0.96   -0.90 0.83 -2.66 1.01  3.68
##                      skew kurtosis   se
## original             0.69     0.03 0.02
## noDIF_liberal       -0.22    -1.06 0.03
## noDIF_conservative   0.05    -0.98 0.03
## anchor_liberal       0.41    -0.49 0.02
## anchor_conservative  0.41    -0.50 0.02
## 
## $`Paragraph comprehension`
## 
##  Descriptive statistics by group 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 6352    0  1   0.15    0.07 0.99 -3.24 1.37  4.61
## noDIF_liberal          2 6352    0  1   0.13    0.16 1.08 -2.78 0.85  3.63
## noDIF_conservative     3 6352    0  1   0.07    0.10 1.00 -2.85 1.19  4.04
## anchor_liberal         4 6352    0  1   0.15    0.07 1.01 -3.16 1.37  4.53
## anchor_conservative    5 6352    0  1   0.14    0.07 1.01 -3.16 1.37  4.53
##                      skew kurtosis   se
## original            -0.54    -0.46 0.01
## noDIF_liberal       -0.97    -0.06 0.01
## noDIF_conservative  -0.62    -0.45 0.01
## anchor_liberal      -0.52    -0.51 0.01
## anchor_conservative -0.53    -0.50 0.01
## ------------------------------------------------------------ 
## group: Hispanic
##                     vars    n  mean   sd median trimmed  mad   min  max range
## original               1 1682 -0.73 1.02  -0.78   -0.76 1.16 -3.24 1.37  4.61
## noDIF_liberal          2 1682 -0.70 1.15  -0.72   -0.66 1.37 -2.78 0.85  3.63
## noDIF_conservative     3 1682 -0.72 1.05  -0.78   -0.75 1.21 -2.85 1.19  4.04
## anchor_liberal         4 1682 -0.96 0.87  -0.96   -0.97 1.04 -3.23 0.69  3.92
## anchor_conservative    5 1682 -0.95 0.88  -0.94   -0.96 1.04 -3.23 0.72  3.94
##                      skew kurtosis   se
## original             0.22    -0.79 0.02
## noDIF_liberal       -0.11    -1.17 0.03
## noDIF_conservative   0.15    -0.89 0.03
## anchor_liberal       0.02    -0.91 0.02
## anchor_conservative  0.02    -0.90 0.02
## 
## $`Mathematics knowledge`
## 
##  Descriptive statistics by group 
## group: White
##                     vars    n mean sd median trimmed  mad   min  max range
## original               1 6352    0  1  -0.19   -0.06 1.06 -2.44 2.12  4.56
## noDIF_liberal          2 6352    0  1  -0.01    0.02 1.16 -2.09 1.54  3.64
## noDIF_conservative     3 6352    0  1  -0.11   -0.03 1.15 -2.23 1.85  4.08
## anchor_liberal         4 6352    0  1  -0.19   -0.06 1.05 -2.45 2.14  4.59
## anchor_conservative    5 6352    0  1  -0.19   -0.06 1.05 -2.44 2.14  4.58
##                      skew kurtosis   se
## original             0.45    -0.71 0.01
## noDIF_liberal       -0.06    -0.97 0.01
## noDIF_conservative   0.20    -0.92 0.01
## anchor_liberal       0.47    -0.69 0.01
## anchor_conservative  0.47    -0.69 0.01
## ------------------------------------------------------------ 
## group: Hispanic
##                     vars    n  mean   sd median trimmed  mad   min  max range
## original               1 1682 -0.55 0.85  -0.75   -0.66 0.65 -2.24 2.12  4.37
## noDIF_liberal          2 1682 -0.54 0.89  -0.65   -0.59 0.78 -2.09 1.54  3.64
## noDIF_conservative     3 1682 -0.53 0.85  -0.71   -0.61 0.67 -2.23 1.85  4.08
## anchor_liberal         4 1682 -0.64 0.71  -0.79   -0.71 0.57 -2.06 1.39  3.45
## anchor_conservative    5 1682 -0.61 0.70  -0.77   -0.69 0.57 -2.02 1.41  3.42
##                     skew kurtosis   se
## original            1.09     0.83 0.02
## noDIF_liberal       0.50    -0.23 0.02
## noDIF_conservative  0.85     0.41 0.02
## anchor_liberal      0.94     0.31 0.02
## anchor_conservative 0.94     0.31 0.02

#number of DIF items by test
map(nlsy_1g_dif_by_test_hw, function(x) {
  x$DIF_stats %>% select(p, p_adj) %>% {colSums(. < .05)}
})

## $`Arithmetic reasoning`
##     p p_adj 
##    15     7 
## 
## $`Word knowledge`
##     p p_adj 
##    29    22 
## 
## $`Paragraph comprehension`
##     p p_adj 
##    10     6 
## 
## $`Mathematics knowledge`
##     p p_adj 
##    17    11

#use non-DIF items from each of the 4 testings
nlsy_1g_dif_by_test_hw_noDIF_items = map(nlsy_1g_dif_by_test_hw, ~.$DIF_stats %>% filter(p_adj > .05) %>% pull(item)) %>% do.call(what = c)
length(nlsy_1g_dif_by_test_hw_noDIF_items)

## [1] 59

#fit again
nlsy_1g_dif_by_test_hw_noDIF_items_fit = mirt(nlsy_items_hw[nlsy_1g_dif_by_test_hw_noDIF_items], model = 1)

## 
Iteration: 1, Log-Lik: -229099.311, Max-Change: 2.79982
Iteration: 2, Log-Lik: -225711.779, Max-Change: 1.39142
Iteration: 3, Log-Lik: -224994.626, Max-Change: 0.43834
Iteration: 4, Log-Lik: -224735.935, Max-Change: 0.21048
Iteration: 5, Log-Lik: -224609.800, Max-Change: 0.28441
Iteration: 6, Log-Lik: -224518.307, Max-Change: 0.26778
Iteration: 7, Log-Lik: -224441.854, Max-Change: 0.26298
Iteration: 8, Log-Lik: -224379.099, Max-Change: 0.11382
Iteration: 9, Log-Lik: -224330.169, Max-Change: 0.15167
Iteration: 10, Log-Lik: -224293.121, Max-Change: 0.09521
Iteration: 11, Log-Lik: -224264.830, Max-Change: 0.11119
Iteration: 12, Log-Lik: -224242.605, Max-Change: 0.08139
Iteration: 13, Log-Lik: -224224.911, Max-Change: 0.08754
Iteration: 14, Log-Lik: -224210.410, Max-Change: 0.07361
Iteration: 15, Log-Lik: -224198.337, Max-Change: 0.07217
Iteration: 16, Log-Lik: -224188.046, Max-Change: 0.06934
Iteration: 17, Log-Lik: -224179.145, Max-Change: 0.06311
Iteration: 18, Log-Lik: -224171.325, Max-Change: 0.06236
Iteration: 19, Log-Lik: -224164.820, Max-Change: 0.05672
Iteration: 20, Log-Lik: -224159.121, Max-Change: 0.07218
Iteration: 21, Log-Lik: -224154.044, Max-Change: 0.06162
Iteration: 22, Log-Lik: -224149.405, Max-Change: 0.07065
Iteration: 23, Log-Lik: -224145.131, Max-Change: 0.05816
Iteration: 24, Log-Lik: -224141.149, Max-Change: 0.06656
Iteration: 25, Log-Lik: -224139.090, Max-Change: 0.01551
Iteration: 26, Log-Lik: -224133.760, Max-Change: 0.05483
Iteration: 27, Log-Lik: -224129.997, Max-Change: 0.04985
Iteration: 28, Log-Lik: -224128.326, Max-Change: 0.03449
Iteration: 29, Log-Lik: -224123.524, Max-Change: 0.05016
Iteration: 30, Log-Lik: -224120.992, Max-Change: 0.04437
Iteration: 31, Log-Lik: -224119.499, Max-Change: 0.05196
Iteration: 32, Log-Lik: -224117.019, Max-Change: 0.04589
Iteration: 33, Log-Lik: -224114.863, Max-Change: 0.01240
Iteration: 34, Log-Lik: -224112.573, Max-Change: 0.04617
Iteration: 35, Log-Lik: -224110.651, Max-Change: 0.03909
Iteration: 36, Log-Lik: -224108.933, Max-Change: 0.04498
Iteration: 37, Log-Lik: -224107.998, Max-Change: 0.01375
Iteration: 38, Log-Lik: -224105.628, Max-Change: 0.04102
Iteration: 39, Log-Lik: -224104.037, Max-Change: 0.03670
Iteration: 40, Log-Lik: -224103.273, Max-Change: 0.01194
Iteration: 41, Log-Lik: -224101.138, Max-Change: 0.00675
Iteration: 42, Log-Lik: -224099.190, Max-Change: 0.00637
Iteration: 43, Log-Lik: -224089.731, Max-Change: 0.01402
Iteration: 44, Log-Lik: -224088.731, Max-Change: 0.00499
Iteration: 45, Log-Lik: -224087.890, Max-Change: 0.00473
Iteration: 46, Log-Lik: -224083.828, Max-Change: 0.00904
Iteration: 47, Log-Lik: -224083.353, Max-Change: 0.00379
Iteration: 48, Log-Lik: -224082.943, Max-Change: 0.00412
Iteration: 49, Log-Lik: -224080.942, Max-Change: 0.00355
Iteration: 50, Log-Lik: -224080.730, Max-Change: 0.00287
Iteration: 51, Log-Lik: -224080.537, Max-Change: 0.00279
Iteration: 52, Log-Lik: -224079.585, Max-Change: 0.00213
Iteration: 53, Log-Lik: -224079.485, Max-Change: 0.00203
Iteration: 54, Log-Lik: -224079.393, Max-Change: 0.00195
Iteration: 55, Log-Lik: -224078.942, Max-Change: 0.00117
Iteration: 56, Log-Lik: -224078.896, Max-Change: 0.00113
Iteration: 57, Log-Lik: -224078.854, Max-Change: 0.00114
Iteration: 58, Log-Lik: -224078.647, Max-Change: 0.00094
Iteration: 59, Log-Lik: -224078.631, Max-Change: 0.00077
Iteration: 60, Log-Lik: -224078.614, Max-Change: 0.00074
Iteration: 61, Log-Lik: -224078.528, Max-Change: 0.00063
Iteration: 62, Log-Lik: -224078.517, Max-Change: 0.00060
Iteration: 63, Log-Lik: -224078.507, Max-Change: 0.00058
Iteration: 64, Log-Lik: -224078.458, Max-Change: 0.00016
Iteration: 65, Log-Lik: -224078.455, Max-Change: 0.00016
Iteration: 66, Log-Lik: -224078.452, Max-Change: 0.00017
Iteration: 67, Log-Lik: -224078.436, Max-Change: 0.00097
Iteration: 68, Log-Lik: -224078.426, Max-Change: 0.00012
Iteration: 69, Log-Lik: -224078.424, Max-Change: 0.00013
Iteration: 70, Log-Lik: -224078.414, Max-Change: 0.00088
Iteration: 71, Log-Lik: -224078.408, Max-Change: 0.00013
Iteration: 72, Log-Lik: -224078.406, Max-Change: 0.00011
Iteration: 73, Log-Lik: -224078.403, Max-Change: 0.00068
Iteration: 74, Log-Lik: -224078.398, Max-Change: 0.00010
Iteration: 75, Log-Lik: -224078.397, Max-Change: 0.00010
Iteration: 76, Log-Lik: -224078.394, Max-Change: 0.00058
Iteration: 77, Log-Lik: -224078.391, Max-Change: 0.00011
Iteration: 78, Log-Lik: -224078.390, Max-Change: 0.00009

nlsy_1g_dif_by_test_hw_noDIF_items_fit_scores = fscores(nlsy_1g_dif_by_test_hw_noDIF_items_fit, full.scores = T, full.scores.SE = T)

#gap
nlsy_hw$g_noDIF2 = nlsy_1g_dif_by_test_hw_noDIF_items_fit_scores[, 1] %>% as.vector() %>% standardize(focal_group = (nlsy_hw$SIRE == "White"))

GG_denhist(nlsy_hw, "g_noDIF2", "SIRE") +
  scale_fill_discrete("Race")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

describeBy(nlsy_hw$g_noDIF2, nlsy_hw$SIRE)

## 
##  Descriptive statistics by group 
## group: White
##    vars    n mean sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 6352    0  1  -0.08   -0.03 1.07 -2.32 2.33  4.65 0.28    -0.62 0.01
## ------------------------------------------------------------ 
## group: Hispanic
##    vars    n  mean   sd median trimmed  mad   min  max range skew kurtosis   se
## X1    1 1682 -0.71 0.85  -0.86   -0.79 0.77 -2.41 2.33  4.75 0.94     0.65 0.02

#reliability
empirical_rxx(nlsy_1g_dif_by_test_hw_noDIF_items_fit_scores)

##    F1 
## 0.933

marginal_rxx(nlsy_1g_dif_by_test_hw_noDIF_items_fit)

## [1] 0.933

Meta

#versions
write_sessioninfo()

## R version 4.0.3 (2020-10-10)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Linux Mint 19.3
## 
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
##  [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=en_US.UTF-8   
##  [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] stats4    stats     graphics  grDevices datasets  utils     methods  
## [8] base     
## 
## other attached packages:
##  [1] renv_0.12.0           furrr_0.2.0           future_1.19.1        
##  [4] lavaan_0.6-7          mirt_1.32.1           rms_6.0-1            
##  [7] SparseM_1.78          polycor_0.7-10        haven_2.3.1          
## [10] kirkegaard_2020-11-08 metafor_2.4-0         Matrix_1.2-18        
## [13] psych_2.0.9           magrittr_1.5          assertthat_0.2.1     
## [16] weights_1.0.1         mice_3.11.0           gdata_2.18.0         
## [19] Hmisc_4.4-1           Formula_1.2-4         survival_3.2-7       
## [22] lattice_0.20-41       forcats_0.5.0         stringr_1.4.0        
## [25] dplyr_1.0.2           purrr_0.3.4           readr_1.4.0          
## [28] tidyr_1.1.2           tibble_3.0.4          ggplot2_3.3.2        
## [31] tidyverse_1.3.0       pacman_0.5.1         
## 
## loaded via a namespace (and not attached):
##   [1] readxl_1.3.1          backports_1.1.10      plyr_1.8.6           
##   [4] GPArotation_2014.11-1 splines_4.0.3         listenv_0.8.0        
##   [7] TH.data_1.0-10        digest_0.6.26         htmltools_0.5.0      
##  [10] fansi_0.4.1           checkmate_2.0.0       cluster_2.1.0        
##  [13] openxlsx_4.2.2        globals_0.13.1        modelr_0.1.8         
##  [16] matrixStats_0.57.0    sandwich_3.0-0        jpeg_0.1-8.1         
##  [19] colorspace_1.4-1      blob_1.2.1            rvest_0.3.6          
##  [22] xfun_0.18             crayon_1.3.4          jsonlite_1.7.1       
##  [25] lme4_1.1-23           zoo_1.8-8             glue_1.4.2           
##  [28] gtable_0.3.0          emmeans_1.5.1         MatrixModels_0.4-1   
##  [31] sjmisc_2.8.5          sjstats_0.18.0        car_3.0-10           
##  [34] dcurver_0.9.1         abind_1.4-5           scales_1.1.1         
##  [37] mvtnorm_1.1-1         DBI_1.1.0             Rcpp_1.0.5           
##  [40] performance_0.5.0     xtable_1.8-4          htmlTable_2.1.0      
##  [43] tmvnsim_1.0-2         foreign_0.8-79        htmlwidgets_1.5.2    
##  [46] httr_1.4.2            RColorBrewer_1.1-2    ellipsis_0.3.1       
##  [49] pkgconfig_2.0.3       farver_2.0.3          nnet_7.3-14          
##  [52] dbplyr_1.4.4          tidyselect_1.1.0      labeling_0.4.2       
##  [55] rlang_0.4.8           effectsize_0.3.3      munsell_0.5.0        
##  [58] multilevel_2.6        cellranger_1.1.0      tools_4.0.3          
##  [61] cli_2.1.0             generics_0.0.2        sjlabelled_1.1.7     
##  [64] broom_0.5.6           evaluate_0.14         yaml_2.2.1           
##  [67] knitr_1.30            fs_1.5.0              zip_2.1.1            
##  [70] nlme_3.1-150          quantreg_5.74         xml2_1.3.2           
##  [73] psychometric_2.2      compiler_4.0.3        rstudioapi_0.11      
##  [76] curl_4.3              png_0.1-7             reprex_0.3.0         
##  [79] statmod_1.4.35        pbivnorm_0.6.0        stringi_1.5.3        
##  [82] parameters_0.8.6      nloptr_1.2.2.2        vegan_2.5-6          
##  [85] permute_0.9-5         vctrs_0.3.4           pillar_1.4.6         
##  [88] lifecycle_0.2.0       pwr_1.3-0             estimability_1.3     
##  [91] data.table_1.13.2     insight_0.9.6         conquer_1.0.2        
##  [94] R6_2.4.1              latticeExtra_0.6-29   gridExtra_2.3        
##  [97] rio_0.5.16            codetools_0.2-18      polspline_1.1.19     
## [100] boot_1.3-25           MASS_7.3-53           gtools_3.8.2         
## [103] withr_2.3.0           mnormt_2.0.2          Deriv_4.1.0          
## [106] multcomp_1.4-14       mgcv_1.8-33           bayestestR_0.7.2     
## [109] parallel_4.0.3        hms_0.5.3             grid_4.0.3           
## [112] rpart_4.1-15          minqa_1.2.4           rmarkdown_2.5        
## [115] carData_3.0-4         lubridate_1.7.9       base64enc_0.1-3

#data out
ves %>% write_rds("data/data_out_ves.rds", compress = "xz")
nlsy %>% write_rds("data/data_out_nlsy.rds", compress = "xz")