# loading required packages
library(ltm)
library(psych)
library(CTT)
library(itemanalysis)
# read in data file
data("dichotomous")
dat <- dichotomous
options(scipen = 999)
# Math subscale
math <- dat[,1:10]
# Check cases with missing data
sum(is.na(math))
## [1] 526
# removing case with missing data
math.no.na <- na.omit(math)
# Scoring multiple-choice items to generate dichotomous data
key <- c("A","D","C","B","C","B","C","D","A","D")
scored <- CTT::score(math.no.na, key, output.scored=TRUE)
# Extract dichotomous responses
math.dich <- as.data.frame(scored$scored)
# Compute total score
math.dich$total <- apply(math.dich, 1, sum)
# Item analysis
psych::alpha(math.dich[,1:10])
##
## Reliability analysis
## Call: psych::alpha(x = math.dich[, 1:10])
##
## raw_alpha std.alpha G6(smc) average_r S/N ase mean sd median_r
## 0.63 0.65 0.63 0.15 1.8 0.0074 0.68 0.21 0.15
##
## 95% confidence boundaries
## lower alpha upper
## Feldt 0.61 0.63 0.64
## Duhachek 0.61 0.63 0.64
##
## Reliability if an item is dropped:
## raw_alpha std.alpha G6(smc) average_r S/N alpha se var.r med.r
## item1 0.60 0.63 0.60 0.16 1.7 0.0079 0.0036 0.15
## item2 0.62 0.64 0.62 0.16 1.8 0.0075 0.0030 0.15
## item3 0.58 0.60 0.57 0.14 1.5 0.0083 0.0023 0.14
## item4 0.62 0.64 0.61 0.16 1.7 0.0076 0.0034 0.15
## item5 0.61 0.63 0.61 0.16 1.7 0.0077 0.0032 0.15
## item6 0.59 0.62 0.59 0.15 1.6 0.0080 0.0032 0.15
## item7 0.61 0.63 0.61 0.16 1.7 0.0077 0.0035 0.15
## item8 0.59 0.60 0.58 0.14 1.5 0.0082 0.0024 0.14
## item9 0.61 0.63 0.61 0.16 1.7 0.0077 0.0034 0.15
## item10 0.58 0.60 0.58 0.14 1.5 0.0083 0.0023 0.14
##
## Item statistics
## n raw.r std.r r.cor r.drop mean sd
## item1 5764 0.47 0.47 0.36 0.29 0.72 0.45
## item2 5764 0.43 0.41 0.28 0.22 0.54 0.50
## item3 5764 0.56 0.58 0.53 0.41 0.82 0.38
## item4 5764 0.45 0.43 0.30 0.24 0.58 0.49
## item5 5764 0.46 0.45 0.33 0.26 0.66 0.47
## item6 5764 0.50 0.51 0.42 0.33 0.74 0.44
## item7 5764 0.47 0.45 0.33 0.26 0.47 0.50
## item8 5764 0.52 0.57 0.50 0.40 0.87 0.34
## item9 5764 0.46 0.44 0.32 0.25 0.55 0.50
## item10 5764 0.55 0.57 0.51 0.40 0.82 0.38
##
## Non missing response frequency for each item
## 0 1 miss
## item1 0.28 0.72 0
## item2 0.46 0.54 0
## item3 0.18 0.82 0
## item4 0.42 0.58 0
## item5 0.34 0.66 0
## item6 0.26 0.74 0
## item7 0.53 0.47 0
## item8 0.13 0.87 0
## item9 0.45 0.55 0
## item10 0.18 0.82 0
IA1 <-ltm::descript(math.dich[,1:10])
IA1
##
## Descriptive statistics for the 'math.dich[, 1:10]' data-set
##
## Sample:
## 10 items and 5764 sample units; 0 missing values
##
## Proportions for each level of response:
## 0 1 logit
## item1 0.2837 0.7163 0.9264
## item2 0.4627 0.5373 0.1495
## item3 0.1778 0.8222 1.5311
## item4 0.4186 0.5814 0.3284
## item5 0.3357 0.6643 0.6825
## item6 0.2569 0.7431 1.0619
## item7 0.5324 0.4676 -0.1300
## item8 0.1315 0.8685 1.8877
## item9 0.4454 0.5546 0.2195
## item10 0.1782 0.8218 1.5288
##
##
## Frequencies of total scores:
## 0 1 2 3 4 5 6 7 8 9 10
## Freq 22 67 159 266 389 556 809 1040 1131 888 437
##
##
## Point Biserial correlation with Total Score:
## Included Excluded
## item1 0.4739 0.2872
## item2 0.4340 0.2186
## item3 0.5555 0.4134
## item4 0.4467 0.2357
## item5 0.4558 0.2562
## item6 0.5028 0.3273
## item7 0.4689 0.2587
## item8 0.5242 0.3956
## item9 0.4627 0.2524
## item10 0.5455 0.4014
##
##
## Cronbach's alpha:
## value
## All Items 0.6264
## Excluding item1 0.6030
## Excluding item2 0.6204
## Excluding item3 0.5793
## Excluding item4 0.6160
## Excluding item5 0.6104
## Excluding item6 0.5942
## Excluding item7 0.6107
## Excluding item8 0.5866
## Excluding item9 0.6121
## Excluding item10 0.5816
##
##
## Pairwise Associations:
## Item i Item j p.value
## 1 2 5 0.000068454720995
## 2 4 9 0.000000000084088
## 3 5 7 0.000000000003804
## 4 2 9 0.000000000000794
## 5 2 4 0.000000000000720
## 6 2 6 0.000000000000146
## 7 5 9 0.000000000000017
## 8 4 7 0.000000000000015
## 9 2 7 0.000000000000006
## 10 4 5 0.000000000000002
IA2 <- CTT::itemAnalysis(math.dich[,1:10])
IA2$itemReport
## itemName itemMean pBis bis alphaIfDeleted
## 1 item1 0.7163428 0.2872076 0.3822021 0.6029544
## 2 item2 0.5373005 0.2185598 0.2743609 0.6204166
## 3 item3 0.8221721 0.4133883 0.6070073 0.5792550
## 4 item4 0.5813671 0.2357169 0.2977029 0.6160138
## 5 item5 0.6642956 0.2562270 0.3318574 0.6103664
## 6 item6 0.7430604 0.3272913 0.4435998 0.5941805
## 7 item7 0.4675573 0.2586720 0.3245880 0.6106904
## 8 item8 0.8684941 0.3956367 0.6270355 0.5866187
## 9 item9 0.5546495 0.2524451 0.3174810 0.6121342
## 10 item10 0.8218251 0.4014348 0.5891809 0.5815800
# What is the mean item difficulties?
mean(IA2$itemReport$itemMean)
## [1] 0.6777065
IA3 <- psychometric::item.exam(math.dich[,1:10], discrim=T)
IA3
## Sample.SD Item.total Item.Tot.woi Difficulty Discrimination
## item1 0.4508115 0.4739252 0.2872076 0.7163428 0.4591359
## item2 0.4986500 0.4339562 0.2185598 0.5373005 0.4705882
## item3 0.3824010 0.5555382 0.4133883 0.8221721 0.4143675
## item4 0.4933778 0.4467196 0.2357169 0.5813671 0.4934930
## item5 0.4722771 0.4558265 0.2562270 0.6642956 0.4726705
## item6 0.4369837 0.5028496 0.3272913 0.7430604 0.4622592
## item7 0.4989896 0.4688565 0.2586720 0.4675573 0.5497137
## item8 0.3379821 0.5242145 0.3956367 0.8684941 0.3342009
## item9 0.4970476 0.4627098 0.2524451 0.5546495 0.5262884
## item10 0.3826931 0.5455200 0.4014348 0.8218251 0.4185320
## Item.Criterion Item.Reliab Item.Rel.woi Item.Validity
## item1 NA 0.2136324 0.1294652 NA
## item2 NA 0.2163735 0.1089754 NA
## item3 NA 0.2124199 0.1580664 NA
## item4 NA 0.2203824 0.1162874 NA
## item5 NA 0.2152578 0.1209996 NA
## item6 NA 0.2197180 0.1430085 NA
## item7 NA 0.2339342 0.1290635 NA
## item8 NA 0.1771598 0.1337065 NA
## item9 NA 0.2299688 0.1254663 NA
## item10 NA 0.2087486 0.1536130 NA
# Item Characteristic Curves of Items
par(mfrow=c(1,2))
CTT :: cttICC(score = math.dich$total, itemVector = math.dich[,1], colTheme="spartans", cex=1.5)
CTT :: cttICC(score = math.dich$total, itemVector = math.dich[,2], colTheme="spartans", cex=1.5)
CTT :: cttICC(score = math.dich$total, itemVector = math.dich[,3], colTheme="spartans", cex=1.5)
CTT :: cttICC(score = math.dich$total, itemVector = math.dich[,4], colTheme="spartans", cex=1.5)
CTT :: cttICC(score = math.dich$total, itemVector = math.dich[,5], colTheme="spartans", cex=1.5)
CTT :: cttICC(score = math.dich$total, itemVector = math.dich[,6], colTheme="spartans", cex=1.5)
CTT :: cttICC(score = math.dich$total, itemVector = math.dich[,7], colTheme="spartans", cex=1.5)
CTT :: cttICC(score = math.dich$total, itemVector = math.dich[,8], colTheme="spartans", cex=1.5)
CTT :: cttICC(score = math.dich$total, itemVector = math.dich[,9], colTheme="spartans", cex=1.5)
CTT :: cttICC(score = math.dich$total, itemVector = math.dich[,10], colTheme="spartans", cex=1.5)
# Distractor/Option Analysis
distractorAnalysis(items = math.no.na, key = key, nGroups = 4, pTable = T)
## $item1
## correct key n rspP pBis discrim lower mid50
## A * A 4129 0.71634282 0.2872076 0.5486286 0.4174092 0.68685776
## B B 438 0.07598890 -0.3508862 -0.1644885 0.1720356 0.07247161
## C C 466 0.08084663 -0.4149051 -0.2026014 0.2056203 0.07247161
## D D 731 0.12682165 -0.3397622 -0.1815387 0.2049349 0.16819903
## mid75 upper
## A 0.85764810 0.966037736
## B 0.03801945 0.007547170
## C 0.02475685 0.003018868
## D 0.07957560 0.023396226
##
## $item2
## correct key n rspP pBis discrim lower mid50
## A A 829 0.1438237 -0.2646521 -0.1039032 0.1740918 0.1730665
## B B 857 0.1486815 -0.3517244 -0.1905642 0.2350925 0.1773932
## C C 981 0.1701943 -0.4301121 -0.2925659 0.3235093 0.1871282
## D * D 3097 0.5373005 0.2185598 0.5870332 0.2673064 0.4624121
## mid75 upper
## A 0.1432361 0.07018868
## B 0.1122900 0.04452830
## C 0.1078691 0.03094340
## D 0.6366048 0.85433962
##
## $item3
## correct key n rspP pBis discrim lower mid50
## A A 260 0.04510756 -0.3545343 -0.1336532 0.1336532 0.03136831
## B B 337 0.05846634 -0.4023818 -0.1684700 0.1699794 0.03948080
## C * C 4739 0.82217210 0.4133883 0.5094531 0.4852639 0.87560844
## D D 428 0.07425399 -0.4239762 -0.2073299 0.2111035 0.05354246
## mid75 upper
## A 0.006189213 0.000000000
## B 0.012378426 0.001509434
## C 0.967285588 0.994716981
## D 0.014146773 0.003773585
##
## $item4
## correct key n rspP pBis discrim lower mid50
## A A 799 0.1386190 -0.3380814 -0.1771950 0.2254969 0.1535965
## B * B 3351 0.5813671 0.2357169 0.5829441 0.2940370 0.5229854
## C C 792 0.1374046 -0.3336456 -0.1701408 0.2131597 0.1595457
## D D 822 0.1426093 -0.3978884 -0.2356083 0.2673064 0.1638724
## mid75 upper
## A 0.10786914 0.04830189
## B 0.70114943 0.87698113
## C 0.11405836 0.04301887
## D 0.07692308 0.03169811
##
## $item5
## correct key n rspP pBis discrim lower mid50
## A A 427 0.0740805 -0.3899011 -0.1801834 0.1884853 0.06057328
## B B 657 0.1139833 -0.3585149 -0.1821470 0.2138451 0.11736074
## C * C 3829 0.6642956 0.2562270 0.5609192 0.3598355 0.64359113
## D D 851 0.1476405 -0.3443062 -0.1985888 0.2378341 0.17847485
## mid75 upper
## A 0.02564103 0.008301887
## B 0.07603890 0.031698113
## C 0.79045093 0.920754717
## D 0.10786914 0.039245283
##
## $item6
## correct key n rspP pBis discrim lower mid50
## A A 414 0.07182512 -0.4033796 -0.1917043 0.1939685 0.05840995
## B * B 4283 0.74306037 0.3272913 0.5509698 0.4256340 0.73931855
## C C 768 0.13324080 -0.3874644 -0.2148538 0.2337217 0.16441320
## D D 299 0.05187370 -0.3519660 -0.1444117 0.1466758 0.03785830
## mid75 upper
## A 0.01768347 0.002264151
## B 0.88505747 0.976603774
## C 0.08664898 0.018867925
## D 0.01061008 0.002264151
##
## $item7
## correct key n rspP pBis discrim lower mid50
## A A 1089 0.1889313 -0.4307031 -0.2996894 0.3351611 0.2195782
## B B 1045 0.1812977 -0.3260854 -0.1837640 0.2426319 0.2266090
## C * C 2695 0.4675573 0.2586720 0.6738790 0.1857437 0.3585722
## D D 935 0.1622137 -0.3301125 -0.1904256 0.2364633 0.1952407
## mid75 upper
## A 0.1299735 0.03547170
## B 0.1715296 0.05886792
## C 0.5499558 0.85962264
## D 0.1485411 0.04603774
##
## $item8
## correct key n rspP pBis discrim lower mid50
## A A 158 0.02741152 -0.2980341 -0.08766201 0.08841672 0.01297999
## B B 348 0.06037474 -0.4245712 -0.19122687 0.19122687 0.03028664
## C C 252 0.04371964 -0.3810814 -0.13838168 0.13913639 0.01892915
## D * D 5006 0.86849410 0.3956367 0.41727055 0.58122001 0.93780422
## mid75 upper
## A 0.003536693 0.000754717
## B 0.011494253 0.000000000
## C 0.011494253 0.000754717
## D 0.973474801 0.998490566
##
## $item9
## correct key n rspP pBis discrim lower mid50
## A * A 3197 0.55464955 0.2524451 0.6216809 0.2666210 0.46619794
## B B 955 0.16568355 -0.3997614 -0.2438946 0.2823852 0.19848567
## C C 531 0.09212353 -0.3693713 -0.1902564 0.2008225 0.09626825
## D D 1081 0.18754337 -0.3329315 -0.1875298 0.2501714 0.23904813
## mid75 upper
## A 0.67992927 0.88830189
## B 0.11052166 0.03849057
## C 0.04067197 0.01056604
## D 0.16887710 0.06264151
##
## $item10
## correct key n rspP pBis discrim lower mid50
## A A 202 0.03504511 -0.3565783 -0.1172036 0.1172036 0.01622499
## B B 435 0.07546842 -0.4190325 -0.2055510 0.2063057 0.05786912
## C C 390 0.06766135 -0.3952812 -0.1715504 0.1768334 0.05678745
## D * D 4737 0.82182512 0.4014348 0.4943050 0.4996573 0.86911844
## mid75 upper
## A 0.0008841733 0.000000000
## B 0.0229885057 0.000754717
## C 0.0176834660 0.005283019
## D 0.9584438550 0.993962264
itemanalysis1(math.no.na, key=key, options = c("A","B","C","D"), ngroup = 10,
correction = TRUE)
## ************************************************************************
## itemanalysis: An R package for Classical Test Theory Item Analysis
##
## Cengiz Zopluoglu
##
## University of Oregon
## College of Education
##
## cen.zop@gmail.com
##
## Please report any programming bug or problem you experience to improve the code.
## *************************************************************************
## Processing Date: Sat Apr 15 11:51:48 2023
##
## ITEM STATISTICS
##
## Item Difficulty Item Threshold Point-Biserial Biserial
## item1 0.716 -0.572 0.287 0.382
## item2 0.537 -0.094 0.219 0.274
## item3 0.822 -0.924 0.413 0.607
## item4 0.581 -0.205 0.236 0.298
## item5 0.664 -0.424 0.256 0.332
## item6 0.743 -0.653 0.327 0.444
## item7 0.468 0.081 0.259 0.325
## item8 0.868 -1.119 0.396 0.627
## item9 0.555 -0.137 0.252 0.317
## item10 0.822 -0.922 0.401 0.589
##
##
##
## DISTRACTOR SELECTION PROPORTIONS
##
## A B C D Missing
## item1 0.716 0.076 0.081 0.127 0
## item2 0.144 0.149 0.170 0.537 0
## item3 0.045 0.058 0.822 0.074 0
## item4 0.139 0.581 0.137 0.143 0
## item5 0.074 0.114 0.664 0.148 0
## item6 0.072 0.743 0.133 0.052 0
## item7 0.189 0.181 0.468 0.162 0
## item8 0.027 0.060 0.044 0.868 0
## item9 0.555 0.166 0.092 0.188 0
## item10 0.035 0.075 0.068 0.822 0
##
##
##
## DISTRACTOR Point-Biserial
##
## A B C D
## item1 0.287 -0.157 -0.226 -0.078
## item2 -0.007 -0.106 -0.183 0.219
## item3 -0.203 -0.234 0.413 -0.233
## item4 -0.090 0.236 -0.086 -0.159
## item5 -0.213 -0.125 0.256 -0.071
## item6 -0.223 0.327 -0.124 -0.196
## item7 -0.187 -0.064 0.259 -0.084
## item8 -0.173 -0.247 -0.228 0.396
## item9 0.252 -0.146 -0.183 -0.046
## item10 -0.227 -0.225 -0.209 0.401
##
##
##
## DISTRACTOR Biserial
##
## A B C D
## item1 0.382 -0.292 -0.412 -0.125
## item2 -0.011 -0.163 -0.271 0.274
## item3 -0.444 -0.470 0.607 -0.435
## item4 -0.141 0.298 -0.135 -0.246
## item5 -0.399 -0.206 0.332 -0.110
## item6 -0.420 0.444 -0.195 -0.408
## item7 -0.271 -0.094 0.325 -0.126
## item8 -0.447 -0.491 -0.504 0.627
## item9 0.317 -0.219 -0.321 -0.067
## item10 -0.540 -0.417 -0.401 0.589
##
##
##
## $item.stat
## Item Difficulty Item Threshold Point-Biserial Biserial
## item1 0.7163428 -0.57201123 0.2872076 0.3822021
## item2 0.5373005 -0.09363510 0.2185598 0.2743609
## item3 0.8221721 -0.92367453 0.4133883 0.6070073
## item4 0.5813671 -0.20539211 0.2357169 0.2977029
## item5 0.6642956 -0.42421538 0.2562270 0.3318574
## item6 0.7430604 -0.65280926 0.3272913 0.4435998
## item7 0.4675573 0.08141175 0.2586720 0.3245880
## item8 0.8684941 -1.11930089 0.3956367 0.6270355
## item9 0.5546495 -0.13741737 0.2524451 0.3174810
## item10 0.8218251 -0.92234287 0.4014348 0.5891809
##
## $dist.sel
## A B C D Missing
## item1 0.71634282 0.07598890 0.08084663 0.12682165 0
## item2 0.14382373 0.14868147 0.17019431 0.53730049 0
## item3 0.04510756 0.05846634 0.82217210 0.07425399 0
## item4 0.13861901 0.58136711 0.13740458 0.14260930 0
## item5 0.07408050 0.11398334 0.66429563 0.14764053 0
## item6 0.07182512 0.74306037 0.13324080 0.05187370 0
## item7 0.18893130 0.18129771 0.46755725 0.16221374 0
## item8 0.02741152 0.06037474 0.04371964 0.86849410 0
## item9 0.55464955 0.16568355 0.09212353 0.18754337 0
## item10 0.03504511 0.07546842 0.06766135 0.82182512 0
##
## $dist.disc
## A B C D
## item1 0.287207557 -0.15739047 -0.22639318 -0.07826719
## item2 -0.007421678 -0.10587128 -0.18282103 0.21855980
## item3 -0.203013659 -0.23369633 0.41338827 -0.23305151
## item4 -0.090241766 0.23571691 -0.08614857 -0.15856421
## item5 -0.213431746 -0.12508244 0.25622697 -0.07146401
## item6 -0.223030784 0.32729125 -0.12372650 -0.19558756
## item7 -0.187333164 -0.06407833 0.25867201 -0.08421074
## item8 -0.172641806 -0.24697122 -0.22836598 0.39563669
## item9 0.252445120 -0.14642218 -0.18326970 -0.04617808
## item10 -0.227052731 -0.22477014 -0.20897648 0.40143482
##
## $dist.disc2
## A B C D
## item1 0.38220207 -0.29169594 -0.4118335 -0.12525288
## item2 -0.01148935 -0.16250278 -0.2713042 0.27436089
## item3 -0.44365449 -0.46972724 0.6070073 -0.43495533
## item4 -0.14104775 0.29770294 -0.1349610 -0.24600980
## item5 -0.39862124 -0.20608772 0.3318574 -0.10988829
## item6 -0.42049298 0.44359984 -0.1954102 -0.40843256
## item7 -0.27117439 -0.09365914 0.3245880 -0.12645058
## item8 -0.44656925 -0.49140434 -0.5042163 0.62703545
## item9 0.31748097 -0.21872111 -0.3207867 -0.06696006
## item10 -0.53978849 -0.41744004 -0.4012916 0.58918086
##
## $plots
## $plots[[1]]
##
## $plots[[2]]
##
## $plots[[3]]
##
## $plots[[4]]
##
## $plots[[5]]
##
## $plots[[6]]
##
## $plots[[7]]
##
## $plots[[8]]
##
## $plots[[9]]
##
## $plots[[10]]
# Score transformation
math.dich$tscore <- round(CTT::score.transform(math.dich$total, mu.new = 50,sd.new = 10,normalize = T)$new.scores,0)
# percentile ranks
math.dich$p.score <- round(CTT::score.transform(math.dich$total, mu.new = 50,
sd.new = 10,normalize = T)$p.scores,3)
math <- math.dich[order(math.dich$total, decreasing = F),]
norm <- data.frame(rawscore=math.dich$total, Tscore=math.dich$tscore, p.score= math.dich$p.score)
norm <- norm[order(norm$rawscore, decreasing = F),]
table(norm$rawscor, norm$Tscore)
##
## 21 27 31 35 38 42 45 50 54 60 68
## 0 22 0 0 0 0 0 0 0 0 0 0
## 1 0 67 0 0 0 0 0 0 0 0 0
## 2 0 0 159 0 0 0 0 0 0 0 0
## 3 0 0 0 266 0 0 0 0 0 0 0
## 4 0 0 0 0 389 0 0 0 0 0 0
## 5 0 0 0 0 0 556 0 0 0 0 0
## 6 0 0 0 0 0 0 809 0 0 0 0
## 7 0 0 0 0 0 0 0 1040 0 0 0
## 8 0 0 0 0 0 0 0 0 1131 0 0
## 9 0 0 0 0 0 0 0 0 0 888 0
## 10 0 0 0 0 0 0 0 0 0 0 437
table(norm$rawscor, 100*norm$p.score) # percentile ranks for each raw score
##
## 0.2 1 2.9 6.6 12.3 20.5 32.3 48.4 67.2 84.7 96.2
## 0 22 0 0 0 0 0 0 0 0 0 0
## 1 0 67 0 0 0 0 0 0 0 0 0
## 2 0 0 159 0 0 0 0 0 0 0 0
## 3 0 0 0 266 0 0 0 0 0 0 0
## 4 0 0 0 0 389 0 0 0 0 0 0
## 5 0 0 0 0 0 556 0 0 0 0 0
## 6 0 0 0 0 0 0 809 0 0 0 0
## 7 0 0 0 0 0 0 0 1040 0 0 0
## 8 0 0 0 0 0 0 0 0 1131 0 0
## 9 0 0 0 0 0 0 0 0 0 888 0
## 10 0 0 0 0 0 0 0 0 0 0 437
par(mfrow=c(1,2))
hist(math.dich$total)
hist(math.dich$tscore)
