#get required library
library(readxl)
library(MatchIt)
library(Matching)
## Loading required package: MASS
## ##
## ## Matching (Version 4.9-7, Build Date: 2020-02-05)
## ## See http://sekhon.berkeley.edu/matching for additional documentation.
## ## Please cite software as:
## ## Jasjeet S. Sekhon. 2011. ``Multivariate and Propensity Score Matching
## ## Software with Automated Balance Optimization: The Matching package for R.''
## ## Journal of Statistical Software, 42(7): 1-52.
## ##
library(rbounds)
library(MASS)
library(dplyr)
##
## Attaching package: 'dplyr'
## The following object is masked from 'package:MASS':
##
## select
## The following objects are masked from 'package:stats':
##
## filter, lag
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
library(janitor)
##
## Attaching package: 'janitor'
## The following objects are masked from 'package:stats':
##
## chisq.test, fisher.test
library(cobalt)
## cobalt (Version 4.2.4, Build Date: 2020-11-05 17:30:21 UTC)
##
## Attaching package: 'cobalt'
## The following object is masked from 'package:MatchIt':
##
## lalonde
library(Zelig)
## Loading required package: survival
library(ggplot2)
##
## Attaching package: 'ggplot2'
## The following object is masked from 'package:Zelig':
##
## stat
library(devtools)
## Loading required package: usethis
#devtools::install_github("Ngendahimana/sensitivityR5", force=TRUE)
library(SensitivityR5)
set.seed(123)
#load dataframe
df <- read_excel("02.SeedSectorData.xlsx") %>% clean_names() %>% remove_empty()
## value for "which" not specified, defaulting to c("rows", "cols")
#change treatment variable to boolean
df$treatment = ifelse(df$affiliation == "Af", 1, 0)
head(df)
## # A tibble: 6 x 27
## hhid seed_collection fgroup affiliation ky ka ak av_age under6
## <dbl> <dbl> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 0 Ky Af 1 0 0 56.7 0
## 2 2 1 Ky Af 1 0 0 19.6 0
## 3 3 1 Ky Af 1 0 0 31.6 0
## 4 4 1 Ky Af 1 0 0 22.2 0.333
## 5 5 0 Ky Af 1 0 0 18.9 0
## 6 6 0 Ky Af 1 0 0 18.3 0.333
## # … with 18 more variables: over60 <dbl>, hh_size <dbl>, prop_eat <dbl>,
## # prop_all <dbl>, too_weak <dbl>, too_alone <dbl>, paid_people <dbl>,
## # volunteers <dbl>, i1 <dbl>, i2 <dbl>, i3 <dbl>, i4 <dbl>, i5 <dbl>,
## # u1 <dbl>, u2 <dbl>, u3 <dbl>, u4 <dbl>, treatment <dbl>
#Replication of Table 4
#Unmatched difference
MatchBalance(treatment ~ av_age + under6 + over60 + hh_size + prop_eat + prop_all + too_weak + too_alone + paid_people + volunteers, data = df)
##
## ***** (V1) av_age *****
## before matching:
## mean treatment........ 24.572
## mean control.......... 26.082
## std mean diff......... -13.953
##
## mean raw eQQ diff..... 1.7103
## med raw eQQ diff..... 1.5
## max raw eQQ diff..... 7.3333
##
## mean eCDF diff........ 0.063621
## med eCDF diff........ 0.05754
## max eCDF diff........ 0.18849
##
## var ratio (Tr/Co)..... 0.94869
## T-test p-value........ 0.35111
## KS Bootstrap p-value.. 0.068
## KS Naive p-value...... 0.077098
## KS Statistic.......... 0.18849
##
##
## ***** (V2) under6 *****
## before matching:
## mean treatment........ 0.10729
## mean control.......... 0.11033
## std mean diff......... -2.3419
##
## mean raw eQQ diff..... 0.021744
## med raw eQQ diff..... 0
## max raw eQQ diff..... 0.25
##
## mean eCDF diff........ 0.039265
## med eCDF diff........ 0.029762
## max eCDF diff........ 0.095238
##
## var ratio (Tr/Co)..... 0.74449
## T-test p-value........ 0.88137
## KS Bootstrap p-value.. 0.448
## KS Naive p-value...... 0.80024
## KS Statistic.......... 0.095238
##
##
## ***** (V3) over60 *****
## before matching:
## mean treatment........ 0.051576
## mean control.......... 0.067999
## std mean diff......... -10.672
##
## mean raw eQQ diff..... 0.015454
## med raw eQQ diff..... 0
## max raw eQQ diff..... 0.25
##
## mean eCDF diff........ 0.028439
## med eCDF diff........ 0.027778
## max eCDF diff........ 0.053571
##
## var ratio (Tr/Co)..... 0.77676
## T-test p-value........ 0.49282
## KS Bootstrap p-value.. 0.622
## KS Naive p-value...... 0.99942
## KS Statistic.......... 0.053571
##
##
## ***** (V4) hh_size *****
## before matching:
## mean treatment........ 5.7083
## mean control.......... 5.8651
## std mean diff......... -7.1747
##
## mean raw eQQ diff..... 0.20833
## med raw eQQ diff..... 0
## max raw eQQ diff..... 2
##
## mean eCDF diff........ 0.019015
## med eCDF diff........ 0.014881
## max eCDF diff........ 0.047619
##
## var ratio (Tr/Co)..... 0.85144
## T-test p-value........ 0.63833
## KS Bootstrap p-value.. 0.92
## KS Naive p-value...... 0.99995
## KS Statistic.......... 0.047619
##
##
## ***** (V5) prop_eat *****
## before matching:
## mean treatment........ 0.22969
## mean control.......... 0.18328
## std mean diff......... 19.859
##
## mean raw eQQ diff..... 0.052583
## med raw eQQ diff..... 0.028571
## max raw eQQ diff..... 0.16667
##
## mean eCDF diff........ 0.067378
## med eCDF diff........ 0.056548
## max eCDF diff........ 0.19444
##
## var ratio (Tr/Co)..... 1.4552
## T-test p-value........ 0.15579
## KS Bootstrap p-value.. 0.034
## KS Naive p-value...... 0.062566
## KS Statistic.......... 0.19444
##
##
## ***** (V6) prop_all *****
## before matching:
## mean treatment........ 0.37364
## mean control.......... 0.35169
## std mean diff......... 10.301
##
## mean raw eQQ diff..... 0.036733
## med raw eQQ diff..... 0.03869
## max raw eQQ diff..... 0.125
##
## mean eCDF diff........ 0.050725
## med eCDF diff........ 0.03373
## max eCDF diff........ 0.16468
##
## var ratio (Tr/Co)..... 0.84093
## T-test p-value........ 0.50099
## KS Bootstrap p-value.. 0.068
## KS Naive p-value...... 0.16652
## KS Statistic.......... 0.16468
##
##
## ***** (V7) too_weak *****
## before matching:
## mean treatment........ 0.54167
## mean control.......... 0.34921
## std mean diff......... 38.357
##
## mean raw eQQ diff..... 0.19444
## med raw eQQ diff..... 0
## max raw eQQ diff..... 1
##
## mean eCDF diff........ 0.09623
## med eCDF diff........ 0.09623
## max eCDF diff........ 0.19246
##
## var ratio (Tr/Co)..... 1.099
## T-test p-value........ 0.0092172
##
##
## ***** (V8) too_alone *****
## before matching:
## mean treatment........ 0.58333
## mean control.......... 0.30159
## std mean diff......... 56.75
##
## mean raw eQQ diff..... 0.27778
## med raw eQQ diff..... 0
## max raw eQQ diff..... 1
##
## mean eCDF diff........ 0.14087
## med eCDF diff........ 0.14087
## max eCDF diff........ 0.28175
##
## var ratio (Tr/Co)..... 1.1609
## T-test p-value........ 0.00012753
##
##
## ***** (V9) paid_people *****
## before matching:
## mean treatment........ 0.27778
## mean control.......... 0.57143
## std mean diff......... -65.104
##
## mean raw eQQ diff..... 0.29167
## med raw eQQ diff..... 0
## max raw eQQ diff..... 1
##
## mean eCDF diff........ 0.14683
## med eCDF diff........ 0.14683
## max eCDF diff........ 0.29365
##
## var ratio (Tr/Co)..... 0.82413
## T-test p-value........ 3.6911e-05
##
##
## ***** (V10) volunteers *****
## before matching:
## mean treatment........ 0.5
## mean control.......... 0.56349
## std mean diff......... -12.61
##
## mean raw eQQ diff..... 0.055556
## med raw eQQ diff..... 0
## max raw eQQ diff..... 1
##
## mean eCDF diff........ 0.031746
## med eCDF diff........ 0.031746
## max eCDF diff........ 0.063492
##
## var ratio (Tr/Co)..... 1.0225
## T-test p-value........ 0.39285
##
##
## Before Matching Minimum p.value: 3.6911e-05
## Variable Name(s): paid_people Number(s): 9
#using nearest neighbor matching
nn.mout <- matchit(treatment ~ av_age + under6 + over60 + hh_size + prop_eat + prop_all + too_weak + too_alone + paid_people + volunteers, data = df, estimand = "ATT", method = "nearest")
#print matching improvement
summary(nn.mout)
##
## Call:
## matchit(formula = treatment ~ av_age + under6 + over60 + hh_size +
## prop_eat + prop_all + too_weak + too_alone + paid_people +
## volunteers, data = df, method = "nearest", estimand = "ATT")
##
## Summary of balance for all data:
## Means Treated Means Control SD Control Mean Diff eQQ Med eQQ Mean
## distance 0.5010 0.2851 0.1996 0.2158 0.2195 0.2177
## av_age 24.5721 26.0823 11.1122 -1.5102 1.5000 1.7103
## under6 0.1073 0.1103 0.1504 -0.0030 0.0000 0.0217
## over60 0.0516 0.0680 0.1746 -0.0164 0.0000 0.0155
## hh_size 5.7083 5.8651 2.3676 -0.1567 0.0000 0.2083
## prop_eat 0.2297 0.1833 0.1937 0.0464 0.0286 0.0526
## prop_all 0.3736 0.3517 0.2324 0.0220 0.0387 0.0367
## too_weak 0.5417 0.3492 0.4786 0.1925 0.0000 0.1944
## too_alone 0.5833 0.3016 0.4608 0.2817 0.0000 0.2778
## paid_people 0.2778 0.5714 0.4968 -0.2937 0.0000 0.2917
## volunteers 0.5000 0.5635 0.4979 -0.0635 0.0000 0.0556
## eQQ Max
## distance 0.3094
## av_age 7.3333
## under6 0.2500
## over60 0.2500
## hh_size 2.0000
## prop_eat 0.1667
## prop_all 0.1250
## too_weak 1.0000
## too_alone 1.0000
## paid_people 1.0000
## volunteers 1.0000
##
##
## Summary of balance for matched data:
## Means Treated Means Control SD Control Mean Diff eQQ Med eQQ Mean
## distance 0.5010 0.4117 0.1732 0.0893 0.1085 0.0895
## av_age 24.5721 25.9644 11.2775 -1.3923 1.3859 1.7806
## under6 0.1073 0.1066 0.1418 0.0007 0.0000 0.0204
## over60 0.0516 0.0801 0.2008 -0.0285 0.0000 0.0285
## hh_size 5.7083 5.6944 2.3052 0.0139 0.0000 0.2361
## prop_eat 0.2297 0.2241 0.2095 0.0056 0.0244 0.0388
## prop_all 0.3736 0.3742 0.2451 -0.0005 0.0357 0.0402
## too_weak 0.5417 0.4583 0.5018 0.0833 0.0000 0.0833
## too_alone 0.5833 0.4722 0.5027 0.1111 0.0000 0.1111
## paid_people 0.2778 0.3750 0.4875 -0.0972 0.0000 0.0972
## volunteers 0.5000 0.5417 0.5018 -0.0417 0.0000 0.0417
## eQQ Max
## distance 0.1462
## av_age 7.3333
## under6 0.1250
## over60 0.3333
## hh_size 2.0000
## prop_eat 0.1667
## prop_all 0.2500
## too_weak 1.0000
## too_alone 1.0000
## paid_people 1.0000
## volunteers 1.0000
##
## Percent Balance Improvement:
## Mean Diff. eQQ Med eQQ Mean eQQ Max
## distance 58.6414 50.5820 58.9043 52.7421
## av_age 7.8060 7.6058 -4.1146 0.0000
## under6 75.7211 0.0000 6.2699 50.0000
## over60 -73.6408 0.0000 -84.5221 -33.3333
## hh_size 91.1392 0.0000 -13.3333 0.0000
## prop_eat 87.8630 14.5833 26.1692 0.0000
## prop_all 97.6103 7.6923 -9.3026 -100.0000
## too_weak 56.7010 0.0000 57.1429 0.0000
## too_alone 60.5634 0.0000 60.0000 0.0000
## paid_people 66.8919 0.0000 66.6667 0.0000
## volunteers 34.3750 0.0000 25.0000 0.0000
##
## Sample sizes:
## Control Treated
## All 126 72
## Matched 72 72
## Unmatched 54 0
## Discarded 0 0
bal.tab(nn.mout)
## Call
## matchit(formula = treatment ~ av_age + under6 + over60 + hh_size +
## prop_eat + prop_all + too_weak + too_alone + paid_people +
## volunteers, data = df, method = "nearest", estimand = "ATT")
##
## Balance Measures
## Type Diff.Adj
## distance Distance 0.4283
## av_age Contin. -0.1286
## under6 Contin. 0.0057
## over60 Contin. -0.1853
## hh_size Contin. 0.0064
## prop_eat Contin. 0.0241
## prop_all Contin. -0.0025
## too_weak Binary 0.0833
## too_alone Binary 0.1111
## paid_people Binary -0.0972
## volunteers Binary -0.0417
##
## Sample sizes
## Control Treated
## All 126 72
## Matched 72 72
## Unmatched 54 0
#plot balance between covariates before and after matching
love.plot(nn.mout, abs=TRUE, var.order = "adjusted", line = TRUE, stats= c("ks.statistics", "abs difference"))
#calculate treatment effect using simulation
z.out <- zelig(i1 ~ treatment + av_age + under6 + over60 + hh_size + prop_eat + prop_all + too_weak + too_alone + paid_people + volunteers, data = match.data(nn.mout), model = "ls")
## Warning: `tbl_df()` is deprecated as of dplyr 1.0.0.
## Please use `tibble::as_tibble()` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_warnings()` to see where this warning was generated.
## Warning: `group_by_()` is deprecated as of dplyr 0.7.0.
## Please use `group_by()` instead.
## See vignette('programming') for more help
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_warnings()` to see where this warning was generated.
## How to cite this model in Zelig:
## R Core Team. 2007.
## ls: Least Squares Regression for Continuous Dependent Variables
## in Christine Choirat, Christopher Gandrud, James Honaker, Kosuke Imai, Gary King, and Olivia Lau,
## "Zelig: Everyone's Statistical Software," https://zeligproject.org/
x.out <- setx(z.out, data = match.data(nn.out, "treat"), cond = TRUE)
s.out <- sim(z.out, x = x.out)
summary(s.out)
##
## sim x :
## -----
## ev
## mean sd 50% 2.5% 97.5%
## 1 0.5063324 0.03262313 0.5071452 0.4419819 0.5688817
## pv
## mean sd 50% 2.5% 97.5%
## [1,] 0.5046454 0.3854788 0.5127219 -0.2549563 1.291935
#using genetic matching
gen.mout <- matchit(treatment ~ av_age + under6 + over60 + hh_size + prop_eat + prop_all + too_weak + too_alone + paid_people + volunteers, data = df, estimand = "ATT", method = "genetic", print.level = 0, pop.size = 1000, unif.seed=112, int.seed=112)
## Loading required namespace: rgenoud
#print matching improvement
summary(gen.mout)
##
## Call:
## matchit(formula = treatment ~ av_age + under6 + over60 + hh_size +
## prop_eat + prop_all + too_weak + too_alone + paid_people +
## volunteers, data = df, method = "genetic", estimand = "ATT",
## print.level = 0, pop.size = 1000, unif.seed = 112, int.seed = 112)
##
## Summary of balance for all data:
## Means Treated Means Control SD Control Mean Diff eQQ Med eQQ Mean
## distance 0.5010 0.2851 0.1996 0.2158 0.2195 0.2177
## av_age 24.5721 26.0823 11.1122 -1.5102 1.5000 1.7103
## under6 0.1073 0.1103 0.1504 -0.0030 0.0000 0.0217
## over60 0.0516 0.0680 0.1746 -0.0164 0.0000 0.0155
## hh_size 5.7083 5.8651 2.3676 -0.1567 0.0000 0.2083
## prop_eat 0.2297 0.1833 0.1937 0.0464 0.0286 0.0526
## prop_all 0.3736 0.3517 0.2324 0.0220 0.0387 0.0367
## too_weak 0.5417 0.3492 0.4786 0.1925 0.0000 0.1944
## too_alone 0.5833 0.3016 0.4608 0.2817 0.0000 0.2778
## paid_people 0.2778 0.5714 0.4968 -0.2937 0.0000 0.2917
## volunteers 0.5000 0.5635 0.4979 -0.0635 0.0000 0.0556
## eQQ Max
## distance 0.3094
## av_age 7.3333
## under6 0.2500
## over60 0.2500
## hh_size 2.0000
## prop_eat 0.1667
## prop_all 0.1250
## too_weak 1.0000
## too_alone 1.0000
## paid_people 1.0000
## volunteers 1.0000
##
##
## Summary of balance for matched data:
## Means Treated Means Control SD Control Mean Diff eQQ Med eQQ Mean
## distance 0.5010 0.4819 0.2013 0.0191 0.0642 0.0638
## av_age 24.5721 23.7111 7.8980 0.8610 1.0250 2.2937
## under6 0.1073 0.1027 0.1311 0.0046 0.0000 0.0211
## over60 0.0516 0.0294 0.0921 0.0222 0.0000 0.0265
## hh_size 5.7083 5.6667 1.8335 0.0417 0.0000 0.3750
## prop_eat 0.2297 0.2103 0.1965 0.0194 0.0542 0.0551
## prop_all 0.3736 0.3426 0.1947 0.0310 0.0264 0.0430
## too_weak 0.5417 0.5278 0.5056 0.0139 0.0000 0.0250
## too_alone 0.5833 0.5833 0.4993 0.0000 0.0000 0.0750
## paid_people 0.2778 0.2778 0.4536 0.0000 0.0000 0.1000
## volunteers 0.5000 0.5972 0.4967 -0.0972 0.0000 0.1000
## eQQ Max
## distance 0.1345
## av_age 28.0000
## under6 0.1250
## over60 0.6667
## hh_size 2.0000
## prop_eat 0.1429
## prop_all 0.3333
## too_weak 1.0000
## too_alone 1.0000
## paid_people 1.0000
## volunteers 1.0000
##
## Percent Balance Improvement:
## Mean Diff. eQQ Med eQQ Mean eQQ Max
## distance 91.1630 70.7565 70.7145 56.5246
## av_age 42.9863 31.6667 -34.1105 -281.8182
## under6 -51.2115 0.0000 3.0029 50.0000
## over60 -35.0465 0.0000 -71.3980 -166.6667
## hh_size 73.4177 0.0000 -80.0000 0.0000
## prop_eat 58.2213 -89.5833 -4.8222 14.2857
## prop_all -41.2275 31.7949 -17.0505 -166.6667
## too_weak 92.7835 0.0000 87.1429 0.0000
## too_alone 100.0000 0.0000 73.0000 0.0000
## paid_people 100.0000 0.0000 65.7143 0.0000
## volunteers -53.1250 0.0000 -80.0000 0.0000
##
## Sample sizes:
## Control Treated
## All 126 72
## Matched 40 72
## Unmatched 86 0
## Discarded 0 0
print(bal.tab(gen.mout))
## Call
## matchit(formula = treatment ~ av_age + under6 + over60 + hh_size +
## prop_eat + prop_all + too_weak + too_alone + paid_people +
## volunteers, data = df, method = "genetic", estimand = "ATT",
## print.level = 0, pop.size = 1000, unif.seed = 112, int.seed = 112)
##
## Balance Measures
## Type Diff.Adj
## distance Distance 0.0915
## av_age Contin. 0.0796
## under6 Contin. 0.0354
## over60 Contin. 0.1441
## hh_size Contin. 0.0191
## prop_eat Contin. 0.0830
## prop_all Contin. 0.1455
## too_weak Binary 0.0139
## too_alone Binary 0.0000
## paid_people Binary 0.0000
## volunteers Binary -0.0972
##
## Sample sizes
## Control Treated
## All 126. 72
## Matched (ESS) 25.66 72
## Matched (Unweighted) 40. 72
## Unmatched 86. 0
#plot balance between covariates before and after matching
love.plot(gen.mout, abs=TRUE, var.order = "adjusted", line = TRUE, stats = "ks.statistics")
#calculate treatment effect using simulation
gen.z.out <- zelig(i1 ~ treatment + av_age + under6 + over60 + hh_size + prop_eat + prop_all + too_weak + too_alone + paid_people + volunteers, data = match.data(gen.mout), model = "ls")
## How to cite this model in Zelig:
## R Core Team. 2007.
## ls: Least Squares Regression for Continuous Dependent Variables
## in Christine Choirat, Christopher Gandrud, James Honaker, Kosuke Imai, Gary King, and Olivia Lau,
## "Zelig: Everyone's Statistical Software," https://zeligproject.org/
gen.x.out <- setx(gen.z.out, data = match.data(gen.mout, "treat"), cond = TRUE)
gen.s.out <- sim(gen.z.out, x = gen.x.out)
summary(gen.s.out)
##
## sim x :
## -----
## ev
## mean sd 50% 2.5% 97.5%
## 1 0.595964 0.0377861 0.5963168 0.5181893 0.6712928
## pv
## mean sd 50% 2.5% 97.5%
## [1,] 0.5660769 0.4077106 0.5815421 -0.2683721 1.350448
#sensitivity analysis for genetic matching
pens2(x = gen.mout, y="i1",Gamma = 5, GammaInc = 0.1, est = 0.5987271)
##
## Rosenbaum Sensitivity Test for Wilcoxon Signed Rank P-Value
##
## Unconfounded estimate .... 0
##
## Gamma Lower bound Upper bound
## 1.0 0 0.0000
## 1.1 0 0.0000
## 1.2 0 0.0000
## 1.3 0 0.0000
## 1.4 0 0.0000
## 1.5 0 0.0000
## 1.6 0 0.0000
## 1.7 0 0.0001
## 1.8 0 0.0001
## 1.9 0 0.0002
## 2.0 0 0.0003
## 2.1 0 0.0005
## 2.2 0 0.0008
## 2.3 0 0.0011
## 2.4 0 0.0015
## 2.5 0 0.0021
## 2.6 0 0.0028
## 2.7 0 0.0037
## 2.8 0 0.0047
## 2.9 0 0.0060
## 3.0 0 0.0074
## 3.1 0 0.0091
## 3.2 0 0.0109
## 3.3 0 0.0131
## 3.4 0 0.0154
## 3.5 0 0.0181
## 3.6 0 0.0210
## 3.7 0 0.0241
## 3.8 0 0.0276
## 3.9 0 0.0313
## 4.0 0 0.0352
## 4.1 0 0.0395
## 4.2 0 0.0439
## 4.3 0 0.0487
## 4.4 0 0.0537
## 4.5 0 0.0590
## 4.6 0 0.0645
## 4.7 0 0.0702
## 4.8 0 0.0762
## 4.9 0 0.0824
## 5.0 0 0.0888
##
## Note: Gamma is Odds of Differential Assignment To
## Treatment Due to Unobserved Factors
##
#sensitivity analysis for nearest neighbor matching
pens2(x = nn.mout, y="i1",Gamma = 7, GammaInc = 0.1, est = 0.5057401)
##
## Rosenbaum Sensitivity Test for Wilcoxon Signed Rank P-Value
##
## Unconfounded estimate .... 0
##
## Gamma Lower bound Upper bound
## 1.0 0 0.0000
## 1.1 0 0.0000
## 1.2 0 0.0000
## 1.3 0 0.0000
## 1.4 0 0.0000
## 1.5 0 0.0000
## 1.6 0 0.0000
## 1.7 0 0.0000
## 1.8 0 0.0000
## 1.9 0 0.0000
## 2.0 0 0.0000
## 2.1 0 0.0001
## 2.2 0 0.0001
## 2.3 0 0.0001
## 2.4 0 0.0002
## 2.5 0 0.0003
## 2.6 0 0.0004
## 2.7 0 0.0005
## 2.8 0 0.0006
## 2.9 0 0.0008
## 3.0 0 0.0010
## 3.1 0 0.0013
## 3.2 0 0.0016
## 3.3 0 0.0019
## 3.4 0 0.0023
## 3.5 0 0.0028
## 3.6 0 0.0033
## 3.7 0 0.0039
## 3.8 0 0.0045
## 3.9 0 0.0052
## 4.0 0 0.0059
## 4.1 0 0.0068
## 4.2 0 0.0076
## 4.3 0 0.0086
## 4.4 0 0.0096
## 4.5 0 0.0107
## 4.6 0 0.0119
## 4.7 0 0.0131
## 4.8 0 0.0144
## 4.9 0 0.0158
## 5.0 0 0.0172
## 5.1 0 0.0188
## 5.2 0 0.0204
## 5.3 0 0.0220
## 5.4 0 0.0237
## 5.5 0 0.0255
## 5.6 0 0.0274
## 5.7 0 0.0293
## 5.8 0 0.0313
## 5.9 0 0.0334
## 6.0 0 0.0355
## 6.1 0 0.0377
## 6.2 0 0.0399
## 6.3 0 0.0422
## 6.4 0 0.0446
## 6.5 0 0.0470
## 6.6 0 0.0495
## 6.7 0 0.0520
## 6.8 0 0.0545
## 6.9 0 0.0572
## 7.0 0 0.0598
##
## Note: Gamma is Odds of Differential Assignment To
## Treatment Due to Unobserved Factors
##