The R Package equateIRT: A Tutorial
1 Functionalities
The equateIRT package computes:
- Direct equating coefficients (between two forms with common items).
- Chain equating coefficients (through a chain of forms with common items in pairs).
- Average (bisector) equating coefficients (between two forms connected through more than one path).
- Equated scores with true score equating and observed score equating methods.
- Standard errors of all equating coefficients and equated scores.
- Test for DIF and tests for drifts.
2 Data preparation
Load the package equateIRT and the data
Estimate a two parameter logistic model for 5 data sets with the R package mirt
library("mirt")
m1 <- mirt(data2pl[[1]], SE = TRUE)
m2 <- mirt(data2pl[[2]], SE = TRUE)
m3 <- mirt(data2pl[[3]], SE = TRUE)
m4 <- mirt(data2pl[[4]], SE = TRUE)
m5 <- mirt(data2pl[[5]], SE = TRUE)Create an object of class modIRT (since package versione
2.5.0 it is possible to skip the import.mirt function)
mlist<- list(m1, m2, m3, m4, m5)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(est.mods = mlist, names = test, display = FALSE)
coef(mod2pl$test1)[1:5]## Dffclt.I1 Dffclt.I2 Dffclt.I3 Dffclt.I4 Dffclt.I5
## 0.058224616 0.028019255 0.073234265 0.415936521 -0.006686047The linkage plan
## [,1] [,2] [,3] [,4] [,5]
## [1,] 20 10 0 0 10
## [2,] 10 20 10 0 0
## [3,] 0 10 20 10 0
## [4,] 0 0 10 20 10
## [5,] 10 0 0 10 20A graphic of the linkage plan with package sna
library(sna)
par(mar=c(0, 0, 0, 0))
set.seed(6)
gplot(lplan, displaylabels = TRUE, vertex.sides = 4, vertex.cex = 5, vertex.rot =45, usearrows = FALSE, label.pos = 5, label.cex = 1, vertex.col = 0)
Linkage plan
3 Direct equating coefficients
Estimation of direct equating coefficients between Forms 1 and 2 using the mean-mean method.
NOTE: Item parameters are converted to the scale of Form 2.
## Direct equating coefficients
## Method: mean-mean
## Link: test1.test2## Link: test1.test2
## Method: mean-mean
## Equating coefficients:
## Estimate StdErr
## A 1.21061 0.028998
## B -0.14242 0.028106Estimation of all direct equating coefficients between forms with common items using the mean-mean method
## Direct equating coefficients
## Method: mean-mean
## Links:
## test1.test2
## test1.test5
## test2.test1
## test2.test3
## test3.test2
## test3.test4
## test4.test3
## test4.test5
## test5.test1
## test5.test4Direct equating coefficients for Forms 1 and 2
## Link: test1.test2
## Method: mean-mean
## Equating coefficients:
## Estimate StdErr
## A 1.21061 0.028998
## B -0.14242 0.0281064 Chain equating coefficients
Estimation of all chain equating coefficients of length 4
## Chain equating coefficients
## Method: mean-mean
## Paths:
## test4.test5.test1.test2
## test3.test2.test1.test5
## test5.test1.test2.test3
## test4.test3.test2.test1
## test5.test4.test3.test2
## test1.test2.test3.test4
## test1.test5.test4.test3
## test2.test3.test4.test5
## test2.test1.test5.test4
## test3.test4.test5.test1Chain equating coefficients for path {1, 2, 3, 4}
## Path: test1.test2.test3.test4
## Method: mean-mean
## Equating coefficients:
## Estimate StdErr
## A 1.25371 0.046538
## B -0.49825 0.038497Estimation of all chain equating coefficients of length 4 from Form 1 to Form 4
## Chain equating coefficients
## Method: mean-mean
## Paths:
## test1.test2.test3.test4## Path: test1.test2.test3.test4
## Method: mean-mean
## Equating coefficients:
## Estimate StdErr
## A 1.25371 0.046538
## B -0.49825 0.038497Estimation of chain equating coefficient for path {1, 5, 4}
pth <- paste("test", c(1,5,4), sep = "")
chainec154 <- chainec(direclist = direclist2pl, pths = pth)
summary(chainec154)## Path: test1.test5.test4
## Method: mean-mean
## Equating coefficients:
## Estimate StdErr
## A 1.15964 0.033424
## B -0.40028 0.033072NOTE: Item parameters are converted to the scale of Form 4.
5 Average equating coefficients
When two forms are linked through more than one path, the equating coefficients can be averaged using the bisector method. Estimation of bisector equating coefficients:
ecall <- c(cec14, chainec154)
fec <- bisectorec(ecall = ecall, weighted = TRUE, unweighted = TRUE)
fec## Bisector and weighted bisector equating coefficients
## Method: mean-mean
##
## Link: test1.test4
## Paths:
## test1.test2.test3.test4
## test1.test5.test4## Link: test1.test4
## Method: mean-mean
## Equating coefficients:
## Path Estimate StdErr
## A test1.test2.test3.test4 1.25371 0.046538
## A test1.test5.test4 1.15964 0.033424
## A bisector 1.20559 0.030635
## A weighted bisector 1.18974 0.029330
## B test1.test2.test3.test4 -0.49825 0.038497
## B test1.test5.test4 -0.40028 0.033072
## B bisector -0.44814 0.029917
## B weighted bisector -0.43163 0.029704Extract the equating coefficients
## link path A B
## 1 test1.test4 test1.test2.test3.test4 1.253712 -0.4982537
## 2 test1.test4 test1.test5.test4 1.159638 -0.4002830
## 3 test1.test4 bisector 1.205588 -0.4481367
## 4 test1.test4 weighted bisector 1.189738 -0.4316305Extract item parameters of two forms being equated in the original scale and item parameters of the first form converted to the scale of the second form.
## Item test1 test4 test1.as.test4
## 1 Dffclt.I1 0.058224616 NA -0.36235841
## 2 Dffclt.I10 0.654862838 NA 0.34748486
## 3 Dffclt.I2 0.028019255 NA -0.39829488
## 4 Dffclt.I21 NA -0.18966258 NA
## 5 Dffclt.I22 NA -0.57151991 NA
## 6 Dffclt.I23 NA -0.97068963 NA
## 7 Dffclt.I24 NA 0.28221916 NA
## 8 Dffclt.I25 NA 0.02656655 NA
## 9 Dffclt.I26 NA -0.12088864 NA
## 10 Dffclt.I27 NA 0.47589388 NA
## 11 Dffclt.I28 NA -0.01063699 NA
## 12 Dffclt.I29 NA -0.93598686 NA
## 13 Dffclt.I3 0.073234265 NA -0.34450086
## 14 Dffclt.I30 NA 0.54314189 NA
## 15 Dffclt.I31 0.522747833 NA 0.19030260
## 16 Dffclt.I32 0.831399476 NA 0.55751724
## 17 Dffclt.I33 0.575285842 NA 0.25280907
## 18 Dffclt.I34 -0.342019714 NA -0.83854437
## 19 Dffclt.I35 0.428922604 NA 0.07867514
## 20 Dffclt.I36 0.852091712 NA 0.58213558
## 21 Dffclt.I37 0.456475784 NA 0.11145621
## 22 Dffclt.I38 -0.137534728 NA -0.59526078
## 23 Dffclt.I39 0.519128781 NA 0.18599687
## 24 Dffclt.I4 0.415936521 NA 0.06322510
## 25 Dffclt.I40 0.841940468 NA 0.57005826
## 26 Dffclt.I41 NA -1.12849656 NA
## 27 Dffclt.I42 NA -0.77000076 NA
## 28 Dffclt.I43 NA 1.08701579 NA
## 29 Dffclt.I44 NA 0.88830215 NA
## 30 Dffclt.I45 NA 0.22917144 NA
## 31 Dffclt.I46 NA -0.78046634 NA
## 32 Dffclt.I47 NA 0.13591938 NA
## 33 Dffclt.I48 NA -0.21787300 NA
## 34 Dffclt.I49 NA -0.05849482 NA
## 35 Dffclt.I5 -0.006686047 NA -0.43958511
## 36 Dffclt.I50 NA -0.06296350 NA
## 37 Dffclt.I6 -0.773019644 NA -1.35132145
## 38 Dffclt.I7 0.155039212 NA -0.24717439
## 39 Dffclt.I8 0.337353040 NA -0.03026867
## 40 Dffclt.I9 -0.182437547 NA -0.64868338
## 41 Dscrmn.I1 1.076092086 NA 0.90447806
## 42 Dscrmn.I10 1.341200299 NA 1.12730710
## 43 Dscrmn.I2 1.123453456 NA 0.94428629
## 44 Dscrmn.I21 NA 0.96336363 NA
## 45 Dscrmn.I22 NA 1.01240417 NA
## 46 Dscrmn.I23 NA 1.05633835 NA
## 47 Dscrmn.I24 NA 0.87356846 NA
## 48 Dscrmn.I25 NA 1.06908275 NA
## 49 Dscrmn.I26 NA 1.10250771 NA
## 50 Dscrmn.I27 NA 1.04526491 NA
## 51 Dscrmn.I28 NA 0.93408343 NA
## 52 Dscrmn.I29 NA 0.93004199 NA
## 53 Dscrmn.I3 1.091380106 NA 0.91732796
## 54 Dscrmn.I30 NA 1.11220053 NA
## 55 Dscrmn.I31 1.483887762 NA 1.24723892
## 56 Dscrmn.I32 1.300806488 NA 1.09335525
## 57 Dscrmn.I33 1.353845457 NA 1.13793562
## 58 Dscrmn.I34 1.381906844 NA 1.16152181
## 59 Dscrmn.I35 1.272436496 NA 1.06950967
## 60 Dscrmn.I36 1.008690352 NA 0.84782549
## 61 Dscrmn.I37 1.288728736 NA 1.08320364
## 62 Dscrmn.I38 1.473055586 NA 1.23813424
## 63 Dscrmn.I39 1.342341968 NA 1.12826670
## 64 Dscrmn.I4 1.281925510 NA 1.07748539
## 65 Dscrmn.I40 1.457139147 NA 1.22475614
## 66 Dscrmn.I41 NA 1.03344365 NA
## 67 Dscrmn.I42 NA 0.96422272 NA
## 68 Dscrmn.I43 NA 0.99182420 NA
## 69 Dscrmn.I44 NA 0.95219518 NA
## 70 Dscrmn.I45 NA 1.13447404 NA
## 71 Dscrmn.I46 NA 1.18533478 NA
## 72 Dscrmn.I47 NA 1.04662444 NA
## 73 Dscrmn.I48 NA 0.94374714 NA
## 74 Dscrmn.I49 NA 0.80777460 NA
## 75 Dscrmn.I5 1.007880221 NA 0.84714455
## 76 Dscrmn.I50 NA 1.05217297 NA
## 77 Dscrmn.I6 0.949042894 NA 0.79769054
## 78 Dscrmn.I7 1.057666073 NA 0.88899061
## 79 Dscrmn.I8 1.201282017 NA 1.00970284
## 80 Dscrmn.I9 0.976907845 NA 0.821111626 Equated scores
Equated scores with the true score equating method
## The following scores are not attainable: 0## theta test4 test1.as.test4 StdErr
## 1 -3.2018628 1 1.015953 5.003132e-02
## 2 -2.4342926 2 1.955948 6.878631e-02
## 3 -1.9548627 3 2.893737 7.874430e-02
## 4 -1.5906884 4 3.841937 8.359716e-02
## 5 -1.2874493 5 4.805130 8.512002e-02
## 6 -1.0206725 6 5.784930 8.446458e-02
## 7 -0.7769609 7 6.781477 8.251528e-02
## 8 -0.5479067 8 7.794013 8.003024e-02
## 9 -0.3275850 9 8.821124 7.769419e-02
## 10 -0.1113201 10 9.860811 7.609765e-02
## 11 0.1050414 11 10.910495 7.564081e-02
## 12 0.3256710 12 11.966967 7.639765e-02
## 13 0.5553007 13 13.026319 7.802189e-02
## 14 0.7999529 14 14.083854 7.975624e-02
## 15 1.0681829 15 15.133999 8.052101e-02
## 16 1.3736059 16 16.170197 7.900210e-02
## 17 1.7410355 17 17.184780 7.366228e-02
## 18 2.2254147 18 18.168749 6.256879e-02
## 19 3.0012603 19 19.111255 4.258250e-02
## 20 42.1302143 20 20.000000 4.009435e-15Equated scores with the observed score equating method
## test4 test1.as.test4 StdErr
## 1 0 -0.01421594 0.03617689
## 2 1 0.95379681 0.05480849
## 3 2 1.90979310 0.06631698
## 4 3 2.86661488 0.07347523
## 5 4 3.83082127 0.07759881
## 6 5 4.80579296 0.07950051
## 7 6 5.79315712 0.07984123
## 8 7 6.79342490 0.07915320
## 9 8 7.80635915 0.07786227
## 10 9 8.83092394 0.07643819
## 11 10 9.86541737 0.07532277
## 12 11 10.90798519 0.07475095
## 13 12 11.95636186 0.07472299
## 14 13 13.00728219 0.07500067
## 15 14 14.05690943 0.07510454
## 16 15 15.10136220 0.07434567
## 17 16 16.13633708 0.07187445
## 18 17 17.15705121 0.06680167
## 19 18 18.15894672 0.05832150
## 20 19 19.13833584 0.04577955
## 21 20 20.09342215 0.02877984A comparison of equated scores obtained with 2 different chains, bisector and weighted bisector methods.
## theta test4 test1.as.test4 StdErr
## 1 1.741036 17 17.2431 0.09998233## theta test4 test1.as.test4 StdErr
## 1 1.741036 17 17.06712 0.1618202## theta test4 test1.as.test4 StdErr
## 1 1.741036 17 17.15485 0.08281067## theta test4 test1.as.test4 StdErr
## 1 1.741036 17 17.18478 0.073662287 Test for DIF
Load the data
Create a dataset for each group and estimate a 2PL model for each group using the R package mirt
library(mirt)
data1 <- dataDIF[dataDIF$group == 1, 1:20]
data2 <- dataDIF[dataDIF$group == 2, 1:20]
data3 <- dataDIF[dataDIF$group == 3, 1:20]
mod1 <- mirt(data1, SE = TRUE)
mod2 <- mirt(data2, SE = TRUE)
mod3 <- mirt(data3, SE = TRUE)Perform the test for DIF on two groups
##
## Test for Differential Item Functioning
##
## Item parameters tested for DIF: intercept and slope
## Equating method used: mean-mean
## Reference group: T1 Focal group: T2
## Item purification not applied
##
## statistic p.value
## I01 38.605 4.14e-09 ***
## I02 1.954 0.376
## I03 3.888 0.143
## I04 0.025 0.988
## I05 1.753 0.416
## I06 1.291 0.524
## I07 3.433 0.180
## I08 0.181 0.914
## I09 0.686 0.710
## I10 0.517 0.772
## I11 3.237 0.198
## I12 0.891 0.640
## I13 0.464 0.793
## I14 1.296 0.523
## I15 1.591 0.451
## I16 0.962 0.618
## I17 2.856 0.240
## I18 0.159 0.924
## I19 0.627 0.731
## I20 1.861 0.394
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Perform the test for DIF on three groups
##
## Test for Differential Item Functioning
##
## Item parameters tested for DIF: intercept and slope
## Equating method used: mean-mean
## Reference group: T1 Focal groups: T2, T3,
## Item purification not applied
##
## statistic p.value
## I01 164.585 <2e-16 ***
## I02 3.331 0.504
## I03 4.862 0.302
## I04 5.300 0.258
## I05 3.368 0.498
## I06 1.323 0.858
## I07 4.605 0.330
## I08 1.556 0.817
## I09 1.449 0.836
## I10 4.122 0.390
## I11 3.874 0.423
## I12 5.300 0.258
## I13 2.859 0.582
## I14 2.770 0.597
## I15 3.640 0.457
## I16 4.158 0.385
## I17 5.924 0.205
## I18 2.080 0.721
## I19 2.123 0.713
## I20 7.399 0.116
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1It is possible to change the reference group, the equating method used, and to apply purification.
res_diftest3 <- dif.test(est.mods = list(mod1, mod2, mod3), reference = 2,
method = "Haebara", purification = TRUE)
res_diftest3##
## Test for Differential Item Functioning
##
## Item parameters tested for DIF: intercept and slope
## Equating method used: Haebara
## Reference group: T2 Focal groups: T1, T3,
## Item purification applied. Significance level = 0.05
##
## statistic p.value
## I01 168.026 <2e-16 ***
## I02 2.553 0.635
## I03 2.921 0.571
## I04 4.263 0.372
## I05 3.395 0.494
## I06 0.648 0.958
## I07 2.796 0.592
## I08 2.796 0.593
## I09 1.668 0.796
## I10 2.405 0.662
## I11 1.972 0.741
## I12 4.116 0.391
## I13 1.721 0.787
## I14 2.381 0.666
## I15 4.668 0.323
## I16 2.148 0.709
## I17 4.457 0.348
## I18 1.301 0.861
## I19 2.900 0.575
## I20 3.497 0.478
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 18 Tests for drifts
The identity test performs a statistical test to verify if the chain equating coefficients from one form to itself are A=1 and B=0.
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
direclist3pl <- alldirec(mods = mod3pl, method = "Haebara")
pth3 <- paste("test", c(1:5,1), sep = "")
chainec_circle <- chainec(direclist = direclist3pl, pths = pth3)
summary(chainec_circle)## Path: test1.test2.test3.test4.test5.test1
## Method: Haebara
## Equating coefficients:
## Estimate StdErr
## A 1.0868759 0.063585
## B 0.0022126 0.041833##
## Identity test
##
## path: test1.test2.test3.test4.test5.test1
## statistic = 2.349279, df = 2, p-value = 0.3089304The null hypothesis A=1 and B=0 is not rejected.
It is also possible to performs a statistical test to verify if the chain equating coefficients that link the same two forms are equal.
In the following example test 1 and 5 are linked through two different paths giving two different pairs of equating coefficients. The example uses the 3PL models and the Haebara method, though other options are possible.
pth3 <- paste("test", 1:5, sep = "")
chainec3 <- chainec(direclist = direclist3pl, pths = pth3)
ecall <- c(chainec3, direclist3pl["test1.test5"])
summary(chainec3)## Path: test1.test2.test3.test4.test5
## Method: Haebara
## Equating coefficients:
## Estimate StdErr
## A 1.06595 0.056590
## B -0.50368 0.042982## Link: test1.test5
## Method: Haebara
## Equating coefficients:
## Estimate StdErr
## A 1.00935 0.029052
## B -0.51886 0.027863##
## Scale drift test
##
## link: test1.test5
## statistic = 1.623208, df = 2, p-value = 0.4441452The null hypothesis of equality of the equating coefficients is not rejected.