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.
2 Data preparation
Load the package equateIRT and the data
library("equateIRT")
data("data2pl", package = "equateIRT")Estimate a two parameter logistic model for 5 data sets with the R package ltm
library("ltm")
m1 <- ltm(data2pl[[1]] ~ z1)
m2 <- ltm(data2pl[[2]] ~ z1)
m3 <- ltm(data2pl[[3]] ~ z1)
m4 <- ltm(data2pl[[4]] ~ z1)
m5 <- ltm(data2pl[[5]] ~ z1)Extract the item parameter estimates and the covariance matrices
estm1 <- import.ltm(m1, display = FALSE)
estm2 <- import.ltm(m2, display = FALSE)
estm3 <- import.ltm(m3, display = FALSE)
estm4 <- import.ltm(m4, display = FALSE)
estm5 <- import.ltm(m5, display = FALSE)
estm1$coef[1:3, ]## (Intercept) z1
## I1 -0.06213808 1.076155
## I2 -0.03090993 1.122379
## I3 -0.07939847 1.091369estm1$var[1:3, 1:3]## [,1] [,2] [,3]
## [1,] 0.0012285184 0.0002460322 0.0002391000
## [2,] 0.0002460322 0.0012628923 0.0002495126
## [3,] 0.0002391000 0.0002495126 0.0012407430Create a list of coefficients and covariance matrices
estc <- list(estm1$coef, estm2$coef, estm3$coef, estm4$coef, estm5$coef)
estv <- list(estm1$var, estm2$var, estm3$var, estm4$var, estm5$var)
test <- paste("test", 1:5, sep = "")Create an object of class modIRT
mod2pl <- modIRT(coef = estc, var = estv, names = test, display = FALSE)
coef(mod2pl$test1)[1:5]## Dffclt.I1 Dffclt.I2 Dffclt.I3 Dffclt.I4 Dffclt.I5
## 0.05774085 0.02753964 0.07275128 0.41568210 -0.00716265The linkage plan
lplan<-linkp(coef = estc)
lplan## [,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.
l12 <- direc(mods = mod2pl, which = c(1,2), method = "mean-mean")
l12## Direct equating coefficients
## Method: mean-mean
## Link: test1.test2summary(l12)## Link: test1.test2
## Method: mean-mean
## Equating coefficients:
## Estimate StdErr
## A 1.21004 0.028981
## B -0.14199 0.028127Estimation of all direct equating coefficients between forms with common items using the mean-mean method
direclist2pl <- alldirec(mods = mod2pl, method = "mean-mean")
direclist2pl## 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
summary(direclist2pl, link="test1.test2")## Link: test1.test2
## Method: mean-mean
## Equating coefficients:
## Estimate StdErr
## A 1.21004 0.028981
## B -0.14199 0.0281274 Chain equating coefficients
Estimation of all chain equating coefficients of length 4
cec4 <- chainec(r = 4, direclist = direclist2pl)
cec4## 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.test1summary(cec4, path="test1.test2.test3.test4")## Path: test1.test2.test3.test4
## Method: mean-mean
## Equating coefficients:
## Estimate StdErr
## A 1.25323 0.046518
## B -0.49789 0.038505Chain equating coefficients for path {1, 2, 3, 4}
summary(cec4, path="test1.test2.test3.test4")## Path: test1.test2.test3.test4
## Method: mean-mean
## Equating coefficients:
## Estimate StdErr
## A 1.25323 0.046518
## B -0.49789 0.038505Estimation of all chain equating coefficients of length 4 from Form 1 to Form 4
cec14 <- chainec(r = 4, direclist = direclist2pl, f1 = "test1", f2 = "test4")
cec14## Chain equating coefficients
## Method: mean-mean
## Paths:
## test1.test2.test3.test4summary(cec14)## Path: test1.test2.test3.test4
## Method: mean-mean
## Equating coefficients:
## Estimate StdErr
## A 1.25323 0.046518
## B -0.49789 0.038505Estimation 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 1.15857 0.033390
## B -0.39971 0.033076NOTE: Item parameters are converted to the scale of Form 4.
5 Average equating coefficients
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.test4summary(fec)## Link: test1.test4
## Method: mean-mean
## Equating coefficients:
## Path Estimate StdErr
## A test1.test2.test3.test4 1.25323 0.046518
## A test1.test5.test4 1.15857 0.033390
## A bisector 1.20480 0.030609
## A weighted bisector 1.18885 0.029302
## B test1.test2.test3.test4 -0.49789 0.038505
## B test1.test5.test4 -0.39971 0.033076
## B bisector -0.44766 0.029927
## B weighted bisector -0.43112 0.029713Extract the equating coefficients
eqc(fec)## link path A B
## 1 test1.test4 test1.test2.test3.test4 1.253235 -0.4978899
## 2 test1.test4 test1.test5.test4 1.158568 -0.3997148
## 3 test1.test4 bisector 1.204801 -0.4476611
## 4 test1.test4 weighted bisector 1.188853 -0.4311214Extract 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.
itm(fec, bistype = "weighted")## Item test1 test4 test1.as.test4
## 1 Dffclt.I1 0.05774085 NA -0.36247602
## 2 Dffclt.I10 0.65469024 NA 0.34720876
## 3 Dffclt.I2 0.02753964 NA -0.39838080
## 4 Dffclt.I21 NA -0.18987389 NA
## 5 Dffclt.I22 NA -0.57178646 NA
## 6 Dffclt.I23 NA -0.97083469 NA
## 7 Dffclt.I24 NA 0.28202761 NA
## 8 Dffclt.I25 NA 0.02635384 NA
## 9 Dffclt.I26 NA -0.12110226 NA
## 10 Dffclt.I27 NA 0.47566134 NA
## 11 Dffclt.I28 NA -0.01084706 NA
## 12 Dffclt.I29 NA -0.93620431 NA
## 13 Dffclt.I3 0.07275128 NA -0.34463082
## 14 Dffclt.I30 NA 0.54298296 NA
## 15 Dffclt.I31 0.52243942 NA 0.18998205
## 16 Dffclt.I32 0.83132905 NA 0.55720626
## 17 Dffclt.I33 0.57523659 NA 0.25275010
## 18 Dffclt.I34 -0.34258115 NA -0.83839982
## 19 Dffclt.I35 0.42869381 NA 0.07853234
## 20 Dffclt.I36 0.85197621 NA 0.58175268
## 21 Dffclt.I37 0.45623368 NA 0.11127318
## 22 Dffclt.I38 -0.13812533 NA -0.59533201
## 23 Dffclt.I39 0.51891939 NA 0.18579726
## 24 Dffclt.I4 0.41568210 NA 0.06306334
## 25 Dffclt.I40 0.84172659 NA 0.56956741
## 26 Dffclt.I41 NA -1.12866114 NA
## 27 Dffclt.I42 NA -0.77022951 NA
## 28 Dffclt.I43 NA 1.08692549 NA
## 29 Dffclt.I44 NA 0.88814831 NA
## 30 Dffclt.I45 NA 0.22894864 NA
## 31 Dffclt.I46 NA -0.78067726 NA
## 32 Dffclt.I47 NA 0.13570645 NA
## 33 Dffclt.I48 NA -0.21808635 NA
## 34 Dffclt.I49 NA -0.05870549 NA
## 35 Dffclt.I5 -0.00716265 NA -0.43963670
## 36 Dffclt.I50 NA -0.06316900 NA
## 37 Dffclt.I6 -0.77378858 NA -1.35104186
## 38 Dffclt.I7 0.15460392 NA -0.24732011
## 39 Dffclt.I8 0.33698282 NA -0.03049849
## 40 Dffclt.I9 -0.18289475 NA -0.64855625
## 41 Dscrmn.I1 1.07615468 NA 0.90520453
## 42 Dscrmn.I10 1.34017474 NA 1.12728428
## 43 Dscrmn.I2 1.12237949 NA 0.94408640
## 44 Dscrmn.I21 NA 0.96336089 NA
## 45 Dscrmn.I22 NA 1.01225882 NA
## 46 Dscrmn.I23 NA 1.05642645 NA
## 47 Dscrmn.I24 NA 0.87351070 NA
## 48 Dscrmn.I25 NA 1.06903268 NA
## 49 Dscrmn.I26 NA 1.10250014 NA
## 50 Dscrmn.I27 NA 1.04533344 NA
## 51 Dscrmn.I28 NA 0.93410270 NA
## 52 Dscrmn.I29 NA 0.93003277 NA
## 53 Dscrmn.I3 1.09136863 NA 0.91800171
## 54 Dscrmn.I30 NA 1.11202742 NA
## 55 Dscrmn.I31 1.48282841 NA 1.24727701
## 56 Dscrmn.I32 1.29985743 NA 1.09337149
## 57 Dscrmn.I33 1.35208875 NA 1.13730572
## 58 Dscrmn.I34 1.38211481 NA 1.16256205
## 59 Dscrmn.I35 1.27117732 NA 1.06924729
## 60 Dscrmn.I36 1.00818083 NA 0.84802852
## 61 Dscrmn.I37 1.28758500 NA 1.08304857
## 62 Dscrmn.I38 1.47309474 NA 1.23908957
## 63 Dscrmn.I39 1.34111442 NA 1.12807469
## 64 Dscrmn.I4 1.28072435 NA 1.07727775
## 65 Dscrmn.I40 1.45641060 NA 1.22505575
## 66 Dscrmn.I41 NA 1.03348633 NA
## 67 Dscrmn.I42 NA 0.96419052 NA
## 68 Dscrmn.I43 NA 0.99167808 NA
## 69 Dscrmn.I44 NA 0.95212573 NA
## 70 Dscrmn.I45 NA 1.13452428 NA
## 71 Dscrmn.I46 NA 1.18530385 NA
## 72 Dscrmn.I47 NA 1.04662208 NA
## 73 Dscrmn.I48 NA 0.94373225 NA
## 74 Dscrmn.I49 NA 0.80774381 NA
## 75 Dscrmn.I5 1.00731869 NA 0.84730333
## 76 Dscrmn.I50 NA 1.05234303 NA
## 77 Dscrmn.I6 0.94866983 NA 0.79797100
## 78 Dscrmn.I7 1.05732762 NA 0.88936820
## 79 Dscrmn.I8 1.20065598 NA 1.00992846
## 80 Dscrmn.I9 0.97710839 NA 0.821892026 Equated scores
Equated scores with the true score equating method
score(fec, bistype = "weighted")## The following scores are not attainable: 0## theta test4 test1.as.test4 StdErr
## 1 -3.2020971 1 1.015003 5.001185e-02
## 2 -2.4345213 2 1.954738 6.878198e-02
## 3 -1.9550888 3 2.892492 7.875379e-02
## 4 -1.5909128 4 3.840790 8.361628e-02
## 5 -1.2876722 5 4.804165 8.514419e-02
## 6 -1.0208940 6 5.784196 8.448979e-02
## 7 -0.7771809 7 6.780997 8.253848e-02
## 8 -0.5481250 8 7.793792 8.004946e-02
## 9 -0.3278013 9 8.821147 7.770849e-02
## 10 -0.1115340 10 9.861057 7.610701e-02
## 11 0.1048303 11 10.910935 7.564599e-02
## 12 0.3254635 12 11.967568 7.639989e-02
## 13 0.5550977 13 13.027045 7.802247e-02
## 14 0.7997556 14 14.084670 7.975604e-02
## 15 1.0679930 15 15.134866 8.052029e-02
## 16 1.3734260 16 16.171075 7.900041e-02
## 17 1.7408695 17 17.185619 7.365858e-02
## 18 2.2252693 18 18.169485 6.256174e-02
## 19 3.0011510 19 19.111786 4.257124e-02
## 20 42.0664330 20 20.000000 4.018669e-15Equated scores with the observed score equating method
score(fec, method = "OSE", bistype = "weighted")## test4 test1.as.test4 StdErr
## 1 0 -0.01489569 0.03615105
## 2 1 0.95282270 0.05477351
## 3 2 1.90871813 0.06629269
## 4 3 2.86555916 0.07346520
## 5 4 3.82986670 0.07760111
## 6 5 4.80499566 0.07951120
## 7 6 5.79255145 0.07985590
## 8 7 6.79302775 0.07916864
## 9 8 7.80617492 0.07787658
## 10 9 8.83094667 0.07644964
## 11 10 9.86563125 0.07533011
## 12 11 10.90836642 0.07475424
## 13 12 11.95688236 0.07472285
## 14 13 13.00791258 0.07499743
## 15 14 14.05761874 0.07509834
## 16 15 15.10211616 0.07433639
## 17 16 16.13709851 0.07186160
## 18 17 17.15778076 0.06678472
## 19 18 18.15960149 0.05830036
## 20 19 19.13886811 0.04575589
## 21 20 20.09377896 0.02876039A comparison of equated scores obtained with 2 different chains, bisector and weighted bisector methods.
score(chainec154, scores = 17)## theta test4 test1.as.test4 StdErr
## 1 1.74087 17 17.24468 0.09997669score(cec4, path = "test1.test2.test3.test4", scores = 17)## theta test4 test1.as.test4 StdErr
## 1 1.74087 17 17.06646 0.1618316score(fec, bistype = "unweighted", scores = 17)## theta test4 test1.as.test4 StdErr
## 1 1.74087 17 17.15531 0.08280613score(fec, bistype = "weighted", scores = 17)## theta test4 test1.as.test4 StdErr
## 1 1.74087 17 17.18562 0.07365858