The equateMultiple package computes:
Data preparation follows the same steps of the equateIRT package.
Load the package equateMultiple and the data
library("equateMultiple")## Caricamento del pacchetto richiesto: equateIRTdata("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
mods <- modIRT(coef = estc, var = estv, names = test, display = FALSE)
coef(mods$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 20Estimation of the equating coefficients using the multiple mean-mean method. Form 1 is the base form.
eqMM <- multiec(mods = mods, base = 1, method = "mean-mean")## Computation of equating coefficients . . . .
## Computation of standard errors . . . .summary(eqMM)## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.84051 0.018648
## A test3 0.84347 0.021334
## A test4 0.83937 0.020694
## A test5 1.02343 0.021524
## B test1 0.00000 0.000000
## B test2 0.10692 0.022427
## B test3 0.20236 0.024039
## B test4 0.36774 0.024111
## B test5 0.50207 0.024026Estimation of the equating coefficients using the multiple mean-geometric mean method.
eqMGM <- multiec(mods = mods, base = 1, method = "mean-gmean")## Computation of equating coefficients . . . .
## Computation of standard errors . . . .summary(eqMGM)## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.83860 0.018695
## A test3 0.84045 0.021383
## A test4 0.83633 0.020746
## A test5 1.02135 0.021591
## B test1 0.00000 0.000000
## B test2 0.10695 0.022410
## B test3 0.20277 0.023938
## B test4 0.36764 0.024044
## B test5 0.50188 0.024001Estimation of the equating coefficients using the multiple item response function method.
eqIRF<-multiec(mods = mods, base = 1, method = "irf")## Computation of equating coefficients . . . .
## Computation of standard errors . . . .summary(eqIRF)## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.83635 0.018353
## A test3 0.83610 0.020919
## A test4 0.82919 0.020173
## A test5 1.01249 0.021184
## B test1 0.00000 0.000000
## B test2 0.10802 0.021770
## B test3 0.20935 0.023031
## B test4 0.37199 0.023090
## B test5 0.49709 0.023555Estimation of the equating coefficients using the multiple item response function method. The initial values are the estimates obtained with the multiple mean-geometric mean method.
eqMGM <- multiec(mods = mods, base = 1, method = "mean-gmean", se = FALSE)## Computation of equating coefficients . . . .eqIRF<-multiec(mods = mods, base = 1, method = "irf", start = eqMGM)## Computation of equating coefficients . . . .
## Computation of standard errors . . . .summary(eqIRF)## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.83635 0.018353
## A test3 0.83610 0.020919
## A test4 0.82919 0.020173
## A test5 1.01249 0.021184
## B test1 0.00000 0.000000
## B test2 0.10802 0.021770
## B test3 0.20935 0.023031
## B test4 0.37199 0.023090
## B test5 0.49709 0.023555Estimation of the equating coefficients using the multiple test response function method.
eqTRF<-multiec(mods = mods, base = 1, method = "trf")## Computation of equating coefficients . . . .
## Computation of standard errors . . . .summary(eqTRF)## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.83681 0.018421
## A test3 0.83744 0.021048
## A test4 0.83154 0.020298
## A test5 1.01646 0.021209
## B test1 0.00000 0.000000
## B test2 0.10639 0.021817
## B test3 0.20580 0.023118
## B test4 0.36873 0.023154
## B test5 0.49499 0.023600The synthetic item parameters and their standard errors can be extracted as follows (using the multiple item response function method).
eqIRF$as## Dscrmn.I1 Dscrmn.I10 Dscrmn.I11 Dscrmn.I12 Dscrmn.I13 Dscrmn.I14 Dscrmn.I15
## 1.0263300 1.3186514 1.0628256 1.1077121 1.3780255 1.1849060 1.0797961
## Dscrmn.I16 Dscrmn.I17 Dscrmn.I18 Dscrmn.I19 Dscrmn.I2 Dscrmn.I20 Dscrmn.I21
## 1.3717438 1.2509322 1.1473431 1.3149445 1.1374079 1.0070348 1.1943277
## Dscrmn.I22 Dscrmn.I23 Dscrmn.I24 Dscrmn.I25 Dscrmn.I26 Dscrmn.I27 Dscrmn.I28
## 1.2399568 1.2972537 1.0413250 1.2803009 1.3294265 1.3129234 1.1536342
## Dscrmn.I29 Dscrmn.I3 Dscrmn.I30 Dscrmn.I31 Dscrmn.I32 Dscrmn.I33 Dscrmn.I34
## 1.1056767 1.0782685 1.3051024 1.4679605 1.3688577 1.3333632 1.4005527
## Dscrmn.I35 Dscrmn.I36 Dscrmn.I37 Dscrmn.I38 Dscrmn.I39 Dscrmn.I4 Dscrmn.I40
## 1.2974183 1.0511445 1.2689048 1.4780647 1.3871910 1.3145141 1.4580333
## Dscrmn.I41 Dscrmn.I42 Dscrmn.I43 Dscrmn.I44 Dscrmn.I45 Dscrmn.I46 Dscrmn.I47
## 1.1895968 1.1991245 1.1820406 1.1438774 1.3954929 1.3527279 1.2618251
## Dscrmn.I48 Dscrmn.I49 Dscrmn.I5 Dscrmn.I50 Dscrmn.I6 Dscrmn.I7 Dscrmn.I8
## 1.1551069 0.9800754 1.0270970 1.2328251 0.9410529 1.0052578 1.1833536
## Dscrmn.I9
## 1.0040658eqIRF$bs## Dffclt.I1 Dffclt.I10 Dffclt.I11 Dffclt.I12 Dffclt.I13 Dffclt.I14
## 0.04680159 0.67020856 0.93109634 0.79043223 0.43683497 0.75463195
## Dffclt.I15 Dffclt.I16 Dffclt.I17 Dffclt.I18 Dffclt.I19 Dffclt.I2
## 0.82951627 0.11667937 0.59893566 0.57922077 0.89350937 0.01160208
## Dffclt.I20 Dffclt.I21 Dffclt.I22 Dffclt.I23 Dffclt.I24 Dffclt.I25
## 0.07364461 0.23644361 -0.09992649 -0.41581481 0.62576552 0.39477998
## Dffclt.I26 Dffclt.I27 Dffclt.I28 Dffclt.I29 Dffclt.I3 Dffclt.I30
## 0.28054816 0.74507324 0.38431305 -0.41217210 0.04639957 0.81854107
## Dffclt.I31 Dffclt.I32 Dffclt.I33 Dffclt.I34 Dffclt.I35 Dffclt.I36
## 0.52952744 0.80600281 0.55941089 -0.36603676 0.45840659 0.85483709
## Dffclt.I37 Dffclt.I38 Dffclt.I39 Dffclt.I4 Dffclt.I40 Dffclt.I41
## 0.48752328 -0.12828412 0.51856681 0.38850704 0.85860534 -0.58246660
## Dffclt.I42 Dffclt.I43 Dffclt.I44 Dffclt.I45 Dffclt.I46 Dffclt.I47
## -0.28920713 1.26110283 1.12164448 0.56075671 -0.31084350 0.47861241
## Dffclt.I48 Dffclt.I49 Dffclt.I5 Dffclt.I50 Dffclt.I6 Dffclt.I7
## 0.20196743 0.27988322 0.01748403 0.34034716 -0.77150306 0.14004614
## Dffclt.I8 Dffclt.I9
## 0.33614495 -0.17864329eqIRF$se.as## Dscrmn.I1 Dscrmn.I10 Dscrmn.I11 Dscrmn.I12 Dscrmn.I13 Dscrmn.I14 Dscrmn.I15
## 0.03495588 0.04187786 0.04221454 0.04287726 0.04899610 0.04464139 0.04231166
## Dscrmn.I16 Dscrmn.I17 Dscrmn.I18 Dscrmn.I19 Dscrmn.I2 Dscrmn.I20 Dscrmn.I21
## 0.04855524 0.04598656 0.04329678 0.04869024 0.03717496 0.03979567 0.04483082
## Dscrmn.I22 Dscrmn.I23 Dscrmn.I24 Dscrmn.I25 Dscrmn.I26 Dscrmn.I27 Dscrmn.I28
## 0.04645440 0.04913040 0.04132664 0.04703964 0.04825891 0.04844480 0.04380390
## Dscrmn.I29 Dscrmn.I3 Dscrmn.I30 Dscrmn.I31 Dscrmn.I32 Dscrmn.I33 Dscrmn.I34
## 0.04393780 0.03598386 0.04846504 0.04237596 0.04101176 0.03931653 0.04122265
## Dscrmn.I35 Dscrmn.I36 Dscrmn.I37 Dscrmn.I38 Dscrmn.I39 Dscrmn.I4 Dscrmn.I40
## 0.03831698 0.03406463 0.03771167 0.04231999 0.04048852 0.04112994 0.04347242
## Dscrmn.I41 Dscrmn.I42 Dscrmn.I43 Dscrmn.I44 Dscrmn.I45 Dscrmn.I46 Dscrmn.I47
## 0.04359194 0.04247471 0.04266961 0.04113018 0.04616402 0.04670080 0.04278476
## Dscrmn.I48 Dscrmn.I49 Dscrmn.I5 Dscrmn.I50 Dscrmn.I6 Dscrmn.I7 Dscrmn.I8
## 0.04032199 0.03641452 0.03503836 0.04209305 0.03478125 0.03458554 0.03819397
## Dscrmn.I9
## 0.03477342eqIRF$se.bs## Dffclt.I1 Dffclt.I10 Dffclt.I11 Dffclt.I12 Dffclt.I13 Dffclt.I14 Dffclt.I15
## 0.02628883 0.02738288 0.03792320 0.03400229 0.02585817 0.03222779 0.03526458
## Dffclt.I16 Dffclt.I17 Dffclt.I18 Dffclt.I19 Dffclt.I2 Dffclt.I20 Dffclt.I21
## 0.02458570 0.02885806 0.02965635 0.03344091 0.02495266 0.02826440 0.02658532
## Dffclt.I22 Dffclt.I23 Dffclt.I24 Dffclt.I25 Dffclt.I26 Dffclt.I27 Dffclt.I28
## 0.02838793 0.03314244 0.03086282 0.02615282 0.02546005 0.02955263 0.02725851
## Dffclt.I29 Dffclt.I3 Dffclt.I30 Dffclt.I31 Dffclt.I32 Dffclt.I33 Dffclt.I34
## 0.03552543 0.02561236 0.03089467 0.02378732 0.02750672 0.02504117 0.02565134
## Dffclt.I35 Dffclt.I36 Dffclt.I37 Dffclt.I38 Dffclt.I39 Dffclt.I4 Dffclt.I40
## 0.02466081 0.03195958 0.02499000 0.02321337 0.02443617 0.02454001 0.02702879
## Dffclt.I41 Dffclt.I42 Dffclt.I43 Dffclt.I44 Dffclt.I45 Dffclt.I46 Dffclt.I47
## 0.03788098 0.03163075 0.03764630 0.03511477 0.02537526 0.03036798 0.02589473
## Dffclt.I48 Dffclt.I49 Dffclt.I5 Dffclt.I50 Dffclt.I6 Dffclt.I7 Dffclt.I8
## 0.02680342 0.02885080 0.02631625 0.02583940 0.03742903 0.02669245 0.02535449
## Dffclt.I9
## 0.02745309Equated scores with the true score equating method
scTSE<-score(eqIRF)## The following scores are not attainable: 0
## The following scores are not attainable: 0
## The following scores are not attainable: 0
## The following scores are not attainable: 0round(scTSE,3)## theta test1 test2.as.test1 StdErr_test2.as.test1 test3.as.test1
## 1 -2.345 1 1.073 0.027 0.784
## 2 -1.662 2 2.072 0.034 1.652
## 3 -1.243 3 3.041 0.039 2.551
## 4 -0.930 4 3.992 0.041 3.470
## 5 -0.673 5 4.933 0.042 4.403
## 6 -0.450 6 5.870 0.042 5.347
## 7 -0.248 7 6.806 0.041 6.301
## 8 -0.061 8 7.742 0.040 7.264
## 9 0.118 9 8.682 0.040 8.236
## 10 0.293 10 9.626 0.041 9.217
## 11 0.466 11 10.576 0.043 10.208
## 12 0.642 12 11.534 0.047 11.208
## 13 0.824 13 12.501 0.052 12.221
## 14 1.018 14 13.481 0.058 13.248
## 15 1.230 15 14.476 0.062 14.293
## 16 1.472 16 15.491 0.065 15.358
## 17 1.764 17 16.533 0.065 16.450
## 18 2.152 18 17.611 0.060 17.577
## 19 2.782 19 18.746 0.044 18.752
## 20 35.219 20 20.000 0.000 20.000
## StdErr_test3.as.test1 test4.as.test1 StdErr_test4.as.test1 test5.as.test1
## 1 0.041 0.933 0.050 0.751
## 2 0.060 1.968 0.071 1.634
## 3 0.070 3.017 0.080 2.563
## 4 0.074 4.062 0.082 3.517
## 5 0.074 5.096 0.080 4.487
## 6 0.071 6.117 0.076 5.467
## 7 0.067 7.124 0.070 6.456
## 8 0.063 8.119 0.065 7.453
## 9 0.060 9.102 0.061 8.456
## 10 0.060 10.075 0.059 9.466
## 11 0.063 11.041 0.059 10.483
## 12 0.068 12.002 0.061 11.508
## 13 0.074 12.960 0.065 12.540
## 14 0.081 13.920 0.069 13.580
## 15 0.087 14.884 0.073 14.629
## 16 0.091 15.858 0.074 15.688
## 17 0.089 16.847 0.073 16.756
## 18 0.081 17.861 0.066 17.836
## 19 0.059 18.909 0.049 18.926
## 20 0.000 20.000 0.000 20.000
## StdErr_test5.as.test1
## 1 0.034
## 2 0.047
## 3 0.052
## 4 0.052
## 5 0.050
## 6 0.047
## 7 0.045
## 8 0.043
## 9 0.041
## 10 0.040
## 11 0.041
## 12 0.042
## 13 0.044
## 14 0.047
## 15 0.050
## 16 0.051
## 17 0.051
## 18 0.047
## 19 0.036
## 20 0.000Equated scores with the observed score equating method, avoiding computation of standard errors
scOSE<-score(eqIRF, method = "OSE", se = FALSE)
round(scOSE,3)## test1 test2.as.test1 test3.as.test1 test4.as.test1 test5.as.test1
## 1 0 0.031 -0.164 -0.024 -0.158
## 2 1 1.032 0.714 0.984 0.725
## 3 2 2.014 1.612 2.014 1.639
## 4 3 2.983 2.525 3.049 2.577
## 5 4 3.941 3.445 4.079 3.532
## 6 5 4.893 4.380 5.102 4.499
## 7 6 5.841 5.327 6.115 5.475
## 8 7 6.788 6.285 7.118 6.460
## 9 8 7.735 7.252 8.110 7.453
## 10 9 8.683 8.227 9.094 8.453
## 11 10 9.635 9.211 10.070 9.458
## 12 11 10.590 10.202 11.041 10.470
## 13 12 11.551 11.203 12.007 11.489
## 14 13 12.519 12.214 12.970 12.515
## 15 14 13.496 13.235 13.935 13.547
## 16 15 14.487 14.269 14.902 14.588
## 17 16 15.489 15.317 15.875 15.636
## 18 17 16.507 16.378 16.857 16.692
## 19 18 17.551 17.451 17.853 17.756
## 20 19 18.626 18.547 18.868 18.828
## 21 20 19.742 19.703 19.906 19.906