The equateMultiple package computes:
Data preparation follows the same steps of the equateIRT package.
Load the package equateMultiple and the data
## Caricamento del pacchetto richiesto: equateIRT
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
mlist<- list(m1, m2, m3, m4, m5)
test <- paste("test", 1:5, sep = "")
mods <- modIRT(est.mods = mlist, names = test, display = FALSE)
The 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 20
Estimation of the equating coefficients using the multiple mean-mean method. Form 1 is the base form.
## Computation of equating coefficients . . . .
## Computation of standard errors . . . .
## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.84001 0.018641
## A test3 0.84285 0.021321
## A test4 0.83876 0.020682
## A test5 1.02323 0.021556
## B test1 0.00000 0.000000
## B test2 0.10723 0.022389
## B test3 0.20275 0.023998
## B test4 0.36789 0.024059
## B test5 0.50312 0.023977
Estimation of the equating coefficients using the multiple mean-geometric mean method.
## Computation of equating coefficients . . . .
## Computation of standard errors . . . .
## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.83813 0.018688
## A test3 0.83986 0.021370
## A test4 0.83575 0.020736
## A test5 1.02115 0.021623
## B test1 0.00000 0.000000
## B test2 0.10726 0.022373
## B test3 0.20316 0.023898
## B test4 0.36779 0.023992
## B test5 0.50293 0.023952
Estimation of the equating coefficients using the multiple item response function method.
## Computation of equating coefficients . . . .
## Computation of standard errors . . . .
## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.83588 0.018346
## A test3 0.83551 0.020907
## A test4 0.82863 0.020163
## A test5 1.01232 0.021216
## B test1 0.00000 0.000000
## B test2 0.10838 0.021732
## B test3 0.20976 0.022989
## B test4 0.37218 0.023038
## B test5 0.49821 0.023505
Estimation 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.
## Computation of equating coefficients . . . .
## Computation of equating coefficients . . . .
## Computation of standard errors . . . .
## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.83588 0.018346
## A test3 0.83551 0.020907
## A test4 0.82863 0.020163
## A test5 1.01232 0.021216
## B test1 0.00000 0.000000
## B test2 0.10838 0.021732
## B test3 0.20976 0.022989
## B test4 0.37218 0.023038
## B test5 0.49821 0.023505
Estimation of the equating coefficients using the multiple test response function method.
## Computation of equating coefficients . . . .
## Computation of standard errors . . . .
## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.83636 0.018414
## A test3 0.83687 0.021036
## A test4 0.83097 0.020288
## A test5 1.01625 0.021242
## B test1 0.00000 0.000000
## B test2 0.10677 0.021781
## B test3 0.20626 0.023079
## B test4 0.36896 0.023105
## B test5 0.49615 0.023550
Estimation of the equating coefficients using the likelihood-based method.
## Computation of equating coefficients . . . .
## Computation of standard errors . . . .
## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 1.00000 0.000000
## A test2 0.83598 0.018307
## A test3 0.83685 0.020879
## A test4 0.83112 0.020097
## A test5 1.01735 0.021281
## B test1 0.00000 0.000000
## B test2 0.10860 0.021723
## B test3 0.21062 0.022994
## B test4 0.37256 0.023027
## B test5 0.49782 0.023502
It is possible to change the base form, that is the form whose item parameter estimates are left unchanged. All other item parameter estimates are converted to the scale of the base form.
## Computation of equating coefficients . . . .
## Computation of standard errors . . . .
## Equating coefficients:
## EQ Form Estimate StdErr
## A test1 0.98302 0.020562
## A test2 0.82178 0.020199
## A test3 0.82268 0.020485
## A test4 0.81696 0.017662
## A test5 1.00000 0.000000
## B test1 -0.48940 0.023486
## B test2 -0.38266 0.023121
## B test3 -0.28236 0.022808
## B test4 -0.12318 0.021301
## B test5 0.00000 0.000000
The synthetic item parameters are an estimate of the item parameters on the scale of the base form, obtained using all estimates available for the same item parameter across all the forms. The synthetic item parameters and their standard errors can be extracted as follows (using the multiple item response function method).
## Dscrmn.I1 Dscrmn.I10 Dscrmn.I11 Dscrmn.I12 Dscrmn.I13 Dscrmn.I14 Dscrmn.I15
## 1.0265253 1.3194832 1.0634221 1.1084510 1.3788564 1.1855599 1.0803419
## Dscrmn.I16 Dscrmn.I17 Dscrmn.I18 Dscrmn.I19 Dscrmn.I2 Dscrmn.I20 Dscrmn.I21
## 1.3726200 1.2517430 1.1480319 1.3158074 1.1382243 1.0076448 1.1950760
## Dscrmn.I22 Dscrmn.I23 Dscrmn.I24 Dscrmn.I25 Dscrmn.I26 Dscrmn.I27 Dscrmn.I28
## 1.2408239 1.2979584 1.0419701 1.2812410 1.3303749 1.3136643 1.1543945
## Dscrmn.I29 Dscrmn.I3 Dscrmn.I30 Dscrmn.I31 Dscrmn.I32 Dscrmn.I33 Dscrmn.I34
## 1.1063735 1.0785502 1.3060424 1.4688876 1.3699594 1.3342125 1.4011431
## Dscrmn.I35 Dscrmn.I36 Dscrmn.I37 Dscrmn.I38 Dscrmn.I39 Dscrmn.I4 Dscrmn.I40
## 1.2981521 1.0517954 1.2697911 1.4786267 1.3881008 1.3154817 1.4585414
## Dscrmn.I41 Dscrmn.I42 Dscrmn.I43 Dscrmn.I44 Dscrmn.I45 Dscrmn.I46 Dscrmn.I47
## 1.1902029 1.2004531 1.1829583 1.1447189 1.3970511 1.3535827 1.2626521
## Dscrmn.I48 Dscrmn.I49 Dscrmn.I5 Dscrmn.I50 Dscrmn.I6 Dscrmn.I7 Dscrmn.I8
## 1.1560787 0.9807114 1.0276615 1.2333594 0.9414937 1.0056828 1.1840444
## Dscrmn.I9
## 1.0042032
## Dffclt.I1 Dffclt.I10 Dffclt.I11 Dffclt.I12 Dffclt.I13 Dffclt.I14
## 0.04729781 0.67038640 0.93113357 0.79050080 0.43713941 0.75476773
## Dffclt.I15 Dffclt.I16 Dffclt.I17 Dffclt.I18 Dffclt.I19 Dffclt.I2
## 0.82963371 0.11717165 0.59912678 0.57943272 0.89351648 0.01210300
## Dffclt.I20 Dffclt.I21 Dffclt.I22 Dffclt.I23 Dffclt.I24 Dffclt.I25
## 0.07415299 0.23688948 -0.09924292 -0.41499245 0.62596997 0.39511469
## Dffclt.I26 Dffclt.I27 Dffclt.I28 Dffclt.I29 Dffclt.I3 Dffclt.I30
## 0.28096278 0.74521506 0.38465839 -0.41131364 0.04689779 0.81859219
## Dffclt.I31 Dffclt.I32 Dffclt.I33 Dffclt.I34 Dffclt.I35 Dffclt.I36
## 0.52979689 0.80606548 0.55956721 -0.36526934 0.45865817 0.85492089
## Dffclt.I37 Dffclt.I38 Dffclt.I39 Dffclt.I4 Dffclt.I40 Dffclt.I41
## 0.48778059 -0.12764922 0.51879390 0.38880128 0.85877342 -0.58159689
## Dffclt.I42 Dffclt.I43 Dffclt.I44 Dffclt.I45 Dffclt.I46 Dffclt.I47
## -0.28819961 1.26082476 1.12148675 0.56097371 -0.31006416 0.47890518
## Dffclt.I48 Dffclt.I49 Dffclt.I5 Dffclt.I50 Dffclt.I6 Dffclt.I7
## 0.20248015 0.28033084 0.01797425 0.34072018 -0.77064798 0.14049864
## Dffclt.I8 Dffclt.I9
## 0.33649415 -0.17811176
## Dscrmn.I1 Dscrmn.I10 Dscrmn.I11 Dscrmn.I12 Dscrmn.I13 Dscrmn.I14 Dscrmn.I15
## 0.03495975 0.04191803 0.04224192 0.04290795 0.04903040 0.04467058 0.04233749
## Dscrmn.I16 Dscrmn.I17 Dscrmn.I18 Dscrmn.I19 Dscrmn.I2 Dscrmn.I20 Dscrmn.I21
## 0.04859053 0.04602021 0.04332686 0.04872432 0.03720922 0.03982246 0.04486314
## Dscrmn.I22 Dscrmn.I23 Dscrmn.I24 Dscrmn.I25 Dscrmn.I26 Dscrmn.I27 Dscrmn.I28
## 0.04649240 0.04916097 0.04135682 0.04707868 0.04829875 0.04847777 0.04383667
## Dscrmn.I29 Dscrmn.I3 Dscrmn.I30 Dscrmn.I31 Dscrmn.I32 Dscrmn.I33 Dscrmn.I34
## 0.04397104 0.03598696 0.04850613 0.04241811 0.04104517 0.03937279 0.04123979
## Dscrmn.I35 Dscrmn.I36 Dscrmn.I37 Dscrmn.I38 Dscrmn.I39 Dscrmn.I4 Dscrmn.I40
## 0.03835338 0.03409166 0.03774838 0.04232501 0.04052315 0.04116937 0.04347621
## Dscrmn.I41 Dscrmn.I42 Dscrmn.I43 Dscrmn.I44 Dscrmn.I45 Dscrmn.I46 Dscrmn.I47
## 0.04362829 0.04252692 0.04270864 0.04116369 0.04621696 0.04673421 0.04281988
## Dscrmn.I48 Dscrmn.I49 Dscrmn.I5 Dscrmn.I50 Dscrmn.I6 Dscrmn.I7 Dscrmn.I8
## 0.04036263 0.03644209 0.03506514 0.04212137 0.03479529 0.03460417 0.03821698
## Dscrmn.I9
## 0.03477178
## Dffclt.I1 Dffclt.I10 Dffclt.I11 Dffclt.I12 Dffclt.I13 Dffclt.I14 Dffclt.I15
## 0.02626137 0.02732356 0.03786443 0.03394265 0.02580696 0.03217098 0.03520837
## Dffclt.I16 Dffclt.I17 Dffclt.I18 Dffclt.I19 Dffclt.I2 Dffclt.I20 Dffclt.I21
## 0.02454754 0.02880217 0.02960223 0.03337780 0.02491868 0.02822900 0.02654187
## Dffclt.I22 Dffclt.I23 Dffclt.I24 Dffclt.I25 Dffclt.I26 Dffclt.I27 Dffclt.I28
## 0.02835816 0.03312636 0.03080688 0.02610118 0.02541336 0.02949214 0.02720872
## Dffclt.I29 Dffclt.I3 Dffclt.I30 Dffclt.I31 Dffclt.I32 Dffclt.I33 Dffclt.I34
## 0.03550641 0.02558258 0.03083072 0.02372639 0.02743385 0.02498080 0.02563614
## Dffclt.I35 Dffclt.I36 Dffclt.I37 Dffclt.I38 Dffclt.I39 Dffclt.I4 Dffclt.I40
## 0.02460116 0.03189395 0.02493596 0.02319156 0.02436926 0.02448540 0.02694958
## Dffclt.I41 Dffclt.I42 Dffclt.I43 Dffclt.I44 Dffclt.I45 Dffclt.I46 Dffclt.I47
## 0.03787620 0.03160015 0.03757152 0.03504277 0.02531162 0.03034852 0.02583944
## Dffclt.I48 Dffclt.I49 Dffclt.I5 Dffclt.I50 Dffclt.I6 Dffclt.I7 Dffclt.I8
## 0.02675857 0.02880551 0.02628429 0.02579272 0.03742373 0.02665738 0.02530837
## Dffclt.I9
## 0.02743718
Equated scores with the true score equating method
## 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: 0
## theta test1 test2.as.test1 StdErr_test2.as.test1 test3.as.test1
## 1 -2.344 1 1.073 0.027 0.783
## 2 -1.661 2 2.072 0.034 1.651
## 3 -1.242 3 3.041 0.039 2.551
## 4 -0.929 4 3.992 0.041 3.469
## 5 -0.672 5 4.933 0.042 4.402
## 6 -0.449 6 5.870 0.042 5.346
## 7 -0.248 7 6.806 0.041 6.301
## 8 -0.060 8 7.743 0.040 7.264
## 9 0.119 9 8.682 0.040 8.236
## 10 0.293 10 9.626 0.041 9.217
## 11 0.467 11 10.576 0.043 10.208
## 12 0.642 12 11.534 0.047 11.209
## 13 0.825 13 12.501 0.052 12.222
## 14 1.018 14 13.481 0.058 13.249
## 15 1.230 15 14.476 0.062 14.293
## 16 1.472 16 15.491 0.065 15.358
## 17 1.764 17 16.532 0.065 16.450
## 18 2.152 18 17.611 0.060 17.578
## 19 2.781 19 18.746 0.044 18.753
## 20 35.210 20 20.000 0.000 20.000
## StdErr_test3.as.test1 test4.as.test1 StdErr_test4.as.test1 test5.as.test1
## 1 0.041 0.932 0.050 0.750
## 2 0.060 1.967 0.071 1.633
## 3 0.070 3.016 0.080 2.562
## 4 0.074 4.061 0.082 3.516
## 5 0.074 5.096 0.080 4.486
## 6 0.071 6.116 0.076 5.467
## 7 0.067 7.124 0.070 6.456
## 8 0.063 8.118 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 12.961 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 16.848 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 0.000
Equated scores with the observed score equating method, avoiding computation of standard errors
## test1 test2.as.test1 test3.as.test1 test4.as.test1 test5.as.test1
## 1 0 0.031 -0.164 -0.025 -0.158
## 2 1 1.032 0.714 0.983 0.725
## 3 2 2.014 1.611 2.013 1.639
## 4 3 2.982 2.524 3.048 2.576
## 5 4 3.941 3.444 4.079 3.531
## 6 5 4.893 4.379 5.102 4.498
## 7 6 5.841 5.327 6.115 5.475
## 8 7 6.788 6.285 7.117 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 10.203 11.041 10.470
## 13 12 11.551 11.203 12.007 11.489
## 14 13 12.519 12.214 12.971 12.515
## 15 14 13.497 13.235 13.935 13.547
## 16 15 14.487 14.269 14.902 14.588
## 17 16 15.489 15.317 15.875 15.637
## 18 17 16.507 16.378 16.858 16.693
## 19 18 17.551 17.451 17.854 17.757
## 20 19 18.626 18.548 18.868 18.829
## 21 20 19.742 19.703 19.906 19.907