1 Functionalities

The equateMultiple package computes:

Equating coefficients between multiple forms.
Synthetic item parameters (sort of mean of the item parameter estimates from different forms).
Standard errors of the equating coefficients and the synthetic item parameters.

2 Data preparation

Data preparation follows the same steps of the equateIRT package.

Load the package equateMultiple and the data

library("equateMultiple")

## Caricamento del pacchetto richiesto: equateIRT

data("data2pl", package = "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

lplan<-linkp(mods = mods)
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   20

3 Multiple equating coefficients

Estimation of the equating coefficients using the multiple mean-mean method. Form 1 is the base form.

eqMM <- multiec(mods = mods, 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  0.84001 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.

eqMGM <- multiec(mods = mods, 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  0.83813 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.

eqIRF<-multiec(mods = mods, 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  0.83588 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.

eqMGM <- multiec(mods = mods, method = "mean-gmean", se = FALSE)

## Computation of equating coefficients  .  .  .  .

eqIRF<-multiec(mods = mods, 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  0.83588 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.

eqTRF<-multiec(mods = mods, 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  0.83636 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.

eqLIK <- multiec(mods = mods, method = "lik")

## Computation of equating coefficients  .  .  .  .   
## Computation of standard errors  .  .  .  .

summary(eqLIK)

## 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.

eqLIK <- multiec(mods = mods, method = "lik", base = 5)

## Computation of equating coefficients  .  .  .  .   
## Computation of standard errors  .  .  .  .

summary(eqLIK)

## 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

4 Synthetic item parameters

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).

eqIRF$as

##  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

eqIRF$bs

##   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

eqIRF$se.as

##  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

eqIRF$se.bs

##  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

5 Equated scores

Equated 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: 0

round(scTSE,3)

##     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

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         -0.025         -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

The R Package equateMultiple: A Tutorial

1 Functionalities

2 Data preparation

3 Multiple equating coefficients

4 Synthetic item parameters

5 Equated scores