# 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)``````

``````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(``