# 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 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 1.091369``````
``estm1\$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 0.0012407430``````

Create 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 -0.00716265``````

``````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   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, 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 0.024026``````

Estimation 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 0.024001``````

Estimation 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 0.023555``````

Estimation of the equating coefficients using the multiple item response function method. The initial values are the estimates obtained with the multiple