direc: Direct Equating Coefficients

Description

Calculates direct equating coefficients and standard errors using IRT methods.

Usage

direc(mods, which, mod1, mod2, method = "mean-mean", suff1 = ".1", suff2 = ".2",
   D = 1, quadrature = TRUE, nq = 30, items.select = NULL)

Value

An object of class eqc with components

tab1: item parameters of the first form.
tab2: item parameters of the second form.
tab: Data frame containing item names (Item), item parameters of the first form (e.g. test1), item parameters of the second form (e.g. test2), and item parameters of the first form converted in the scale of the second form (e.g. test1.as.test2).
var12: covariance matrix of item parameters of the first and the second form (only items used for equating).
varFull: list of covariance matrices of the item parameters (one matrix for each form)
partial: partial derivatives of equating coefficients A and B with respect to item parameters.
A: equating coefficient A.
B: equating coefficient B.
varAB: covariance matrix of the equating coefficients.
commonitem: list of length 2 containing the names of common item parameters and the names of the common items selected for equating.
suffixes: suffixes used to identify the forms.
ni: number of common items.
nis: number of common items selected for equating.
forms: names of equated forms.
method: the equating method used.
itmp: number of item parameters of the IRT model.

Arguments

mods: an object of the class modIRT containing item parameter coefficients and their covariance matrix of the forms to be equated.
which: which forms to equate. Can be specified by name or number.
mod1: deprecated; please use mods instead. An object of the class modIRT containing item parameter coefficients and their covariance matrix of the first form.
mod2: deprecated; please use mods instead. An object of the class modIRT containing item parameter coefficients and their covariance matrix of the second form.
method: the equating method to be used. This should be one of "mean-mean", "mean-gmean", "mean-sigma", "Haebara" or "Stocking-Lord".
suff1: suffix to identify the first form to be equated.
suff2: suffix to identify the second form to be equated.
D: constant D of the IRT model used to estimate item parameters. See below for more details.
quadrature: logical; if TRUE the Gauss-Hermite quadrature is used to approximate the integral in the function that is minimized in the Haebara and Stocking-Lord methods. If FALSE the integral is replaced with a sum over 40 equally spaced values ranging from -4 to 4 with an increment of 0.05 and weights equal to one for all values.
nq: number of quadrature points used for the Gauss-Hermite quadrature if quadrature is TRUE.
items.select: optional character vector including the names of the items to use for equating.

Author

Michela Battauz

Details

Equating coefficients perform the conversion from the scale of the first form to the scale of the second form.

In the three-parameter logistic model the probability of a positive response on item $i$ is $$\pi_i = c_i + (1 - c_i) \frac{\exp[D a_i (\theta - b_i)]}{1 + \exp[D a_i (\theta - b_i)]}.$$ where $a_i$ is the item discrimination parameter, $b_i$ is the item difficulty parameter, $c_i$ is the item guessing parameter and $\theta$ is the latent ability. The constant $D$ can be specified using argument D of the direc function. The two-parameter logistic model can be obtained by setting $c_i$ equal to zero, the one-parameter logistic model can be obtained by setting $c_i$ equal to zero and $a_i$ costant across items, while the Rasch model can be obtained by setting $c_i$ equal to zero and $a_i$ equal to 1.

The type of IRT model does not need to be specified as it is obtained from arguments mod1 and mod2.

References

Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. Journal of Statistical Software, 68, 1--22.

Kolen, M.J. and Brennan, R.L. (2014). Test equating, scaling, and linking: methods and practices, 3nd ed., New York: Springer

Ogasawara, H. (2000). Asymptotic standard errors of IRT equating coefficients using moments. Economic Review (Otaru University of Commerce), 51, 1--23.

Ogasawara, H. (2001). Standard Errors of Item Response Theory Equating/Linking by Response Function Methods. Applied Psychological Measurement, 25, 53--67.

Examples

Run this code


# the following code can be used to start from item response data

if (FALSE) {
library("mirt")
data("data2pl")

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)

mlist<- list(m1,m2,m3,m4,m5)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(est.mods = mlist, names = test, display = FALSE)

# two-parameter logistic model
# direct equating coefficients between forms 2 and 3 using the Haebara method
l23 <- direc(mods = mod2pl, which = c(2,3), method = "Haebara")
summary(l23)
}

# ===========================================================================
# the following code uses item parameter estimates previously obtained

# three-parameter logistic model
# direct equating coefficients between forms 1 and 2 using the Stocking-Lord method
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)
l12 <- direc(mods = mod3pl, which = c("test1", "test2"), method = "Stocking-Lord")
summary(l12)

# two-parameter logistic model
# direct equating coefficients between forms 1 and 5 using the Haebara method
data(est2pl)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)
l15 <- direc(mods = mod2pl, which = c(1,5), method = "Haebara")
summary(l15)

# Rasch model
# direct equating coefficients between forms 5 and 4 using the mean-mean method
data(estrasch)
test <- paste("test", 1:5, sep = "")
modrasch <- modIRT(coef = estrasch$coef, var = estrasch$var, names = test, 
	display = FALSE)
l54 <- direc(mods = modrasch, which = c(5,4), method = "mean-mean")
summary(l54)

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