equateIRT (version 2.0)

direc: Direct Equating Coefficients

Description

Calculates direct equating coefficients and standard errors using IRT methods.

Usage

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

Arguments

mod1

an object of the class modIRT containing item parameter coefficients and their covariance matrix of the first form.

mod2

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.

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.

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 1 containing the names of common item parameters.

ni

number of common items.

forms

names of equated forms.

method

the equating method used.

itmp

number of item parameters of the IRT model.

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.

See Also

eqc, itm, modIRT, summary.eqc

Examples

# 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(mod1 = mod3pl[1], mod2 = mod3pl[2], 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(mod1 = mod2pl[1], mod2 = mod2pl[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(mod1 = modrasch[5], mod2 = modrasch[4], method = "mean-mean")
summary(l54)