equateIRT (version 2.0)

modIRT: Estimated Coefficients and Covariance Matrix of IRT Models

Description

Creates an object of the class modIRT containing estimated coefficients and covariance matrices of IRT models. Rasch, one-parameter logistic, two-parameter logistic and three-parameter logistic models are included.

Usage

modIRT(coef, var = NULL, names = NULL, ltparam = TRUE, lparam = TRUE, 
	display = TRUE, digits = 2)

Arguments

coef

list of matrices (one for each form) containing item parameter estimates. Guessing, difficulty and discrimination parameters should strictly be given in this order and they are contained in different columns of the matrix. The name of the rows of each matrix should be the name of the item.

var

list of matrices (one for each form) containing the covariance matrix of item parameter estimates. They should be given in the same order of coefficients.

names

character vector containing the names of the forms. This should have the same length of coef and var. If NULL, the names of the forms are assigned by function modIRT.

ltparam

logical; if TRUE the latent trait parameterization is used for difficulty parameters and the modIRT function performs a transformation of item parameters to return them in the usual IRT parameterization. Set to FALSE to avoid transformations. See below for more details.

lparam

logical; if TRUE the logistic parameterization is used for guessing parameters and the modIRT function performs a transformation of item parameters to return them in the usual IRT parameterization. Set to FALSE to avoid transformations. See below for more details.

display

logical; if TRUE coefficients and standard errors are printed.

digits

integer indicating the number of decimal places to be used if display is TRUE.

Value

An object of class modIRT consisting in a list with length equal to the number of forms containing lists with components

coefficients

item parameter estimates.

var

covariance matrix of item parameter estimates.

itmp

number of item parameters of the IRT model. This is 1 for the Rasch model, 2 for the one-parameter logistic model, 2 for the two-parameter logistic model and 3 for the three-parameter logistic model.

Details

ltparam and lparam refers the the parameterization used by the software used to estimate item parameters. The R package ltm, and the programs IRTPRO and flexMIRT use these parameterizations. If ltparam is TRUE the latent trait parameterization is used. Under this parameterization, the three-parameter logistic model is as follows $$\pi_i = c_i + (1 - c_i) \frac{\exp(\beta_{1i} + \beta_{2i} z)}{1 + \exp(\beta_{1i} + \beta_{2i} z)},$$ where $\pi_i$ denotes the conditional probability of responding correctly to the $i$th item given $z$, $c_i$ denotes the guessing parameter, $\beta_{1i}$ is the easiness parameter, $\beta_{2i}$ is the discrimination parameter, and $z$ denotes the latent ability. The two-parameter logistic model, the one-parameter logistic model and the Rasch model present the same formulation. 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 $\beta_{2i}$ costant across items, while the Rasch model can be obtained by setting $c_i$ equal to zero and $\beta_{2i}$ equal to 1. If lparam is TRUE the guessing parameters are given under this parameterization $$c_i = \frac{\exp(c_i^*)}{1+\exp(c_i^*)}. $$

The modIRT function returns parameter estimates under the usual IRT parameterization, that 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 $D a_i = \beta_{2i}$, $b_i = -\beta_{1i}/\beta_{2i}$ and $\theta = z$. If ltparam or lparam are TRUE, the covariance matrix is calculated using the delta method. If item parameters are already given under the usual IRT parameterization, arguments ltparam and lparam should be set to FALSE.

References

Battauz, M. (2015). equateIRT: An R Package for IRT Test Equating. Journal of Statistical Software, 68, 1--22. Bartholomew, D., Knott, M. and Moustaki, I. (2011) Latent Variable Models and Factor Analysis: a Unified Approach, 3rd ed. Wiley. Rizopoulos, D. (2006). ltm: an R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1--25.

See Also

direc, import.ltm

Examples

# three-parameter logistic model
data(est3pl)
test <- paste("test", 1:5, sep = "")
mod3pl <- modIRT(coef = est3pl$coef, var = est3pl$var, names = test, display = FALSE)

# two-parameter logistic model
data(est2pl)
test <- paste("test", 1:5, sep = "")
mod2pl <- modIRT(coef = est2pl$coef, var = est2pl$var, names = test, display = FALSE)

# Rasch model
data(estrasch)
test <- paste("test", 1:5, sep = "")
modrasch <- modIRT(coef = estrasch$coef, var = estrasch$var, names = test, 
	display = FALSE)

# one-parameter logistic model imported from the R package ltm
library(ltm)
mod1pl <- rasch(LSAT)
summary(mod1pl)
est.mod1pl <- import.ltm(mod1pl)
mod1pl.ltm <- modIRT(coef = list(est.mod1pl$coef), var = list(est.mod1pl$var), digits = 4)