IRT.linearCFA: Linear Approximation of a Confirmatory Factor Analysis

In TAM: Test Analysis Modules

Linear Approximation of a Confirmatory Factor Analysis

Description

This function approximates a fitted item response model by a linear confirmatory factor analysis. I.e., given item response functions, the expectation E(X_i | \theta_1, \ldots, \theta_D) is linearly approximated by a_{i1} \theta _1 + \ldots + a_{iD} \theta_D. See Vermunt and Magidson (2005) for details.

Usage

IRT.linearCFA( object, group=1)

## S3 method for class 'IRT.linearCFA'
summary(object,  ...)

Arguments

object	Fitted item response model for which the IRT.expectedCounts method is defined.
group	Group identifier which defines the selected group.
...	Further arguments to be passed.

Value

A list with following entries

loadings	Data frame with factor loadings. Mlat and SDlat denote the model-implied item mean and standard deviation. The values ResidVar and h2 denote residual variances and item communality.
stand.loadings	Data frame with standardized factor loadings.
M.trait	Mean of factors
SD.trait	Standard deviations of factors

References

Vermunt, J. K., & Magidson, J. (2005). Factor Analysis with categorical indicators: A comparison between traditional and latent class approaches. In A. Van der Ark, M.A. Croon & K. Sijtsma (Eds.), New Developments in Categorical Data Analysis for the Social and Behavioral Sciences (pp. 41-62). Mahwah: Erlbaum

See Also

See tam.fa for confirmatory factor analysis in TAM.

Examples

## Not run: 
library(lavaan)

#############################################################################
# EXAMPLE 1: Two-dimensional confirmatory factor analysis data.Students
#############################################################################

data(data.Students, package="CDM")
# select variables
vars <- scan(nlines=1, what="character")
    sc1 sc2 sc3 sc4 mj1 mj2 mj3 mj4
dat <- data.Students[, vars]

# define Q-matrix
Q <- matrix( 0, nrow=8, ncol=2 )
Q[1:4,1] <- Q[5:8,2] <- 1

#*** Model 1: Two-dimensional 2PL model
mod1 <- TAM::tam.mml.2pl( dat, Q=Q, control=list( nodes=seq(-4,4,len=12) ) )
summary(mod1)

# linear approximation CFA
cfa1 <- TAM::IRT.linearCFA(mod1)
summary(cfa1)

# linear CFA in lavaan package
lavmodel <- "
    sc=~ sc1+sc2+sc3+sc4
    mj=~ mj1+mj2+mj3+mj4
    sc1 ~ 1
    sc ~~ mj
    "
mod1b <- lavaan::sem( lavmodel, data=dat, missing="fiml", std.lv=TRUE)
summary(mod1b, standardized=TRUE, fit.measures=TRUE )

#############################################################################
# EXAMPLE 2: Unidimensional confirmatory factor analysis data.Students
#############################################################################

data(data.Students, package="CDM")
# select variables
vars <- scan(nlines=1, what="character")
    sc1 sc2 sc3 sc4
dat <- data.Students[, vars]

#*** Model 1: 2PL model
mod1 <- TAM::tam.mml.2pl( dat )
summary(mod1)

# linear approximation CFA
cfa1 <- TAM::IRT.linearCFA(mod1)
summary(cfa1)

# linear CFA
lavmodel <- "
    sc=~ sc1+sc2+sc3+sc4
    "
mod1b <- lavaan::sem( lavmodel, data=dat, missing="fiml", std.lv=TRUE)
summary(mod1b, standardized=TRUE, fit.measures=TRUE )

## End(Not run)

TAM documentation built on Aug. 29, 2025, 5:32 p.m.