Functional Unidimensional Item Response Model

Estimates the functional unidimensional item response model for dichotomous data (Ip, Molenberghs, Chen, Goegebeur & De Boeck, 2013). Either the IRT model is estimated using a probit link and employing tetrachoric correlations or item discriminations and intercepts of a pre-estimated multidimensional IRT model are provided as input.

Usage

f1d.irt(dat=NULL, nnormal=1000, nfactors=3, A=NULL, intercept=NULL,
    mu=NULL, Sigma=NULL, maxiter=100, conv=10^(-5), progress=TRUE)

Arguments

dat: Data frame with dichotomous item responses
nnormal: Number of $\theta_p$ grid points for approximating the normal distribution
nfactors: Number of dimensions to be estimated
A: Matrix of item discriminations (if the IRT model is already estimated)
intercept: Vector of item intercepts (if the IRT model is already estimated)
mu: Vector of estimated means. In the default it is assumed that all means are zero.
Sigma: Estimated covariance matrix. In the default it is the identity matrix.
maxiter: Maximum number of iterations
conv: Convergence criterion
progress: Display progress? The default is TRUE.

Details

The functional unidimensional item response model (F1D model) for dichotomous item responses is based on a multidimensional model with a link function $g$ (probit or logit): $$ P( X_{pi}=1 | \bold{\theta}_p )= g( \sum_d a_{id} \theta_{pd} - d_i ) $$ It is assumed that $\bold{\theta}_p$ is multivariate normally distribution with a zero mean vector and identity covariance matrix.

The F1D model estimates unidimensional item response functions such that $$ P( X_{pi}=1 | \theta_p^\ast ) \approx g \left( a_{i}^\ast \theta_{p}^\ast - d_i^\ast \right) $$ The optimization function $F$ minimizes the deviations of the approximation equations $$ a_{i}^\ast \theta_{p}^\ast - d_i^\ast \approx \sum_d a_{id} \theta_{pd} - d_i $$ The optimization function $F$ is defined by $$ F( \{ a_i^\ast, d_i^\ast \}_i, \{ \theta_p^\ast \}_p )= \sum_p \sum_i w_p ( a_{id} \theta_{pd} - d_i- a_{i}^\ast \theta_{p}^\ast + d_i^\ast )^2 \rightarrow Min! $$ All items $i$ are equally weighted whereas the ability distribution of persons $p$ are weighted according to the multivariate normal distribution (using weights $w_p$). The estimation is conducted using an alternating least squares algorithm (see Ip et al. 2013 for a different algorithm). The ability distribution $\theta_p^\ast$ of the functional unidimensional model is assumed to be standardized, i.e. does have a zero mean and a standard deviation of one.

Value

A list with following entries:

item: Data frame with estimated item parameters: Item intercepts for the functional unidimensional $a_{i}^\ast$ (ai.ast) and the ('ordinary') unidimensional (ai0) item response model. The same holds for item intercepts $d_{i}^\ast$ (di.ast and di0 respectively).
person: Data frame with estimated $\theta_p^\ast$ distribution. Locations are theta.ast with corresponding probabilities in wgt.
A: Estimated or provided item discriminations
intercept: Estimated or provided intercepts
dat: Used dataset
tetra: Object generated by tetrachoric2 if dat is specified as input. This list entry is useful for applying greenyang.reliability.

References

Ip, E. H., Molenberghs, G., Chen, S. H., Goegebeur, Y., & De Boeck, P. (2013). Functionally unidimensional item response models for multivariate binary data. Multivariate Behavioral Research, 48, 534-562.

Examples

#############################################################################
# EXAMPLE 1: Dataset Mathematics data.math | Exploratory multidimensional model
#############################################################################
data(data.math)
dat <- ( data.math$data )[, -c(1,2) ] # select Mathematics items

#****
# Model 1: Functional unidimensional model based on original data

#++ (1) estimate model with 3 factors
mod1 <- sirt::f1d.irt( dat=dat, nfactors=3)

#++ (2) plot results
     par(mfrow=c(1,2))
# Intercepts
plot( mod1$item$di0, mod1$item$di.ast, pch=16, main="Item Intercepts",
        xlab=expression( paste( d[i], " (Unidimensional Model)" )),
        ylab=expression( paste( d[i], " (Functional Unidimensional Model)" )))
abline( lm(mod1$item$di.ast ~ mod1$item$di0), col=2, lty=2 )
# Discriminations
plot( mod1$item$ai0, mod1$item$ai.ast, pch=16, main="Item Discriminations",
        xlab=expression( paste( a[i], " (Unidimensional Model)" )),
        ylab=expression( paste( a[i], " (Functional Unidimensional Model)" )))
abline( lm(mod1$item$ai.ast ~ mod1$item$ai0), col=2, lty=2 )
     par(mfrow=c(1,1))

#++ (3) estimate bifactor model and Green-Yang reliability
gy1 <- sirt::greenyang.reliability( mod1$tetra, nfactors=3 )

if (FALSE) {
#****
# Model 2: Functional unidimensional model based on estimated multidimensional
#          item response model

#++ (1) estimate 2-dimensional exploratory factor analysis with 'smirt'
I <- ncol(dat)
Q <- matrix( 1, I,2 )
Q[1,2] <- 0
variance.fixed <- cbind( 1,2,0 )
mod2a <- sirt::smirt( dat, Qmatrix=Q, irtmodel="comp", est.a="2PL",
                variance.fixed=variance.fixed, maxiter=50)
#++ (2) input estimated discriminations and intercepts for
#       functional unidimensional model
mod2b <- sirt::f1d.irt( A=mod2a$a, intercept=mod2a$b )

#############################################################################
# EXAMPLE 2: Dataset Mathematics data.math | Confirmatory multidimensional model
#############################################################################

data(data.math)
library(TAM)

# dataset
dat <- data.math$data
dat <- dat[, grep("M", colnames(dat) ) ]
# extract item informations
iteminfo <- data.math$item
I <- ncol(dat)
# define Q-matrix
Q <- matrix( 0, nrow=I, ncol=3 )
Q[ grep( "arith", iteminfo$domain ), 1 ] <- 1
Q[ grep( "Meas", iteminfo$domain ), 2 ] <- 1
Q[ grep( "geom", iteminfo$domain ), 3 ] <- 1

# fit three-dimensional model in TAM
mod1 <- TAM::tam.mml.2pl(  dat, Q=Q, control=list(maxiter=40, snodes=1000) )
summary(mod1)

# specify functional unidimensional model
intercept <- mod1$xsi[, c("xsi") ]
names(intercept) <- rownames(mod1$xsi)
fumod1 <- sirt::f1d.irt( A=mod1$B[,2,], intercept=intercept, Sigma=mod1$variance)
fumod1$item
}