f1d.irt: Functional Unidimensional Item Response Model
In sirt: Supplementary Item Response Theory Models

f1d.irt

R Documentation

Functional Unidimensional Item Response Model

Description

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 )

## Not run: 
#****
# 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

## End(Not run)

sirt documentation built on May 29, 2024, 8:43 a.m.