sirt (version 1.5-0)

f1d.irt: Functional Unidimensional Item Response Model

Description

Estimates the functional unidimensional item response model for dichotomous data (Ip et al., 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 discrminations (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.

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.

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.

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.

See Also

For estimation of bifactor models and Green-Yang reliability based on tetrachoric correlations see greenyang.reliability. For estimation of bifactor models based on marginal maximum likelihood (i.e. full information maximum likelihood) see the tam.fa function in the TAM package.

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 <- f1d.irt( dat =dat , nfactors=3)

#++ (2) plot results
     par(mfrow=c(1,2))
library(MASS)     
# 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 )
abline( MASS::rlm(mod1$item$di.ast ~ mod1$item$di0) , col=3 , lty=3 )
# 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 )
abline( MASS::rlm(mod1$item$ai.ast ~ mod1$item$ai0) , col=3 , lty=3 )
     par(mfrow=c(1,1))

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

#****
# 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 <- 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 <- 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 <- f1d.irt( A = mod1$B[,2,] , intercept=intercept , Sigma= mod1$variance)
fumod1$item