sirt (version 1.5-0)

testlet.marginalized: Marginal Item Parameters from a Testlet (Bifactor) Model

Description

This function computes marginal item parameters of a general factor if item parameters from a testlet (bifactor) model are provided as an input (see Details).

Usage

testlet.marginalized(tam.fa.obj=NULL ,a1=NULL, d1=NULL, testlet=NULL, 
      a.testlet=NULL, var.testlet=NULL)

Arguments

tam.fa.obj

Optional object of class tam.fa generated by tam.fa from the TAM package.

a1

Vector of item discriminations of general factor

d1

Vector of item intercepts of general factor

testlet

Integer vector of testlet (bifactor) identifiers (must be integers between 1 to $T$).

a.testlet

Vector of testlet (bifactor) item discriminations

var.testlet

Vector of testlet (bifactor) variances

Value

• A data frame containing all input item parameters and marginal item intercept $d_i^\ast$ (d1_marg) and marginal item slope $a_i^\ast$ (a1_marg).

Details

A testlet (bifactor) model is assumed to be estimated: $$P(X_{pit} = 1 | \theta_{p} , u_{pt} ) = invlogit( a_{i1} \theta_p + a_t u_{pt} - d_{i} )$$ with $Var( u_{pt} )=\sigma_t^2$. This multidimensional item response model with locally independent items is equivalent to a unidimensional IRT model with locally dependent items (Ip, 2010). Marginal item parameters $a_i^\ast$ and $d_i^\ast$ are obtained according to the response equation $$P(X_{pit} = 1 | \theta_{p}^\ast ) = invlogit( a_{i}^\ast \theta_p^\ast - d_{i}^\ast )$$ Calculation details can be found in Ip (2010).

References

Ip, E. H. (2010). Empirically indistinguishable multidimensional IRT and locally dependent unidimensional item response models. British Journal of Mathematical and Statistical Psychology, 63, 395-416.

See Also

For estimating a testlet (bifactor) model see tam.fa (TAM).

Examples

#############################################################################
# EXAMPLE 1: Small numeric example for Rasch testlet model
#############################################################################

# Rasch testlet model with 9 items contained into 3 testlets
# the third testlet has essentially no dependence and therefore
# no testlet variance
testlet <- rep( 1:3 , each=3 )
a1 <- rep(1 , 9 )   # item slopes first dimension
d1 <- rep( c(-1.25,0,1.5) , 3 ) # item intercepts
a.testlet <- rep( 1 , 9 )  # item slopes testlets
var.testlet <- c( .8 , .2 , 0 )  # testlet variances

# apply function
res <- testlet.marginalized( a1=a1 , d1=d1 , testlet=testlet , 
            a.testlet=a.testlet , var.testlet=var.testlet )
round( res , 2 )
  ##    item testlet a1    d1 a.testlet var.testlet a1_marg d1_marg
  ##  1    1       1  1 -1.25         1         0.8    0.89   -1.11
  ##  2    2       1  1  0.00         1         0.8    0.89    0.00
  ##  3    3       1  1  1.50         1         0.8    0.89    1.33
  ##  4    4       2  1 -1.25         1         0.2    0.97   -1.21
  ##  5    5       2  1  0.00         1         0.2    0.97    0.00
  ##  6    6       2  1  1.50         1         0.2    0.97    1.45
  ##  7    7       3  1 -1.25         1         0.0    1.00   -1.25
  ##  8    8       3  1  0.00         1         0.0    1.00    0.00
  ##  9    9       3  1  1.50         1         0.0    1.00    1.50

#############################################################################
# EXAMPLE 2: Dataset reading
#############################################################################

library(TAM)
data(data.read)
resp <- data.read
maxiter <-  100

# Model 1: Rasch testlet model with 3 testlets
dims <- substring( colnames(resp),1,1 )  # define dimensions
mod1 <- TAM::tam.fa( resp=resp , irtmodel="bifactor1" , dims=dims , 
               control=list(maxiter=maxiter) )
# marginal item parameters
res1 <- testlet.marginalized( mod1 )

#***
# Model 2: estimate bifactor model but assume that items 3 and 5 do not load on
#           specific factors
dims1 <- dims
dims1[c(3,5)] <- NA
mod2 <- TAM::tam.fa( resp=resp , irtmodel="bifactor2" , dims=dims1 , 
              control=list(maxiter=maxiter) ) 
res2 <- testlet.marginalized( mod2 )
res2