Marginal Item Parameters from a Testlet (Bifactor) Model

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::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

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).

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).

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.

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 <- sirt::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

if (FALSE) {
#############################################################################
# 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 <- sirt::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 <- sirt::testlet.marginalized( mod2 )
res2
}