Simulate from Generalized Logistic Item Response Model

This function simulates dichotomous item responses from a generalized logistic item response model (Stukel, 1988). The four-parameter logistic item response model (Loken & Rulison, 2010) is a special case. See rasch.mml2 for more details.

Usage

sim.raschtype(theta, b, alpha1=0, alpha2=0, fixed.a=NULL,
    fixed.c=NULL, fixed.d=NULL)

Arguments

theta: Unidimensional ability vector \(\theta\)
b: Vector of item difficulties \(b\)
alpha1: Parameter \(\alpha_1\) in generalized logistic link function
alpha2: Parameter \(\alpha_2\) in generalized logistic link function
fixed.a: Vector of item slopes \(a\)
fixed.c: Vector of lower item asymptotes \(c\)
fixed.d: Vector of lower item asymptotes \(d\)

Details

The class of generalized logistic link functions contain the most important link functions using the specifications (Stukel, 1988):

logistic link function: \(\alpha_1=0\) and \(\alpha_2=0\)
probit link function: \(\alpha_1=0.165\) and \(\alpha_2=0.165\)
loglog link function: \(\alpha_1=-0.037\) and \(\alpha_2=0.62\)
cloglog link function: \(\alpha_1=0.62\) and \(\alpha_2=-0.037\)

See pgenlogis for exact transformation formulas of the mentioned link functions.

Value

Data frame with simulated item responses

References

Loken, E., & Rulison, K. L. (2010). Estimation of a four-parameter item response theory model. British Journal of Mathematical and Statistical Psychology, 63, 509-525.

Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83, 426-431.

Examples

#############################################################################
## EXAMPLE 1: Simulation of data from a Rasch model (alpha_1=alpha_2=0)
#############################################################################

set.seed(9765)
N <- 500    # number of persons
I <- 11     # number of items
b <- seq( -2, 2, length=I )
dat <- sirt::sim.raschtype( stats::rnorm( N ), b )
colnames(dat) <- paste0( "I", 1:I )