LSMjml (version 0.6.0)

LSMsim: Simulating Data according to the Latent Space Item Response Model

Description

This function simulates data according to the Latent Space Item Response Model (LSIRM) with an R dimensional latent space and binary and/or ordinal item scores.

Usage

LSMsim(N, nit, ndim_z, nc=NULL, theta=NULL, b=NULL, z=NULL, w=NULL, gamma=NULL)

Value

An list containing:

X

the simulated data

par

a list containing the true parameter values, including wt and zt, the rotated matrices of \(w_{ir}\) and \(z_{pr}\) parameters.

Arguments

N

Sample size

nit

Number of items

ndim_z

Number of dimensions of the latent space, R.

nc

Vector of length nit containng the number of response categories for each item. If NULL (the default) all items are simulated to be binary

theta

N-dimensional vector of true person intercepts \(\theta_p\), if NULL these are drawn from a standard normal distribution

b

nit-dimensional vector of true item intercepts \(b_p\), if NULL these are drawn from a uniform distribution

z

N by ndim_z matrix of true latent space person coordinates \(z_{pr}\), if NULL these are drawn from a standard normal distribution

w

nit by ndim_z matrix of true latent space item coordinates \(w_{ir}\), if NULL these are drawn from a standard normal distribution

gamma

a weight parameter for the Euclidean distances (see details), if NULL gamma=1

Author

Dylan Molenaar

Details

Data is simulated according to the original LSIRM by Jeon et al. (2021): $$\text{logit}(E(X_{pi})) = \theta_p + b_i - \gamma(\Sigma_{r=1}^{R} (z_{pr}-w_{ir})^2)^{1/2}$$ In LSMfit, \(\gamma\) is fixed to one as it is not identified in a joint maximum likelihood framework (see Molenaar & Jeon, submitted). However, for data simulation, \(\gamma\) can be used to change the strength of the effect of \(z_{pr}\) and \(w_{ir}\).

References

Jeon, M., Jin, I. H., Schweinberger, M., & Baugh, S. (2021). Mapping unobserved item–respondent interactions: A latent space item response model with interaction map. Psychometrika, 86(2), 378-403. doi:10.1007/s11336-021-09762-5

Molenaar, D., & Jeon, M.J. (in press). Joint maximum likelihood estimation of latent space item response models. Psychometrika.

See Also

LSMfit for fitting LSIRM models using joint maximum likelihood.

Examples

 # data sim with 1000 subjects and 20 items according to 2 dimensional
 # latent space model (R=2) with both binary and ordinal items
 set.seed(1111)
 N=1000
 nit=20
 ndim_z=2
 nc=sample(c(2,3,5),nit,replace=TRUE)    # mix of 2, 3, and 5 point scales
 dat_obj=LSMsim(N,nit,ndim_z,nc=nc)
 X=dat_obj$X
 zt=dat_obj$par$zt   # rotated z
 wt=dat_obj$par$wt   # rotated w

 #fit model
 results=LSMfit(X,2)

 #plot the parameter recovery results
 oldpar=par(mfrow=c(2,2))

 s_p=sign(cor(results$z,zt))          # to correct for sign switches in the plots
 s_i=sign(cor(results$w,wt))

 plot(s_p[1,1]*zt[,1],results$z[,1]); abline(0,1)
 plot(s_p[2,2]*zt[,2],results$z[,2]); abline(0,1)
 plot(s_i[1,1]*wt[,1],results$w[,1]); abline(0,1)
 plot(s_i[2,2]*wt[,2],results$w[,2]); abline(0,1)

 par(oldpar)