sim.gdina: Simulation of the GDINA model

Description

The function sim.gdina.prepare creates necessary design matrices Mj, Aj and necc.attr. In most cases, only the list of item parameters delta must be modified by the user when applying the simulation function sim.gdina. The distribution of latent classes $\alpha$ is represented by an underlying multivariate normal distribution $\alpha^\ast$ for which a mean vector thresh.alpha and a covariance matrix cov.alpha must be specified. Alternatively, a matrix of skill classes alpha can be given as an input.

Note that this version of sim.gdina only works for dichotomous attributes.

Usage

sim.gdina(n, q.matrix, delta, link = "identity",  thresh.alpha=NULL,  cov.alpha=NULL, alpha=NULL, Mj, Aj, necc.attr) 
sim.gdina.prepare( q.matrix )

Arguments

Number of persons

q.matrix

Q-matrix (see sim.din)

delta

List with $J$ entries where $J$ is the number of items. Every list element corresponds to the parameter of an item.

link

Link function. Choices are identity (default), logit and log.

thresh.alpha

Vector of thresholds (means) of $\alpha^\ast$

cov.alpha

Covariance matrix of $\alpha^\ast$

alpha

Matrix of skill classes if they should not be simulated

Design matrix, see gdina

necc.attr

List with $J$ entries containing necessary attributes for each item

Value

data: Simulated item responses
alpha: Data frame with simulated attributes
q.matrix: Used Q-matrix
delta: Used delta item parameters
Aj: Design matrices $A_j$
Mj: Design matrices $M_j$
link: Used link function

References

de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179--199.

Examples

Run this code

#############################################################################
# EXAMPLE 1: Simulating the GDINA model
#############################################################################

n <- 50             # number of persons
# define Q-matrix
q.matrix <- matrix(  c(1,1,0 , 0,1,1, 1,0,1, 1,0,0,
    0,0,1, 0,1,0,  1,1,1,  0,1,1, 0,1,1) , ncol=3 , byrow=TRUE)
# thresholds for attributes alpha^\ast
thresh.alpha <- c( .65 , 0 , -.30 )
# covariance matrix for alpha^\ast
cov.alpha <- matrix(1,3,3)
cov.alpha[1,2] <- cov.alpha[2,1] <- .4
cov.alpha[1,3] <- cov.alpha[3,1] <- .6
cov.alpha[3,2] <- cov.alpha[2,3] <- .8

# prepare design matrix by applying sim.gdina.prepare function
rp <- sim.gdina.prepare( q.matrix )
delta <- rp$delta
necc.attr <- rp$necc.attr
Aj <- rp$Aj
Mj <- rp$Mj
# define delta parameters
# intercept - main effects - second order interactions - ...
str(delta)  # => modify the delta parameter list which contains only zeroes as default
##   List of 9
##    $ : num [1:4] 0 0 0 0
##    $ : num [1:4] 0 0 0 0
##    $ : num [1:4] 0 0 0 0
##    $ : num [1:2] 0 0
##    $ : num [1:2] 0 0
##    $ : num [1:2] 0 0
##    $ : num [1:8] 0 0 0 0 0 0 0 0
##    $ : num [1:4] 0 0 0 0
##    $ : num [1:4] 0 0 0 0
delta[[1]] <- c( .2 , .1 , .15 , .4 )
delta[[2]] <- c( .2 , .3 , .3 , -.2 )
delta[[3]] <- c( .2 , .2 , .2 , 0 )
delta[[4]] <- c( .15 , .6 )
delta[[5]] <- c( .1 , .7 )
delta[[6]] <- c( .25 , .65 )
delta[[7]] <- c( .25 , .1 , .1 , .1 , 0 , 0 , 0 , .25 )
delta[[8]] <- c( .2 , 0 , .3 , -.1 )
delta[[9]] <- c( .2 , .2 , 0 , .3 )

#******************************************
# Now, the "real simulation" starts
sim.res <- sim.gdina( n=n, q.matrix =q.matrix, delta=delta, link = "identity", 
                thresh.alpha=thresh.alpha , cov.alpha=cov.alpha ,
                Mj=Mj , Aj=Aj , necc.attr =necc.attr)
# sim.res$data      # simulated data
# sim.res$alpha     # simulated alpha

## Not run: 
# #############################################################################
# # EXAMPLE 2: Simulation based on already estimated GDINA model for data.ecpe
# #############################################################################
# 
# data(data.ecpe)
# dat <- data.ecpe$data
# q.matrix <- data.ecpe$q.matrix
# 
# #***
# # (1) estimate GDINA model
# mod <- gdina( data=dat[,-1] , q.matrix=q.matrix )
# 
# #***
# # (2) simulate data according to GDINA model
# set.seed(977)
# 
# # prepare design matrix by applying sim.gdina.prepare function
# rp <- sim.gdina.prepare( q.matrix )
# necc.attr <- rp$necc.attr
# 
# # number of subjects to be simulated
# n <- 3000
# # simulate attribute patterns
# probs <- mod$attribute.patt$class.prob   # probabilities
# patt <- mod$attribute.patt.splitted      # response patterns
# alpha <- patt[ sample( 1:(length(probs) ) , n , prob=probs , replace=TRUE) , ]
# 
# # simulate data using estimated item parameters
# sim.res <- sim.gdina( n=n, q.matrix =q.matrix, delta=mod$delta, link = "identity", 
#                 alpha=alpha , Mj=mod$Mj , Aj= mod$Aj , necc.attr = rp$necc.attr)               
# # extract data
# dat <- sim.res$data
# 
# #############################################################################
# # EXAMPLE 3: Simulation based on already estimated RRUM model for data.ecpe
# #############################################################################
# 
# data(data.ecpe)
# dat <- data.ecpe$data
# q.matrix <- data.ecpe$q.matrix
# 
# #***
# # (1) estimate reduced RUM model
# mod <- gdina( data=dat[,-1] , q.matrix=q.matrix , rule="RRUM" )
# summary(mod)
# 
# #***
# # (2) simulate data according to RRUM model
# set.seed(977)
# 
# # prepare design matrix by applying sim.gdina.prepare function
# rp <- sim.gdina.prepare( q.matrix )
# necc.attr <- rp$necc.attr
# 
# # number of subjects to be simulated
# n <- 5000
# # simulate attribute patterns
# probs <- mod$attribute.patt$class.prob   # probabilities
# patt <- mod$attribute.patt.splitted      # response patterns
# alpha <- patt[ sample( 1:(length(probs) ) , n , prob=probs , replace=TRUE) , ]
# 
# # simulate data using estimated item parameters
# sim.res <- sim.gdina( n=n, q.matrix =q.matrix, delta=mod$delta, link = mod$link , 
#                 alpha=alpha , Mj=mod$Mj , Aj=mod$Aj , necc.attr = rp$necc.attr)               
# # extract data
# dat <- sim.res$data
# ## End(Not run)