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 )

Value

The output of sim.gdina is a list with following entries:

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

The function sim.gdina.prepare possesses the following values as output in a list: delta, necc.attr, Aj and Mj.

Arguments

n: 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
Mj: Design matrix, see gdina
Aj: Design matrix, see gdina
necc.attr: List with \(J\) entries containing necessary attributes for each item

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 <- CDM::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 <- CDM::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

if (FALSE) {
#############################################################################
# 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 <- CDM::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 <- CDM::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 <- CDM::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
#############################################################################

dat <- CDM::data.ecpe$data
q.matrix <- CDM::data.ecpe$q.matrix

#***
# (1) estimate reduced RUM model
mod <- CDM::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 <- CDM::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 <- CDM::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
}