vcov: Asymptotic Covariance Matrix, Standard Errors and Confidence Intervals

Description

Computes the asymptotic covariance matrix for din objects. The covariance matrix is computed using the empirical cross-product approach (see Paek & Cai, 2014).

In addition, an S3 method IRT.se is defined which produces an extended output including vcov and confint.

Usage

# S3 method for din
vcov(object, extended=FALSE, infomat=FALSE,ind.item.skillprobs=TRUE,
       ind.item=FALSE, diagcov=FALSE, h=.001,...)
# S3 method for din
confint(object, parm, level=.95, extended=FALSE,
      ind.item.skillprobs=TRUE, ind.item=FALSE, diagcov=FALSE, h=.001, ... )
IRT.se(object, ...)
# S3 method for din
IRT.se( object, extended=FALSE, parm=NULL, level=.95,
      infomat=FALSE, ind.item.skillprobs=TRUE, ind.item=FALSE,
      diagcov=FALSE, h=.001, ... )

Value

coef: A vector of parameters.

vcov: A covariance matrix. The corresponding coefficients can be extracted as the attribute coef from this object.

IRT.se: A data frame containing coefficients, standard errors and confidence intervals for all parameters.

Arguments

object: An object inheriting from class din.
extended: An optional logical indicating whether the covariance matrix should be calculated for an extended set of parameters (estimated and derived parameters).
infomat: An optional logical indicating whether the information matrix instead of the covariance matrix should be the output.
ind.item.skillprobs: Optional logical indicating whether the covariance between item parameters and skill class probabilities are assumed to be zero.
ind.item: Optional logical indicating whether covariances of item parameters between different items are zero.
diagcov: Optional logical indicating whether all covariances between estimated parameters are set to zero.
h: Parameter used for numerical differentiation for computing the derivative of the log-likelihood function.
parm: Vector of parameters. If it is missing, then for all estimated parameters a confidence interval is calculated.
level: Confidence level
...: Additional arguments to be passed.

References

Paek, I., & Cai, L. (2014). A comparison of item parameter standard error estimation procedures for unidimensional and multidimensional item response theory modeling. Educational and Psychological Measurement, 74(1), 58-76.

Examples

Run this code

if (FALSE) {
#############################################################################
# EXAMPLE 1: DINA model sim.dina
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

dat <- sim.dina
q.matrix <- sim.qmatrix

#****** Model 1: DINA Model
mod1 <- CDM::din( dat, q.matrix=q.matrix, rule="DINA")
# look into parameter table of the model
mod1$partable
# covariance matrix
covmat1 <- vcov(mod1 )
# extract coefficients
coef(mod1)
# extract standard errors
sqrt( diag( covmat1))
# compute confidence intervals
confint( mod1, level=.90 )
# output table with standard errors
IRT.se( mod1, extended=TRUE )

#****** Model 2: Constrained DINA Model

# fix some slipping parameters
constraint.slip <- cbind( c(2,3,5), c(.15,.20,.25) )
# set some skill class probabilities to zero
zeroprob.skillclasses <- c(2,4)
# estimate model
mod2 <- CDM::din( dat, q.matrix=q.matrix, guess.equal=TRUE,
     constraint.slip=constraint.slip, zeroprob.skillclasses=zeroprob.skillclasses)
# parameter table
mod2$partable
# freely estimated coefficients
coef(mod2)
# covariance matrix (estimated parameters)
vmod2a <- vcov(mod2)
sqrt( diag( vmod2a))        # standard errors
colnames( vmod2a )
names( attr( vmod2a, "coef") )    # extract coefficients

# covariance matrix (more parameters, extended=TRUE)
vmod2b <- vcov(mod2, extended=TRUE)
sqrt( diag( vmod2b))
attr( vmod2b, "coef")
# attach standard errors to parameter table
partable2 <- mod2$partable
partable2 <- partable2[ ! duplicated( partable2$parnames ), ]
partable2 <- data.frame( partable2, "se"=sqrt( diag( vmod2b)) )
partable2

# confidence interval for parameter "skill1" which is not in the model
#   cannot be calculated!
confint(mod2, parm=c( "skill1", "all_guess" ) )
# confidence interval for only some parameters
confint(mod2, parm=paste0("prob_skill", 1:3 ) )

# compute only information matrix
infomod2 <- vcov(mod2, infomat=TRUE)
}