MCMC Estimation of the Two-Parameter Normal Ogive Item Response Model

This function estimates the Two-Parameter normal ogive item response model by MCMC sampling (Johnson & Albert, 1999, p. 195ff.).

Usage

mcmc.2pno(dat, weights=NULL, burnin=500, iter=1000, N.sampvalues=1000,
      progress.iter=50, save.theta=FALSE)

Arguments

dat: Data frame with dichotomous item responses
weights: An optional vector with student sample weights
burnin: Number of burnin iterations
iter: Total number of iterations
N.sampvalues: Maximum number of sampled values to save
progress.iter: Display progress every progress.iter-th iteration. If no progress display is wanted, then choose progress.iter larger than iter.
save.theta: Should theta values be saved?

Details

The two-parameter normal ogive item response model with a probit link function is defined by $$ P(X_{pi}=1 | \theta_p )=\Phi ( a_i \theta_p - b_i ) \quad, \quad \theta_p \sim N(0,1) $$ Note that in this implementation non-informative priors for the item parameters are chosen (Johnson & Albert, 1999, p. 195ff.).

Value

A list of class mcmc.sirt with following entries:

mcmcobj: Object of class mcmc.list
summary.mcmcobj: Summary of the mcmcobj object. In this summary the Rhat statistic and the mode estimate MAP is included. The variable PercSEratio indicates the proportion of the Monte Carlo standard error in relation to the total standard deviation of the posterior distribution.
burnin: Number of burnin iterations
iter: Total number of iterations
a.chain: Sampled values of $a_i$ parameters
b.chain: Sampled values of $b_i$ parameters
theta.chain: Sampled values of $\theta_p$ parameters
deviance.chain: Sampled values of Deviance values
EAP.rel: EAP reliability
person: Data frame with EAP person parameter estimates for $\theta_p$ and their corresponding posterior standard deviations
dat: Used data frame
weights: Used student weights
...: Further values

References

Johnson, V. E., & Albert, J. H. (1999). Ordinal Data Modeling. New York: Springer.

Examples

if (FALSE) {
#############################################################################
# EXAMPLE 1: Dataset Reading
#############################################################################
data(data.read)
# estimate 2PNO with MCMC with 3000 iterations and 500 burn-in iterations
mod <- sirt::mcmc.2pno( dat=data.read, iter=3000, burnin=500 )
# plot MCMC chains
plot( mod$mcmcobj, ask=TRUE )
# write sampled chains into codafile
mcmclist2coda( mod$mcmcobj, name="dataread_2pno" )
# summary
summary(mod)

#############################################################################
# EXAMPLE 2
#############################################################################
# simulate data
N <- 1000
I <- 10
b <- seq( -1.5, 1.5, len=I )
a <- rep( c(1,2), I/2 )
theta1 <- stats::rnorm(N)
dat <- sirt::sim.raschtype( theta=theta1, fixed.a=a, b=b )

#***
# Model 1: estimate model without weights
mod1 <- sirt::mcmc.2pno( dat, iter=1500, burnin=500)
mod1$summary.mcmcobj
plot( mod1$mcmcobj, ask=TRUE )

#***
# Model 2: estimate model with weights
# define weights
weights <- c( rep( 5, N/4 ), rep( .2, 3/4*N ) )
mod2 <- sirt::mcmc.2pno( dat, weights=weights, iter=1500, burnin=500)
mod1$summary.mcmcobj
}