sirt: Supplementary Item Response Theory Models

mcmc.2pno.ml

R Documentation

Random Item Response Model / Multilevel IRT Model

Description

This function enables the estimation of random item models and multilevel (or hierarchical) IRT models (Chaimongkol, Huffer & Kamata, 2007; Fox & Verhagen, 2010; van den Noortgate, de Boeck & Meulders, 2003; Asparouhov & Muthen, 2012; Muthen & Asparouhov, 2013, 2014). Dichotomous response data is supported using a probit link. Normally distributed responses can also be analyzed. See Details for a description of the implemented item response models.

Usage

mcmc.2pno.ml(dat, group, link="logit", est.b.M="h", est.b.Var="n",
    est.a.M="f", est.a.Var="n", burnin=500, iter=1000,
    N.sampvalues=1000, progress.iter=50, prior.sigma2=c(1, 0.4),
    prior.sigma.b=c(1, 1), prior.sigma.a=c(1, 1), prior.omega.b=c(1, 1),
    prior.omega.a=c(1, 0.4), sigma.b.init=.3 )

Arguments

`dat`	Data frame with item responses.
`group`	Vector of group identifiers (e.g. classes, schools or countries)
`link`	Link function. Choices are `"logit"` for dichotomous data and `"normal"` for data under normal distribution assumptions
`est.b.M`	Estimation type of `b_i` parameters: `n` - non-hierarchical prior distribution, i.e. `\omega_b` is set to a very high value and is not estimated `h` - hierarchical prior distribution with estimated distribution parameters `\mu_b` and `\omega_b`
`est.b.Var`	Estimation type of standard deviations of item difficulties `b_i`. `n` – no estimation of the item variance, i.e. `\sigma_{b,i}` is assumed to be zero `i` – item-specific standard deviation of item difficulties `j` – a joint standard deviation of all item difficulties is estimated, i.e. `\sigma_{b,1}=\ldots=\sigma_{b,I}=\sigma_b`
`est.a.M`	Estimation type of `a_i` parameters: `f` - no estimation of item slopes, i.e all item slopes `a_i` are fixed at one `n` - non-hierarchical prior distribution, i.e. `\omega_a=0` `h` - hierarchical prior distribution with estimated distribution parameter `\omega_a`
`est.a.Var`	Estimation type of standard deviations of item slopes `a_i`. `n` – no estimation of the item variance `i` – item-specific standard deviation of item slopes `j` – a joint standard deviation of all item slopes is estimated, i.e. `\sigma_{a,1}=\ldots=\sigma_{a,I}=\sigma_a`
`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`.
`prior.sigma2`	Prior for Level 2 standard deviation `\sigma_{L2}`
`prior.sigma.b`	Priors for item difficulty standard deviations `\sigma_{b,i}`
`prior.sigma.a`	Priors for item difficulty standard deviations `\sigma_{a,i}`
`prior.omega.b`	Prior for `\omega_b`
`prior.omega.a`	Prior for `\omega_a`
`sigma.b.init`	Initial standard deviation for `\sigma_{b,i}` parameters

Details

For dichotomous item responses (link="logit") of persons p in group j on item i, the probability of a correct response is defined as

P( X_{pji}=1 | \theta_{pj} )=\Phi ( a_{ij} \theta_{pj} - b_{ij} )

The ability \theta_{pj} is decomposed into a Level 1 and a Level 2 effect

\theta_{pj}=u_j + e_{pj} \quad, \quad u_j \sim N ( 0, \sigma_{L2}^2 ) \quad, \quad e_{pj} \sim N ( 0, \sigma_{L1}^2 )

In a multilevel IRT model (or a random item model), item parameters are allowed to vary across groups:

b_{ij} \sim N( b_i, \sigma^2_{b,i} ) \quad, \quad a_{ij} \sim N( a_i, \sigma^2_{a,i} )

In a hierarchical IRT model, a hierarchical distribution of the (main) item parameters is assumed

b_{i} \sim N( \mu_b, \omega^2_{b} ) \quad, \quad a_{i} \sim N( 1, \omega^2_{a} )

Note that for identification purposes, the mean of all item slopes a_i is set to one. Using the arguments est.b.M, est.b.Var, est.a.M and est.a.Var defines which variance components should be estimated.

For normally distributed item responses (link="normal"), the model equations remain the same except the item response model which is now written as

X_{pji}=a_{ij} \theta_{pj} - b_{ij} + \varepsilon_{pji} \quad, \quad \varepsilon_{pji} \sim N( 0, \sigma^2_{res,i} )

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.
`ic`	Information criteria (DIC)
`burnin`	Number of burnin iterations
`iter`	Total number of iterations
`theta.chain`	Sampled values of `\theta_{pj}` parameters
`theta.chain`	Sampled values of `u_{j}` parameters
`deviance.chain`	Sampled values of Deviance values
`EAP.rel`	EAP reliability
`person`	Data frame with EAP person parameter estimates for `\theta_pj` and their corresponding posterior standard deviations
`dat`	Used data frame
`...`	Further values

References

Asparouhov, T. & Muthen, B. (2012). General random effect latent variable modeling: Random subjects, items, contexts, and parameters. http://www.statmodel.com/papers_date.shtml.

Chaimongkol, S., Huffer, F. W., & Kamata, A. (2007). An explanatory differential item functioning (DIF) model by the WinBUGS 1.4. Songklanakarin Journal of Science and Technology, 29, 449-458.

Fox, J.-P., & Verhagen, A.-J. (2010). Random item effects modeling for cross-national survey data. In E. Davidov, P. Schmidt, & J. Billiet (Eds.), Cross-cultural Analysis: Methods and Applications (pp. 467-488), London: Routledge Academic.

Muthen, B. & Asparouhov, T. (2013). New methods for the study of measurement invariance with many groups. http://www.statmodel.com/papers_date.shtml

Muthen, B. & Asparouhov, T. (2014). Item response modeling in Mplus: A multi-dimensional, multi-level, and multi-timepoint example. In W. Linden & R. Hambleton (2014). Handbook of item response theory: Models, statistical tools, and applications. http://www.statmodel.com/papers_date.shtml

van den Noortgate, W., De Boeck, P., & Meulders, M. (2003). Cross-classification multilevel logistic models in psychometrics. Journal of Educational and Behavioral Statistics, 28, 369-386.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Dataset Multilevel data.ml1 - dichotomous items
#############################################################################
data(data.ml1)
dat <- data.ml1[,-1]
group <- data.ml1$group
# just for a try use a very small number of iterations
burnin <- 50 ; iter <- 100

#***
# Model 1: 1PNO with no cluster item effects
mod1 <- sirt::mcmc.2pno.ml( dat, group, est.b.Var="n", burnin=burnin, iter=iter )
summary(mod1)    # summary
plot(mod1,layout=2,ask=TRUE) # plot results
# write results to coda file
mcmclist2coda( mod1$mcmcobj, name="data.ml1_mod1" )

#***
# Model 2: 1PNO with cluster item effects of item difficulties
mod2 <- sirt::mcmc.2pno.ml( dat, group, est.b.Var="i", burnin=burnin, iter=iter )
summary(mod2)
plot(mod2, ask=TRUE, layout=2 )

#***
# Model 3: 2PNO with cluster item effects of item difficulties but
#          joint item slopes
mod3 <- sirt::mcmc.2pno.ml( dat, group, est.b.Var="i", est.a.M="h",
              burnin=burnin, iter=iter )
summary(mod3)

#***
# Model 4: 2PNO with cluster item effects of item difficulties and
#          cluster item effects with a jointly estimated SD
mod4 <- sirt::mcmc.2pno.ml( dat, group, est.b.Var="i", est.a.M="h",
                est.a.Var="j", burnin=burnin, iter=iter )
summary(mod4)

#############################################################################
# EXAMPLE 2: Dataset Multilevel data.ml2 - polytomous items
#            assuming a normal distribution for polytomous items
#############################################################################
data(data.ml2)
dat <- data.ml2[,-1]
group <- data.ml2$group
# set iterations for all examples (too few!!)
burnin <- 100 ; iter <- 500

#***
# Model 1: no intercept variance, no slopes
mod1 <- sirt::mcmc.2pno.ml( dat=dat, group=group, est.b.Var="n",
             burnin=burnin, iter=iter, link="normal",  progress.iter=20  )
summary(mod1)

#***
# Model 2a: itemwise intercept variance, no slopes
mod2a <- sirt::mcmc.2pno.ml( dat=dat, group=group, est.b.Var="i",
            burnin=burnin, iter=iter,link="normal",  progress.iter=20  )
summary(mod2a)

#***
# Model 2b: homogeneous intercept variance, no slopes
mod2b <- sirt::mcmc.2pno.ml( dat=dat, group=group, est.b.Var="j",
              burnin=burnin, iter=iter,link="normal",  progress.iter=20  )
summary(mod2b)

#***
# Model 3: intercept variance and slope variances
#          hierarchical item and slope parameters
mod3 <- sirt::mcmc.2pno.ml( dat=dat, group=group,
               est.b.M="h", est.b.Var="i", est.a.M="h", est.a.Var="i",
               burnin=burnin, iter=iter,link="normal",  progress.iter=20  )
summary(mod3)

#############################################################################
# EXAMPLE 3: Simulated random effects model | dichotomous items
#############################################################################
set.seed(7698)

#*** model parameters
sig2.lev2 <- .3^2   # theta level 2 variance
sig2.lev1 <- .8^2   # theta level 1 variance
G <- 100            # number of groups
n <- 20             # number of persons within a group
I <- 12             # number of items
#*** simuate theta
theta2 <- stats::rnorm( G, sd=sqrt(sig2.lev2) )
theta1 <- stats::rnorm( n*G, sd=sqrt(sig2.lev1) )
theta  <- theta1 + rep( theta2, each=n )
#*** item difficulties
b <- seq( -2, 2, len=I )
#*** define group identifier
group <- 1000 + rep(1:G, each=n )
#*** SD of group specific difficulties for items 3 and 5
sigma.item <- rep(0,I)
sigma.item[c(3,5)] <- 1
#*** simulate group specific item difficulties
b.class <- sapply( sigma.item, FUN=function(sii){ stats::rnorm( G, sd=sii ) } )
b.class <- b.class[ rep( 1:G,each=n ), ]
b <- matrix( b, n*G, I, byrow=TRUE ) + b.class
#*** simulate item responses
m1 <- stats::pnorm( theta - b )
dat <- 1 * ( m1 > matrix( stats::runif( n*G*I ), n*G, I ) )

#*** estimate model
mod <- sirt::mcmc.2pno.ml( dat, group=group, burnin=burnin, iter=iter,
            est.b.M="n", est.b.Var="i", progress.iter=20)
summary(mod)
plot(mod, layout=2, ask=TRUE )

## End(Not run)

alexanderrobitzsch/sirt documentation built on Sept. 8, 2024, 2:45 a.m.