mcmc.2pno.ml: 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). 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_

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

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 wi

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 estim

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

Value

A list of class mcmc.sirt with following entries:
mcmcobjObject of class mcmc.list
summary.mcmcobjSummary 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.
icInformation criteria (DIC)
burninNumber of burnin iterations
iterTotal number of iterations
theta.chainSampled values of $\theta_{pj}$ parameters
theta.chainSampled values of $u_{j}$ parameters
deviance.chainSampled values of Deviance values
EAP.relEAP reliability
personData frame with EAP person parameter estimates for $\theta_pj$ and their corresponding posterior standard deviations
datUsed data frame
...Further values

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

References

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. 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

Run this code

#############################################################################
# EXAMPLE 1: Dataset Multilevel 1
#############################################################################
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 <- 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 <- 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 <- 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 <- 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 2
#############################################################################
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 <- 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 <- 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 <- 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 <- 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)

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