DPrasch: Bayesian analysis for a semiparametric Rasch model
In DPpackage: Bayesian Nonparametric Modeling in R

Description Usage Arguments Details Value Author(s) References See Also Examples

This function generates a posterior density sample for a semiparametric Rasch model, using a DP or a MDP prior for the distribution of the random effects.

DPrasch(y,prior,mcmc,offset,state,status,
        grid=seq(-10,10,length=1000),data=sys.frame(sys.parent()),
        compute.band=FALSE)

`y`	a matrix giving the data for which the Rasch Model is to be fitted.
`prior`	a list giving the prior information. The list includes the following parameter: `a0` and `b0` giving the hyperparameters for prior distribution of the precision parameter of the Dirichlet process prior, `alpha` giving the value of the precision parameter (it must be specified if `a0` is missing), `mub` and `Sb` giving the hyperparameters of the normal prior distribution for the mean of the normal baseline distribution, `mu` giving the mean of the normal baseline distribution (is must be specified if `mub` and `Sb` are missing), `tau1` and `tau2` giving the hyperparameters for the prior distribution of variance of the normal baseline distribution, `sigma2` giving the variance of the normal baseline distribution (is must be specified if `tau1` and `tau2` are missing), and `beta0` and `Sbeta0` giving the hyperparameters of the normal prior distribution for the difficulty parameters.
`mcmc`	a list giving the MCMC parameters. The list must include the following integers: `nburn` giving the number of burn-in scans, `nskip` giving the thinning interval, `nsave` giving the total number of scans to be saved, and `ndisplay` giving the number of saved scans to be displayed on screen (the function reports on the screen when every `ndisplay` iterations have been carried out).
`offset`	this can be used to specify an a priori known component to be included in the linear predictor during the fitting.
`state`	a list giving the current value of the parameters. This list is used if the current analysis is the continuation of a previous analysis.
`status`	a logical variable indicating whether this run is new (`TRUE`) or the continuation of a previous analysis (`FALSE`). In the latter case the current value of the parameters must be specified in the object `state`.
`grid`	grid points where the density estimate is evaluated. The default is seq(-10,10,length=1000).
`data`	data frame.
`compute.band`	logical variable indicating whether the confidence band for the CDF must be computed.

This generic function fits a semiparametric Rasch model as in San Martin et al. (2011), where the linear predictor is modeled as follows:

etaij = thetai - betaj, i=1,…,n, j=1,…,k

thetai | G ~ G

G | alpha, G0 ~ DP(alpha G0)

beta | beta0, Sbeta0 ~ N(beta0,Sbeta0)

where, G0 = N(theta| mu, sigma2). To complete the model specification, independent hyperpriors are assumed,

alpha | a0, b0 ~ Gamma(a0,b0)

mu | mub, Sb ~ N(mub,Sb)

sigma2^-1 | tau1, tau2 ~ Gamma(tau1/2,tau2/2)

The precision or total mass parameter, alpha, of the DP prior can be considered as random, having a gamma distribution, Gamma(a0,b0), or fixed at some particular value. When alpha is random the method described by Escobar and West (1995) is used. To let alpha to be fixed at a particular value, set a0 to NULL in the prior specification.

Each of the parameters of the baseline distribution, mu and sigma2 can be considered as random or fixed at some particular value. In the first case, a Mixture of Dirichlet Process is considered as a prior for the distribution of the random effects. To let sigma2 to be fixed at a particular value, set tau1 to NULL in the prior specification. To let mu to be fixed at a particular value, set mub to NULL in the prior specification.

The computational implementation of the model is based on the marginalization of the DP and on the use of MCMC methods for nonconjugate priors (see, MacEachern and Muller, 1998; Neal, 2000). Specifically, the algorithm 8 with m=1 of Neal (2000), is considered in the DPraschpoisson function. In this case, the full conditional distributions for the difficulty parameters and in the resampling step of random effects are generated through the Metropolis-Hastings algorithm with a IWLS proposal (see, West, 1985 and Gamerman, 1997).

The functionals parameters are sampled using the epsilon-DP approximation proposed by Muliere and Tardella (1998), with epsilon=0.01.

An object of class DPrasch representing the Rasch model fit. Generic functions such as print, plot, and summary have methods to show the results of the fit. The results include beta, mu, sigma2, the precision parameter alpha, and the number of clusters.

The function DPrandom can be used to extract the posterior mean of the random effects.

The list state in the output object contains the current value of the parameters necessary to restart the analysis. If you want to specify different starting values to run multiple chains set status=TRUE and create the list state based on this starting values. In this case the list state must include the following objects:

`ncluster`	an integer giving the number of clusters.
`alpha`	giving the value of the precision parameter.
`b`	a vector of dimension nsubjects giving the value of the random effects for each subject.
`bclus`	a vector of dimension nsubjects giving the value of the random effects for each clusters (only the first `ncluster` are considered to start the chain).
`ss`	an interger vector defining to which of the `ncluster` clusters each subject belongs.
`beta`	giving the value of the difficulty parameters.
`mu`	giving the mean of the normal baseline distributions.
`sigma2`	giving the variance of the normal baseline distributions.

Alejandro Jara <atjara@uc.cl>

Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588.

Gamerman, D. (1997) Sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing, 7: 57-68.

MacEachern, S. N. and Muller, P. (1998) Estimating mixture of Dirichlet Process Models. Journal of Computational and Graphical Statistics, 7 (2): 223-338.

Muliere, P. and Tardella, L. (1998) Approximating distributions of random functionals of Ferguson-Dirichlet priors. The Canadian Journal of Statistics, 26(2): 283-297.

Neal, R. M. (2000) Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9:249-265.

San Martin, E., Jara, A., Rolin, J.-M., and Mouchart, M. (2011) On the Bayesian nonparametric generalization of IRT-type models. Psychometrika (To appear)

West, M. (1985) Generalized linear models: outlier accomodation, scale parameter and prior distributions. In Bayesian Statistics 2 (eds Bernardo et al.), 531-558, Amsterdam: North Holland.

DPrandom, FPTrasch

## Not run: 
    ####################################
    # A simulated Data Set
    ####################################
      nsubject <- 200
      nitem <- 40
      
      y <- matrix(0,nrow=nsubject,ncol=nitem)
      dimnames(y) <- list(paste("id",seq(1:nsubject)), 
                          paste("item",seq(1,nitem)))

      
      ind <- rbinom(nsubject,1,0.5)
      theta <- ind*rnorm(nsubject,1,0.25)+(1-ind)*rnorm(nsubject,3,0.25)
      beta <- c(0,seq(-1,3,length=nitem-1))
      true.cdf <- function(grid)
      {
         0.5*pnorm(grid,1,0.25)+0.5*pnorm(grid,3,0.25) 
      }  
      for(i in 1:nsubject)
      {
         for(j in 1:nitem)
         {
            eta<-theta[i]-beta[j]         
            mean<-exp(eta)/(1+exp(eta))
            y[i,j]<-rbinom(1,1,mean)
         }
      }

    # Prior information

      beta0 <- rep(0,nitem-1)
      Sbeta0 <- diag(1000,nitem-1)

      prior <- list(alpha=1,
                    tau1=6.02,
                    tau2=2.02,
                    mub=0,
                    Sb=100,
                    beta0=beta0,
                    Sbeta0=Sbeta0)

    # Initial state
      state <- NULL

    # MCMC parameters

      nburn <- 5000
      nsave <- 5000
      nskip <- 0
      ndisplay<- 1000
      mcmc <- list(nburn=nburn,
                   nsave=nsave,
                   nskip=nskip,
                   ndisplay=ndisplay)

    # Fit the model
      fit1 <- DPrasch(y=y,prior=prior,mcmc=mcmc, 
                      state=state,status=TRUE,grid=seq(-1,5,0.01),
                      compute.band=TRUE)

    # CDF estimate and truth
      plot(fit1$grid,true.cdf(fit1$grid),type="l",lwd=2,col="red",
           xlab=expression(theta),ylab="CDF")
      lines(fit1$grid,fit1$cdf,lwd=2,col="blue")
      lines(fit1$grid,fit1$cdf.l,lwd=2,col="blue",lty=2)
      lines(fit1$grid,fit1$cdf.u,lwd=2,col="blue",lty=2)

    # Summary with HPD and Credibility intervals
      summary(fit1)
      summary(fit1,hpd=FALSE)

    # Plot model parameters 
    # (to see the plots gradually set ask=TRUE)
      plot(fit1,ask=FALSE)
      plot(fit1,ask=FALSE,nfigr=2,nfigc=2)	

    # Extract random effects
    
      DPrandom(fit1)
      plot(DPrandom(fit1))
      DPcaterpillar(DPrandom(fit1))

## End(Not run)