R: Bayesian analysis for a semiparametric Rasch model

DPMrasch {DPpackage} | R Documentation

## Bayesian analysis for a semiparametric Rasch model

### Description

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

### Usage

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

### Arguments

`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: `N` giving the truncation of the Dirichlet process prior, `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), `m0` and `s0` giving the hyperparameters of the normal prior distribution for the mean, `mub`, of the normal baseline distribution, `mub` giving the mean of the baseline distribution (it must be specified if `s0` is missing), `taub1` and `taub2` giving the hyperparameters of the inverted gamma prior distribution for the variance, `sigmab`, of the baseline distribution, `sigmab` giving the variance of the baseline distribution (is must be specified if `taub1` is missing), `tauk1` giving the hyperparameter for the prior distribution of variance of the normal kernel, and `taus1` and `taus2` giving th hyperparameters of the gamma distribution for `tauk2`, `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 density and CDF must be computed.

### Details

This generic function fits a semiparametric Rasch model as in San Martin et al. (2011), where

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

thetai | G ~ int N(mu,sigma)G(d mu, d sigma)

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

G | alpha, G0 ~ DP(alpha G0)

where G0 = N(mu| mub, sigmab)IG(sigma|tauk1,tauk2). To complete the model specification, independent hyperpriors are assumed,

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

mub | m0, s0 ~ N(m0,s0)

sigma_b^-2 | taub1, taub2 ~ Gamma(taub1/2,taub2/2)

tauk2 | taus1, taus2 ~ Gamma(taus1/2,taus2/2)

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. To let alpha to be fixed at a particular value, set a0 to NULL in the prior specification.

The computational implementation of the model is based on the finite approximation for DP proposed by Ishwaran and James (2002). 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).

### Value

An object of class `DPMrasch` 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`, `mub`, `sigmab`, `sigmak2`, 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:

`alpha` | giving the value of the precision parameter.
`b` | a vector of dimension nsubjects giving the value of the random effects for each subject.
`beta` | giving the value of the difficulty parameters.
`ncluster` | an integer giving the number of clusters.
`ss` | an interger vector defining to which of the `ncluster` clusters each subject belongs.
`muclus` | a vector of dimension N giving the value of the normal means.
`sigmaclus` | a vector of dimension N giving the value of the normal variances.
`mub` | giving the mean of the normal baseline distributions.
`sigmab` | giving the variance of the normal baseline distributions.
`tauk2` | giving the parameter of the inverse-gamma prior for the normal kernel variances.
`wdp` | giving the vector of DP weights.
`vdp` | giving the vector of stick-breaking beta random variables used to create the DP weights.

### Author(s)

Alejandro Jara <atjara@uc.cl>

### References

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

Ishwaran, H. and James, L.F. (2002) Approximate Dirichlet process computing finite normal mixtures: smoothing and prior information. Journal of Computational and Graphical Statistics, 11: 508-532.

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.

### See Also

`DPrandom`, `DPrasch`, `FPTrasch`

### Examples

    ## Not run:
        ####################################
        # A simulated Data Set
        ####################################
          nsubject <- 250
          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,sqrt(0.25))+
                   (1-ind)*rnorm(nsubject,2,sqrt(0.065))
          beta <- c(0,seq(-3,3,length=nitem-1))

          true.density <- function(grid)
          {
                0.5*dnorm(grid,-1,sqrt(0.25))+0.5*dnorm(grid,2,sqrt(0.065))
          }

          true.cdf <- function(grid)
          {
                0.5*pnorm(grid,-1,sqrt(0.25))+0.5*pnorm(grid,2,sqrt(0.065))
          }

          for(i in 1:nsubject)
          {
             for(j in 1:nitem)
             {
                eta <- theta[i]-beta[j]
                prob <- exp(eta)/(1+exp(eta))
                y[i,j] <- rbinom(1,1,prob)
             }
          }

        # Prior information

          beta0 <- rep(0,nitem-1)
          Sbeta0 <- diag(100,nitem-1)

          prior <- list(N=50,
                              alpha=1,
                              taub1=6.01,
                              taub2=2.01,
                              taus1=6.01,
                              taus2=2.01,
                              tauk1=6.01,
                              m0=0,
                              s0=100,
                              beta0=beta0,
                              Sbeta0=Sbeta0)

        # Initial state
          state <- NULL

        # MCMC parameters

          nburn <- 4000
          nsave <- 4000
          nskip <- 0
          ndisplay <- 100
          mcmc <- list(nburn=nburn,
                                nsave=nsave,
                                nskip=nskip,
                                ndisplay=ndisplay)

        # Fit the model
          fit1 <- DPMrasch(y=y,prior=prior,mcmc=mcmc,
                           state=state,status=TRUE,grid=seq(-3,4,0.01))

          plot(fit1$grid,fit1$dens.m,type="l",lty=1,col="red",
               xlim=c(-3,4),ylim=c(0,0.8))
          lines(fit1$grid,true.density(fit1$grid),
                lty=2,col="blue")

          plot(fit1$grid,fit1$cdf.m,type="l",lty=1,col="red")
          lines(fit1$grid,true.cdf(fit1$grid),lty=2,col="blue")

        # 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)

[Package DPpackage version 1.1-7.4 Index]