DPraschpoisson           package:DPpackage           R Documentation

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_D_e_s_c_r_i_p_t_i_o_n:

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

_U_s_a_g_e:

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

_A_r_g_u_m_e_n_t_s:

       y: a matrix giving the data for which the Rasch Poisson 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), code{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, code{sigma} giving the standard
          deviation 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.

_D_e_t_a_i_l_s:

     This generic function fits a semiparametric Rasch Poisson model as
     in San Martin et al. (2006), 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 sigma 
     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 sigma 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
     fully 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).

_V_a_l_u_e:

     An object of class 'DPraschpoisson' representing the Rasch Poisson
     model fit. Generic functions such as 'print', 'plot', and
     'summary' have methods to show the results of the fit. The results
     include  'beta', 'mu', 'sigma', 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.

   sigma: giving the std of the normal baseline distributions.

_A_u_t_h_o_r(_s):

     Alejandro Jara <Alejandro.JaraVallejos@med.kuleuven.be>

_R_e_f_e_r_e_n_c_e_s:

     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.

     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. (2006) On
     the Analysis of Bayesian Semiparametric IRT-type Models. In
     preparation.

     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.

_S_e_e _A_l_s_o:

     'DPrandom', 'FPTraschpoisson'

_E_x_a_m_p_l_e_s:

     ## Not run: 
         ####################################
         # A simulated Data Set
         ####################################
           nsubject<-200
           nitem<-5
           
           y<-matrix(0,nrow=nsubject,ncol=nitem)
           
           ind<-rbinom(nsubject,1,0.5)
           theta<-ind*rnorm(nsubject,-3,0.25)+(1-ind)*rnorm(nsubject,3,0.25)
           beta<-c(0,seq(-1,1,length=nitem-1))

           for(i in 1:nsubject)
           {
              for(j in 1:nitem)
              {
                 eta<-theta[i]-beta[j]         
                 mean<-exp(eta)
                 y[i,j]<-rpois(1,mean)
              }
           }
      
         # Prior information

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

           prior<-list(alpha=1,sigma=1,mu=0,beta0=beta0,
                       Sbeta0=Sbeta0)

         # Initial state
           state <- NULL

         # MCMC parameters

           nburn<-5000
           nsave<-25000
           nskip<-20
           ndisplay<-1000
           mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip,
                        ndisplay=ndisplay)

         # Fit the model
           fit1<-DPraschpoisson(y=y,prior=prior,mcmc=mcmc, 
                                state=state,status=TRUE)

         # 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)

