DPolmm               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 ordinal linear mixed model using a Dirichlet
     Process or a Mixture of Dirichlet process prior for the
     distribution of the random effects.

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

     DPolmm(fixed,random,prior,mcmc,state,status,data=sys.frame(sys.parent()),
           na.action=na.fail)

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

   fixed: a two-sided linear formula object describing the
          fixed-effects part of the model, with the response on the
          left of a '~' operator and the terms, separated by '+'
          operators, on the right.

  random: a one-sided formula of the form '~z1+...+zn | g', with 
          'z1+...+zn' specifying the model for the random effects and 
          'g' the grouping variable. The random effects formula will be
          repeated for all levels of grouping.

   prior: a list giving the prior information. The list include 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' and 'b0'
          are missing, see details below), 'nu0' and 'Tinv' giving the
          hyperparameters of the  inverted Wishart prior distribution
          for the scale matrix of the normal baseline distribution,
          'sigma' giving the value of the covariance matrix of the
          centering distribution (it must be specified if  'nu0' and
          'tinv' are missing), 'mub' and 'Sb' giving the
          hyperparameters  of the normal prior distribution for the
          mean of the normal baseline distribution, 'mu' giving the
          value of the mean of the centering distribution (it must be
          specified if  'mub' and 'Sb' are missing), 'beta0' and
          'Sbeta0' giving the  hyperparameters of the normal prior
          distribution for the fixed effects (must be specified only if
          fixed effects are considered in the model).  

    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).

   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'.

    data: data frame.

na.action: a function that indicates what should happen when the data
          contain 'NA's. The default action ('na.fail') causes 
          'DPolmm' to print an error message and terminate if there are
          any incomplete observations.

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

     This generic function fits an ordinal linear mixed-effects model
     with a probit link (see, e.g., Molenberghs and Verbeke, 2005):


           Yij = k, if gammak-1 <= Wij < gammak, k=1,...,K


 Wij | betaF, betaR , bi ~ N(Xij betaF + Zij betaR + Zij bi, 1), i=1,...,N, j=1,...,ni


                            thetai | G ~ G


                     G | alpha, G0 ~ DP(alpha G0)


     where, thetai = betaR + bi, beta = betaF, and G0=N(theta| mu,
     Sigma). To complete the model  specification, independent
     hyperpriors are assumed,

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


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


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


                      Sigma | nu0, T ~ IW(nu0,T)


     A uniform prior is used for the cutoff points. Note that the
     inverted-Wishart prior is parametrized such that E(Sigma)=
     T^{-1}/(nu0-q-1).

     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.

     The computational implementation of the model is based on the
     marginalization of the 'DP' and on the use of MCMC methods for
     conjugate priors (Escobar, 1994; Escobar and West, 1998). The
     betaR parameters are sampled using the epsilon-DP approximation
     proposed by Muliere and Tardella (1998), with epsilon=0.01.

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

     An object of class 'DPolmm' representing the linear mixed-effects
     model fit. Generic functions such as 'print', 'plot', 'summary',
     and 'anova' have methods to show the results of the fit.  The
     results include 'betaR', 'betaF', 'mu', the elements of  'Sigma',
     the cutoff points, '\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 matrix of dimension (nsubjects)*(nrandom effects) giving
          the value of the random effects for each subject.

   bclus: a matrix of dimension (nsubjects)*(nrandom effects) giving
          the value of the random effects for each clusters (only the
          first 'ncluster'  are considered to start the chain).

  cutoff: a real vector defining the cutoff points. Note that the first
          cutoff must be fixed to 0 in this function.

      ss: an interger vector defining to which of the 'ncluster'
          clusters each subject belongs.

    beta: giving the value of the fixed effects.

      mu: giving the mean of the normal baseline distributions.

   sigma: giving the variance matrix 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. (1994) Estimating Normal Means with a Dirichlet
     Process Prior, Journal of the American Statistical Association,
     89: 268-277.

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

     Escobar, M.D. and West, M. (1998) Computing Bayesian Nonparametric
     Hierarchical Models, in Practical Nonparametric and Semiparametric
     Bayesian Statistics,  eds: D. Dey, P. Muller, D. Sinha, New York:
     Springer-Verlag, pp. 1-22.

     Molenberghs, G. and Verbeke, G. (2005). Models for discrete
     longitudinal data, New York: Springer-Verlag.

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

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

     'DPrandom', 'DPlmm'  , 'DPglmm',  'DPMglmm', 'DPMlmm', 'DPMolmm'

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

     ## Not run: 

         # Schizophrenia Data
           data(psychiatric)
           attach(psychiatric)

         # MCMC parameters

           nburn<-5000
           nsave<-10000
           nskip<-10
           ndisplay<-100
           mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip,ndisplay=ndisplay)

         # Initial state
           state <- NULL

         # Prior information

           tinv<-diag(10,1)
           prior<-list(alpha=1,nu0=4.01,tinv=tinv,mub=rep(0,1),Sb=diag(100,1),
                       beta0=rep(0,3),Sbeta0=diag(1000,3))

         # Fitting the model

           fit1<-DPolmm(fixed=imps79o~sweek+tx+sweek*tx,random=~1|id,prior=prior,
                        mcmc=mcmc,state=state,status=TRUE)
           fit1

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

         # Plot model parameters
           plot(fit1)

         # Plot an specific model parameter
           plot(fit1,ask=FALSE,nfigr=1,nfigc=2,param="sigma-(Intercept)")    
           plot(fit1,ask=FALSE,nfigr=1,nfigc=2,param="ncluster")     

         # Extract random effects
         
           DPrandom(fit1)
           DPrandom(fit1,centered=TRUE)

         # Extract predictive information of random effects
         
           DPrandom(fit1,predictive=TRUE)
           DPrandom(fit1,centered=TRUE,predictive=TRUE)
           
           plot(DPrandom(fit1,predictive=TRUE))
           plot(DPrandom(fit1,centered=TRUE,predictive=TRUE))
     ## End(Not run)

