DPMglmm              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 generalized linear mixed model using a Dirichlet
     Process Mixture of Normals prior for the distribution of the
     random effects.

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

     DPMglmm(fixed,random,family,offset,n,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.

  family: a description of the error distribution and link function to
          be used in the model. This can be a character string naming a
          family function, a family function or the result of a call to
          a family function. The families(links) considered by 
          'DPglmm' so far are binomial(logit), binomial(probit),
          Gamma(log), and poisson(log). The gaussian(identity) case is 
          implemented separately in the function 'DPlmm'.

  offset: this can be used to specify an a priori known component to be
          included in the linear predictor during the fitting (only for
          poisson and gamma models).

       n: this can be used to indicate the total number of cases in a
          binomial model (only implemented for the logistic link). If
          it is not specified the response variable must be binary.

   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 kernel, 'mb' and 'Sb'
          giving the hyperparameters  of the normal prior distribution
          for the mean of the normal baseline distribution,'nub' and
          'Tbinv' giving the hyperparameters of the  inverted Wishart
          prior distribution for the scale matrix of the normal
          baseline distribution, '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) and, 'tau1' and 'tau2' giving the
          hyperparameters for the  prior distribution for the inverse
          of the dispersion parameter of  the Gamma model  (they must
          be specified only if the Gamma model is considered).

    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, 'ndisplay' giving the
          number of saved scans to be displayed on the screen (the
          function reports  on the screen when every 'ndisplay'
          iterations have been carried out), 'tune1' giving the
          positive Metropolis tuning parameter for the  precision
          parameter of the Gamma model (the default value is 1.1).

   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 
          'DPMglmm' to print an error message and terminate if there
          are any incomplete observations.

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

     This generic function fits a generalized linear mixed-effects
     model, where the linear predictor is modeled as follows:

            etai = Xi betaF + Zi betaR + Zi bi, i=1,...,n


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


          G | alpha, mub, Sigmab ~ DP(alpha N(mub, Sigmab))


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

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


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


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


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


                     Sigma | nub, Tb ~ IW(nub,Tb)


     Note that the inverted-Wishart prior is parametrized such that
     E(Sigma)= Tinv/(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 inverse of the dispersion parameter of the Gamma model is
     modeled using 'gamma' distribution, Gamma(tau1/2,tau2/2). 

     The computational implementation of the model is based on the
     marginalization of the 'DP' and the MCMC is model-specific. 

     For the 'poisson', 'Gamma', and 'binomial(logit)', MCMC methods
     for nonconjugate  priors (see, MacEachern and Muller, 1998; Neal,
     2000) are used. Specifically, the algorithm 8  with 'm=1' of Neal
     (2000), is considered in  the 'DPMglmm' function. In this case,
     the fully conditional distributions for fixed 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).

     For the 'binomial(probit)' the following latent variable
     representation is used:

                     yij = I(wij > 0), j=1,...,ni


      wij | beta, thetai, lambdai ~ N(Xij beta + Zij thetai, 1)


     In this case, the computational implementation of the model is
     based on the marginalization of the 'DP' and on the use of MCMC
     methods for conjugate priors  for a collapsed state described by
     MacEachern (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 'DPMglmm' representing the generalized 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', 'sigma2e', 
     'Sigma', 'mub', the elements of 'Sigmab', '\alpha', and the 
     number of clusters.

     The function 'DPMrandom' 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.

      mu: a matrix of dimension (nsubjects)*(nrandom effects) giving
          the value of the means of the normal kernel for each cluster
          (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 fixed effects.

   sigma: giving the variance matrix of the normal kernel.

     mub: giving the mean of the normal baseline distributions.

  sigmab: giving the variance matrix of the normal baseline
          distributions.

     phi: giving the dispersion parameter for the Gamma model (if
          needed).

_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. (1998) Computational Methods for Mixture of
     Dirichlet Process Models, in Practical Nonparametric and
     Semiparametric Bayesian Statistics,  eds: D. Dey, P. Muller, D.
     Sinha, New York: Springer-Verlag, pp. 1-22.

     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.

     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:

     'DPMrandom', 'DPMlmm'   , 'DPMolmm', 'DPlmm'    , 'DPglmm' ,
     'DPolmm'

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

     ## Not run: 
         # Respiratory Data Example

           data(indon)
           attach(indon)

           baseage2<-baseage**2
           follow<-age-baseage
           follow2<-follow**2 

         # Prior information

           prior<-list(alpha=1,
                       nu0=4.01,
                       tinv=diag(1,1),
                       nub=4.01,
                       tbinv=diag(1,1),
                       mb=rep(0,1),
                       Sb=diag(1000,1),
                       beta0=rep(0,9),
                       Sbeta0=diag(1000,9))

         # 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 Probit model
           fit1<-DPMglmm(fixed=infect~gender+height+cosv+sinv+xero+baseage+
                         baseage2+follow+follow2,
                         random=~1|id,family=binomial(probit),
                         prior=prior,mcmc=mcmc,state=state,status=TRUE)

         # Fit the Logit model
           fit2<-DPMglmm(fixed=infect~gender+height+cosv+sinv+xero+baseage+
                         baseage2+follow+follow2,random=~1|id,
                         family=binomial(logit),
                         prior=prior,mcmc=mcmc,state=state,status=TRUE)

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

           summary(fit2)
           summary(fit2,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)      

         # Plot an specific model parameter (to see the plots gradually set ask=TRUE)
           plot(fit1,ask=FALSE,nfigr=1,nfigc=2,param="baseage")      
           plot(fit1,ask=FALSE,nfigr=1,nfigc=2,param="ncluster")     
     ## End(Not run)

