DPpackage (version 1.1-7.4)

DPMglmm: Bayesian analysis for a semiparametric generalized linear mixed model using a DPM of normals

Description

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.

Usage

DPMglmm(fixed,random,family,offset,n,prior,mcmc,state,status,
      data=sys.frame(sys.parent()),na.action=na.fail)

Arguments

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 NAs. The default action (na.fail) causes DPMglmm to print an error message and terminate if there are any incomplete observations.

Value

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).

Details

This generic function fits a generalized linear mixed-effects model, where the linear predictor is modeled as follows: $$\eta_i = X_i \beta_F + Z_i \beta_R + Z_i b_i, i=1,\ldots,n$$ $$\theta_i | G,\Sigma \sim \int N(\mu,\Sigma) G(d\mu)$$ $$G | \alpha, \mu_b,\Sigma_b \sim DP(\alpha N(\mu_b,\Sigma_b))$$

where, \(\theta_i = \beta_R + b_i\) , \(\beta = \beta_F\), and \(G_0 = N(\theta| \mu, \Sigma)\). To complete the model specification, independent hyperpriors are assumed, $$\beta | \beta_0, S_{\beta_0} \sim N(\beta_0,S_{\beta_0})$$ $$\Sigma | \nu_0, T \sim IW(\nu_0,T)$$ $$\alpha | a_0, b_0 \sim Gamma(a_0,b_0)$$ $$\mu_b | m_b, S_b \sim N(m_b,S_b)$$ $$\Sigma_b | \nu_b, Tb \sim IW(\nu_b,Tb)$$

Note that the inverted-Wishart prior is parametrized such that \(E(\Sigma)= T^{-1}/(\nu_0-q-1)\).

The precision or total mass parameter, \(\alpha\), of the DP prior can be considered as random, having a gamma distribution, \(Gamma(a_0,b_0)\), 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, \(a_0\) to NULL in the prior specification.

The inverse of the dispersion parameter of the Gamma model is modeled using gamma distribution, \(\Gamma(\tau_1/2,\tau_2/2)\).

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: $$y_{ij} = I(w_{ij} > 0), j=1,\ldots,n_i$$ $$w_{ij} | \beta, \theta_i, \lambda_i \sim N(X_{ij} \beta + Z_{ij} \theta_i, 1)$$

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

References

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.

See Also

DPMrandom, DPMlmm , DPMolmm, DPlmm , DPglmm , DPolmm, PTlmm , PTglmm, PTolmm

Examples

# 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")	
# }