DPglmm {DPpackage}R Documentation

Performs a Bayesian analysis for a semiparametric generalized linear mixed model

Description

This function generates a posterior density sample for a semiparametric generalized linear mixed model.

Usage


DPglmm(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), 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 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 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, see details below), nu0 and Tinv giving the hyperparameters of the inverted Wishart prior distribution for the scale matrix of the normal baseline distribution, mub and Sb giving the hyperparameters of the normal prior distribution for the mean of the normal baseline distribution and, 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 the 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 NAs. The default action (na.fail) causes DPglmm to print an error message and terminate if there are any incomplete observations.

Details

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

eta_i = X_i β_F + Z_i β_R + Z_i b_i, i=1,...,n

theta_i | G sim G

G | α, G_0 sim DP(α G_0)

where, theta_i = β_R + b_i , β = β_F, and G_0 = N(theta| μ, Σ). To complete the model specification, independent hyperpriors are assumed,

α | a_0, b_0 sim Gamma(a_0,b_0)

β | β_0, S_{β_0} sim N(β_0,S_{β_0})

μ | μ_b, S_b sim N(mu_b,S_b)

Σ | nu_0, T sim IW(nu_0,T)

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

The precision or total mass parameter, α, 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 α is random the method described by Escobar and West (1995) is used. To let α to be fixed at a particular value set, a_0 to NULL in the prior specification.

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

For the poisson and binomial (when the total number of cases, n, is specified) 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 DPglmm 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 conditonal bernoulli models, binomial(probit) and binomial(logit), (i.e., the total number of cases is not specified) the following latent variable representation is used:

y_{ij} = I(w_{ij} > 0), j=1,...,n_i

w_{ij} | β, theta_i, λ_i sim N(X_{ij} β + Z_{ij} theta_i, λ_{ij})

In the probit case, λ_{ij}=1 while in the logit case (see, Andrews and Mallows, 1974),

λ_{ij} = (2 phi_{ij})^2

phi_{ij} sim KS

where KS refers to the Kolmogorov Smirnov distribution. 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 (Escobar, 1994; Escobar and West, 1998). For the logistic model, the full conditional distribution of the scale parameters, λ_{ij} are sampled via the rejection sampling algorithm as proposed by Holmes and Held (2006).

Value

An object of class DPglmm representing the generalized linear mixed-effects model fit. Generic functions such as print, plot, and summary have methods to show the results of the fit. The results include betaR, betaF, mu, the elements of 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 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).
ss an interger vector defining to which of the ncluster clusters each subject belongs.
beta giving the value of the fixed effects.
betar giving the mean value of the random effects.
mu giving the mean of the normal baseline distributions.
sigma giving the variance matrix of the normal baseline distributions.

Author(s)

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

References

Andrews, D. and Mallows, C. (1974) Scale mixture of normal distributions. Journal of the Royal Statistical Society, B, 36: 99-102.

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.

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

Holmes, C.C. and Held, L. (2006) Bayesian auxiliary variable models for binary and multinomial regression. Bayesian Analysis, 1: 145-168.

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.

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, DPlmm

Examples

## Not run: 
    # Respiratory Data Example

      data(indon)
      attach(indon)

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

    # Prior information

      beta0<-rep(0,9)
      Sbeta0<-diag(1000,9)
      tinv<-diag(1,1)
      prior<-list(a0=2,b0=0.1,nu0=4,tinv=tinv,mub=rep(0,1),Sb=diag(1000,1),
                  beta0=beta0,Sbeta0=Sbeta0)

    # Initial state
      state <- NULL

    # MCMC parameters

      nburn<-5
      nsave<-100
      nskip<-5
      ndisplay<-100
      mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip,ndisplay=ndisplay)

    # Fit the Probit model
      fit1<-DPglmm(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<-DPglmm(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)

[Package DPpackage version 1.0-0 Index]