R: Bayesian analysis for a semiparametric generalized linear...

PTglmm {DPpackage} | R Documentation

## Bayesian analysis for a semiparametric generalized linear mixed model using a MMPT

### Description

This function generates a posterior density sample for a semiparametric generalized linear mixed model, using a Mixture of Multivariate Polya Trees prior for the distribution of the random effects.

### Usage

    PTglmm(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 `PTglmm` so far are binomial(logit), binomial(probit), Gamma(log), and poisson(log). The gaussian(identity) case is implemented separately in the function `PTlmm`.
`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 Polya Tree (PT) 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), `M` giving the finite level of the PT prior to be considered, and `frstlprob` a logical variable indicating whether the first level probabilities of the PT are fixed or not (the default is FALSE), `tau1` and `tau2` giving the hyperparameters for the prior distribution for the inverse of the precision parameter of the Gamma model (they must be specified only if the Gamma model is considered), and `typep` indicating whether the type of decomposition of the centering covariance matrix is random (1) or not (0).
`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 screen (the function reports on the screen when every `ndisplay` iterations have been carried out), `nbase` giving the number scans to be performed before the parameters of the centering distribution and the precision parameter are updated (i.e., the update of this parameters is invoked only once in every `nbase` scans) (the default value is 1), `tune1`, `tune2`, `tune3`, `tune4` and `tune5` giving the Metropolis tuning parameter for the baseline mean, variance, precision parameter, partition and dispersion parameter (only for the Gamma mode), respectively. If `tune1`, `tune2`, `tune3` or `tune4` are not specified or negative, an adpative Metropolis algorithm is performed. If `tune5` is not specified, a default value of 1.1 is assumed. Finally, the integer `samplef` indicates whether the functional parameters must be sample (1) or not (0).
`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 `PTglmm` to print an error message and terminate if there are any incomplete observations.

### Details

This generic function fits a generalized linear mixed-effects model using a Mixture of Multivariate Polya Trees prior (see, Lavine 1992; 1994, for details about univariate PT) for the distribution of the random effects as described in Jara, Hanson and Lesaffre (2009). The linear predictor is modeled as follows:

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

thetai | G ~ G

G | alpha,mu,Sigma,O ~ PT^M(Pi^{mu,Sigma,O},A)

where, thetai = betaR + bi, beta = betaF, and O is an orthogonal matrix defining the decomposition of the centering covariance matrix. As in Hanson (2006), the PT prior is centered around the N_d(mu,Sigma) distribution. However, we consider the class of partitions Pi^{mu,Sigma, O}. The partitions starts with base sets that are Cartesian products of intervals obtained as quantiles from the standard normal distribution. A multivariate location-scale transformation theta=mu+Sigma^{1/2} z is applied to each base set yielding the final sets. Here Sigma^{1/2}=T'O', where T is the unique upper triangular Cholesky matrix of Sigma. The family A={alphae: e \in E*}, where E*=U_{m=0}^{M} E_d^m, with E_d and E_m the d-fold product of E=\{0,1\} and the the m-fold product of E_d, respectively. The family A was specified as alpha{e1 … em}=α m^2.

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)

O ~ Haar(q)

Note that the inverted-Wishart prior is parametrized such that E(Sigma)= Tinv/(nu0-q-1).

The precision parameter, alpha, of the `PT` prior can be considered as random, having a `gamma` distribution, Gamma(a0,b0), or fixed at some particular value.

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 `PT` as discussed in Jara, Hanson and Lesaffre (2009).

### Value

An object of class `PTglmm` 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`, the dispersion parameter of the Gamma model, and `ortho`.

The function `PTrandom` 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:

`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.
`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.
`phi` | giving the precision parameter for the Gamma model (if needed).
`ortho` | giving the orthogonal matrix `H`, used in the decomposition of the covariance matrix.

### Author(s)

Alejandro Jara <atjara@uc.cl>

Tim Hanson <hansont@stat.sc.edu>

### References

Hanson, T. (2006) Inference for Mixtures of Finite Polya Trees. Journal of the American Statistical Association, 101: 1548-1565.

Jara, A., Hanson, T., Lesaffre, E. (2009) Robustifying Generalized Linear Mixed Models using a New Class of Mixtures of Multivariate Polya Trees. Journal of Computational and Graphical Statistics, 18(4): 838-860.

Lavine, M. (1992) Some aspects of Polya tree distributions for statistical modelling. The Annals of Statistics, 20: 1222-11235.

Lavine, M. (1994) More aspects of Polya tree distributions for statistical modelling. The Annals of Statistics, 22: 1161-1176.

### See Also

`PTrandom`, `PTglmm` , `PTolmm`, `DPMglmm`, `DPMlmm`, `DPMolmm`, `DPlmm` , `DPglmm`, `DPolmm`

### 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,
                        M=4,
                        frstlprob=FALSE,
                        nu0=4,
                        tinv=diag(1,1),
                        mub=rep(0,1),
                        Sb=diag(1000,1),
                        beta0=rep(0,9),
                        Sbeta0=diag(10000,9))

        # Initial state
          state <- NULL

        # MCMC parameters

          nburn <- 5000
          nsave <- 5000
          nskip <- 20
          ndisplay <- 100
          mcmc <- list(nburn=nburn,
                       nsave=nsave,
                       nskip=nskip,
                       ndisplay=ndisplay,
                       tune1=0.5,tune2=0.5,
                       samplef=1)

        # Fitting the Logit model
          fit1 <- PTglmm(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)

          fit1

          plot(PTrandom(fit1,predictive=TRUE))

        # 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
          PTrandom(fit1)
          PTrandom(fit1,centered=TRUE)

        # Extract predictive information of random effects
          PTrandom(fit1,predictive=TRUE)

        # Predictive marginal and joint distributions
          plot(PTrandom(fit1,predictive=TRUE))

        # Fitting the Probit model
          fit2 <- PTglmm(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)
          fit2

        # Plot model parameters (to see the plots gradually set ask=TRUE)
          plot(fit2,ask=FALSE)
          plot(fit2,ask=FALSE,nfigr=2,nfigc=2)

        # Extract random effects
          PTrandom(fit2)

        # Extract predictive information of random effects
          PTrandom(fit2,predictive=TRUE)

        # Predictive marginal and joint distributions
          plot(PTrandom(fit2,predictive=TRUE))

    ## End(Not run)

[Package DPpackage version 1.1-7.4 Index]