DPMglmm {DPpackage} | R Documentation |
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.
DPMglmm(fixed,random,family,offset,n,prior,mcmc,state,status, data=sys.frame(sys.parent()),na.action=na.fail)
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. |
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.
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). |
Alejandro Jara <ajarav@udec.cl>
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.
DPMrandom
,
DPMlmm
, DPMolmm
,
DPlmm
, DPglmm
, DPolmm
,
PTlmm
, PTglmm
, PTolmm
## 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)