DPglmm {DPpackage} | R Documentation |
This function generates a posterior density sample for a semiparametric generalized linear mixed model.
DPglmm(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),
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 NA s. The default action (na.fail ) causes
DPglmm 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:
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).
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. |
Alejandro Jara <Alejandro.JaraVallejos@med.kuleuven.be>
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.
## 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)