Description Usage Arguments Details Value Author(s) References See Also Examples
This function generates a posterior density sample for a semiparametric generalized linear mixed model using a Dirichlet Process or a Mixture of Dirichlet process prior for the distribution of the random effects.
1 2 |
fixed |
a two-sided linear formula object describing the
fixed-effects part of the model, with the response on the
left of a |
random |
a one-sided formula of the form |
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
|
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 includes the following
parameter: |
mcmc |
a list giving the MCMC parameters. The list must include
the following integers: |
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 ( |
data |
data frame. |
na.action |
a function that indicates what should happen when the data
contain |
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 ~ G
G | alpha, G0 ~ DP(alpha G0)
where, thetai = betaR + bi , beta = betaF, and G0 = N(theta| mu, Sigma). 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)
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 binomial(logit)
, poisson
, and Gamma
, 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 model 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 (Escobar, 1994;
Escobar and West, 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 DPglmm
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
, mu
, the elements of Sigma
, the precision parameter
alpha
, the number of clusters, and the dispersion parameter of the Gamma model.
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 |
ss |
an interger vector defining to which of the |
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). |
Alejandro Jara <atjara@uc.cl>
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.
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.
DPrandom
,
DPlmm
, DPolmm
,
DPMlmm
,DPMglmm
, DPMolmm
,
PTlmm
, PTglmm
, PTolmm
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | ## 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 <- 5000
nsave <- 5000
nskip <- 0
ndisplay <- 1000
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)
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