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 Mixture of Multivariate Polya Trees 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 include 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 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).
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 |
Alejandro Jara <atjara@uc.cl>
Tim Hanson <hansont@stat.sc.edu>
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.
PTrandom
,
PTglmm
, PTolmm
,
DPMglmm
, DPMlmm
, DPMolmm
,
DPlmm
, DPglmm
, DPolmm
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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | ## 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)
|
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.