Description Usage Arguments Details Value Author(s) References See Also Examples
This function generates a posterior density sample for a semiparametric linear mixed model using a Dirichlet Process Mixture of Normals prior for the distribution of the random effects.
1 2 3 |
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 |
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 linear mixed-effects model (Verbeke and Molenberghs, 2000):
yi ~ N(Xi betaF + Zi betaR + Zi bi, sigma2e Ini), i=1,…,n
thetai | G, Sigma ~ int N(mu,Sigma)G(d mu)
G | alpha, mub, Sigmab ~ DP(alpha N(mub, Sigmab))
sigma2e^-1 | tau1, tau2 ~ Gamma(tau1/2,tau2/2)
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)= T^{-1}/(nu0-q-1).
The precision or total mass parameter, α, 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 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 of 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 DPMlmm
representing the 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 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. |
mu |
a matrix of dimension (nsubjects)*(nrandom effects) giving the value of the means
of the normal kernel for each cluster (only the first |
ss |
an interger vector defining to which of the |
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. |
sigma2e |
giving the error variance. |
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.
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.
Muliere, P. and Tardella, L. (1998) Approximating distributions of random functionals of Ferguson-Dirichlet priors. The Canadian Journal of Statistics, 26(2): 283-297.
Verbeke, G. and Molenberghs, G. (2000). Linear mixed models for longitudinal data, New York: Springer-Verlag.
DPMrandom
,
DPMglmm
, DPMolmm
,
DPlmm
, DPglmm
, DPolmm
,
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 | ## Not run:
# School Girls Data Example
data(schoolgirls)
attach(schoolgirls)
# Prior information
prior<-list(alpha=1,
tau1=0.01,tau2=0.01,
nu0=4.01,
tinv=diag(10,2),
nub=4.01,
tbinv=diag(10,2),
mb=rep(0,2),
Sb=diag(1000,2))
# Initial state
state <- NULL
# MCMC parameters
nburn<-5000
nsave<-10000
nskip<-20
ndisplay<-1000
mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip,ndisplay=ndisplay)
# Fit the model: First run
fit1<-DPMlmm(fixed=height~1,random=~age|child,prior=prior,mcmc=mcmc,
state=state,status=TRUE)
fit1
# Fit the model: Continuation
state<-fit1$state
fit2<-DPMlmm(fixed=height~1,random=~age|child,prior=prior,mcmc=mcmc,
state=state,status=FALSE)
fit2
# Summary with HPD and Credibility intervals
summary(fit2)
summary(fit2,hpd=FALSE)
# Plot model parameters
# (to see the plots gradually set ask=TRUE)
plot(fit2,ask=FALSE)
plot(fit2,ask=FALSE,nfigr=2,nfigc=2)
# Plot an specific model parameter
# (to see the plots gradually set ask=TRUE)
plot(fit2,ask=FALSE,nfigr=1,nfigc=2,param="sigma-(Intercept)")
plot(fit2,ask=FALSE,nfigr=1,nfigc=2,param="ncluster")
## End(Not run)
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