GLMM_MCMC: MCMC estimation of generalized linear mixed model with mixtures in the distributions.

Description

THIS FUNCTION IS BEING DEVELOPED AND ORDINARY USERS ARE NOT RECOMMENDED TO USE IT. This function runs MCMC for a generalized linear mixed model with possibly several response variables and possibly normal mixtures in the distributions of random effects.

Usage

GLMM_MCMC(y, dist="gaussian", id, x, z, random.intercept,
          prior.beta, init.beta,                      
          scale.b,    prior.b,   init.b,
          prior.eps,  init.eps,
          nMCMC=c(burn=10, keep=10, thin=1, info=10),
          tuneMCMC=list(beta=1, b=1),
          store=c(b=FALSE), keep.chains=TRUE)
## S3 method for class 'GLMM_MCMC':
print(x, \dots)

Arguments

vector, matrix or data frame with responses. If y is vector then there is only one response in the model. If y is matrix or data frame then each column gives values of one response. Missing values are allowed.

dist

character (vector) which determines distribution (and a link function) for each response variable. Possible values are: gaussian for gaussian (normal) distribution (with identity link), binomial(logit) for

vector which determines longitudinally or otherwise dependent observations. If not given then it is assumed that there are no clusters and all observations of one response are independent.

matrix or a list of matrices with covariates (intercept not included) for fixed effects. If there is more than one response, this must always be a list. Note that intercept in included in all models. Use a character value empty as

matrix or a list of matrices with covariates (intercept not included) for random effects. If there is more than one response, this must always be a list. Note that random intercept is specified using the argument random.intercept.

random.intercept

logical (vector) which determines for which responses random intercept should be included.

prior.beta

a list which specifies prior distribution for fixed effects (not the means of random effects). The prior distribution is normal and the user can specify the mean and variances. The list prior.b can have the components listed below

init.beta

a numeric vector with initial values of fixed effects (not the means of random effects). A sensible value is determined using the maximum-likelihood fits (using lmer functions) and does not hav

scale.b

a list specifying how to scale the random effects during the MCMC. A sensible value is determined using the maximum-likelihood fits (using lmer functions) and does not have to be given by the u

prior.b

a list which specifies prior distribution for (shifted and scaled) random effects. The prior is in principle a normal mixture (being a simple normal distribution if we restrict the number of mixture components to be equal to one). The

init.b

a list with initial values for parameters related to the distribution of random effects and random effects themselves. Sensible initial values are determined by the function itself and do not have to be given by the user.

prior.eps

a list specifying prior distributions for error terms for continuous responses. The list prior.eps can have the components listed below. For all components, a sensible value leading to weakly informative prior distribution can

init.eps

a list with initial values for parameters related to the distribution of error terms of continuous responses. The list init.eps can have the components listed below. For all components, a sensible value can be determined by the fu

nMCMC

numeric vector of length 4 giving parameters of the MCMC simulation. Its components may be named (ordering is then unimportant) as: [object Object],[object Object],[object Object],[object Object] In total $(M_{burn} + M_{keep}) \cdot M_{thin}$

tuneMCMC

a list with tuning scale parameters for proposal distribution of fixed and random effects. It is used only when there are some discrete response profiles. The components of the list have the following meaning: [object Object],[object Objec

store

logical vector indicating whether the chains of parameters should be stored. Its components may be named (ordering is then unimportant) as: [object Object]

keep.chains

logical. If FALSE, only summary statistics are returned in the resulting object. This might be useful in the model searching step to save some memory.

...

additional arguments passed to the default print method.

Value

An object of class GLMM_MCMC. It can have the following components (some of them may be missing according to the context of the model):
iterindex of the last iteration performed.
nMCMCused value of the argument nMCMC.
dista character vector of length R corresponding to the dist argument.
Ra two component vector giving the number of continuous responses and the number of discrete responses.
pa numeric vector of length R giving the number of non-intercept beta parameters for each response.
qa numeric vector of length R giving the number of non-intercept random effects for each response.
fixed.intercepta logical vector of length R which indicates inclusion of fixed intercept for each response.
random.intercepta logical vector of length R which indicates inclusion of random intercept for each response.
lbetalength of the vector of fixed effects.
dimbdimension of the distribution of random effects.
prior.betaa list containing the used value of the argument prior.beta.
prior.ba list containing the used value of the argument prior.b.
prior.epsa list containing the used value of the argument prior.eps.
init.betaa numeric vector with the used value of the argument init.beta.
init.ba list containing the used value of the argument init.b.
init.epsa list containing the used value of the argument init.eps.
state.betaa numeric vector with the last sampled value of fixed effects $\beta$. It can be used as argument init.beta to restart MCMC.
state.ba list with the last sampled values of parameters related to the distribution of random effects. It has components named b, K, w, mu, Sigma, Li, Q, gammaInv, r. It can be used as argument init.b to restart MCMC.
state.epsa list with the last sampled values of parameters related to the distribution of residuals of continuous responses. It has components named sigma, gammaInv. It can be used as argument init.eps to restart MCMC.
prop.accept.betaacceptance proportion from the Metropolis-Hastings algorithm for fixed effects (separately for each response type). Note that the acceptance proportion is equal to one for continuous responses since the Gibbs algorithm is used there.
prop.accept.bacceptance proportion from the Metropolis-Hastings algorithm for random effects (separately for each cluster). Note that the acceptance proportion is equal to one for models with continuous responses only since the Gibbs algorithm is used there.
scale.ba list containing the used value of the argument scale.b.
poster.mean.etaa data.frame with columns labeled fixed and random holding posterior means for fixed effect part of the linear predictor and the random effect part of the linear predictor. In each column, there are first all values for the first response, then all values for the second response etc.
poster.mean.profilea data.frame with columns labeled b1, ..., bq, LogL, Logpb with posterior means of random effects for each cluster and posterior means of $\log(L)$ (log-likelihood given random effects) and $\log\bigl{p(\boldsymbol{b})\bigr}$ for each cluster.
poster.mean.w_ba numeric vector with posterior means of mixture weights after re-labeling. It is computed only if $K_b$ is fixed and even then I am not convinced that these are useful posterior summary statistics. In any case, they should be used with care.
poster.mean.mu_ba matrix with posterior means of mixture means after re-labeling. It is computed only if $K_b$ is fixed and even then I am not convinced that these are useful posterior summary statistics. In any case, they should be used with care.
poster.mean.Q_ba list with posterior means of mixture inverse variances after re-labeling. It is computed only if $K_b$ is fixed and even then I am not convinced that these are useful posterior summary statistics. In any case, they should be used with care.
poster.mean.Sigma_ba list with posterior means of mixture variances after re-labeling. It is computed only if $K_b$ is fixed and even then I am not convinced that these are useful posterior summary statistics. In any case, they should be used with care.
poster.mean.Li_ba list with posterior means of Cholesky decompositions of mixture inverse variances after re-labeling. It is computed only if $K_b$ is fixed and even then I am not convinced that these are useful posterior summary statistics. In any case, they should be used with care.
poster.comp.prob1a matrix which is present in the output object if the number of mixture components in the distribution of random effects is fixed and equal to $K$. In that case, poster.comp.prob1 is a matrix with $K$ columns and $I$ rows ($I$ is the number of subjects defining the longitudinal profiles or correlated observations) with estimated posterior component probabilities -- posterior means of the components of the underlying 0/1 allocation vector. These can be used for possible clustering of the subjects based on the longitudinal profiles.
poster.comp.prob2a matrix which is present in the output object if the number of mixture components in the distribution of random effects is fixed and equal to $K$. In that case, poster.comp.prob2 is a matrix with $K$ columns and $I$ rows ($I$ is the number of subjects defining the longitudinal profiles or correlated observations) with estimated posterior component probabilities -- posterior mean over model parameters including random effects. These can be used for possible clustering of the subjects based on the longitudinal profiles.
summ.betaa matrix with posterior summary statistics for fixed effects.
summ.b.Meana matrix with posterior summary statistics for means of random effects.
summ.b.SDCorra matrix with posterior summary statistics for standard deviations of random effects and correlations of each pair of random effects.
summ.sigma_epsa matrix with posterior summary statistics for standard deviations of the error terms in the (mixed) models of continuous responses.
freqK_bfrequency table for the MCMC sample of the number of mixture components in the distribution of the random effects.
propK_bposterior probabilities for the numbers of mixture components in the distribution of random effects.
K_bnumeric vector with a chain for $K_b$ (number of mixture components in the distribution of random effects).
w_bnumeric vector or matrix with a chain for $w_b$ (mixture weights for the distribution of random effects). It is a matrix with $K_b$ columns when $K_b$ is fixed. Otherwise, it is a vector with weights put sequentially after each other.
mu_bnumeric vector or matrix with a chain for $\mu_b$ (mixture means for the distribution of random effects). It is a matrix with $dimb\cdot K_b$ columns when $K_b$ is fixed. Otherwise, it is a vector with means put sequentially after each other.
Q_bnumeric vector or matrix with a chain for lower triangles of $\boldsymbol{Q}_b$ (mixture inverse variances for the distribution of random effects). It is a matrix with $\frac{dimb(dimb+1)}{2}\cdot K_b$ columns when $K_b$ is fixed. Otherwise, it is a vector with lower triangles of $\boldsymbol{Q}_b$ matrices put sequentially after each other.
Sigma_bnumeric vector or matrix with a chain for lower triangles of $\Sigma_b$ (mixture variances for the distribution of random effects). It is a matrix with $\frac{dimb(dimb+1)}{2}\cdot K_b$ columns when $K_b$ is fixed. Otherwise, it is a vector with lower triangles of $\Sigma_b$ matrices put sequentially after each other.
Li_bnumeric vector or matrix with a chain for lower triangles of Cholesky decompositions of $\boldsymbol{Q}_b$ matrices. It is a matrix with $\frac{dimb(dimb+1)}{2}\cdot K_b$ columns when $K_b$ is fixed. Otherwise, it is a vector with lower triangles put sequentially after each other.
gammaInv_bmatrix with $dimb$ columns with a chain for inverses of the hyperparameter $\boldsymbol{\gamma}_b$.
order_bnumeric vector or matrix with order indeces of mixture components in the distribution of random effects. It is a matrix with $K_b$ columns when $K_b$ is fixed. Otherwise it is a vector with orders put sequentially after each other.
rank_bnumeric vector or matrix with rank indeces of mixture components in the distribution of random effects. It is a matrix with $K_b$ columns when $K_b$ is fixed. Otherwise it is a vector with ranks put sequentially after each other.
mixture_bdata.frame with columns labeled b.Mean.*, b.SD.*, b.Corr.*.* containing the chains for the means, standard deviations and correlations of the distribution of the random effects based on a normal mixture at each iteration.
ba matrix with the MCMC chains for random effects. It is included only if store[b] is TRUE.
betanumeric vector or matrix with the MCMC chain(s) for fixed effects.
sigma_epsnumeric vector or matrix with the MCMC chain(s) for standard deviations of the error terms in the (mixed) models for continuous responses.
gammaInv_epsmatrix with $dimb$ columns with MCMC chain(s) for inverses of the hyperparameter $\boldsymbol{\gamma}_b$.

Details

See accompanying paper ($\mbox{Kom\'arek}$ et al., 2010).

References

$\mbox{Kom\'{a}rek, A.}$, Hansen, B. E., Kuiper, E. M. M., van Buuren, H. R., and Lesaffre, E. (2010). Discriminant analysis using a multivariate linear mixed model with a normal mixture in the random effects distribution. Statistics in Medicine. To appear.

Examples

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