DPlmm {DPpackage}R Documentation

Performs a Bayesian analysis for a semiparametric linear mixed model

Description

This function generates a posterior density sample for a semiparametric linear mixed model.

Usage


DPlmm(fixed,random,prior,mcmc,state,status,data=sys.frame(sys.parent()),
      na.action=na.fail)

Arguments

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.
prior a list giving the prior information. The list include 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 and b0 are 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, 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) and, tau1 and tau2 giving the hyperparameters for the prior distribution of the error variance.
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 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 NAs. The default action (na.fail) causes DPlmm to print an error message and terminate if there are any incomplete observations.

Details

This generic function fits a linear mixed-effects model (Verbeke and Molenberghs, 2000):

y_i sim N(X_i β_F + Z_i β_R + Z_i b_i, σ^2_e I_{n_i}), i=1,...,n

theta_i | G sim G

G | α, G_0 sim DP(α G_0)

σ^{-2}_e | tau_1, tau_2 sim Gamma(tau_1/2,tau_2/2)

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(μ_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 on the use of MCMC methods for conjugate priors (Escobar, 1994; Escobar and West, 1998).

Value

An object of class DPlmm representing the 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, sigma2e, mu, the elements of Sigma, \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.
sigma2e giving the error variance.

Author(s)

Alejandro Jara <Alejandro.JaraVallejos@med.kuleuven.be>

References

Escobar, M.D. (1994) Estimating Normal Means with a Dirichlet Process Prior, Journal of the American Statistical Association, 89: 268-277.

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.

Verbeke, G. and Molenberghs, G. (2000). Linear mixed models for longitudinal data, New York: Springer-Verlag.

See Also

DPrandom, DPglmm

Examples

## Not run: 
    # School Girls Data Example

      data(schoolgirls)
      attach(schoolgirls)

    # Prior information

      tinv<-diag(100,2)
      prior<-list(a0=0.01,b0=0.01,nu0=2,tau1=0.001,tau2=0.001,
      tinv=tinv,mub=rep(0,2),Sb=diag(10000,2))

    # Initial state
      state <- NULL

    # MCMC parameters

      nburn<-100
      nsave<-50
      nskip<-10
      ndisplay<-10
      mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip,ndisplay=ndisplay)

    # Fit the model: First run
    
      fit1<-DPlmm(fixed=height~1,random=~age|child,prior=prior,mcmc=mcmc,
                  state=state,status=TRUE)
      fit1

    # Fit the model: Continuation
      state<-fit1$state     
    
      fit2<-DPlmm(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)

[Package DPpackage version 1.0-0 Index]