DPpackage (version 1.1-7.4)

PTlmm: Bayesian analysis for a semiparametric linear mixed model using a MMPT

Description

This function generates a posterior density sample for a semiparametric linear mixed model, using a Mixture of Multivariate Polya Trees prior for the distribution of the random effects.

Usage

PTlmm(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 Polya Tree (PT) 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, sigma giving the value of the covariance matrix of the centering distribution (it must be specified if nu0 and tinv are missing), mub and Sb giving the hyperparameters of the normal prior distribution for the mean of the normal baseline distribution, mu giving the value of the mean of the centering distribution (it must be specified if mub and Sb are missing), 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), tau1 and tau2 giving the hyperparameters for the prior distribution of the error variance, M giving the finite level of the PT prior to be considered, frstlprob a logical variable indicating whether the first level probabilities of the PT are fixed or not (the default is FALSE), and typepr indicating whether the type of decomposition of the centering covariance matrix is random (1) or not (0).

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, 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), nbase giving the number scans to be performed before the parameters of the centering distribution and the precision parameter are updated (i.e., the update of this parameters is invoked only once in every nbase scans) (the default value is 1), tune1, tune2, and tune3, giving the Metropolis tuning parameter for the baseline mean, variance, and precision parameter, respectively. If tune1, tune2, or tune3 are not specified or negative, an adpative Metropolis algorithm is performed. Finally, the integer samplef indicates whether the functional parameters must be sample (1) or not (0).

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 PTlmm to print an error message and terminate if there are any incomplete observations.

Value

An object of class PTlmm 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 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.

sigma2e

giving the error variance.

ortho

giving the orthogonal matrix H, used in the decomposition of the covariance matrix.

Details

This generic function fits a 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): $$y_i \sim N(X_i \beta_F + Z_i \beta_R + Z_i b_i, \sigma^2_e I_{n_i}), i=1,\ldots,n$$ $$\theta_i | G \sim G$$ $$G | \alpha,\mu,\Sigma,O \sim PT^M(\Pi^{\mu,\Sigma,O},\mathcal{A})$$ $$\sigma^{-2}_e | \tau_1, \tau_2 \sim Gamma(\tau_1/2,\tau_2/2)$$

where, \(\theta_i = \beta_R + b_i\), \(\beta = \beta_F\), and \(O\) is an orthogonal matrix defining the decomposition of the centering covariance matrix. As in Hanson (2006), the PT prior is centered around a \(N_d(\mu,\Sigma)\) 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 \(\mathcal{A}=\{\alpha_e: e \in E^{*}\}\), where \(E^{*}=\bigcup_{m=0}^{M} E_d^m\), with \(E_d\) and \(E_d^m\) the \(d\)-fold product of \(E=\{0,1\}\) and the the \(m\)-fold product of \(E_d\), respectively. The family \(\mathcal{A}\) was specified as \(\alpha_{e_1 \ldots e_m}=\alpha m^2\).

To complete the model specification, independent hyperpriors are assumed, $$\alpha | a_0, b_0 \sim Gamma(a_0,b_0)$$ $$\beta | \beta_0, S_{\beta_0} \sim N(\beta_0,S_{\beta_0})$$ $$\mu | \mu_b, S_b \sim N(\mu_b,S_b)$$ $$\Sigma | \nu_0, T \sim IW(\nu_0,T)$$ $$O \sim Haar(q)$$

Note that the inverted-Wishart prior is parametrized such that \(E(\Sigma)= T^{-1}/(\nu_0-q-1)\).

The precision or total mass parameter, \(\alpha\), of the DP prior can be considered as random, having a gamma distribution, \(Gamma(a_0,b_0)\), or fixed at some particular value.

The computational implementation of the model is based on the marginalization of the PT as descried in Jara, Hanson and Lesaffre (2009).

References

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.

See Also

PTrandom, PTglmm , PTolmm, DPMglmm, DPMlmm, DPMolmm, DPlmm , DPglmm, DPolmm

Examples

# NOT RUN {
    # School Girls Data Example
      data(schoolgirls)
      attach(schoolgirls)

    # Prior information
      prior <- list(a0=5,b0=1,
                    M=4,
                    typepr=1,
                    frstlprob=FALSE,
                    tau1=0.01,tau2=0.01,
                    nu0=4.01,
                    tinv=diag(10,2),
                    mub=rep(0,2),
                    Sb=diag(1000,2))

    # Initial state
      state <- NULL

    # MCMC parameters

      nburn <- 10000
      nsave <- 10000
      nskip <- 20
      ndisplay <- 1000
      mcmc <- list(nburn=nburn,nsave=nsave,nskip=nskip,ndisplay=ndisplay,
                   tune1=1.5,tune2=1.1,samplef=1)

    # Fitting the model
    
      fit1 <- PTlmm(fixed=height~1,random=~age|child,prior=prior,mcmc=mcmc,
                    state=state,status=TRUE)
      fit1

    # Summary with HPD and Credibility intervals
      summary(fit1)
      summary(fit1,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="sigma-(Intercept)")	

    # Random effects information
      PTrandom(fit1)
      
    # Predictive marginal and joint distributions
      plot(PTrandom(fit1,predictive=TRUE))
# }