DPpackage (version 1.1-0)

PTolmm: Bayesian analysis for a semiparametric ordinal linear mixed model using a MMPT

Description

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

Usage

PTolmm(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 + opera

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

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 (

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

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 specifie

data

data frame.

na.action

a function that indicates what should happen when the data contain NAs. The default action (na.fail) causes PTolmm to print an error message and terminate if there are any

Value

• An object of class PTolmm 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, 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.
• cutoff
  a real vector defining the cutoff points. Note that the first cutoff must be fixed at 0 in this function.
• 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.
• ortho
  giving the orthogonal matrix H, used in the decomposition of the covariance matrix.

Details

This generic function fits an ordinal linear mixed-effects model with a probit link and 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 Lessaffre (2009): $$Y_{ij} = k, \mbox{\ if } \gamma_{k-1} \leq W_{ij} < \gamma_{k}, k=1,\ldots,K$$ $$W_{ij} \mid \beta_F, \beta_R , b_i \sim N(X_{ij} \beta_F + Z_{ij} \beta_R + Z_{ij} b_i, 1), i=1,\ldots,N, j=1,\ldots,n_i$$ $$\theta_i | G \sim G$$ $$G | \alpha,\mu,\Sigma,O \sim PT^M(\Pi^{\mu,\Sigma,O},\mathcal{A})$$ 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, $$\alpha | a_0, b_0 \sim Gamma(a_0,b_0)$$ $$\beta | \beta_0, S_{\beta_0} \sim N(\beta_0,S_{\beta_0})$$ $$\mu | \mu_b, S_b \sim N(\mu_b,S_b)$$ $$\Sigma | \nu_0, T \sim IW(\nu_0,T)$$ $$O \sim Haar(q)$$ A uniform prior is used for the cutoff points. 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 Lessaffre (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 Mixture of Multivariate Polya Trees. Journal of Computational and Graphical Statistics (To Appear). 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, PTlmm , PTglmm, DPMglmm, DPMlmm, DPMolmm, DPlmm , DPglmm, DPolmm

Examples

# Schizophrenia Data
      data(psychiatric)
      attach(psychiatric)

    # Prior information
      prior <- list(M=4,
                    frstlprob=FALSE,
                    alpha=1,
                    nu0=4.01,
                    tinv=diag(1,1),
                    mub=rep(0,1),
                    Sb=diag(100,1),
                    beta0=rep(0,3),
                    Sbeta0=diag(1000,3))

    # MCMC parameters
      mcmc <- list(nburn=10000,
                   nsave=10000,
                   nskip=20,
                   ndisplay=100,
                   samplef=1)

    # Initial state
      state <- NULL

    # Fitting the model
      fit1 <- PTolmm(fixed=imps79o~sweek+tx+sweek*tx,random=~1|id,prior=prior,
                     mcmc=mcmc,state=state,status=TRUE)
      fit1

    # Summary with HPD and Credibility intervals
      summary(fit1)
      summary(fit1,hpd=FALSE)

    # Plot model parameters
      plot(fit1)

    # Plot an specific model parameter
      plot(fit1,ask=FALSE,nfigr=1,nfigc=2,param="sigma-(Intercept)")	

    # Extract random effects
      PTrandom(fit1)

    # Extract predictive information of random effects
      aa<-PTrandom(fit1,predictive=TRUE)
      aa

    # Predictive marginal and joint distributions      
      plot(aa)