PTmeta: Bayesian analysis for a semiparametric linear mixed effects meta-analysis model using a MPT

Description

This function generates a posterior density sample for a semiparametric linear mixed effects meta-analysis model using a Polya Tree or a Mixture of Polya Trees prior for the distribution of the random effects.

Usage

PTmeta(formula,prior,mcmc,state,status,data=sys.frame(sys.parent()),
       na.action=na.fail)

Arguments

formula

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. Both effect and variance must be included in the LHS of the formula object

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 prior, alpha giving the value of the precision parameter (it must be specified if a0 and b0 are missing, see details below), tau1 and tau2 giving the hyperparameters for the prior distribution of the variance of the centering distribution, sigma giving the value of the variance of the centering distribution (it must be specified if tau1 and tau2 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), and 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), M giving the finite level of the PT prior to be considered, and frstlprob a logical variable indicating whether the first level probabilities of the PT are fixed or not (the default is FALSE) (see, details).

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

Value

An object of class PTmeta 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 beta, mu, sigma2, and alpha.

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

a vector of dimension (nsubjects) giving the value of the random effects for each subject.

beta

giving the value of the fixed effects.

giving the mean of the normal baseline distributions.

sigma2

giving the variance of the normal baseline distributions.

Details

This generic function fits a semiparametric linear mixed effects meta-analysis model using a Polya tree prior on the distribution (see, Lavine (1992; 1994) and Hanson (2006) for details about PT) on the distribution of the random effects:

$$y_i \sim N(\theta_i +X_i \beta, \sigma^2_{ei}), i=1,\ldots,n$$ $$\theta_i | G \sim G$$ $$G | \alpha,\mu,\sigma \sim PT(\Pi^{\mu,\sigma},\textit{A})$$

where the PT prior is centered around a $N(\mu,\sigma^2)$ distribution. If frstlprob is equal to TRUE, $\mu=0$ and a median zero PT prior is considered (see, Branscum and Hanson, 2008).

To complete the model specification, independent hyperpriors are assumed, $$\alpha | a_0, b_0 \sim Gamma(a_0,b_0)$$ $$\beta | \beta_0, S_{\beta_0} \sim N(\beta_0,S_{\beta_0})$$ $$\mu | \mu_b, S_b \sim N(\mu_b,S_b)$$ $$\sigma^{-2} | \tau_1, \tau_2 \sim Gamma(\tau_1/2,\tau_2/2)$$

The precision parameter, $\alpha$, of the PT 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 and on the MCMC algorihtms described in Hanson (2006) and Jara, Hanson and Lesaffre (2009).

The average effect is sampled using the method of composition described in Jara, Hanson and Lesaffre (2009).

References

Branscum, A. and Hanson, T. (2008) Bayesian nonparametric meta-analysis using Polya tree mixture models. Biometrics, 64: 825-833.

Christensen, R., Hanson, T. Jara, A.. 2008. Parametric Nonparametric Statistics: An Introduction to Mixtures of Finite Polya Trees Models. The American Statistician, 62: 296-306.

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.

Examples

Run this code

# NOT RUN {
    ##################################################################    
    # Data on the effectiveness of silver sulfadiazine coating
    # on venous catheters for preventing bacterial colonisation of 
    # the catheter and bloodstream infection. 
    # Veenstra D et al (1998) "Efficacy of Antiseptic Impregnated 
    # Central Venous Catheters in Preventing Nosocomial Infections: 
    # A Meta-analysis" JAMA 281:261-267. 
    #
    # Note that -Inf and Inf have been replaced by NA.
    ##################################################################    
    
      studies <- c("Tennenberg","Maki","vanHeerden",
                    "Hannan","Bach(a)","Bach(b)",
                    "Heard","Collins","Ciresi","Ramsay",
                    "Trazzera","George")    

      logOR <- c(-1.5187189,-0.7136877,-1.3217558,-0.1910552,
                  NA,-2.2005195,-0.5057461,-2.3538784,-0.3643810,
                  -0.5371429,-0.7608058,-2.1400662)
       
      varlogOR <- c(0.4157541,0.2632550,0.6739189,0.3727788,NA,
                    0.7623470,0.2306169,0.7477891,0.3645463,0.2291839,
                    0.3561542,0.5190489)^2

      names(logOR) <- studies
      names(varlogOR) <- studies
      y <- cbind(logOR,varlogOR)
      colnames(y) <- c("logOR","varlogOR")


    # Initial state
      state <- NULL

    # MCMC parameters

      nburn<-20000
      nsave<-10000
      nskip<-20
      ndisplay<-100
      mcmc <- list(nburn=nburn,
                   nsave=nsave,
                   nskip=nskip,
                   ndisplay=ndisplay)

    # Prior information 1: non-median zero PT

      prior1<-list(alpha=1,
                   tau1=20,
                   tau2=10,
                   mub=0,
                   Sb=100,
                   M=4)

    # Prior information 2: median zero PT

      prior2<-list(alpha=1,
                   tau1=20,
                   tau2=10,
                   mub=0,
                   Sb=100,
                   M=4,
                   frstlprob=TRUE,
                   Sbeta0=diag(1000,1),
                   beta0=rep(0,1))
                   

    # Fitting the models

      fit1<-PTmeta(formula=y~1,prior=prior1,mcmc=mcmc,
                   state=state,status=TRUE)
      fit1


      fit2<-PTmeta(formula=y~1,prior=prior2,mcmc=mcmc,
                   state=state,status=TRUE)
      fit2

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

      summary(fit2)
      summary(fit2,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(fit2,ask=FALSE)
      plot(fit2,ask=FALSE,nfigr=2,nfigc=2)	
# }

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