R: Bayesian analysis for a dependent semiparametric two...

LDDPtwopl {DPpackage} | R Documentation

## Bayesian analysis for a dependent semiparametric two parameters logistic model

### Description

This function generates a posterior density sample for a semiparametric two parameters logistic model, using a LDDP mixture of normals prior for the distribution of the random effects.

### Usage

    LDDPtwopl(formula,prior,mcmc,
              state,status,
              grid=seq(-10,10,length=1000),
              zpred,data=sys.frame(sys.parent()),
              compute.band=FALSE)

### Arguments

`formula` | a two-sided linear formula object describing the model fit, with the response on the left of a `~` operator and the terms, separated by `+` operators, on the right. The design matrix is used to model the distribution of the response in the LDPP mixture of normals model.
`prior` | a list giving the prior information. The list includes 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` is missing), `m0` and `S0` giving the hyperparameters of the normal prior distribution for the mean of the normal baseline distribution, `mub` giving the mean of the normal baseline distribution of the regression coefficients (is must be specified if `m0` is missing), `nu` and `psiinv` giving the hyperparameters of the inverted Wishart prior distribution for the scale matrix, `sigmab`, of the baseline distribution, `sigmab` giving the variance of the baseline distribution (is must be specified if `nu` is missing), `tau1` giving the hyperparameter for the prior distribution of variance of the normal kernel, and `taus1` and `taus2` giving th hyperparameters of the gamma distribution for `tau2`, `beta0` and `Sbeta0` giving the hyperparameters of the normal prior distribution for the difficulty parameters, and `r1` and `r2` giving the hyperparameters of the lognormal prior distribution for the discrimination parameters.
`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`.
`grid` | grid points where the density estimate is evaluated. The default is seq(-10,10,length=1000).
`zpred` | a matrix giving the covariate values where the predictive density is evaluated.
`data` | data frame.
`compute.band` | logical variable indicating whether the confidence band for the density and CDF must be computed.

### Details

This generic function fits a linear dependent semiparametric two parameters logistic model:

etaij = gammaj(thetai - betaj), i=1,…,n, j=1,…,k

beta | beta0, Sbeta0 ~ N(beta0,Sbeta0)

gammaj | r1, r2 ~ LN(r1,r2), j=2,…,k

thetai | fXi ~ fXi

fXi = \int N(Xi alphac, sigma2) G(d alphac d sigma2)

G | alpha, G0 ~ DP(alpha G0)

where, G0 = N(alphac| mub, sb)Gamma(sigma^-2|tau1/2,tau2/2). To complete the model specification, the following independent hyperpriors are assumed,

alpha | a0, b0 ~ Gamma(a0,b0)

mub | m0, S0 ~ N(m0,S0)

sb | nu, psi ~ IW(nu,psi)

tau2 ~ Gamma(tau2 | taus1, taus2 ~ Gamma(taus1/2,taus2/2)

Note that the inverted-Wishart prior is parametrized such that if A ~ IWq(nu, psi) then E(A)= psiinv/(nu-q-1).

Note also that the LDDP model is a natural and simple extension of the the ANOVA DDP model discussed in in De Iorio et al. (2004). The same model is used in Mueller et al.(2005) as the random effects distribution in a repeated measurements model.

The precision or total mass parameter, alpha, of the `DP` prior can be considered as random, having a `gamma` distribution, Gamma(a0,b0), or fixed at some particular value. When alpha is random the method described by Escobar and West (1995) is used. To let alpha to be fixed at a particular value, set a0 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 non-conjugate DPM models (see, e.g, MacEachern and Muller, 1998; Neal, 2000).

### Value

An object of class `LDDPtwopl` representing the LDDP mixture of normals two parameters logistic model. Generic functions such as `print`, `plot`, and `summary` have methods to show the results of the fit. The results include `beta`, `gamma`, `mub`, `sb`, `tau2`, the precision parameter `alpha`, and the number of clusters.

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:

`b` | a vector of dimension nsubjects giving the value of the random effects for each subject.
`beta` | giving the value of the difficulty parameters.
`gamma` | giving the value of the discrimination parameters.
`alphaclus` | a matrix of dimension (number of subject + 100) times the number of columns in the design matrix, giving the regression coefficients for each cluster (only the first `ncluster` are considered to start the chain).
`sigmaclus` | a vector of dimension (number of subjects + 100) giving the variance of the normal kernel for each cluster (only the first `ncluster` are considered to start the chain).
`alpha` | giving the value of the precision parameter.
`mub` | giving the mean of the normal baseline distributions.
`sb` | giving the covariance matrix the normal baseline distributions.
`ncluster` | an integer giving the number of clusters.
`ss` | an interger vector defining to which of the `ncluster` clusters each subject belongs.
`tau2` | giving the value of the tau2 parameter.

### Author(s)

Alejandro Jara <atjara@uc.cl>

### References

De Iorio, M., Muller, P., Rosner, G., and MacEachern, S. (2004), An ANOVA model for dependent random measures," Journal of the American Statistical Association, 99(465): 205-215.

De Iorio, M., Johnson, W., Muller, P., and Rosner, G.L. (2009) Bayesian Nonparametric Nonproportional Hazards Survival Modeling. Biometrics, To Appear.

Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588.

Farina, P., Quintana, E., San Martin, E., Jara, A. (2009). A Dependent Semiparametric Rasch Model for the Analysis of Chilean Educational Data. In preparation.

MacEachern, S. N. and Muller, P. (1998) Estimating mixture of Dirichlet Process Models. Journal of Computational and Graphical Statistics, 7 (2): 223-338.

Mueller, P., Rosner, G., De Iorio, M., and MacEachern, S. (2005). A Nonparametric Bayesian Model for Inference in Related Studies. Applied Statistics, 54 (3), 611-626.

Neal, R. M. (2000). Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9: 249-265.

### See Also

`LDDPrasch`

### Examples

    ## Not run:
        ####################################
        # A simulated Data Set
        ####################################

          grid <- seq(-4,4,0.01)

          dtrue1 <- function(grid)
          {
             0.6*dnorm(grid,-1,0.4)+
             0.3*dnorm(grid,0,0.5)+
             0.1*dnorm(grid,1,0.5)
          }

          dtrue2 <- function(grid)
          {
             0.5*dnorm(grid,-1,0.5)+
             0.5*dnorm(grid,1,0.5)
          }

          dtrue3 <- function(grid)
          {
             0.1*dnorm(grid,-1,0.5)+
             0.3*dnorm(grid,0,0.5)+
             0.6*dnorm(grid,1,0.4)
          }

          rtrue1 <- function(n)
          {
              ind <- sample(x=c(1,2,3),
              size=n,replace =TRUE,
              prob =c(0.6,0.3,0.1))
              x1 <- rnorm(n,-1,0.4)
              x2 <- rnorm(n, 0,0.5)
              x3 <- rnorm(n, 1,0.5)
              x <- rep(0,n)
              x[ind==1] <- x1[ind==1]
              x[ind==2] <- x2[ind==2]
              x[ind==3] <- x3[ind==3]
              return(x)
          }

          rtrue2 <- function(n)
          {
              ind <- sample(x=c(1,2),
              size=n,replace=TRUE,
              prob =c(0.5,0.5))
              x1 <- rnorm(n,-1,0.5)
              x2 <- rnorm(n, 1,0.5)
              x <- rep(0,n)
              x[ind==1] <- x1[ind==1]
              x[ind==2] <- x2[ind==2]
              return(x)
          }

          rtrue3 <- function(n)
          {
              ind <- sample(x=c(1,2,3),
              size=n,replace=TRUE,
              prob =c(0.1,0.3,0.6))
              x1 <- rnorm(n,-1,0.5)
              x2 <- rnorm(n, 0,0.5)
              x3 <- rnorm(n, 1,0.4)
              x <- rep(0,n)
              x[ind==1] <- x1[ind==1]
              x[ind==2] <- x2[ind==2]
              x[ind==3] <- x3[ind==3]
              return(x)
          }

          b1 <- rtrue1(n=200)
          hist(b1,prob=TRUE,xlim=c(-4,4),ylim=c(0,0.7))
          lines(grid,dtrue1(grid))

          b2 <- rtrue2(n=200)
          hist(b2,prob=TRUE,xlim=c(-4,4),ylim=c(0,0.7))
          lines(grid,dtrue2(grid))

          b3 <- rtrue3(n=200)
          hist(b3,prob=TRUE,xlim=c(-4,4),ylim=c(0,0.7))
          lines(grid,dtrue3(grid))

          nsubject <- 600
          theta <- c(b1,b2,b3)
          trt <- as.factor(c(rep(1,200),rep(2,200),rep(3,200)))
          nitem <- 40

          y <- matrix(0,nrow=nsubject,ncol=nitem)
          dimnames(y)<-list(paste("id",seq(1:nsubject)),
                            paste("item",seq(1,nitem)))

          beta <- c(0,sample(seq(-1.8,1.8,length=nitem-1)))
          gam <- c(1,sample(seq(0.2,1.4,length=nitem-1)))

          for(i in 1:nsubject)
          {
             for(j in 1:nitem)
             {
                eta <- gam[j]*(theta[i]-beta[j])
                prob <- exp(eta)/(1+exp(eta))
                y[i,j] <- rbinom(1,1,prob)
             }
          }

       ##############################
       # design's prediction matrix
       ##############################

         zpred <- matrix(c(1,0,0,
                           1,1,0,
                           1,0,1),nrow=3,ncol=3,byrow=TRUE)

       ###########################
       # prior
       ###########################

         prior <- list(alpha=1,
                       beta0=rep(0,nitem-1),
                       Sbeta0=diag(1000,nitem-1),
                       r1=0,
                       r2=1,
                       mu0=rep(0,3),
                       S0=diag(100,3),
                       tau1=6.01,
                       taus1=6.01,
                       taus2=2.01,
                       nu=5,
                       psiinv=diag(1,3))

       ###########################
       # mcmc
       ###########################
         mcmc <- list(nburn=5000,
                      nskip=3,
                      ndisplay=200,
                      nsave=5000)

       ###########################
       # fitting the model
       ###########################

         fitLDDP <-  LDDPtwopl(formula=y ~ trt,
                               prior=prior,
                               mcmc=mcmc,
                               state=NULL,
                               status=TRUE,
                               zpred=zpred,
                               grid=grid,compute.band=TRUE)

         fitLDDP

         summary(fitLDDP)

       #########################################
       # plots
       #########################################
         plot(fitLDDP)

         plot(fitLDDP,param="prediction")

       #########################################
       # plot the estimated and true densities
       #########################################

         par(cex=1.5,mar=c(4.1, 4.1, 1, 1))
         plot(fitLDDP$grid,fitLDDP$dens.m[1,],xlim=c(-4,4),ylim=c(0,0.8),
              type="l",lty=1,lwd=3,xlab="Ability",ylab="density",col=1)
         lines(fitLDDP$grid,fitLDDP$dens.u[1,],lty=2,lwd=3,col=1)
         lines(fitLDDP$grid,fitLDDP$dens.l[1,],lty=2,lwd=3,col=1)
         lines(grid,dtrue1(grid),lwd=3,col="red",lty=3)

         par(cex=1.5,mar=c(4.1, 4.1, 1, 1))
         plot(fitLDDP$grid,fitLDDP$dens.m[2,],xlim=c(-4,4),ylim=c(0,0.8),
              type="l",lty=1,lwd=3,xlab="Ability",ylab="density",col=1)
         lines(fitLDDP$grid,fitLDDP$dens.u[2,],lty=2,lwd=3,col=1)
         lines(fitLDDP$grid,fitLDDP$dens.l[2,],lty=2,lwd=3,col=1)
         lines(grid,dtrue2(grid),lwd=3,col="red",lty=3)

         par(cex=1.5,mar=c(4.1, 4.1, 1, 1))
         plot(fitLDDP$grid,fitLDDP$dens.m[3,],xlim=c(-4,4),ylim=c(0,0.8),
              type="l",lty=1,lwd=3,xlab="Ability",ylab="density",col=1)
         lines(fitLDDP$grid,fitLDDP$dens.u[3,],lty=2,lwd=3,col=1)
         lines(fitLDDP$grid,fitLDDP$dens.l[3,],lty=2,lwd=3,col=1)
         lines(grid,dtrue3(grid),lwd=3,col="red",lty=3)

       #########################################
       # Extract random effects
       #########################################
         DPrandom(fitLDDP)
         plot(DPrandom(fitLDDP))
         DPcaterpillar(DPrandom(fitLDDP))

    ## End(Not run)

[Package DPpackage version 1.1-7.4 Index]