LDDPtwopl: 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.

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:

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.

giving the covariance matrix the normal baseline distributions.

ncluster

an integer giving the number of clusters.

an interger vector defining to which of the ncluster clusters each subject belongs.

tau2

giving the value of the tau2 parameter.

Details

This generic function fits a linear dependent semiparametric two parameters logistic model: $$\eta_{ij} = \gamma_j(\theta_i - \beta_j), i=1,\ldots,n, j=1,\ldots,k$$ $$\beta | \beta_0, S_{\beta_0} \sim N(\beta_0,S_{\beta_0})$$ $$\gamma_j | r_1, r_2 \sim LN(r_1,r_2), j=2,\ldots,k$$ $$\theta_i | f_{X_i} \sim f_{X_i}$$ $$f_{X_i} = \int N(X_i \alpha_c, \sigma^2) G(d \alpha_c d \sigma^2)$$ $$G | \alpha, G_0 \sim DP(\alpha G_0)$$ where, $G_0 = N(\alpha_c| \mu_b, s_b)\Gamma(\sigma^{-2}|\tau_1/2,\tau_2/2)$. To complete the model specification, the following independent hyperpriors are assumed, $$\alpha | a_0, b_0 \sim Gamma(a_0,b_0)$$ $$\mu_b | m_0, S_0 \sim N(m_0,S_0)$$ $$s_b | \nu, \Psi \sim IW(\nu,\Psi)$$ $$\tau_2 | \tau_{s1}, \tau_{s2} \sim Gamma(\tau_{s1}/2,\tau_{s2}/2)$$

Note that the inverted-Wishart prior is parametrized such that if $A \sim IW_q(\nu, \psi)$ then $E(A)= \psi^{-1}/(\nu-q-1)$.

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, $Gamma(a_0,b_0)$, 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 $a_0$ 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 non-conjugate DPM models (see, e.g, MacEachern and Muller, 1998; Neal, 2000).

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.

Examples

Run this code

# 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))

# }

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