R: Linear dependent DP model for conditional ROC curve...

LDDProc {DPpackage} | R Documentation

## Linear dependent DP model for conditional ROC curve estimation.

### Description

This function generates a posterior density sample for a Linear Dependent Dirichlet Process Mixture of Normals model for conditional ROC curve estimations.

### Usage

    LDDProc(y.d,z.d,y.nond,z.nond,
            zpred.d,zpred.nond=NULL,prior.d,prior.nond=NULL,
            mcmc,state,status,ngrid=100,
            grid=NULL,compute.band=FALSE,type.band="PD",
            data=sys.frame(sys.parent()),na.action=na.fail,
            work.dir=NULL)

### Arguments

`y.d` | a vector giving the responses for the diseased group.
`z.d` | a matrix giving the design matrix for the diseased group.
`y.nond` | a vector giving the responses for the non-diseased group.
`z.nond` | a matrix giving the design matrix for the non-diseased group.
`zpred.d` | a matrix giving the covariate values where the predictive density is evaluated for the diseased group.
`zpred.nond` | a matrix giving the covariate values where the predictive density is evaluated for the non-diseased group. By default, zpred.nond=NULL which means that zpred.nond=zpred.d.
`prior.d` | a list giving the prior information for the diseased group. The list includes the following parameters: `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 `Sbeta0` 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`.
`prior.nond` | a list giving the prior information for the non-diseased group. The list includes the same parameters than `prior.d`. The default specification (prior.nond = NULL) uses `prior.nond`=`prior.d`.
`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).
`state` | a list giving the current value of the parameters for each individual model. 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`.
`ngrid` | integer giving the number of grid points where the conditional density estimates are evaluated. The default is 100.
`grid` | vector of grid points where the conditional density estimate is evaluated. The default value is NULL and the grid is chosen according to the range of the data.
`compute.band` | logical variable indicating whether the credible band for the conditional density and mean function must be computed.
`type.band` | string indication the type of credible band to be computed; if equal to "HPD" or "PD" then the 95 percent pointwise HPD or PD band is computed, respectively.
`data` | data frame.
`na.action` | a function that indicates what should happen when the data contain `NA`s. The default action (`na.fail`) causes `LDDProc` to print an error message and terminate if there are any incomplete observations.
`work.dir` | working directory.

### Details

The generic function fits the model described in Inacio et al. (2012) for conditional ROC curve estimation. Specifically, the function fits independent Linear Dependent Dirichlet Process Mixture of Normals models for the diseased (i=1) and non-diseased (i=2) groups. The conditional ROC curves are obtained from the conditional densities. The model is given by:

yij | fXij ~ fXij

fXi = \int N(Xij beta, sigma2) Gi(d beta d sigma2)

Gi | alphai, G0i ~ DP(alphai G0i)

where, G0 = N(beta| mubi, sbi)Gamma(sigma2|tau1/2,tau2i/2). To complete the model specification, independent hyperpriors are assumed,

alphai | a0i, b0i ~ Gamma(a0i,b0i)

mubi | m0i, Sbeta0i ~ N(m0i,Sbeta0i)

sbi | nui, psii ~ IW(nui,psii)

tau2i | taus1i, taus2i ~ Gamma(taus1i/2,taus2i/2)

The precision or total mass parameters, alphai, of the `DP` prior can be considered as random, having a `gamma` distribution, Gamma(a0i,b0i), or fixed at some particular value.

We refer the reader to the help file associated with the `LDDPdensity` function for more details about the prior specification, parameterizations and computational strategy.

### Value

An object of class `LDDProc` representing the two LDDP mixture of normals model fits. Generic functions such as `print`, `plot`, and `summary` have methods to show the results of the fit. The results for each model include `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. Two different objects are included: `state.d` and `state.nd`. 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, each of the lists included in `state` must include the following objects:

`betaclus` | 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

Inacio, V., Jara, A., Hanson, T.E., de Carvalho, M. (2012) Bayesian nonparametric ROC regression modeling with application to diabetes diagnosis. Technical report.

### See Also

`LDDPdensity`

### Examples

    ## Not run:

        ##############################################################
        # Simulated data example.
        # - Data generated using "perfect" simulation.
        # - one binary predictor.
        # - 250 observations in each
        #   combination of predictor and
        #   status.
        ##############################################################

        # Functions required for simulation

          findq <- function(true.cdf,target,low,upp,
                            epsilon=0.0000001)
          {
               plow <- true.cdf(low)
               pupp <- true.cdf(upp)
               pcenter <- true.cdf((upp+low)/2)
               err <- abs(pcenter-target)
               i <- 0
               while(err > epsilon)
               {
                     i <- i + 1
                     if(target< pcenter)
                     {
                        upp <- (upp+low)/2
                        pupp <- pcenter
                        pcenter <- true.cdf((upp+low)/2)
                        err <- abs(pcenter-target)
                     }
                     if(target>= pcenter)
                     {
                        low <- (upp+low)/2
                        plow <- pcenter
                        pcenter <- true.cdf((upp+low)/2)
                        err <- abs(pcenter-target)
                     }
               }
               return((upp+low)/2)
          }

          true.cdf.nond1 <- function(x)
          {
             pnorm(x,2.1,sqrt(0.0324))
          }

          true.cdf.nond2 <- function(x)
          {
    	 0.5*pnorm(x,1.85,sqrt(0.005))+
             0.5*pnorm(x,2.25,sqrt(0.005))
          }

          true.cdf.d1 <- function(x)
          {
    	 0.5*pnorm(x,1.95,sqrt(0.005))+
             0.5*pnorm(x,2.35,sqrt(0.005))
          }

          true.cdf.d2 <- function(x)
          {
              pnorm(x,2.5,sqrt(0.0324))
          }

        # Simulating the data

          nsim <- 250
          qq <- seq(1,nsim)/(nsim+1)

          y.nond1 <- rep(0,nsim)
          for(i in 1:nsim)
          {
              aa <- findq(true.cdf.nond1,qq[i],
                          low=-6,upp=6,epsilon=0.0000001)
              y.nond1[i] <- aa
          }

          y.nond2 <- rep(0,nsim)
          for(i in 1:nsim)
          {
              aa <- findq(true.cdf.nond2,qq[i],
                          low=-6,upp=6,epsilon=0.0000001)
              y.nond2[i] <- aa
          }
          y.nond <- c(y.nond1,y.nond2)
          trt.nond <- c(rep(0,nsim),rep(1,nsim))

          y.d1 <- rep(0,nsim)
          for(i in 1:nsim)
          {
              aa <- findq(true.cdf.d1,qq[i],
                          low=-6,upp=6,epsilon=0.0000001)
              y.d1[i] <- aa
          }

          y.d2 <- rep(0,nsim)
          for(i in 1:nsim)
          {
              aa <- findq(true.cdf.d2,qq[i],
                          low=-6,upp=6,epsilon=0.0000001)
              y.d2[i] <- aa
          }

          y.d <- c(y.d1,y.d2)
          trt.d <- c(rep(0,nsim),rep(1,nsim))

        # Design matrices

          z.d <- cbind(rep(1,2*nsim),trt.d)
          colnames(z.d) <- c("(Intercept)","trt")
          z.nond <- cbind(rep(1,2*nsim),trt.nond)
          colnames(z.nond) <- c("(Intercept)","trt")

        # design matrix for posterior predictive inference

          zpred <- rbind(c(1,0),c(1,1))

        # Prior information
          prior <- list(a0=10,
                        b0=1,
                        nu=4,
                        m0=rep(0,2),
                        S0=diag(100,2),
                        psiinv=diag(1,2),
                        tau1=6.01,
                        taus1=6.01,
                        taus2=2.01)

        # Initial state
          state <- NULL

        # MCMC parameters

          nburn <- 5000
          nsave <- 5000
          nskip <- 4
          ndisplay <- 500
          mcmc <- list(nburn=nburn,
                       nsave=nsave,
                       nskip=nskip,
                       ndisplay=ndisplay)

        # Fitting the model

          fit1 <- LDDProc(y.d=y.d,z.d=z.d,
                          y.nond=y.nond,z.nond=z.nond,
                          zpred.d=zpred,
                          prior.d=prior,
                          prior.nond=prior,
                          mcmc=mcmc,
                          state=state,
                          status=TRUE,
                          compute.band=TRUE)

          fit1
          summary(fit1)
          plot(fit1)

         # Ploting the conditional
         # ROC curve for x=c(1,0),
         # along with the true curve

           par(cex=1.7,mar=c(4.1, 4.1, 1, 1))

           plot(fit1$rocgrid,fit1$rocp.h[1,],type="l",
                lty=2,lwd=2,ylim=c(0,1),xlim=c(0,1),
                xlab="False positive rate",
                ylab="True positive rate")
           lines(fit1$rocgrid,fit1$rocp.l[1,],lty=2,lwd=2)
           lines(fit1$rocgrid,fit1$rocp.m[1,],lty=1,lwd=2)

           nn <- length(fit1$rocgrid)
           tt <- rep(0,nn)
           for(i in 1:nn)
           {
    	   tt[i] <- findq(true.cdf.nond1,
                              1-fit1$rocgrid[i],
                              low=-6,upp=6,
                              epsilon=0.0000001)
           }
           true.roc1 <- 1.0 - true.cdf.d1(tt)
           lines(fit1$rocgrid,true.roc1,
                 lty=1,lwd=3,col="red")

         # Ploting the conditional
         # ROC curve for x=c(1,1),
         # along with the true curve

           par(cex=1.7,mar=c(4.1, 4.1, 1, 1))

           plot(fit1$rocgrid,fit1$rocp.h[2,],type="l",
                lty=2,lwd=2,ylim=c(0,1),xlim=c(0,1),
                xlab="False positive rate",
                ylab="True positive rate")
           lines(fit1$rocgrid,fit1$rocp.l[2,],lty=2,lwd=2)
           lines(fit1$rocgrid,fit1$rocp.m[2,],lty=1,lwd=2)

           nn <- length(fit1$rocgrid)
           tt <- rep(0,nn)
           for(i in 1:nn)
           {
               tt[i] <- findq(true.cdf.nond2,
                              1-fit1$rocgrid[i],
                              low=-6,upp=6,
                              epsilon=0.0000001)
           }
           true.roc2 <- 1.0 - true.cdf.d2(tt)
           lines(fit1$rocgrid,true.roc2,lty=1,lwd=3,col="red")

    ## End(Not run)

[Package DPpackage version 1.1-7.4 Index]