LDDProc: 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 NAs. The default action (na.fail) causes LDDProc to print an error message and terminate if there are any incomplete observations.

work.dir

working directory.

Value

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.

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: $$y_{ij} | f_{X_{ij}} \sim f_{X_{ij}}$$ $$f_{X_{ij}} = \int N(X_{ij} \beta, \sigma^2) G_i(d \beta d \sigma^2)$$ $$G_i | \alpha_i, G_{0i} \sim DP(\alpha_i G_{0i})$$ where, $G0 = N(beta| mubi, sbi)Gamma(sigma2|tau1/2,tau2i/2)$. To complete the model specification, independent hyperpriors are assumed, $$\alpha_i | a_{0i}, b_{0i} \sim Gamma(a_{0i},b_{0i})$$ $$\mu_{bi} | m_{0i}, S_{\beta_{0i}} \sim N(m_{0i},S_{\beta_{0i}})$$ $$s_{bi} | \nu_i, \Psi_i \sim IW(\nu_i,\Psi_i)$$ $$\tau_{2i} | \tau_{s1i}, \tau_{s2i} \sim Gamma(\tau_{s1i}/2,\tau_{s2i}/2)$$

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.

References

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

Examples

Run this code

## 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)

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