Description Usage Arguments Details Value Author(s) References See Also Examples
This function generates a posterior density sample for a Linear Dependent Dirichlet Process Mixture of Normals model for conditional ROC curve estimations.
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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: |
prior.nond |
a list giving the prior information for the non-diseased group. The list
includes the same parameters than |
mcmc |
a list giving the MCMC parameters. The list must include
the following integers: |
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 ( |
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 |
work.dir |
working directory. |
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.
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 |
sigmaclus |
a vector of dimension (number of subjects + 100) giving the variance of the normal kernel for
each cluster (only the first |
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 |
tau2 |
giving the value of the tau2 parameter. |
Alejandro Jara <atjara@uc.cl>
Inacio, V., Jara, A., Hanson, T.E., de Carvalho, M. (2012) Bayesian nonparametric ROC regression modeling with application to diabetes diagnosis. Technical report.
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##############################################################
# 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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