SpatDensReg {spBayesSurv}

Bayesian Nonparametric Spatially Smoothed Density Estimation

Description

This function provides a Bayesian nonparametric density estimator that changes smoothly in space. The estimator is built from the predictive rule for a marginalized Polya tree (PT), modified so that observations are spatially weighted by their distance from the location of interest.

Usage

SpatDensReg(formula, data, na.action, prior=NULL, state=NULL,
            mcmc=list(nburn=3000, nsave=2000, nskip=0, ndisplay=500),
            permutation=TRUE, fix.theta=TRUE)

Arguments

formula

a formula expression with the response returned by the Surv function in the survival package. It supports right-censoring, left-censoring, interval-censoring, and mixtures of them. Note: for survival data, the input response should be log survival times.

data

a data frame in which to interpret the variables named in the formula argument.

na.action

a missing-data filter function, applied to the model.frame.

prior

a list giving the prior information. The list includes the following parameter: maxL an integer giving the maximum level of Polya trees, (a0, b0) parameters of the gamma prior for the precision parameter alpha, (theta0, V0) mean and variance of the normal prior for the centering distribution parameter vector theta, phiq0 the prior probability of phi=0, phib0 the rate parameter of the exponential prior of phi. The function itself provides all default priors. See Hanson, Zhou, and de Carvalho (2018).

state

a list giving the current value of the parameters. If NULL, all values are provided based on the centering parametric model.

mcmc

a list giving the MCMC parameters. The list must include the following elements: nburn an integer giving the number of burn-in scans, nskip an integer giving the thinning interval, nsave an integer giving the total number of scans to be saved, ndisplay an integer 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).

permutation

flag to indicate whether a random data permutation will be implemented in the beginning of each iterate; default is TRUE.

fix.theta

flag to indicate whether the centering distribution parameters theta=(location, log(scale)) are fixed; default is TRUE indicating fixed.

Value

The SpatDensReg object is a list containing at least the following components:

modelname

the name of the fitted model

terms

the terms object used

call

the matched call

prior

the list of hyperparameters used in all priors.

mcmc

the list of MCMC parameters used

n

the number of row observations used in fitting the model

p

the number of columns in the model matrix

Surv

the Surv object used

X

the n by p design matrix

theta

the 2 by nsave matrix of posterior samples for location and log(scale) involved in the centering distribution of Polya tree prior.

phi

the vector of posterior samples for the phi parameter in the marginal PT definition.

alpha

the vector of posterior samples for the precision parameter alpha in the PT prior.

maxL

the truncation level used in the PT prior.

Surv.cox.snell

the Surv object used for Cox-Snell residual plot. This is not recommended for frailty models, for which please use the function cox.snell.survregbayes.

ratetheta

the acceptance rate in the posterior sampling of theta vector involved in the centering distribution

ratec

the acceptance rate in the posterior sampling of precision parameter alpha involved in the PT prior

ratephi

the acceptance rate in the posterior sampling of phi parameter

initial.values

the list of initial values used for the parameters

BF

the Bayes factor for comparing the spatial model vs. the exchangeable model

Author(s)

Haiming Zhou and Timothy Hanson

References

Hanson, T., Zhou, H., and de Carvalho, V. (2018). Bayesian nonparametric spatially smoothed density estimation. In New Frontiers of Biostatistics and Bioinformatics (pp 87-105). Springer.

Examples

## Simulated data
rm(list=ls())
library(survival)
library(spBayesSurv)
library(coda)

## True conditional density
fiofy_x = function(y, x){
  0.5*dnorm(y, -x, 1)+0.5*dnorm(y, x, 1);
}

## Generate data
n = 200;
x = runif(n, 0, 3)
y = rep(0, n);
uu = runif(n);
for(i in 1:n){
  if(uu[i]<0.5){
    y[i] = rnorm(1, -x[i], 1);
  }else{
    y[i] = rnorm(1, x[i], 1);
  }
}

## right censored 
y1=y;y2=y;
Centime = runif(n, 2, 4); 
delta = (y<=Centime) +0 ; 
length(which(delta==0))/n; ## censoring rate
rcen = which(delta==0);
y1[rcen] = Centime[rcen];
y2[rcen] = NA;
## make a data frame
## Method 1: in the interval-censoring notation: 
## y1 is the left endpoint and y2 is the right endpoint.
## This way we could use Surv(y1, y2, type="interval2")
## Method 2: Because we have right-censored data, 
## we could use y1 as the observed survival times and delta as the indicator. 
## This way we could use Surv(y1, delta). This is the same as above. 
d = data.frame(y1=y1, y2=y2, x=x, delta=delta); 

##-------------fit the model-------------------##
# MCMC parameters
nburn=50; nsave=50; nskip=0;
# Note larger nburn, nsave and nskip should be used in practice.
mcmc=list(nburn=nburn, nsave=nsave, nskip=nskip, ndisplay=50);
prior = list(maxL=4, phiq0=0);
# Note please set 0<phiq0<1 for a valid Bayes factor of testing 
# spatial model vs. exchangeable model.
# If the Bayes factor is not needed, setting phiq0=0 will speed up 
# the computing time about seven times. 
state = list(alpha=1);
ptm<-proc.time()
res1 = SpatDensReg(formula = Surv(y1, delta)~x, data=d, 
                   prior=prior, state=state, mcmc=mcmc, permutation = TRUE, 
                   fix.theta=FALSE);
## Or equivalently formula = Surv(y1, y2, type="interval2") can also be used.
sfit=summary(res1); sfit
systime1=proc.time()-ptm; systime1;
traceplot(mcmc(res1$theta[1,]))
traceplot(mcmc(res1$theta[2,]))
traceplot(mcmc(res1$alpha))
traceplot(mcmc(res1$phi))

## plots
ygrid = seq(-6, 6,length.out=100);
xpred = data.frame(x=c(0,1,2,3)); 
plot(res1, xnewdata=xpred, ygrid=ygrid);