SuperSurvRegBayes {spBayesSurv}

Bayesian Semiparametric Super Survival Model

Description

This function fits a super survival model (Zhang, Hanson and Zhou, 2018). It can fit both Case I and Case II interval censored data, as well as standard right-censored, uncensored, and mixtures of these. The Bernstein Polynomial Prior is used for fitting the baseline survival function.

Usage

SuperSurvRegBayes(formula, data, na.action, dist="lognormal", 
                  mcmc=list(nburn=3000, nsave=2000, nskip=0, ndisplay=500),
                  prior=NULL, state=NULL, truncation_time=NULL, subject.num=NULL, 
                  InitParamMCMC=FALSE, scale.designX=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.

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.

dist

centering distribution for MPT. Choices include "loglogistic", "lognormal", and "weibull".

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).

prior

a list giving the prior information. The list includes: maxL an integer giving the maximum level of MPT, a0 and b0 gamma prior of the precision parameter of MPT, Shat the initial covariance matrix of adaptive M-H for each of the beta_h, beta_o and beta_q, Vhat the initial covariance matrix of adaptive M-H for theta, beta0 and S0 the prior for each of the beta_h, beta_o and beta_q, for which the default is the g-prior, theta0 and V0 the prior for theta.

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.

truncation_time

a vector of left-trucation times with length n.

subject.num

a vector of suject id numbers when time dependent covariates are considered. For example, for subject 1 time dependent covariates are recorded over [0,1), [1,3), and for subject 2 covariates are recorded over [0,2), [2,3), [3,4). Suppose we only have two subjects, i.e. n=2. In this case, we save the data in the long format, set truncation_time=c(0,1,0,2,3) and subject.num=c(1,1,2,2,2).

InitParamMCMC

flag to indicate wheter an initial MCMC will be run based on the centering parametric model, where TRUE indicates yes.

scale.designX

flag to indicate wheter the design matrix X will be centered by column means and scaled by column standard deviations, where TRUE indicates yes. The default is TRUE for improving numerical stability. Even when it is scaled, the reported regression coefficients are in original scales. Note if we want to specify informative priors for regression coefficients, these priors should correspond to scaled predictors when scale.designX=TRUE.

Value

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

modelname

the name of the fitted model

terms

the terms object used

dist

the centering distribution used in the TBP prior on baseline survival function

coefficients

a named vector of coefficients. The last two elements are the estimates of theta1 and theta2 involved in the centering baseline survival function.

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

nsubject

the number of subjects/individuals, which is equal to n in the absence of time-dependent covariates

subject.num

the vector of subject id numbers when time dependent covariates are considered

truncation_time

the vector of left-trucation times

Surv

the Surv object used

X.scaled

the n by p scaled design matrix

X

the n by p orginal design matrix

theta.scaled

the 2 by nsave matrix of posterior samples for theta1 and theta2 involved in the centering baseline survival function. Note that these posterior samples are based scaled design matrix.

alpha

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

maxL

the truncation level used in the TBP prior.

weight

the maxL by nsave matrix of posterior samples for the weights in the TBP prior.

cpo

the length n vector of the stabilized estiamte of CPO; used for calculating LPML

pD

the effective number of parameters involved in DIC

DIC

the deviance information criterion (DIC)

ratetheta

the acceptance rate in the posterior sampling of theta vector involved in the centering baseline survival function

ratebeta

the acceptance rate in the posterior sampling of beta coefficient vector

rateYs

the acceptance rate in the posterior sampling of weight vector involved in the TBP prior

ratec

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

BF

the Bayes factors for testing AFT, PH, PO, AH, EH and YP models.

Author(s)

Haiming Zhou

References

Zhang, J., Hanson, T., and Zhou, H. (2019). Bayes factors for choosing among six common survival models. Lifetime Data Analysis, 25(2): 361-379.

See Also

survregbayes

Examples

#################################################################
# A simulated data based on PH_PO_AFT super model
#################################################################
rm(list=ls())
library(coda)
library(survival)
library(spBayesSurv)

## True coeffs
betaT_h = c(1, 1);
betaT_o = c(0, 0);
betaT_q = c(1, 1);
## Baseline Survival
f0oft = function(t) 0.5*dlnorm(t, -1, 0.5)+0.5*dlnorm(t,1,0.5);
S0oft = function(t) {
  0.5*plnorm(t, -1, 0.5, lower.tail=FALSE)+
    0.5*plnorm(t, 1, 0.5, lower.tail=FALSE)
}
h0oft = function(t) f0oft(t)/S0oft(t);
## The Survival function:
Sioft = function(t,x){
  xibeta_h = sum(x*betaT_h);
  xibeta_o = sum(x*betaT_o);
  xibeta_q = sum(x*betaT_q);
  (1+exp(xibeta_o-xibeta_h+xibeta_q)*
     (1/S0oft(exp(xibeta_q)*t)-1))^(-exp(xibeta_h-xibeta_q));
}
Fioft = function(t,x) 1-Sioft(t,x);
## The inverse for Fioft
Finv = function(u, x) uniroot(function (t) Fioft(t,x)-u, lower=1e-100, 
                              upper=1e100, extendInt ="yes", tol=1e-6)$root
### true plots
tt=seq(1e-10, 4, 0.02);
xpred1 = c(0,0);
xpred2 = c(0,1);
plot(tt, Sioft(tt, xpred1), "l", ylim=c(0,1));
lines(tt, Sioft(tt, xpred2), "l");

##-------------Generate data-------------------##
## generate x 
n = 80; 
x1 = rbinom(n, 1, 0.5); x2 = rnorm(n, 0, 1); X = cbind(x1, x2);
## generate survival times
u = runif(n);
tT = rep(0, n);
for (i in 1:n){
  tT[i] = Finv(u[i], X[i,]);
}

### ----------- right censored -------------###
t1=tT;t2=tT;
Centime = runif(n, 2, 6);
delta = (tT<=Centime) +0 ; length(which(delta==0))/n;
rcen = which(delta==0);
t1[rcen] = Centime[rcen];
t2[rcen] = NA;
## make a data frame
d = data.frame(t1=t1, t2=t2, x1=x1, x2=x2, delta=delta, tT=tT); table(d$delta)/n;

##-------------Fit the model-------------------##
# MCMC parameters
nburn=200; nsave=500; nskip=0; niter = nburn+nsave
mcmc=list(nburn=nburn, nsave=nsave, nskip=nskip, ndisplay=1000);
prior = list(maxL=15, a0=1, b0=1, M=10, q=.9);
ptm<-proc.time()
res1 = SuperSurvRegBayes(formula = Surv(t1, t2, type="interval2")~x1+x2, data=d, 
                          prior=prior, mcmc=mcmc, dist="lognormal");
sfit=summary(res1); sfit;
systime1=proc.time()-ptm; systime1;
par(mfrow = c(3,2))
traceplot(mcmc(res1$beta_h[1,]), main="beta_h for x1");
traceplot(mcmc(res1$beta_h[2,]), main="beta_h for x2");
traceplot(mcmc(res1$beta_o[1,]), main="beta_o for x1");
traceplot(mcmc(res1$beta_o[2,]), main="beta_o for x2");
traceplot(mcmc(res1$beta_q[1,]), main="beta_q for x1");
traceplot(mcmc(res1$beta_q[2,]), main="beta_q for x2");

####################################################################
## Get curves
####################################################################
par(mfrow = c(1,1))
tgrid = seq(1e-10,4,0.2);
xpred = data.frame(x1=c(0,0), x2=c(0,1)); 
plot(res1, xnewdata=xpred, tgrid=tgrid);