compound.reg {compound.Cox}

Compound shrinkage estimation under the Cox model

Description

This function implements the "compound shrinkage estimator" to calculate the regression coefficients of the Cox model, which was proposed by Emura, Chen & Chen (2012). The method is a variant of the Cox partial likelihood estimator such that the regression coefficients are mixed with the univariate Cox regression estimators. The resultant estimator is applicable even when the number of covariates is greater than the number of samples (the p>n setting). The standard errors (SEs) are calculated based on the asymptotic theory (see Emura et al., 2012).

Usage

compound.reg(t.vec, d.vec, X.mat, K = 5, delta_a = 0.025, a_0 = 0, var = FALSE,
plot=TRUE, randomize = FALSE, var.detail = FALSE)

Arguments

t.vec

Vector of survival times (time to either death or censoring)

d.vec

Vector of censoring indicators, 1=death, 0=censoring

X.mat

n by p matrix of covariates, where n is the sample size and p is the number of covariates

K

The number of cross validation folds, K=n corresponds to a leave-one-out cross validation (default=5)

delta_a

The step size for a grid search for the maximum of the cross-validated likelihood (default=0.025)

a_0

The starting value of a grid search for the maximum of the cross-validated likelihood (default=0)

var

If TRUE, the standard deviations and confidence intervals are given (default=FALSE, to reduce the computational cost)

plot

If TRUE, the cross validated likelihood curve and its maximized point are drawn

randomize

If TRUE, randomize the subject ID's so that the subjects in the cross validation folds are randomly chosen. Otherwise, the cross validation folds are constructed in the ascending sequence

var.detail

Detailed information about the covariance matrix, which is mainly used for theoretical purposes. Please consult Takeshi Emura for more details (default=FALSE)

Details

K=5 cross validation is recommended for computational efficiency, though the results appear to be robust against the choice of the number K. If the number of covariates is greater than 200, the computational time becomes very long. In such a case, the univariate pre-selection is recommended to reduce the number of covariates.

Value

a

An optimized value of the shrinkage parameter (0<=a<=1)

beta

Estimated regression coefficients

SE

Standard errors for estimated regression coefficients

Lower95CI

Lower ends of 95 percent confidence intervals (beta_hat-1.96*SE)

Upper95CI

Upper ends of 95 percent confidence intervals (beta_hat+1.96*SE)

Sigma

Covariance matrix for estimated regression coefficients

V

Estimates of the information matrix (-[Hessian of the loglikelihood]/n)

Hessian_CV

Second derivative of the cross-validated likelihood. Normally negative since the cross-validated curve is concave

h_dot

Derivative of Equation (8) of Emura et al. (2012) with respect to a shrinkage parameter "a"

Author(s)

Takeshi Emura & Yi-Hau Chen

References

Emura T, Chen Y-H, Chen H-Y (2012) Survival Prediction Based on Compound Covariate under Cox Proportional Hazard Models. PLoS ONE 7(10): e47627. doi:10.1371/journal.pone.0047627

Examples

### A simulation study ###
n=50 ### sample size
beta_true=c(1,1,0,0,0)
p=length(beta_true) 
t.vec=d.vec=numeric(n)
X.mat=matrix(0,n,p)

set.seed(1)
for(i in 1:n){
  X.mat[i,]=rnorm(p,mean=0,sd=1)
  eta=sum( as.vector(X.mat[i,])*beta_true )
  T=rexp(1,rate=exp(eta))
  C=runif(1,min=0,max=5)
  t.vec[i]=min(T,C)
  d.vec[i]=(T<=C)
}
compound.reg(t.vec,d.vec,X.mat,delta_a=0.1) 
### compare the estimates (beta) with the true value ###
beta_true

### Lung cancer data analysis (Emura et al. 2012 PLoS ONE) ###
data(Lung)
temp=Lung[,"train"]==TRUE
t.vec=Lung[temp,"t.vec"]
d.vec=Lung[temp,"d.vec"]
X.mat=as.matrix( Lung[temp,-c(1,2,3)] )
#compound.reg(t.vec=t.vec,d.vec=d.vec,X.mat=X.mat,delta_a=0.025) # time-consuming process