Variable Selection for Cox's Model with Interval-Censored Data

Description

Perform variable selection for Cox regression model with interval-censored data by using the methods proposed in Zhao et al. (2020a), Wu et al. (2020) and Zhao et al. (2020b). Can deal with both low-dimensional and high-dimensional data.

Usage

CoxICPen(LR = LR,
         x = x,
         lamb = log(nrow(x))/2-2,
         beta.initial = rep(0,ncol(x)),
         pen = "BAR",
         nfold = 5,
         BernD = 3,
         subj.wt = rep(1,nrow(x)))

Arguments

Value

beta: Penalized estimates of the regression coefficients in the Cox's model.

phi: Estimates of the coefficients in Bernstein Polynomials.

logL: Log likelihood function based on current parameter estimates and lambda value.

Lamb0: Estimate of the cumulative baseline hazard function at each observation time point.

cv.out: Cross-validation outcome for each lambda. Will be NULL if cross-validation is not performed.

References

Zhao, H., Wu, Q., Li, G., Sun, J. (2020a). Simultaneous Estimation and Variable Selection for Interval-Censored Data with Broken Adaptive Ridge Regression. Journal of the American Statistical Association. 115(529):204-216.

Wu, Q., Zhao, H., Zhu, L., Sun, J. (2020). Variable Selection for High-dimensional Partly Linear Additive Cox Model with Application to Alzheimer's disease. Statistics in Medicines.39(23):3120-3134.

Zhao, H., Wu, Q., Gilbert, P. B., Chen, Y. Q., Sun, J. (2020b). A Regularized Estimation Approach for Case-cohort Periodic Follow-up Studies with An Application to HIV Vaccine Trials. Biometrical Journal. 62(5):1176-1191.

Examples

# Generate an example data

require(foreach)

n <- 300  # Sample size
p <- 20   # Number of covariates

bet0 <- c(1, -1, 1, -1, rep(0,p-4))  # True values of regression coefficients

set.seed(1)
x.example <- matrix(rnorm(n*p,0,1),n,p)  # Generate covariates matrix

T.example <- c()
for (i in 1:n){
  T.example[i] <- rexp(1,exp(x.example%*%bet0)[i])  # Generate true failure times
}

timep <- seq(0,3,,10)
LR.example <- c()
for (i in 1:n){
  obsT <- timep*rbinom(10,1,0.5)
  if (max(obsT) < T.example[i]) {LR.example <- rbind(LR.example,c(max(obsT), Inf))} else {
    LR.example <- rbind(LR.example,c(max(obsT[obsT<T.example[i]]), min(obsT[obsT>=T.example[i]])))
  }
}  # Generate interval-censored failure times


# Fit Cox's model with penalized estimation

model1 <- CoxICPen(LR = LR.example, x = x.example, lamb = 100, pen = "RIDGE")
beta.initial <- model1$beta

model2 <- CoxICPen(LR = LR.example, x = x.example, beta.initial = beta.initial, pen = "BAR")
model2$beta

#model3 <- CoxICPen(LR = LR.example, x = x.example, lamb = seq(0.1,1,0.1),
#                   beta.initial = beta.initial, pen = "SELO")
#model3$beta

CoxICPen with two sets of covariates

Description

Perform variable selection for Cox regression model with two sets of covariates by using the method in Wu et al. (2020). Variable selection is performed on the possibly high-dimensional covariates x with linear effects. Covariates z with possibly nonlinear effects are always kept in the model.

Usage

CoxICPen.XZ(LR = LR,
            x = x,
            z = z,
            lamb = log(nrow(x))/2-2,
            beta.initial = rep(0,ncol(x)),
            pen = "BAR",
            nfold = 5,
            BernD = 3,
            subj.wt = rep(1,nrow(x)))

Arguments

Value

beta: Penalized estimates of the regression coefficients in the Cox's model.

phi: Estimates of the coefficients in Bernstein Polynomials.

logL: Log likelihood function based on current parameter estimates and lambda value.

Lamb0: Estimate of the cumulative baseline hazard function at each observation time point.

cv.out: Cross-validation outcome for each lambda. Will be NULL if cross-validation is not performed.

f.est.all: A matrix that contains the values of covariates z and the corresponding estimated effects.

References

Wu, Q., Zhao, H., Zhu, L., Sun, J. (2020). Variable Selection for High-dimensional Partly Linear Additive Cox Model with Application to Alzheimer's disease. Statistics in Medicines.39(23):3120-3134.

Examples

# Generate an example data

require(foreach)

n <- 300  # Sample size
p <- 20   # Number of covariates

bet0 <- c(1, -1, 1, -1, rep(0,p-4))  # True values of regression coefficients
f1 <- function(z) sin(2*pi*z)  # True effects of z1
f2 <- function(z) cos(2*pi*z)  # True effects of z2
set.seed(1)
x.example <- matrix(rnorm(n*p,0,1),n,p)  # Generate x covariates matrix
z.example <- cbind(runif(n,0,1),runif(n,0,1))  # Generate z covariates matrix

T.example <- c()
for (i in 1:n){
  T.example[i] <- rexp(1,exp(x.example%*%bet0+
    f1(z.example[,1])+f2(z.example[,2]))[i])  # Generate true failure times
}

timep <- seq(0,3,,10)
LR.example <- c()
for (i in 1:n){
  obsT <- timep*rbinom(10,1,0.5)
  if (max(obsT) < T.example[i]) {LR.example <- rbind(LR.example,c(max(obsT), Inf))} else {
    LR.example <- rbind(LR.example,c(max(obsT[obsT<T.example[i]]), min(obsT[obsT>=T.example[i]])))
  }
}  # Generate interval-censored failure times


# Fit Cox's model with penalized estimation

model1 <- CoxICPen.XZ(LR = LR.example, x = x.example, z = z.example, lamb = 100, pen = "RIDGE")
beta.initial <- model1$beta

model2 <- CoxICPen.XZ(LR = LR.example, x = x.example, z = z.example, 
                      beta.initial = beta.initial, pen = "BAR")
model2$beta

# Plots of covariate effects of z

par(mfrow=c(1,2))

plot(model2$f.est.all$z1, model2$f.est.all$f1, type="l", ylim=c(-1,2),
     xlab="z1", ylab="f1")
lines(model2$f.est.all$z1, f1(model2$f.est.all$z1), col="blue")
legend("topright", col=c("black","blue"), lty=rep(1,2), c("Estimate", "True"))

plot(model2$f.est.all$z2, model2$f.est.all$f2, type="l", ylim=c(-1,2),
     xlab="z2", ylab="f2")
lines(model2$f.est.all$z2, f2(model2$f.est.all$z2), col="blue")
legend("topright", col=c("black","blue"), lty=rep(1,2), c("Estimate", "True"))