I am trying to calculate P-values associated with point estimates obtained from a Cox PH model with time-varying coefficients. The function that I have written does not provide the correct P-values. I will illustrate this by making use of the NCCTG Lung Cancer Data from the survival package.
# Setup
require(survival)
# Effect of Karnofsky score, linear
fit <- coxph(Surv(time/365.25, status == 2) ~ ph.karno + tt(ph.karno),
lung, tt=function(x, t, ...) {x*t})
The function:
# Same function but now with a P-value in the output
calculate.timeDependentHazard.P <- function(model,time) {
index.1 <- which(names(model$coef)=="ph.karno")
index.2 <- which(names(model$coef)=="tt(ph.karno)")
coef <- model$coef[c(index.1,index.2)]
var <- rbind(c(model$var[index.1,index.1],model$var[index.1,index.2]),
c(model$var[index.2,index.1],model$var[index.2,index.2]))
var.at.time <- t(c(1,time)) %*% var %*% c(1,time)
hazard.at.time <- t(c(1,time)) %*% coef
lower.95 <- hazard.at.time - 1.96*sqrt(var.at.time)
upper.95 <- hazard.at.time + 1.96*sqrt(var.at.time)
z.at.time <- hazard.at.time/(sqrt(var.at.time))
p.value <- pnorm(-abs(z.at.time))
results <- c(exp(c(hazard.at.time,lower.95,upper.95)),p.value)
names(results) <- c("hazard ratio","95% lower","95% upper","P.value")
options(scipen = 999)
results
}
# Point estimates after 1.05*365.25 = 383.5 days of follow-up
calculate.timeDependentHazard.P(fit,1.05)
The output:
> calculate.timeDependentHazard.P(fit,1.05)
hazard ratio 95% lower 95% upper P.value
0.98913256 0.97654719 1.00188013 0.04721342
Apparently, the P-value should be >.05 but somehow it is not. The P-values calculated via this approach seem to be too low. Anyone who can discover the flaw?
It seems like you want a two sided alternative so multiply pnorm(-abs(z.at.time)) by two. I.e., do 2*pnorm(-abs(z.at.time)).