There is no obvious way of how to deal with survival variables as covariates in imputation models. Options discussed in (White and Royston 2009) include:
By \(\text{surv}(t)\), we denote the Nelson-Aalen survival estimate at each value of \(t\). The third option seems attractive as it explicitly deals with censoring information. We provide some additional details on it in the example.
For illustration, we use data from a randomized two-arm trial about lung cancer. The aim is to estimate the treatment effect of “trt” with reliable inference using Cox regression. We add missing values in the covariates “age”, “karno”, and “diagtime”.
Let’s estimate the covariate adjusted treatment effect using the following steps:
library(missRanger)
library(survival)
library(mice)
set.seed(65)
head(veteran)
# trt celltype time status karno diagtime age prior
# 1 1 squamous 72 1 60 7 69 0
# 2 1 squamous 411 1 70 5 64 10
# 3 1 squamous 228 1 60 3 38 0
# 4 1 squamous 126 1 60 9 63 10
# 5 1 squamous 118 1 70 11 65 10
# 6 1 squamous 10 1 20 5 49 0
# 1. Calculate Nelson-Aalen survival probabilities for each time point
nelson_aalen <- survfit(Surv(time, status) ~ 1, data = veteran) |>
summary(times = unique(veteran$time))
nelson_aalen <- nelson_aalen[c("time", "surv")]
# Add it to the original data set
veteran2 <- merge(veteran, nelson_aalen, all.x = TRUE, by = "time")
# Add missing values to make things tricky
veteran2 <- generateNA(veteran2, p = c(age = 0.1, karno = 0.1, diagtime = 0.1))
# 2. Generate 20 complete data sets, representing "time" and "status" by "surv"
filled <- replicate(
20,
missRanger(veteran2, . ~ . - time - status, verbose = 0, pmm.k = 10, num.trees = 100),
simplify = FALSE
)
# 3. Run a Cox regression for each of the completed data sets
models <- lapply(filled, function(x) coxph(Surv(time, status) ~ . - surv, x))
# 4. Pool the results by mice
summary(pooled_fit <- pool(models))
# term estimate std.error statistic df p.value
# 1 trt 0.231154077 0.214672763 1.0767741 105.1514 2.840454e-01
# 2 celltypesmallcell 0.805824737 0.285571376 2.8217980 114.1273 5.634607e-03
# 3 celltypeadeno 1.130585762 0.306698637 3.6863084 113.3636 3.506786e-04
# 4 celltypelarge 0.340627347 0.296740520 1.1478963 103.4753 2.536583e-01
# 5 karno -0.030623274 0.005653790 -5.4164149 106.3603 3.806255e-07
# 6 diagtime 0.001273007 0.009102230 0.1398566 108.7518 8.890320e-01
# 7 age -0.005587627 0.009379064 -0.5957554 105.3053 5.526170e-01
# 8 prior 0.005174395 0.023433186 0.2208148 112.4847 8.256369e-01
# Compare with the results on the original data
summary(coxph(Surv(time, status) ~ ., veteran))$coefficients
# coef exp(coef) se(coef) z Pr(>|z|)
# trt 2.946028e-01 1.3425930 0.207549604 1.419433313 1.557727e-01
# celltypesmallcell 8.615605e-01 2.3668512 0.275284474 3.129709606 1.749792e-03
# celltypeadeno 1.196066e+00 3.3070825 0.300916994 3.974738536 7.045662e-05
# celltypelarge 4.012917e-01 1.4937529 0.282688638 1.419553530 1.557377e-01
# karno -3.281533e-02 0.9677173 0.005507757 -5.958020093 2.553121e-09
# diagtime 8.132051e-05 1.0000813 0.009136062 0.008901046 9.928981e-01
# age -8.706475e-03 0.9913313 0.009300299 -0.936149992 3.491960e-01
# prior 7.159360e-03 1.0071850 0.023230538 0.308187441 7.579397e-01