There is no obvious way of how to deal with survival variables as covariables in imputation models.
Options discussed in (White and Royston 2009) include:
Use both status variable \(s\) and (censored) time variable \(t\)
\(s\) and \(\log(t)\)
\(\text{surv}(t)\), and, optionally \(s\)
By \(\text{surv}(t)\), we denote the Nelson-Aalen survival estimate at each value of \(t\). The third option is the most elegant one 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. Unfortunately, we have added missing values in the covariables “age”, “karno”, and “diagtime”.
A reasonable way to estimate the covariable adjusted treatment effect is the following:
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 <- summary(
survfit(Surv(time, status) ~ 1, data = veteran),
times = unique(veteran$time)
)[c("time", "surv")]
nelson_aalen <- data.frame(nelson_aalen)
# Add it to the original data set
veteran2 <- merge(veteran, nelson_aalen, all.x = TRUE)
# Add missing values to make things tricky
veteran2 <- generateNA(veteran2, p = c(age = 0.1, karno = 0.1, diagtime = 0.1))
# 2. Generate 20 complete data sets, representing "time" and "status" by "surv"
filled <- replicate(
20,
missRanger(
veteran2, . ~ . - time - status,
verbose = 0,
pmm.k = 3,
num.trees = 25
),
simplify = FALSE
)
# 3. Run a Cox regression for each of the completed data sets
models <- lapply(filled, function(x) coxph(Surv(time, status) ~ . - surv, x))
# 4. Pool the results by mice
summary(pooled_fit <- pool(models))
# term estimate std.error statistic df p.value
# 1 trt 0.245855250 0.212810467 1.1552780 108.72929 2.505091e-01
# 2 celltypesmallcell 0.805233656 0.284424937 2.8310937 114.17088 5.483657e-03
# 3 celltypeadeno 1.110172771 0.307570269 3.6094931 111.91588 4.603422e-04
# 4 celltypelarge 0.328227283 0.291163500 1.1272954 109.30510 2.620862e-01
# 5 karno -0.031838682 0.005663349 -5.6218824 112.60325 1.390333e-07
# 6 diagtime 0.002775351 0.009382270 0.2958081 86.61582 7.680847e-01
# 7 age -0.007843577 0.009293988 -0.8439410 107.86917 4.005701e-01
# 8 prior 0.003165245 0.023501821 0.1346809 111.84783 8.931063e-01
# Compare with the results on the original data
summary(coxph(Surv(time, status) ~ ., veteran))$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