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)
surv(t), and, optionally s
By 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:
mice (Buuren and Groothuis-Oudshoorn 2011).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$time)
)[c("time", "surv")]
nelson_aalen <- data.frame(nelson_aalen)
# Add it to the original data set
veteran2 <- merge(veteran, nelson_aalen, all.x = TRUE)
# Add missing values to make things tricky
veteran2 <- generateNA(veteran2, p = c(age = 0.1, karno = 0.1, diagtime = 0.1))
# 2. Generate 20 complete data sets, representing "time" and "status" by "surv"
filled <- replicate(20, missRanger(veteran2, . ~ . - time - status,
verbose = 0, pmm.k = 3, num.trees = 25), simplify = FALSE)
# 3. Run a Cox regression for each of the completed data sets
models <- lapply(filled, function(x) coxph(Surv(time, status) ~ . - surv, x))
# 4. Pool the results by mice
summary(pooled_fit <- pool(models))
#> term estimate std.error statistic df p.value
#> 1 trt 0.238408881 0.213416156 1.1171079 108.30303 2.664203e-01
#> 2 celltypesmallcell 0.801088770 0.286383107 2.7972626 112.17712 6.066665e-03
#> 3 celltypeadeno 1.134351839 0.308977998 3.6713030 110.65791 3.731780e-04
#> 4 celltypelarge 0.327092592 0.291069423 1.1237614 109.29555 2.635765e-01
#> 5 karno -0.031250557 0.005786704 -5.4004073 99.60711 4.529695e-07
#> 6 diagtime 0.002889092 0.009020319 0.3202872 106.17585 7.493801e-01
#> 7 age -0.007620985 0.009632902 -0.7911411 97.27459 4.307864e-01
#> 8 prior 0.003954604 0.023537476 0.1680131 111.98360 8.668760e-01
# On the original
summary(coxph(Surv(time, status) ~ ., veteran))$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