Censored Variables

2023-04-28

How to deal with censored variables?

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:

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.

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:

  1. Add Nelson-Aalen survival estimates “surv” to the dataset.
  2. Use “surv” as well as the covariables to impute missing values in the covariables multiple times.
  3. Perform the intended Cox regression for each of the imputed data sets.
  4. Pool their results by Rubin’s rule (Rubin 1987), using package {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

# 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

References

Buuren, Stef van, and Karin Groothuis-Oudshoorn. 2011. “Mice: Multivariate Imputation by Chained Equations in r.” Journal of Statistical Software, Articles 45 (3): 1–67. https://doi.org/10.18637/jss.v045.i03.
Rubin, D. B. 1987. Multiple Imputation for Nonresponse in Surveys. Wiley Series in Probability and Statistics. Wiley.
White, Ian R., and Patrick Royston. 2009. “Imputing Missing Covariate Values for the Cox Model.” Statistics in Medicine 28 (15): 1982–98. https://doi.org/10.1002/sim.3618.