Extrapolation before imputation reduces bias when imputing censored covariates

This repository contains R code and simulation data to reproduce results from the manuscript by Lotspeich and Garcia (2022+).

For the imputeCensRd package, which implements the conditional mean imputation approaches from the paper, can be found in its own repo here.

Each of the "Script (Run Simulations)" files is coded to run 1 replication of each setting for demonstration. Per the NOTES at the bottom of the scripts, some more time-intensive simulations were run in parallel.

Tables

Table 1. Simulation results for Weibull $X$ from the full cohort analysis and conditional mean imputation (CMI) approaches.

• Script (Run Simulations)
• Script (Make Table)
• Data (Simulation Results)

Table S1. Simulation results for Weibull $X$ independent of $Z$ from the full cohort analysis (i.e., where all $n$ observations had uncensored $X$) and conditional mean imputation (CMI) approaches.

• Script (Run Simulations)
• Script (Make Table)
• Data (Simulation Results)

Table S2. Simulation results for log-normal $X$ from the full cohort analysis (i.e., where all $n$ observations had uncensored $X$) and conditional mean imputation (CMI) approaches.

• Script (Run Simulations)
• Script (Make Table)
• Data (Simulation Results)

Figures

Figure S1. Illustration of the four extrapolation methods for a step survival function $\widehat{S}(t)$ in simulated data.

• Script (Make Figure)

Figure S2. We explored light ($\sim 17%$), heavy ($\sim 49%$), and extra heavy ($\sim 82%$) censoring in Weibull $X$, induced by generating $C$ from an exponential distribution with rates $= 0.5$, $2.9$, and $20$, respectively.

• Script (Make Figure)

Figure S3. With Weibull $X$, extrapolating Breslow's estimator $\widehat{S}_0(t)$ beyond the largest uncensored value $\widetilde{X}$ with the Weibull extension offered the lowest bias and best efficiency for $\hat{\beta}$ in conditional mean imputation with adaptive quadrature.

• Script (Run Simulations)
• Script (Make Figure)
• Data (Simulation Results)

Figure S4. With log-normal $X$, extrapolating Breslow's estimator $\widehat{S}_0(t)$ beyond the largest uncensored value $\widetilde{X}$ with any of the three extrapolation methods offered similar bias and efficiency for $\hat{\beta}$ in conditional mean imputation with adaptive quadrature.

• Script (Run Simulations)
• Script (Make Figure)
• Data (Simulation Results)

Figure S5. Interpolating Breslow's estimator $\widehat{S}_0(t)$ between uncensored values with either of the two interpolation methods offered similar bias and efficiency for $\hat{\beta}$ in conditional mean imputation with adaptive quadrature.

• Script (Run Simulations)
• Script (Make Figure)
• Data (Simulation Results)

Figure S6. Extrapolating Breslow's estimator $\widehat{S}_0(t)$ beyond the largest uncensored value $\widetilde{X}$ with any of the three extrapolation methods offered similar bias and efficiency for $\hat{\beta}$ in conditional mean imputation with the trapezoidal rule.

• Script (Run Simulations)
• Script (Make Figure)
• Data (Simulation Results)

Figure S7. Due to the Weibull distribution's skewness, higher censoring rates led to smaller values of $W_{(n)}$ (the maximum of the observed covariate), which led to worse performance (i.e., higher bias) when calculating the conditional mean with the trapezoidal rule.

• Script (Make Figure)