A comparison of multiple imputation and fully augmented weighted estimators for Cox regression with missing covariates

Abstract

Several approaches exist for handling missing covariates in the Cox proportional hazards model. The multiple imputation (MI) is relatively easy to implement with various software available and results in consistent estimates if the imputation model is correct. On the other hand, the fully augmented weighted estimators (FAWEs) recover a substantial proportion of the efficiency and have the doubly robust property. In this paper, we compare the FAWEs and the MI through a comprehensive simulation study. For the MI, we consider the multiple imputation by chained equation and focus on two imputation methods: Bayesian linear regression imputation and predictive mean matching. Simulation results show that the imputation methods can be rather sensitive to model misspecification and may have large bias when the censoring time depends on the missing covariates. In contrast, the FAWEs allow the censoring time to depend on the missing covariates and are remarkably robust as long as getting either the conditional expectations or the selection probability correct due to the doubly robust property. The comparison suggests that the FAWEs show the potential for being a competitive and attractive tool for tackling the analysis of survival data with missing covariates.

Figure 1

Percentage bias (PB) and average length (AL) vs. selection probability for the FAWE and the MI methods of the Cox Model. Baseline hazard function is 1, β = (−ln(2), ln(2)), where Z^m, Z^o ∼ N(0, 1), with correlation coefficient of 0.6; uniform censoring time with censoring rate 65%; and selection probability π(X, δ, Z^o) = (1 + exp(a + bδ + cX + dZ^o))⁻¹, where a,b,c,d were selected to generate 30% to 80% of cohort members to have missing Z^m; the cohort size is 250. For FAWE, both π and E were obtained based on (X, δ, Z^o) using the Nadaraya-Watson estimator with normal kernel and bandwidth h = 4σ_Wn^−1/3. For MI, imputation models contained (X, δ, Z^o).

Figure 2

Percentage bias (PB) and average length (AL) vs. correlation for the FAWE and the MI methods of the Cox Model. Baseline hazard function is 1, β = (−ln(2), ln(2)), where Z^m, Z^o ∼ N(0, 1), with correlation coefficients ranging from 0 to 0.7; uniform censoring time with censoring rate 65%; and selection probability π(X, δ, Z^o) = (1 + exp(a + bδ + cX + dZ^o))⁻¹, where a,b,c,d are selected to generate 50% of cohort members to have missing Z^m; the cohort size is 250. For FAWE, both π and E were obtained based on (X, δ, Z^o) using the Nadaraya-Watson estimator with normal kernel and bandwidth h = 4σ_Wn^−1/3. For MI, imputation models contained (X, δ, Z^o).