Cox Regression with Covariates Missing Not at Random

Cox regression with survival-time-dependent missing covariate values

Analysis with time-to-event data in clinical and epidemiological studies often encounters missing covariate values, and the missing at random assumption is commonly adopted, which assumes that missingness depends on the observed data, including the observed outcome which is the minimum of survival and censoring time. However, it is conceivable that in certain settings, missingness of covariate values is related to the survival time but not to the censoring time. This is especially so when covariate missingness is related to an unmeasured variable affected by the patient's illness and prognosis factors at baseline. If this is the case, then the covariate missingness is not at random as the survival time is censored, and it creates a challenge in data analysis. In this article, we propose an approach to deal with such survival-time-dependent covariate missingness based on the well known Cox proportional hazard model. Our method is based on inverse propensity weighting with the propensity estimated by nonparametric kernel regression. Our estimators are consistent and asymptotically normal, and their finite-sample performance is examined through simulation. An application to a real-data example is included for illustration.

Validity of using multiple imputation for "unknown" stage at diagnosis in population-based cancer registry data

Background

The multiple imputation approach to missing data has been validated by a number of simulation studies by artificially inducing missingness on fully observed stage data under a pre-specified missing data mechanism. However, the validity of multiple imputation has not yet been assessed using real data. The objective of this study was to assess the validity of using multiple imputation for “unknown” prostate cancer stage recorded in the New South Wales Cancer Registry (NSWCR) in real-world conditions.

Methods

Data from the population-based cohort study NSW Prostate Cancer Care and Outcomes Study (PCOS) were linked to 2000–2002 NSWCR data. For cases with “unknown” NSWCR stage, PCOS-stage was extracted from clinical notes. Logistic regression was used to evaluate the missing at random assumption adjusted for variables from two imputation models: a basic model including NSWCR variables only and an enhanced model including the same NSWCR variables together with PCOS primary treatment. Cox regression was used to evaluate the performance of MI.

Results

Of the 1864 prostate cancer cases 32.7% were recorded as having “unknown” NSWCR stage. The missing at random assumption was satisfied when the logistic regression included the variables included in the enhanced model, but not those in the basic model only. The Cox models using data with imputed stage from either imputation model provided generally similar estimated hazard ratios but with wider confidence intervals compared with those derived from analysis of the data with PCOS-stage. However, the complete-case analysis of the data provided a considerably higher estimated hazard ratio for the low socio-economic status group and rural areas in comparison with those obtained from all other datasets.

Conclusions

Using MI to deal with “unknown” stage data recorded in a population-based cancer registry appears to provide valid estimates. We would recommend a cautious approach to the use of this method elsewhere.