Analyzing disease recurrence with missing at risk information.

When analyzing time to disease recurrence, we sometimes need to work with data where all the recurrences are recorded, but no information is available on the possible deaths. This may occur when studying diseases of benign nature where patients are only seen at disease recurrences or in poorly-designed registries of benign diseases or medical device implantations without sufficient patient identifiers to obtain their dead/alive status at a later date. When the average time to disease recurrence is long enough in comparison with the expected survival of the patients, statistical analysis of such data can be significantly biased. Under the assumption that the expected survival of an individual is not influenced by the disease itself, general population mortality tables may be used to remove this bias. We show why the intuitive solution of simply imputing the patient's expected survival time does not give unbiased estimates of the usual quantities of interest in survival analysis and further explain that cumulative incidence function analysis does not require additional assumptions on general population mortality. We provide an alternative framework that allows unbiased estimation and introduce two new approaches: an iterative imputation method and a mortality adjusted at risk function. Their properties are carefully studied, with the results supported by simulations and illustrated on a real-world example.