Most cancer registries choose not to rely on cause of death when presenting survival statistics on cancer patients, but instead to look at overall mortality after diagnosis and adjust for the expected mortality in the cohort had they not been diagnosed with cancer. For many years the relative survival (observed survival divided by expected survival) was estimated by the Ederer-II method. More recently statisticians have used the theory of classical competing risks to estimate the net survival – that is the survival that would be observed in cancer patients if it were possible to remove all competing causes of death. Pohar-Perme showed that in general estimators of the relative survival and not consistent for the net survival, and proposed a new consistent estimator of the net survival. Poher-Perme’s estimator can have much larger variance than Ederer-II (and may not be robust). Thus whereas some statisticians have argued that one must use the Poher-Perme estimator because it is the only one that is consistent for the net survival, others have argued that there is a bias-variance trade off and Ederer-II may still be preferred even though it is inconsistent.
We draw analogy from the literature regarding robust estimation of location. If one wants to estimate the mean of a distribution consistently, then it may be difficult to improve on the sample mean. But if one simply wants a measure of location then other estimators are possible and might be preferred to the sample mean. We define a measure of net survival to be a functional satisfying certain equivariance and order conditions. The limits of neither Ederer-II nor Pohar-Perme satisfy our definition of being an invariant measure of net survival. We introduce two families of functionals that do satisfy our definition. Consideration of minimum variance and robustness then allows us to select a single member of each family as the preferred measure of net survival.
Noting that in a homogeneous population the relative survival and the net survival are identical and correspond to the survival of the excess hazard, we can then view our functionals of weighted averages of stratum-specific relative-survival, net-survival or excess-hazards. These can be viewed as standardised estimators with standardising weights that are time-dependent. The preferred measures use weights that depend on the numbers at risk in each stratum from a standard population as a function of time. For example, when the strata are defined by age at diagnosis, the standardising weights will depend on the age-specific prevalence of the cancer in the standard population.
We show through simulation that, unlike both Ederer-II and Pohar-Perme, our estimators are invariant and robust under changing population structures, and also that they are consistent and reasonably efficient. Although our estimator does not (consistently) estimate the (marginal) net hazard it performs as well or better than both the crude and standardised versions of both Ederer-II and Pohar-Perme in all simulations.
Joint work with Adam Brentnall