Non-parametric estimation of the case fatality ratio with competing risks data: an application to Severe Acute Respiratory Syndrome (SARS).

For diseases with some level of associated mortality, the case fatality ratio measures the proportion of diseased individuals who die from the disease. In principle, it is straightforward to estimate this quantity from individual follow-up data that provides times from onset to death or recovery. In particular, in a competing risks context, the case fatality ratio is defined by the limiting value of the sub-distribution function, F(1)(t) = Pr(T <or=t and J = 1), associated with death, as t --> infinity, where T denotes the time from onset to death (J = 1) or recovery (J = 2). When censoring is present, however, estimation of F(1)(infinity) is complicated by the possibility of little information regarding the right tail of F(1), requiring use of estimators of F(1)(t(*)) or F(1)(t(*))/(F(1)(t(*))+F(2)(t(*))) where t(*) is large, with F(2)(t) = Pr(T <or=t and J = 2) being the analogous sub-distribution function associated with recovery. With right censored data, the variability of such estimators increases as t(*) increases, suggesting the possibility of using estimators at lower values of t(*) where bias may be increased but overall mean squared error be smaller. These issues are investigated here for non-parametric estimators of F(1) and F(2). The ideas are illustrated on case fatality data for individuals infected with Severe Acute Respiratory Syndrome (SARS) in Hong Kong in 2003.