Approximate Reciprocal Relationship Between Two Cause-Specific Hazard Ratios in COVID-19 Data With Mutually Exclusive Events

Cetin, S.; Ulgen, A.; Li, W.; Sivgin, H.

COVID-19 survival data presents a special situation where not only the time-to-event period is short, but also the two events or outcome types, death and release from hospital, are mutually exclusive, leading to two cause-specific hazard ratios (csHR_d and csHR_r). The eventual mortality/release outcome can also be analyzed by logistic regression to obtain odds-ratio (OR). We have the following three empirical observations concerning csHR_d, csHR_r and OR: (1) The magnitude of OR is an upper limit of the csHR_d: |log(OR)| >= |log(csHR_d)|. This relationship between OR and HR might be understood from the definition of the two quantities; (2) csHR_d and csHR_r point in opposite directions: log(csHR_d) log(csHR_r) < 0; This relation is a direct consequence of the nature of the two events; and (3) there is a tendency for a reciprocal relation between csHR_d and csHR_r: csHR_d ~ 1/csHR_r. Though an approximate reciprocal trend between the two hazard ratios is in indication that the same factor causing faster death also lead to slow recovery by a similar mechanism, and vice versa, a quantitative relation between csHR_d and csHR_r in this context is not obvious. These resutls may help future analyses of COVID-19 data, in particular if the deceased samples are lacking.