Approximate reciprocal relationship between two cause-specific hazard ratios in COVID-19 data with mutually exclusive events

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 is also analyzed by logistic regression to obtain odds-ratio (OR). We have the following three empirical observations: (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) similar to 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 results may help future analyses of data from COVID-19 or other similar diseases, in particular if the deceased patients are lacking, whereas surviving patients are abundant.