Frequentist and Bayesian analysis methods for case series data and application to early outpatient COVID-19 treatment case series of high risk patients

When confronted with a public health emergency, significant innovative treatment protocols can sometimes be discovered by medical doctors at the front lines based on repurposed medications. We propose a very simple hybrid statistical framework for analyzing the case series of patients treated with such new protocols, that enables a comparison with our prior knowledge of expected outcomes, in the absence of treatment. The goal of the proposed methodology is not to provide a precise measurement of treatment efficacy, but to establish the existence of treatment efficacy,  in order to facilitate the binary decision of whether the treatment protocol should be adopted  on an emergency basis. The methodology consists of a frequentist component that compares a treatment group against the  probability of an adverse outcome in the absence of treatment, and calculates an efficacy threshold that has to be exceeded by this  probability, in order to control the corresponding $p$-value, and reject the null hypothesis. The  efficacy threshold is further adjusted with a Bayesian technique, in order to also control the false positive rate. A selection bias threshold is then calculated from the efficacy threshold to control for random selection bias. Exceeding the efficacy threshold establishes efficacy by the preponderance of evidence, and exceeding the more demanding selection bias threshold establishes efficacy by the clear and convincing evidentiary standard. The combined techniques are applied to case series of high-risk COVID-19 outpatients, that were treated using the early Zelenko protocol and the more enhanced McCullough protocol. The resulting efficacy thresholds are then compared against our prior knowledge of mortality and hospitalization rates of untreated high-risk COVID-19 patients, as reported in the research literature.