Frequentist and Bayesian analysis methods for case series data and application to early outpatient COVID-19 treatment case series of high risk patients
Ph.D Eleftherios Gkioulekas, MD, MPH Peter A Mccullough, MD Vladimir Zelenko
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.
Author contributions EG conceptualized the mathematical framework, analyzed the case series data, and wrote the first draft of the paper. PAC and VZ contributed to the literature review, as well as writing and editorial changes to the manuscript. VZ contributed copies of his three public letters [16, 17, 24] , attached to our supplementary material document [25] , containing some of the consecutive case series data analyzed in the manuscript. VZ also contributed data on how his treatment protocol evolved over time during 2020. All authors read and approved the final version of the manuscript.
Ethics approval and consent to participate Not applicable.
Conflict of interest The authors declare no conflicts of interest. Peter A. Mc-Cullough is serving as the editor-in-chief of this journal. Appendix A: Exact Fisher test in the limit of an infinite control group Let N be the total number of patients in the treatment group, let a be the number of patients with an adverse outcome (hospitalization or death) in the treatment group, let M be the total number of patients in the control group, and let b be the number of patients in the control group with an adverse outcome. In this appendix we will show that in the limit of an infinite control group (M, b) with x = b/M, the p-value p(N, a, M, b) obtained from the two-tail exact Fisher test converges to p(N, a, x). In the exact Fisher test, we assume that N, M, and a + b, are fixed numbers, and under the null hypothesis, we also..
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