Example 1
Does inhaled anosine improve lung function in severe ZOVID?

Most research questions are best addressed by estimating a "focal parameter" that relates as directly as possible to each question and by quantifying the statistical uncertainty about that estimate.

When the research question is motivated by a hypothesis, then we can go on to assess how much that focal estimate supports or undermines the hypothesis.

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most research questions are best addressed by estimating a "focal parameter" that relates as directly as possible to each question and by quantifying the statistical uncertainty about that estimate.

In frequentism, we obtain estimates and confidence intervals for that parameter, and, when the research question is a bona fide hypothesis, we can compute and try to interpret a p-value.

In Bayesianism, we set a sound prior distribution for the focal parameter and then let the data revise that prior to give us a posterior distribution. From that posterior, we can compute compute a posterior mean or median, a set of posterior quantiles (giving us, say, 80%, 90%, 95% and/or 99% probability/credible intervals), and posterior probabilities of our choosing. Critically, plotting the prior and posterior density functions gives us rich images, which tap into our native visual processing abilities.

Note. This example is fiction, but mirrors a real trial I collaborated in designing in 2021 to treat COVID-19.

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Investigators. Brandon Pike, MD, the study's lead investigator, an expert in pulmonary O2 and CO2 exchange. Cybill Varing, MS, an innovative and modern Bayesian statistical scientist.

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Subjects. Patients so severely ill with ZOVID that they had acute respiratory distress syndrome (ARDS), a condition with a mortality rate that exceeds 45% under high-quality standard care. They were put on ventilators immediately and treated within four hours under study protocol.

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Treatment question. Does aerosolized inhaled anosine (fictitious) improve lung function in these patients? Whether the patient survives is not the whole story. A survivor may still have permanently weakened lung function, a sequela in "long ZOVID."

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Primary outcome measure. The PaO2/FiO2 ratio (real) is a "hard" measure of oxygen uptake; let's just call it the O2ratio. Higher the better. Dr. Pike was adamant that by measuring the O2ratio at baseline and 5 days later, a straightforward algorithm reliably and validly distinguishes patients as having markedly improved (+) or not markedly improved (-) over this super-critical time period.

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Statistical endpoint. So, this boils down to a binary outcome. Ms. Varing is typically wary about dichotomizing a continuous measure, but comes to endorse it in this case.

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Research goal. Demonstrate "sufficient" clinical efficacy: P = Prob[+ | treated] > 0.60. If this study used a frequentist structure, the null hypothesis would be H0: P < 0.60.

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Planning scenarios. To shape her statistical planning, Ms. Varing engaged Dr. Pike in an "elicitation session." She asked: If this study had a control group in which patients were given high-quality standard care, what proportion would have "markedly improved" O2ratios over the first five days? Let's call this P0 = Prob[+ | not treated]. This generated considerable discussion and clarification, but a scenario emerged. Given the new Omega variant of ZOVID, there was little good data so far, but P0 = 0.30 "sounds about right," and P0 = 0.45 is "almost surely" close to the upper limit. Dr. Pike also said that he believed that the inhaled anosine treatment was likely 80% efficacious, Prob[+ |  treated] = 0.80, a conjecture necessary for the sample-size analysis discussed below.

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Bayesian approach. The study conforms neatly with the ordinary beta-binomial model. For a given prior distribution, P0 ~ beta(a0, b0), and m1 patients who markedly improve and m2 who do not, the posterior distribution is P1 ~ beta(a0+m1, b0+m2).

Two priors, both legitimate but markedly different, will be used: