Suppose that a pharmaceutical company has asked you to work as a consultant on their COVID-19 vaccine project. They are planning to conduct a study to demonstrate that their vaccine candidate is more effective than the one Pfizer and BioNTech announced in early November 2020, which was stated to have efficacy of 90% against placebo in participants without prior evidence of SARSCoV-2 infection. They currently plan to enroll 2,000 participants, randomizing half to receive their vaccine candidate and half to receive the vaccine candidate from Pfizer and BioNTech. They are interested in detecting a difference in proportions of at least 5% on the outcome of whether a patient becomes infected with SARS-CoV-2. The results of the trial will be analyzed with a two-sided test. Previous dose-testing trials have been conducted to determine the lowest possible dose of virus sufficient for infecting most of those exposed; this will be the dosage used in the current study. Participants in the study will be healthy volunteers ages 18-30 years who consent to being deliberately exposed to the SARS-CoV-2 virus in a controlled setting and have screened negative for risk factors associated with severe COVID-19. Following exposure, any participants who test positive for the virus will be immediately treated with an antiviral drug. Studies that involve deliberate exposure of participants to an infectious disease are commonly referred to as challenge trials. You have been asked to conduct a simulation study to assess whether the study team has chosen an appropriate sample size. Based on discussions with the study team, you decide it seems feasible that they have identified a more effective vaccine and assign probability 0.75 to the alternative hypothesis.

Suppose the study team conducts the trial with the sample size and obtains statistically significant results at the = 0.05 level. Compute the estimated probability that the alternative hypothesis is true, given the observed data. Explain why this estimated probability changes if you have less confidence in the effectiveness of the new vaccine candidate.