STA 145 Bayesian Statistical Inference – Exercises – midterm
Instructor: Jairo Fúquene-Pati o, Term: Spring/2022
Exercise
Consider the following statistical analysis of the SARS-CoV-2 RNA Vaccine and complete the sections:
Bayesian interpretation and Sensitivity analysis. Please attach the R code.
Reference: Pfizer, A., 2020. Phase 1/2/3, Placebo-Controlled, Randomized, Observer-Blind, Dose-Finding
Study to Evaluate the Safety, Tolerability, Immunogenicity, and Efficacy of SARS-CoV-2 RNA Vaccine
Candidates Against COVID-19 in Healthy Individuals. (attached)
Intervention: When participants are dosed at the site, they will receive study intervention directly from
the investigator or designee, under medical supervision. The date and time of each dose administered in the
clinic will be recorded in the source documents and recorded in the CRF. The dose of study intervention and
study participant identification will be confirmed at the time of dosing by a member of the study site staff
other than the person administering the study intervention.
Aim of study: The purpose of the study is to rapidly describe the safety, tolerability, and immunogenicity
of 2 BNT162 RNA-based COVID-19 vaccine candidates against COVID-19, and the efficacy of 1 candidate,
in healthy individuals. There are currently no licensed vaccines to prevent infection with SARS-CoV-2 or
COVID-19. Given the global crisis of COVID-19 and fast expansion of the disease in the United States and
elsewhere, the rapid development of an effective vaccine is of utmost importance.
Study design: This is a multicenter, multinational, Phase 1/2/3, randomized, placebo-controlled, observer-
blind, dose-finding, vaccine candidate–selection, and efficacy study in healthy individuals.
Outcome measure: Let piv and pic be the population probabilities that a vaccinated subject or a subject in
the control group, respectively, fall ill to Covid-19. Then the population vaccine efficacy is given by
V E = 1 pivpic
Assume a prior distribution for a parameter
θ = (1 V E)/(2 V E)
Plugging in the definition of the population vaccine efficacy VE, θ is given by
θ = pivpic + pic
Biontech/Pfizer assigned the same number of subjects to the treatment and control group. θ denotes the
probability that a subject who fell ill with Covid-19 is from the treatment group, while 1 θ is the probability
that the subject is from the control group.
Biontech/Pfizer stated as interim success criterion that the posterior probability of an efficacy below 30%
(corresponding to θ > 0.4118) is smaller than 2.5%.
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Prospective Bayesian analysis : No, it was carried out after the trial reported its results.
Prior distribution: a prior distribution for the possible values of the unknown vaccine effect. A minimally
informative Beta prior, Beta(a=0.700102, b=1), is proposed for
θ = (1 V E)/(2 V E)
The prior is centered at θ = 0.4118 (VE=30%) which can be considered pessimistic. The prior allows
considerable uncertainty; the 95% interval for θ is (0.005, 0.964) and the corresponding 95% interval for VE
is (-26.2, 0.995).
Loss function or demands: None specified.
Computation/software: Conjugate Beta/Binomial analysis.
Bayesian interpretation: (exercise)
Let θ be the probability that a subject who fell ill with Covid-19 is from the treatment group and 1 θ the
probability that the subject is from the control group. Assuming that 94 subjects fell ill to Covid-19 (with a
sample efficacy above 90%) and at most 8 of those 94 subjects were vaccinated.
a. Consider a Beta prior for θ: p(θ)=Beta(a=0.700102, b=1), where a and b are the shape parameters of
the Beta distribution. Plot the prior, likelihood and posterior as function of θ.
b. Compute the posterior probability of having a value of θ > 0.4118.
c. Compute a 95% credible (Quantile-based interval and the highest posterior density region) intervals
and a confidence interval for θ.
d. Plot the posterior predictive density. If a new sample of 94 subjects with Covid-19 is taken, how many
subjects were vaccinated .
e. Conclusions.
Sensitivity analysis: (exercise)
For a wide range of θ0 = a/(a+ b) and w = a+ b conduct a sensitivity analysis. Therefore explore the effect
of θ0 (and therefore VE) and w via contour plots of posterior quantities.
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