ST440/540 – Mid-term exam 1 - Due February 26
THIS IS AN EXAM - DO NOT DISCUSS THE PROBLEM WITH ANYONE (INCLUDING
OTHER STUDENTS OR THE TA)! If you have questions, please email me.
Developing a COVID-19 vaccine is one of the most important scientific endeavors our of
lifetime. In this exam, you will conduct a Bayesian analysis of the data produced by the
clinical trial of the Moderna vaccine1. The data2 are:
Placebo group Vaccine group
Infected Participants Infected Participants
All patients 185 14,073 11 14,134
White 144 8,916 10 9,023
Communities of color 41 5,132 1 5,088
Let θg ∈ (0, 1) be the infection probability in group g ∈ {placebo, vaccine}. That is, a
randomly-selected person given the placebo has probability θplacebo of contracting the virus,
and a randomly-selected person given the vaccine has probability θvaccine of contracting the
virus. The efficacy of the vaccine if often defined as
E = 1− θvaccine
θplacebo
.
Using these data, answer the following questions:
1. Placebo analysis: Using all 14,073 patients in the placebo group, state a reasonable
likelihood for the data and an uninformative conjugate prior distribution for θplacebo.
What are the main assumptions in your model and do you think they are valid?
2. Derivation: Using the model from the previous question, derive (i.e., a 3-4 line math-
ematical derivation) the posterior distribution of θplacebo. Your final answer should be
a distribution, e.g., θplacebo|Y ∼ Poisson(Y + 1), and you must show your work.
3. Efficacy analysis: Using all 14,073+14,134 patients, perform a Bayesian analysis
of the efficacy of the vaccine, E. Are the results sensitive to the prior (i.e., do they
change a lot when you change the prior)? Be sure to give all details such as likelihoods,
priors and a clear description of the results you report. Summarize the results in a
clearly-labeled plot and a table.
4. Hypothesis test: According to the protocol2 (Page 14), the stated objective was to
test the hypothesis that the vaccine has efficacy at least 0.70. Carry out a Bayesian
test of this hypothesis. Are the results sensitive to your the prior?
1https://www.modernatx.com/sites/default/files/mRNA-1273-P301-Protocol.pdf
2Taken from Figure 4 of https://www.nejm.org/doi/full/10.1056/NEJMoa2035389
5. Subgroup analysis: Carry out the Bayesian analysis separately for white subjects
and subjects from communities of color, and test whether there is a difference in the
efficacy across these groups. Summarize the results in a clearly-labeled plot and a
table.
Your paper should be written as a professional document with full sentences and paragraphs,
clearly labeled and numbered figures and tables, and few spelling/grammar errors. Organize
your report with five sections, one for each question. You should include enough detail that
another student in class could reproduce your results. Summarize your analysis in a PDF
document that is no more than four pages long (excluding code). Append your code to the
end of this document and submit a single document on moodle.
HAVE FUN!














































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