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贝叶斯代写-ST440/540

时间：2021-02-24

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!

学霸联盟

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!

学霸联盟