Inference for a 2×2 table: an experiment was performed to estimate the effect of beta blockers on...

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Inference for a 2×2 table: an experiment was performed to estimate the effect of beta blockers on mortality of cardiac patients. A group of patients were randomly assigned to treatment and control groups: out of 674 patients receiving the control, 39 died, and out of 680 receiving the treatment, 22 died. Assume that the outcomes are independent and binomially distributed, with probabilities of death of p₀ and p₁ under the control and treatment, respectively.

(a) Set up a non-informative prior distribution on (p₀, p₁) and obtain posterior simulations.

(b) The odds ratio is defined as (p₁/(1−p₁))/(p₀/(1−p₀)). Summarize the posterior distribution for this estimand.

(c) Discuss the sensitivity of your inference to your choice of non-informative prior density.

We return to this example in Section 5.6.

Aditi · Accepted Answer

Solution
Inference for the difference between proportions 
a. The likelihood models are given by binomial distributions
p(y|p ) =    674   p39(1 − p )635	0
39
0
0
 (
1
) (
22
) (
0
) (
1
)p(y|p ) =    680   p22(1 − p )658	
The non-informative uniform prior over [0, 1] is assumed for both p(p0) and p(p1):
p(p0) = p(p1) = 1	
The posterior of p0 is given by
p(p0|y) ∼ Beta(39 + 1, 674 − 39 + 1) = Beta(40,

Inference for a 2×2 table: an experiment was performed to estimate the effect of beta blockers on mortality of cardiac patients. A group of patients were randomly assigned to treatment and control...

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