Bayesian reanalysis of Apolipoprotein A1 Infusions

Apolipoprotein A1 Infusions and Cardiovascular Outcomes after Acute Myocardial Infarction 24

https://www.nejm.org/doi/full/10.1056/NEJMoa2400969

P(θ | D) = (P(D | θ) * P(θ)) / P(D)

marginal likelihood P(D) = ∫ P(D | θ) * P(θ) dθ

Likelihood: P(D | θ) ~ N(μ_l, σ_l^2)

Likelihood ratio: (P(D | θ1))/(P(D | θ2)*)

Bayes factor: (P(D | θ1)* P(θ1))/(P(D | θ2)* P(θ2))

Prior: P(θ) ~ N(μ_p, σ_p^2)

P(θ | D) ∝ P(D | θ) * P(θ)

When we normalize the product of the two PDFs by dividing by the marginal likelihood p(D), the constant term cancels out, and we are left with a normal distribution with mean μ_new and variance σ_new^2.

Prior mean 0, sd 0.5

log(HR) = log(0.93) = -0.0726

SE = (log(1.05) - log(0.81)) / (2 * 1.96) = 0.0644

Prior precision = 1 / (0.35^2) = 8.16

Likelihood precision = 1 / (0.0644^2) = 241.18

Posterior precision = 8.16 + 241.18 = 249.34

Posterior mean = (8.16 * 0 + 241.18 * -0.0726) / 249.34 = -0.0701

Posterior SD = sqrt(1 / 249.34) = 0.0634

Posterior 95% credible interval for the log(HR) is [-0.0701 - 1.96 * 0.0634, -0.0701 + 1.96 * 0.0634] = [-0.1944, 0.0542]

Posterior HR = exp(-0.0701) = 0.932

Posterior 95% credible interval of [exp(-0.1944), exp(0.0542)] = [0.823, 1.056]

Posterior Z = (0+ 0.0701)/0.0634 = 1.106

pnorm(1.106) = 0.87 # prob that posterior log(hr) is less than 0

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