Solution

Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 41 / 2 / a / Solution

{title2=$\widehat{\operatorname{Var}}(\log\widehat{\mathrm{OR}})=a^{-1}+b^{-1}+c^{-1}+d^{-1}$}

For two independent <binomial distributions> with event and nonevent counts $a,b$ and $c,d$, the <odds ratio> is $ad/(bc)$. The <delta method> applied to $\log(p/(1-p))$, whose derivative is $1/[p(1-p)]$, gives a group variance $1/[Np(1-p)]$. Substitution of the fitted proportion gives $1/a+1/b$ in the first group and $1/c+1/d$ in the second. Their independence gives the displayed sum. A <normal approximation> then gives a <confidence interval> on the <log odds ratio> scale; exponentiation produces an odds-ratio interval. Zero cells require another approximation or model.

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 Incoming links: Log odds ratio variance from a two-by-two table