Consider two independent Binomial variables with parameters $(n_1,p_1)$ and $(n_2,p_2)$, say $X$ and $Y$. The odds ratio is then defined by $\phi = \frac{p_1(1-p_2)}{(1-p_1)p_2}$. The minimal sufficient set of statistic $\phi$ is $(X,X+Y)$. A bunch of papers, for example, Numerical Results on Approximate Confidence Limits for the Odds Ratio, stated that "inferences on $\phi$ can be based on the conditional distribution of $X$ given $X+Y$" and derived theories based on this statement. For instance, they estimated the odds ratio $\phi$ by maximizing the conditional probability $P(X=x|X+Y=m)$.

However, I'm not sure why people can conduct inference based on the conditional distribution of one statistic in the minimal sufficient set given the other statistic in the minimal sufficient set.

  • $\begingroup$ $(X, X+Y)$ is the minimal sufficient statistic for $\phi$, but the conditional probability is not independent from $\phi$, $P(X=x|X+Y=m, \phi)$. In the paper you cite, inference about $\phi$ is performed through the analysis of the asymptotic behaviour of the conditional probability. Which part is not clear? $\endgroup$ – user289381 Jul 11 at 17:20

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