# Why inferences on odds ratio can be based on the conditional distribution of X given X+Y

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.

• $(X, X+Y)$ 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? – user289381 Jul 11 at 17:20