[R] Confidence Limits for a Cross-Product Ratio

[Peter Dalgaard]

francogrex wrote:
> Data from Fisher's paper: Confidence Limits for a Cross-Product Ratio.
> 
>> y
>      col1 col2
> [1,]   10    3
> [2,]    2   15
> 
> fisher.test(y)
> 
>         Fisher's Exact Test for Count Data
> 
> data:  y 
> p-value = 0.0005367
> alternative hypothesis: true odds ratio is not equal to 1 
> 95 percent confidence interval:
>    2.753438 300.682787 
> sample estimates:
> odds ratio 
>   21.30533 
> 
> The crude odds ratio in Fisher's paper is 25 and the lower 95%CI is 2.750.
> How come this is different here? Why is the estimate 21.30533 and how is the
> confidence limits calculated (is there a reference for a statistical paper
> other than that of Fisher)?
> 

(R is open source, you know. You _can_ read the code of fisher.test for 
yourself.)

The estimated OR is the conditional MLE in the noncentral hypergeometric 
distribution. This is not equal to the crude OR, a fact that can be 
easily noted by the results being different(!), but it's also mentioned 
in places like Breslow/Day's book on case-control studies.

The CI given is the intersection of the two exact one-sided 0.975 
levels. (Exact in the sense that it looks for the parameter for which 
the p-value is exactly 0.025).

Exact 2-sided intervals are awkward to do (insofar as they can even be 
defined) because of probability mass switching between the tails of the 
distribution.

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