suppose I have $K$ different methods for forecasting a binary random variable, which I test on independent sets of data, resulting in $K$ contingency tables of values $n_{ijk}$ for $i,j=1,2$ and $k=1,2,...,K$. How can I compare these methods based on the contingency tables?  The general case would be nice, but $K=2$ is also very interesting.

I can think of a few approaches:

 - compare the tetrachoric coefficients of the $K$ tables. this would be especially useful if there were something like Fisher's R-to-Z transform for the tetrachoric coefficient.
 - compute the Chi-square statistic on each of the tables, and compare those random variables (I'm not sure if this is a standard problem or not),
 - something like [Goodman's improvement of Stouffer's method][1], but I cannot access this paper, and was hoping for something a little more recent (more likely to have the latest-greatest, plus computer simulations).

any ideas?


  [1]: http://www.jstor.org/pss/2982447 "On methods of comparing contingency tables"