How to do a Fisher's exact $2 \times 2$ test on data over MANY randomized trials? Suppose that we have several randomized trials over two treatment options for cancer, and we take note if cancer recurred after taking the drug or not. As a clear-cut example, I have included a sample dataset with 14 randomized trials over cancer treatment of Tamoxifen and a Control:

For each study, Rec. means if it recurred or not. My question is, could we combine the entire data into a Fisher's exact 2x2 test? In other words, let us sum the first column, then take the second column, sum it, then subtract from the first's sum, to get:
               Recurrence  No Recurrence  
    Tamoxifen  1989        2554  
    Control    2256        2329 

Would a Fisher's exact test off this table be valid? thanks!
 A: What you would normally want to do in this kind of situation is a meta-analysis of some form. The closest equivalent to Fischer's exact test would be to use exact logistic regression stratified by trial. 
But is this really a binomial outcome? I would assume the individual trials might have been more appropriately analyzed using time-to-event methods and it is likely better to combine the results of such analyses rather than artficially turning it into a binomial data problem, which it isn't. 
A: Besides needing time-to-event analysis as mentioned by Bjorn, note that Fisher's "exact" test is not very accurate and should be replaced by the ordinary Pearson $\chi^2$ test if $Y$ is truly binary.
A: You cannot do this. The Fisher test is a test of 2 by 2 data, asymptotically consistent with the OR and corresponding Pearson chi-sq test. Using either of these tests would be wrong because you would be pooling data. So I'll describe exactly why you can't pool these data. Basically, it's because of collapsibility of the OR.
The reason why your approach is not okay is because you are not comparing cases to controls in their respective studies. For instance, suppose inclusion criteria in trial 1 is to have stage 4 disease so that risk of recurrence (or death) is very high. However, suppose inclusion criteria in trial 2 is to have stage 3 or 4 disease so that risk is slightly lower. Even when the odds ratio is constant between the studies (it's usually not), the pooled data will not yield the same results.
As a simple example:
Trial 1 has 10 treated (5 of whom fail) and 10 untreated (7 of whom fail) for an OR of 5 * 3 / (5 * 7) = 3/7. Trial 2 also has 10 treated (3 of whom fail) and 14 untreated (7 of whom fail) for an OR of 3 * 7 / (7 * 7) = 3/7. The pooled OR is given by 8 * 10 / (12 * 14) = 10 / 21. In this case 10/21 > 3/7 which means the effect of intervention is attenuated. This is a general problem with collabsibility: strata specific ORs are stronger than their crude counterparts. This is why you don't collapse them.
A simple way of obtaining a stratified odds ratio for the various studies is to calculate the Mantel Haenszel odds ratio.
This statistic calculates a pooled OR of (5*3/20 + 3*7/24)/(5*7/20 + 7*7/24) = 3/7
This problem persists even in seemingly identical studies with the same inclusion criteria because of heterogeneity between studies. Use a stratified estimator.
