Effect of corrections for multiple testing on sensitivity and specificity Let's imagine we have one variable (factor.to.explain) that we want to explain by 10 other variables using 10 linear model (no interactions computed). We should correct for multiple testing. It is possible that one, some or several of them affects the factor.to.explain.
Note: TP = True Positive, FN = False Negative, FP = False Positive, TN = True Negative


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*How does a Bonferroni correction affects sensitivity (= TP / TP + FN) and specificity (= TN / TN + FP) ?

*How does other corrections for multiple comparisons affects sensitivity and specificity ?

*Do sensitivity and specificity change if I do only 5 or all (10) my linear models ?

*Is there a objective way of deciding what is the best sensitivity to specificity ratio ?*


Specific question (not important if I don't get an answer):


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*What sensitivity and specificity ratio is usually aimed in ecology ?

 A: You are apparently applying classification terminology to hypothesis testing. That's fine but couching the problem in the more traditional type I error/type II error/power terms might help relate this to the multiple testing literature.
Hypothesis tests can be presented in this way:
                      Null hypothesis (H0) is true                H0 is false

Reject H0                         Type I error (FP)      Correct outcome (TP)
Failure to reject H0           Correct outcome (TN)        Type II error (FN)

The type I error rate or significance level, $\alpha$, is the proportion of rejections under the null, i.e. $FP/(FP+TN)$. This is also $1 - specificity$. It makes sense because  a lower significance level means a more stringent test, i.e higher specificity, which is a desirable characteristic.
Instead of the type II error rate, $\beta$, practitioners often talk of “power”, i.e. $1 - \beta$. Power can therefore also be described as $TP/(FN+TP)$, i.e. sensitivity. Here the technical terminology is very intuitive, a more sensitive test is a more powerful test as would everyday English usage suggest. Importantly, power depends on $\alpha$, but not in a simple arithmetic way, because of the usual trade-off between sensitivity and specificity. For a given effect and sample size, lowering $\alpha$ will increase $\beta$ and lower power.
Now, the Bonferroni correction is based on a very basic probability inequality. You simply divide the error level for each individual test by the total number of tests you perform. Each individual test will have a lower $\alpha$ and therefore a higher specificity but also, as I mentioned before, a lower sensitivity. All tests considered together will have at least the specified level of specificity (in fact, most of the time, a higher specificity, which is why it is said to be conservative).
Other multiple comparison procedures also aim at keeping the overall error rate in check (there are some further distinctions between different kind of error rates here, hence the somewhat fuzzy terminology). The Holm procedure for example is as general as the Bonferroni correction but is less conservative so it will provide a lower specificity and a higher sensitivity/power for individual tests. This is a good thing because the starting point is that you were OK with a lower specificity to begin with, so as long as you can control the familywise error rate/overall specificity, you want to increase power/sensitivity as much as possible.
Given the way the Bonferroni correction works, it will obviously be affected by the number of tests you perform. The more tests, the lower the error level for each test. The lower the error level, the higher the specificity but also the lower the sensitivity. If you have 5 tests instead of 10, you will have a lower specificity and a higher sensitivity for each individual test. Other multiple comparison procedures go about it more cleverly but will also generally be impacted in the same way, just not as dramatically.
I don't have anything to say about ecology beside the fact that there is no “objective” way to determine the best sensitivity to specificity ratio (or the ratio of type II/type I error rates). It depends on what you are more concerned about: False negatives or false positives? Erring on the side of caution or not noticing a potentially important effect? If you can attach a monetary cost or a number of lives saved to each outcome in the decision, you could of course go about the problem more systematically.
In some disciplines, despite decades of criticism, current practice suggests that researchers are mainly obsessed with type I errors, systematically setting the specificity at 95%, worrying about multiple comparisons, being very reluctant to interpret or discuss results that are not significant at the conventional error level, etc. whereas power/sensitivity only comes as an afterthought. Interestingly, if people worry about power at all, they only ever consider increasing the sample size or trying to reduce measurement error, never lowering specificity for the sake of sensitivity.
Writing about psychology and given typical effect and sample sizes in this discipline, Rosenthal and Rubin (1985) write that researchers “behave as though Type I errors were from 5 to 95 times more serious than Type II errors depending on whether we choose the .05 (5 to 20) or the .01 (40 to 95) level of significance”. For a type I error rate of 5%, this translates to a sensitivity-to-specificity ratio between 2 and 10.
Rosenthal, R., & Rubin, D.B. (1985). Statistical Analysis: Summarizing Evidence Versus Establishing Facts. Psychological Bulletin, 97 (3), 527-529. 
