statistical analysis - without data dredging I have conducted an experiment on $M$ different cell-types and have recorded an observed, continuous, value (irrelevant what it is) when I exposed the genes for a new drug and when I did not expose them (control group). For each gene, I did $N$ measurements with the drug and $N$ without.
I would now like to statistical test to see for which cell-types the drug is effective. However, I would like to avoid a situation of data dredging. If $M$ is large enough, in comparison to $N$, the probability is high that at least some cell will report a postitve effect of the drug. 
I do not know how to analyze this type of situation. Any help would be most helpful.
Thanks
 A: What you call data dredging is a very well known and tricky issue in statistics.  It is called Multiplicity.  If you test 20 different hypotheses simultaneously, there is a pretty high likelihood that you will have a false positive.  One of the hypotheses will be confirmed at the 0.05 p-value level.  You have two solutions.  
The first solution is to use a Post Hoc test like Tukey's Honestly Significant Difference (HSD) test or Dunnett test.  Dunnett test is also called a Planned Comparison test.  This means it is somewhat less restrictive than a true Post Hoc test because in this case, it factors that among your 20 different hypotheses there may be just a few that you were truly more focused upon from the start. 
The second solution is actually a lot easier.  It consists in adjusting your p-value alpha thresholds to reflect the number of hypotheses you are testing.  So, let's say you initially started with a p-value alpha threshold of 5% and you are testing 10 hypothesis.  Now you adjust your p-value alpha threshold as follows: 5%/10 = 0.5%.  This adjustment is called the Bonferroni test.  There is a second method that uses some compounding and is technically fractionally more accurate called the Sidak test.  In this same example it would be calculated as follows: 1 -(1-5%)^(1/10) = 0.51%.  As you can tell the Sidak test makes very little difference vs. the Bonferroni test.
There is an extensive literature on all the terms and tests used above, including good material at Wikipedia.  As indicated, the Bonferroni and Sidak tests are pretty easy to calculate manually.      
A: The answer by Sympa goes into more traditional multiple hypothesis testing procedures.  I think your situation might be a better fit to the more modern ideas of False Discovery Rate control.  The false discovery rate (FDR) is the (expected) proportion of rejected nulls (that is, accepted research hypothesis) that are in fact false.  Methods controlling FDR will probably have higher power.  To read more about this I will propose the paper

Efron, B. (2004) "Large-scale simultaneous hypothesis testing: the
       choice of a null hypothesis", Jour Amer Stat Assoc, 99, pp.
       96-104

with methods which are implemented in the R package locfdr, (on CRAN).  The help pages for that package will give you more references.  There are also other R packages for FDR control, like PsiHat, or you can search CRAN.
