I think that Bruce's approach is the right one. The critical aspect of this sort of exercise is defining 'not changing', which is a subset of 'not different'. There is, as has been pointed out, no test for sameness, since sameness is an unprovable concept. An absence of evidence is not evidence of absence.

Confidence intervals can give you a post-hoc definition of sameness as 'unlikely to be more different than this', and equivalence testing can allow you to set up prior definitions of what would constitute difference and then set up an appropriately powered study to detect this difference. 

Hi Achim. Rather than bother with statistical tests, I think you just need to look at confidence intervals. It should be possible to establish your confidence that your samples are not deviating more than X from each other. Of course, you will need to decide what constitutes a boring/non-(biologically-)significant change. Confidence intervals are often more useful for interpretation than statistical tests anyway. Rich

Look up equivalence testing.  See below for several of the hits I get when I Google that term.

http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/hypothesis-tests/tests-of-means/why-use-an-equivalence-test/

http://www.graphpad.com/guides/prism/6/statistics/index.htm?stat_testing_for_equivalence_with_c.htm

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3019319/

http://pubs.acs.org/doi/pdf/10.1021/ac053390m

Hi Achim; The paper entitled "Understanding Equivalence and Noninferiority Testing" which was introduced by Bruce has described equivalence concept thoroughly. I am not officially a biostatistician and not an expert in omics but you may find the following explanation helpful too:

Hypothesis testing is to evaluate probability of the observed difference in your data under assumption of no difference. The other side of story is the power of your test which reflects second type of error called beta error. If you have not found a statistically significant difference after correction for multiple testing, then you should calculate the corresponding power of your test for the observed difference. If your test has adequate power, say more than 90%, then you can be sure that a difference between the two sets is "improbable". But bear in mind that we never can be 100% sure, we just rely on probability. It is also very important to consider clinical significance in addition to statistical significance that means paying attention to the differences that are meaningful. You can see how much changes was observed in the item of interest in other analysis or to what extend such changes resulted in a consequence. Confidence interval of mean difference in other studies could be very helpful in this regard. But this is conceptual and does not have a rule and depends on your context; if in a clinical trial you find a pill decreases blood pressure of "subjects" 3 mmHg more than another pill, this hardly has clinical significance; but if in a community trial you can show such decrease between intervention and non-intervention "area", that would be an achievement. 

Another useful link for equivalence testing to add to Bruce's above.

http://www2.sas.com/proceedings/sugi23/Stats/p218.pdf

Dear Achim

I think ANOVA will be suitable to test the significant of your hypothesis , and then you can use post hoc test ( I prefer Tukey ) to determine the significant differences among group means .

Good Luck

I think that Bruce's approach is the right one. The critical aspect of this sort of exercise is defining 'not changing', which is a subset of 'not different'. There is, as has been pointed out, no test for sameness, since sameness is an unprovable concept. An absence of evidence is not evidence of absence.

Confidence intervals can give you a post-hoc definition of sameness as 'unlikely to be more different than this', and equivalence testing can allow you to set up prior definitions of what would constitute difference and then set up an appropriately powered study to detect this difference.