About statistical significance, and the lack thereof

Absence of statistical significance (i.e., p > 0.05) in the results of a frequentist test comparing two samples is often used as evidence of absence of difference, or absence of effect of a treatment, on the measured variable. Such conclusions are often wrong because absence of significance may merely result from a sample size that is too small to reveal an effect. To conclude that there is no meaningful effect of a treatment/condition, it is necessary to use an appropriate statistical approach. For frequentist statistics, a simple tool for this goal is the 'two one-sided t-test,' a form of equivalence test that relies on the a priori definition of a minimal difference considered to be relevant. In other words, the smallest effect size of interest should be established in advance. We present the principles of this test and give examples where it allows correct interpretation of the results of a classical t-test assuming absence of difference. Equivalence tests are also very useful in probing whether certain significant results are also biologically meaningful, because when comparing large samples it is possible to find significant results in both an equivalence test and in a two-sample t-test, assuming no difference as the null hypothesis.