# In the equivalence test (TOST procedure), why is the null hypothesis the presence of a real effect?

In the test for equivalence (or non-inferiority / non-superiority) we define a "minimal effect size" $$d$$, and test that the measured effect is not larger that $$d$$

$$X_1 \sim PDF1$$ $$X_2 \sim PDF2$$ $$TOST: mean(X1 - X2) < |d|$$

The difference from a t-test being that we do assume an effect size $$> 0$$, but we set a boundary under which to neglect this effect size.

However, in the TOST procedure, the null hypothesis claims that the effect size is larger that $$d$$, and the alternative hypothesis is that it is smaller. This is a bit against the usual null hypothesis that assumes randomness, non-relatedness, and absent effect sizes.

If we assume that the treatments that generate $$X_1$$ and $$X_2$$ are meaningless, and both are random variables drawn from the same distribution $$PDF1 = PDF2$$, which is what we would assume doing a t-test, the null of the TOST should also be the absence of an effect (i.e the effect size being smaller than a pre-defined $$d$$).

Why does the TOST procedure have a contra-intuitive null?

## 1 Answer

The working hypothesis is: "the effect is scientifically negligible". This is what the researcher claims. The null hypothesis is simply the contrary, i.e. there is a scientifically non-negligible effect.

I find this very intuitive (in contrast to the fact that no multiple testing issue occurs with the two one-sided tests).

• This makes it sound as if the choice of the null and alternative is an arbitrary choice of whatever the researcher 'claims', and are interchangeable by just re-arranging the wording of the research question. I assume there is a deeper reason why the null is designed that way, related to what can be tested with evidence and what can't. We choose the null in t-testing to be non-relatedness, as is the only claim we can either reject or fail to reject (we can't accept/prove absence of effect in a noisy world). Am I confused about this? – hirschme Apr 23 '20 at 18:22
• Indeed, this is what hypothis testing is about. The researcher claims something, then does a nice study and verifies (or not) his claim by statistics. Another researcher might claim the contrary. His hypotheses would be flipped. Of course this second researcher will need to perform a new study to check his flipped working hypothesis. – Michael M Apr 23 '20 at 19:13
• I think the point is that each test tackles a specific claim, so each test has its specific "type of" null hypothesis. A t-test H0 has to be under random and non-effect assumption, and this can't be flipped. The TOST is designed to test for non-equivalence, so its H0 is based on the assumption of a true effect. Your comment makes sense in this case: yes, hypothesis are interchangeable, but each one then suggests a different and appropriate test. Does this agree with you thoughts? – hirschme Apr 23 '20 at 19:49