# 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?