# What ANOVA non-superiority test corresponds to an omnibus set of TOST equivalence tests?

I am not sure if such a beast exists, but is there some non-superiority test for a oneway ANOVA of $k$ groups, probably based on a $\Delta^{2}$ equivalence threshold, for $m=\left[k(k-1)\right]/2$ post hoc pairwise tests for equivalence (using the two one-sided tests procedure) based on an equivalence threshold of $\Delta$? (Where $\Delta$ is in units of the score being tested; my question could be reframed in terms of an equivalence threshold in units of the test statistic distribution, as in $\varepsilon = \Delta/s_{\bar{x}}$.)

Edit: turning this into self-study

On the one hand, I might argue that the F statistic is a ratio two variances of the variable being tested, and that since variances are in squared units, that I could simply test something like H$_{0}\text{: }F \ge \frac{\bar{n}\Delta^{2}}{s^{2}_{\text{within}}}$, and reject H$_{0}$ if P$(F \le \frac{\bar{n}\Delta^{2}}{s^{2}_{\text{within}}}) < \alpha$. Is this naïve?

On the other hand:

Relationship between t and F tests

$\begin{eqnarray} t_{\nu} & = \frac{\mathcal{N}\left(0,1\right)}{\sqrt{\chi^{2}_{\nu}/\nu}}\text{, where }\nu=N-1.\\ t^{2}_{\nu} & = \frac{\mathcal{N}^{2}\left(0,1\right)}{\chi^{2}_{\nu}/\nu}\\ & = \frac{\chi^{2}_{1}/1}{\chi^{2}_{\nu}/\nu}\\ & = F_{1,\nu}\\ \end{eqnarray}$

TOST for a t test for equivalence, given symmetric equivalence threshold $\Delta$:

$\begin{eqnarray} t_{1\nu} & = & \frac{\Delta - \mathcal{N}\left(0,1\right)}{\sqrt{\chi^{2}_{\nu}/\nu}}\\ & = & \frac{\Delta}{\sqrt{\chi^{2}_{\nu}/\nu}} - \frac{\mathcal{N}\left(0,1\right)}{\sqrt{\chi^{2}_{\nu}/\nu}}\\ t^{2}_{1\nu} & = & \left(\frac{\Delta}{\sqrt{\chi^{2}_{\nu}/\nu}} - \frac{\mathcal{N}\left(0,1\right)}{\sqrt{\chi^{2}_{\nu}/\nu}}\right)^{2}\\ & = & \left(\frac{\Delta}{\sqrt{\chi^{2}_{\nu}/\nu}}\right)^{2} - \frac{2\Delta\mathcal{N}\left(0,1\right)}{\sqrt{\chi^{2}_{\nu}/\nu}} + \left(\frac{\mathcal{N}\left(0,1\right)}{\sqrt{\chi^{2}_{\nu}/\nu}}\right)^{2}\\ & = & \frac{\Delta^{2}}{\chi^{2}_{\nu}/\nu} - \frac{2\Delta\mathcal{N}\left(0,1\right)}{\sqrt{\chi^{2}_{\nu}/\nu}} + \frac{\mathcal{N}^{2}\left(0,1\right)}{\chi^{2}_{\nu}/\nu}\\ & = & \frac{\Delta}{\sqrt{\chi^{2}_{\nu}/\nu}}\left(\frac{\Delta}{\sqrt{\chi^{2}_{\nu}/\nu}} - 2\mathcal{N}\left(0,1\right)\right) + F_{1,\nu} \end{eqnarray}$

But I feel like I am kinda falling down a rabbit hole with this approach, and don't know how to translate that to $k$ number of groups >2.

• Perhaps this is of interest: arxiv.org/abs/1905.11875 Kelley (2007) is also a great reference for this sort of thing. – happydog May 29 at 19:43