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There are established ways to rule out medium/high effects like TOST for two-groups.

But is there a way to rule out medium/high effects in one multiple regression? Maybe using eta-squared? What would be the equivalence bounds? Would there be a non-parametric equivalent for such a method?

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For individual coefficients, you can construct your TOST through confidence intervals. For overall impact, your approach is reasonable: Specify an equivalence margin for R-squared (e.g. everything below 0.05 is like 0). Then calculate an upper confidence bound. If it is below your margin, claim "equivalence".

Since the confidence bound of R-squared is based on non-central F distribution, it could be an idea to use non-parametric approaches, e.g. bootstrap.

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