Can we disregard the whole model? Omnibus non-inferiority testing for $R^{2}$ in multivariable linear regression and $\hatη^{2}$ in ANOVA

Abstract: Determining a lack of association between an outcome variable and a number of different explanatory variables is frequently necessary in order to disregard a proposed model (i.e., to confirm the lack of an association between an outcome and predictors). Despite this, the literature rarely offers information about, or technical recommendations concerning, the appropriate statistical methodology to be used to accomplish this task. This paper introduces non-inferiority tests for ANOVA and linear regression analyses, that correspond to the standard widely used $F$-test for $\hat{\eta}^2$ and $R^{2}$, respectively. A simulation study is conducted to examine the type I error rates and statistical power of the tests, and a comparison is made with an alternative Bayesian testing approach. The results indicate that the proposed non-inferiority test is a potentially useful tool for 'testing the null.'