For simple linear regression (SLR), in order for $R^2$ (the coefficient of determination) to be a meaningful measure, it must be true that $X$ and $Y$ are linearly correlated. Specifically, $R^2=r^2$, where $r$ is Pearson's correlation coefficient.
When we move into the multiple linear regression (MLR) framework, I'm curious how this linearity requirement transfers.
Take, for example, a polynomial model where $\hat y = \hat \beta_0 + \hat \beta_1x+ \hat \beta_2x^2$. In this model, assume the regression assumptions are met (i.e., the data are truly related according to a parabolic curve, so fitting $x^2$ as a predictor allows us to meet the linearity assumption).
Now, $X$ and $Y$ are not linearly related, but $X$ and $X^2$ jointly allow us to correctly model the association with $Y$. So, since we've met the linearity assumption of MLR, $R^2$ is meaningful, correct?
So would the conclusion be that $R^2$ is meaningful if the modeled relationship between $Y$ and the predictors (whatever they may be (i.e., even if they're polynomials)) satisfies the regression assumptions?
If so, we would say: In the case of SLR, this forces the requirement of $X$ and $Y$ being linearly related, but for MLR, the relationship between $\textbf X$ and $Y$ may be curved, as long as the linearity assumption is met.