tl;dr - for OLS regression, does a higher R-squared also imply a higher P-value? Specifically for a single explanatory variable (Y = a + bX + e) but would also be interested to know for n multiple explanatory variables (Y = a + b1X + ... bnX + e).
Context - I'm performing OLS regression on a range of variables and am trying to develop the best explanatory functional form by producing a table containing the R-squared values between the linear, logarithmic, etc., transformations of each explanatory (independent) variable and the response (dependent) variable. This looks a bit like:
Variable name --linear form-- --ln(variable) --exp(variable)-- ...etc
Variable 1 ------- R-squared ----R-squared ----R-squared --
I'm wondering if R-squared is appropriate or if P-values would be better. Presumably there is some relationship, as a more significant relationship would imply higher explanatory power, but not sure if that is true in a rigorous way.