Transformations of Variables

Suppose that you are have a response variable $Y$ and explanatory variables $X_1$, $X_2$ and $X_3$. If we want to use a quadratic transformation for $X_1$, would we still include $X_1$? In other words, would we have:

$$E[Y|X] = \beta_0+\beta_{1}X_{1}^{2} + \beta_{2}X_{2} + \beta_{3}X_{3}$$ or $$E[Y|X] = \beta_0+\beta_{1}X_{1} + \beta_{2}X_{1}^{2} + \beta_{3}X_{2}+\beta_{4}X_{3}$$

If instead we did a logarithmic transformation, then would it just be:

$$E[Y|X] = \beta_0+\beta_{1} \log(X_1) + \beta_{2}X_{2} + \beta_{3}X_{3}$$

• NB the first model imposes a constraint that the minimum or maximum of $Y$ plotted against $X_1$ occurs at exactly $X_1=0$. You'll want to be sure that's sensible. It often isn't, & therefore the first model is ruled out a priori. Mar 11 '14 at 9:50

Either model could work, and which to use depends on why you transformed. If you took the log because $Y$ is expected to be linearly related to $\log X_1$ (e.g. response to log dose of medication), then you would probably just go with $\log X_1$ and drop the untransformed term. On the other hand if you added a quadratic term because it looked like $Y$ was related to $X_1$ in a polynomial fashion (e.g. crop yield vs latitude) you would probably try keeping both $X_1$ and $X_1^2$ and then drop one (or both) if it turned out to not be significant.