# Cubic and linear relationships in multiple regression model

What is the correct way to fit a multiple regression model where I have a combination of cubic and linearly related independent variables?

If I transform the variable showing cubic relationship, how do I transform back the forecasted variable, given that it's the result of both non-transformed and transformed data sets?

• If you don't actually care about the specific parameter values of the functions, and instead want to simply establish whether relationships (linear or non-linear) exist between your variables, I suggest you look at Generalized Additive Modelling, which will flexibly/adaptively accommodate both linear and non-linear effects. – Mike Lawrence Dec 4 '11 at 23:06
• Its not clear what you're asking... why are you transforming your outcome variable? Its often best to include a term linearly as well as using its square and/or cube. – Michael Bishop Dec 5 '11 at 0:53
• I have X1 that shows cubic relationship against Y. However X2 has linear relationship. I want to use a multiple regression model to calculate the coefficients. Should I transform X1 first, then run a multiple linear regression? Or are there other ways to fit X1, X2? If I transform X1 first, how do I transform back the predicted Y variable, given only 1 of the two dependent variables was transformed? – Robert Kubrick Dec 5 '11 at 12:59

One way to fit the model is, as you guess in the comment, to transform $X_1$ first, then run a multiple linear regression. However, you don't have to do any transformation back to the predicted $Y$ value, since the regression is still using the untransformed $Y$ variable as the dependent variable. Let's say you create $Z = X_1^3$, then the regression becomes $Y = \beta_0 + \beta_1 Z + \beta_2 X_2 + e$. The estimates and/or predictions you get out of the model with the transformed $X_1$ are still for $Y$.
There are other ways to fit the model, but the choice between fitting methodologies is, in this case, independent of the transform of $X_1$.
Edit: It has occurred to me that you may have meant a cubic relationship where $Y^3 \propto X_1$. This is equivalent to $Y \propto X_1^{1/3}$, and you can still just transform $X_1$ without having to transform $Y$ at all.
• The relationship isn't linear, so there isn't an appropriate transform. Just run the multiple linear regression with $X, X^2, X^3$ as your right hand side variables. If you want to plot, just plot your estimated $Y$ against $X$. The key is that you can represent $Y$ as a sum of coefficients times some other variables, but the other variables certainly can all be functions of some underlying variable, as in your cubic polynomial. People run that sort of regression all the time, so I wouldn't worry about it. – jbowman Dec 5 '11 at 18:56