Including both transformed and original data (untransformed) in a multivariable linear regression.

Question

This may have a quick response (i.e. don't do it).

Just attended a lecture on multivariable linear regression where the outcome is forced expiratory volume (the amount of air you can push out of your lungs in one go). The explanatory variables are age, height, gender and smoking status. In this example, the lecturer finds that height is skewed and therefore transforms the variable to Height^2.

After performing forward selection he ends up with 5 significant explanatory variables in his model: age, height, gender, smoking status and height^2.

My question is: How can you end up with both height^2 and height in your model. Wouldn't they interact (be highly related)? More broadly, when we transform a variable, should the untransformed variable ever be in the model with the transformed?

Answer 1

Partially answered in comments:

There are two ways to think about it. One is to think of it as an interaction with itself. When we use interactions we generally do include the main effects and the interaction term: e.g. on ucla site. Another is to think of it as a polynomial. There is no reason why the relationship should be linear. If a polynomial fits better then why not use it? The Data Generating Process might be non-linear.

– Toby

BTW in this case think of your lungs as two cylinders that are a function of someone's size. The volume of a cylinder is $πr^2h$. People who are taller probably are also larger in other dimensions, they tend to be right?, so the $r$ will be larger. And since the cylinder is a nonlinear function of $r$ it will be a nonlinear function of size. It seems to make sense to me to include height squared. Write it in logs and you have $\log(πr^2 h) = \log(\pi) + \log(r^2) + \log(h)$. If $h$ and $r$ are both a function of size for which height is a reasonable proxy then this is your model.

– Toby

( This argument seems to lead to log transforms of both response and height )

Using height and height^2 is fitting a quadratic in height, other predictors aside. It could be just a way of catching some curvature in the data. There are even cases, e.g. trajectory of a projectile, in which a quadratic is exactly the shape you expect.

– Nick Cox

That's very helpful. I can't see a biological reason to have height#height^2 as an interaction term? I do, however, see why you'd square height in this specific example. My take on his question is, more generally when we transform non-normal covariates, whether we should include both the transformed and original variables in the model?

– bobmcpop

Toby's argument in comments can be extended this way: often $r$ wil be only approximately proportional to $h$, $r$ wil grow slower (and slower and slower) as $h$ grows, so maybe $r\propto h^\alpha$ for some $0<\alpha<1$ is a better model. That leads to $y (\text{volume})\propto h^{1+2\alpha}$ and taking logarithms on both sides leads to a model in only $\log h$, where we in addition have an theoretical expectation for the value of slope: $1+2\alpha$, so we can see from the fit if this geometrical speculations holds for this case. And no, in this model we do not need $h$ untransformed as an additional term.

Answer 2

If you are concerned about the existence of multicollinearity, it happens when there is a linear relationship among the explanatory variables. You can include powers of the same variable in many models. Transforms can be included with the original variable, as long as they are not linear combinations, which would inflict multicollinearity to the model. Powers obviously are not linear, therefore it would not be a problem here.