# Can you add up effect sizes of two related variables (e.g. age + age$^2$) from a single regression model?

I am interested in the effect of age on outcome Y.

I have two nested linear regression models to test linear and quadratic effects of age:

1. Y= $\beta_0$ + $\beta_1$ some_covariate + $\beta_2$ Age + error
2. Y= $\beta_0$ + $\beta_1$ some_covariate + $\beta_2$ Age + $\beta_3$ Age$^2$ + error

I have calculated η2 (not partial η2) as measures of effect size for the individual predictors.

Let's assume that the effect sizes are as follows:

1. $\eta^2$ age= .05
2. $\eta^2$ age= .03 & age$^2$= .04

For model (1) I could state that the effect size of age is .05.

If I want to make a statement about the effect size of age in the second model, am I allowed to add the two effect sizes up? E.g. the $η^2$ for age in model two is .07?

• I'm a little confused by your models labelled 1. and 2. I understand in general terms what might be intended by Y, "some covariate" and Age, but then what's Xi1, Xi2 and Xi3? You seem to have six variates in your second model and no main effects term. Dec 26 '14 at 23:43
• For clarification I have corrected the models. Dec 27 '14 at 1:21
• Ah, much clearer. I've made a few additional clarifications and done a little formatting Dec 27 '14 at 2:07

In the case of $\eta^2$, it makes some sense to combine the two, since it's essentially a "proportion of variance explained". It would be the overall "effect of age" in the model rather than linear effect of age (corresponding to the sums of squares due to both divided by total SS). That is, if you adjust for everything but age, and then combine the $\text{age|covariates}$ and $\text{age}^2|\text{covariates,age}$ sum of squares in the numerator of your $\eta^2$, that would make some degree of sense (though it would retain the usual problems one gets with using $\eta^2$ a measure of effect size more generally).
• also consider the effects of age and age$^2$ separately (if you do this I strongly suggest fitting them using orthogonal polynomials, so you can discuss the effect size of linear-age and then the additional effect of age$^2$ over linear age in the one model)