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?

  • $\begingroup$ 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. $\endgroup$
    – Glen_b
    Dec 26 '14 at 23:43
  • $\begingroup$ For clarification I have corrected the models. $\endgroup$
    – Vincent
    Dec 27 '14 at 1:21
  • $\begingroup$ Ah, much clearer. I've made a few additional clarifications and done a little formatting $\endgroup$
    – Glen_b
    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).

You might investigate this effect size further several ways

  • 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)

  • consider how effect-size differs at different ages. With a linear model it doesn't matter which age you look at, the effect of age is the same. With a model that's not linear in age the effect of age changes across different ages. This would involve looking at a difference measure of effect size though - you'd be looking at some standardized mean effect at some particular ages.

  • $\begingroup$ "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)" This is exactly what I mean. From what you wrote I understand it is valid to consider the two effect sizes together. $\endgroup$
    – Vincent
    Dec 27 '14 at 1:23
  • 1
    $\begingroup$ "In the case of η2 (though I assume you really mean partial η2 here)" I actually was talking about η2 and not partial η2. I chose η2 and not partial-η2 because η2 gives the amount of explained variance per predictor in your model and is not affected by other predictors in your model (whereas partial-η2 is: stats.stackexchange.com/questions/15958/…). Would you rather report partial η2? $\endgroup$
    – Vincent
    Dec 27 '14 at 1:34
  • $\begingroup$ Thanks. The article linked at that question points out that these terms are neither uniquely nor always clearly defined. I have tried to edit in such a way that it should be clear what I intended - in particular, I've explicitly mentioned what I would mean by 'combine'. $\endgroup$
    – Glen_b
    Dec 27 '14 at 1:59

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