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I am fitting GLMM's (using a binary variable as response variable and continuous variables as explanatory variables [family = binomial(link="logit")]), and I am interested in obtaining the effect sizes for each explanatory variable.

I obtain the effect size value by calculating odds ratios (Effect size in GLMM).

However, I am considering a variable with a linear (a) and quadratic form(a^2). Here is an example of a model: 

model <- x ~ a + I(a^2) + (1|b)

In this case (linear and quadratic forms), 1) is the effect size estimated in the same way (odds ratio), and 2) with the same interpretation?

I can't seem to find information about this topic; do you know of any good literature?

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mto23
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  • The odds ratio is the effect size so perhaps you need to explain what you are trying to do. – mdewey Mar 06 '17 at 14:25
  • @mdewey I was looking for a way to estimate the effect size (which, in this case, should be done using odds ratio) for both terms of a variable, at the same time – mto23 Mar 06 '17 at 14:27

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No. When you have a quadratic (or higher order) term or an interaction, there is no single overall measure of the effect size of a variable because the effect size will vary at different levels. So, you could find the OR at any particular level of the variable, but it will be different at different levels.

The same is true for "regular" logistic regression. See this thread

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Peter Flom
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  • So, if I understood you correctly, I can estimate the odds ratio for all terms (in this case, "a" and "a^2"), but I cannot compare the values at different levels (only within levels)? – mto23 Mar 06 '17 at 14:32
  • Well, but one could define an "effect size function" one could plot, maybe even with a confidence band. Or, if the population disteibution of $x$ is k own, integrate to find a population average effect size ... – kjetil b halvorsen Mar 25 '17 at 21:50
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    You can compare any combinations of levels of the independent variables to any other, but no variable has a single odds ratio. The odds ratio varies. – Peter Flom Mar 26 '17 at 12:25