Interpreting Quadratic Variables in Negative Binomial Regression I want to include a quadratic term for age in my negative binomial model as past work has suggested it may be curvilinear. I know how to interpret a quadratic coefficient in OLS, but am unsure with negative binomial.

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*Is it appropriate to include both the original variable (age) and the quadratic (agequad) in the model?

*How are the coefficients interpreted?

*When plotting predicted probabilities, is the original (age) or quadratic (agequad) used? Note: I use stata's margins command for this.

 A: 
I want to include a quadratic term for age in my negative binomial model as past work has suggested it may be curvilinear.

While I do not doubt the conditional mean is non-linear in age, do you have any specific reason to prefer a quadratic term as opposed to something more flexible such as a natural spline?  Polynomials are great tools, but are rather biased because you can only estimate things which look like quadratic functions (or whatever polynomial you decide to use).  If you require additional flexibility, a spline is a more flexible but still interpretable method.
As to your other questions...

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*Yes, it is more than appropriate, it is expected.  One explanation is found here.


*The coefficients are not directly interpretable, you have to look at the conditional mean in order to interpret the effects of age.  In a negative binomial regression, your model will look like
$$ \log(E(y)) = \beta_0 + \beta_1 x + \beta_2 x^2 $$
The typical "a one unit change in $x$ leads to a $\beta$ unit change in the expectation of the outcome" no longer applies because of the included quadratic term.  You could take a partial derivative with respect to $x$ in order to determine how varying $x$ changes the expectation on the log scale, but my preference is just to plot the estimated function.  If all other predictors are linear, then they will simply move the function up or down.


*Negative Binomial regression does not predict probabilities.  Are you sure you're not talking about logistic regression?  In any case, both age and age squared are used in making predictions.
