# interpretation of the extreme point in a quadratic regression

I am doing a negative binomial regression analysis with panel data. My dependent variable is the number of patents and my independent variable the age of a CEO and its squared term. A u-test in stata gives out an extreme point of 47.54 and an inverse U-shaped relationship.

How do I interpret the extreme point? Can I, for example, say that the optimal age for a CEO is at 47,54 years, and that the innovation output (patents) decreases with every additional year? Also, a minor additional question: how do I correctly interpret the coefficients of age and age_squared?

how do I correctly interpret the coefficients of age and age_squared?

You don't interpret either if these independently. Normally, we interpret coefficients as the expected changes in the outcome resulting in a one unit change in the associated predictor. You can't increase age by one unit without also increasing age squared. The effect is then the partial of the conditional mean with respect to age (Thanks Dave).

How do I interpret the extreme point? Can I, for example, say that the optimal age for a CEO is at 47,54 years, and that the innovation output (patents) decreases with every additional year?

I wouldn't say that exactly. You could say really young and really old CEOs are associated with fewer patents, but making a statement about a precise age at which patents are maximized seems like you're overselling the result.

Additionally, I would think about some lurking third variables you might be missing. Does your analysis mean that if your company hires a CEO in his 40s then it will yield more patents? No, of course not. This would vary by industry, company age, etc etc. Very old CEOs, though very experienced, may not have the energy to run large corps.

• Regarding the first paragraph, remember that (at least for continuous predictors) you get the effect by taking the partial derivative. If you’ve got a squared variable in there, you take the partial derivative like you usually do for a function like $f(x,y)=x+x^2$ (so $f_x=1+2x$, not just a single number).
– Dave
Dec 4, 2020 at 2:58
• @Dave Thanks. I've made an edit Dec 4, 2020 at 3:04

On the question "how do I correctly interpret the coefficients of age and age_squared", I favor an answer based on the specific industry.

For example, new tech companies in social media needs a CEO with a new perspective and acquaintance with current and upcoming trends (so, likely younger, a negative coefficient on the quadratic term).

A computer company that has been around awhile needs a combination of creativity and seasoned marketing skill (so, say middle-aged perhaps, near-zero quadratic coefficient).

Conversely, a cyclic natural resource company needs a CEO who is conservative with a longer-term view (so older, a positive quadratic term).

So, the age_squared variable could be a possible proxy to account for the characteristic average age of the CEO in industry sectors of existing successful (survivors) companies.

[EDIT] Per a comment, explicitly in the case of a negative binomial (NB) regression model where the dependent variable is the number of patents and for independent variables the age of a CEO and its squared term (resulting in an inverse U-shaped relationship), I claim an interpretation from the structural form of the NB regression model, represented as follows:

$${𝜇_𝑖 = exp(ln(𝑡_𝑖) + 𝛽_1X1_𝑖 + 𝛽_2X2_𝑖 + ⋯ + 𝛽_𝑘X𝑘_i)}$$

So implicitly, for a one unit change in the explanatory variable, the difference in the natural logs of expected counts for the response variable is expected to change by the respective regression coefficient, given that other predictor variables in the model are held constant.

Per my prior remarks above, for example, with a negative (or positive) coefficient times a positive change in the independent variable, there is a corresponding multiplicative effect on the response count by a factor less (or greater) than one.

So, my prior comments above on positive/negative effect should be viewed as in a multiplicative corrective sense, which possibly could indeed result 'in an inverse U-shaped relationship' with a quadratic term.

• My question is not about the context or the practical/theoretical reasoning. I just want to know the statistically correct interpretation of the extreme point and of the coefficients in a negative binomial regression.
– Marv
Dec 3, 2020 at 23:06
• Now, an Edit to hopefully more clearly note implications of my above remarks in the particular context of a negative binomial (NB) regression model. Dec 4, 2020 at 2:53