Number of books published in a year (noBook) is my dependent variable and I have independent variables including the age of the author (age).

The coefficient of age is positive and it is significant. When I add age^2 to the model, age and age^2 are significant but the sign of age becomes negative.

Could you please advise why it is like that? And, which model should I consider?

• One the given information, it implies the minimum of the fitted quadratic is at a positive age, but doesn't tell you much of anything else. It's not surprising that the coefficient of a variable changes sign when you add another variable. Jan 27, 2014 at 22:15
• You need to tell us what the coefficients actually are. It would also help to know the range of age. Jan 27, 2014 at 22:46
• Also, I hope you are not using OLS regression. Number of books is a count. Jan 27, 2014 at 22:47
• I am using negative binomial regression, range of age is from 1 to 20 since I have considered the career age of the authors. Jan 27, 2014 at 23:00
• @peter The coefficient of age in the first model is 0.0062546 and then in the second model it changes to -0.0297705 while the coefficient of age^2 is 0.0023998 in the second model. Jan 27, 2014 at 23:02

The fact that the coefficient of age becomes negative doesn't tell you much on its own. In the new model, the predictive effect of the author's age isn't just represented by an expression of the form $\beta\textit{age}$, but by a sum $\beta_{1}\mathit{age}^2+\beta_{2}\mathit{age}$. Assuming that the coefficient of age was positive in your first model, I would expect that over the range that you are considering, this quadratic expression is still increasing (check the derivative $\frac{\partial}{\partial\textit{age}}$ to make sure), indicating that nbooks increases with age, even if $\beta_{2}$ is negative.

(This reasoning still holds even if you have interaction terms, but the partial derivative above will take a little more time to compute, and you may have to consider a few more cases).

• Plot the response against age over the range you're considering if you don't know what a partial derivative is. Jan 28, 2014 at 10:06

OK, in your comment you wrote

The coefficient of age in the first model is 0.0062546 and then in the second model it changes to -0.0297705 while the coefficient of age^2 is 0.0023998 in the second model.

If there are no other variables in the model, then this means that e.g. for a 30 year old author, the predicted number of books in model 1 is

intercept + .0062545*30 = intercept + 0.19


in model 2 it is intercept - 0.02977*30 + .0024*900 = intercept + 1.27

Whereas for 60 year olds the effect of the squared term would be much bigger.

model 1 = intercept + 0.38
model 2 = intercept + 8.46


All this is presuming the modelling is otherwise correct; as I noted, ordinary least squares regression is unlikely to be correct here.