An squared term is significant but the predicted probability is not U-shaped.

[April Kimm]

I use Firth logistic regression to examine the following equations.

Y= a +b1x+b2x2 (squared) but the predicted probability is not U - shaped (the first graph)



Also, I have another equation: Y= c+dx+d2x2 (squared) but the coefficient (d) is negative and statistically significant but it is not inverted U-shaped (the second graph).

I used the following command to calculate predicted probabilities.

estimate store full
margins, at ( duration_squared_c = (1 9)) atmeans expression(invlogit(predict(xb)))
marginsplot


Does anyone know why these are happening?

[Nick Cox]

The appropriate plot is against duration, not its square, but that's a detail. A quadratic being a fair fit doesn't mean that the turning point (minimum or maximum) will occur within the range of your data.

Consider (1 + x)^2 = 1 + 2x + x^2 which has a minimum at -1. The minimum may not occur in the range of the data; that's empirical, not mathematical.

The following sequence of graphs, which you can type directly regardless of data in Stata, makes the point. Naturally as the logit link involves a monotonic transformation the position of a turning point is unchanged by pushing a quadratic through invlogit().

.

  twoway function (1 + 2*x + x^2), ra(1 10)

.  twoway function invlogit(1 + 2*x + x^2), ra(1 10)

.  twoway function invlogit(1 + 2*x + x^2), ra(-10 10)