A squared term is negative and significant but the graph is not.

[April Kimm]

A squared term is negative and statistically significant, suggesting an inverted-U relationship as hypothesized, but the graph for the predicted probability of the variable indicates a linear relationship. What does it mean and what should I do?

[Nick Cox]

Hard to say anything much without any real details, e.g. knowing the command you used or seeing the command output or the graph in question, but if you're fitting a quadratic (i.e. linear and squared terms together), then

1. Both terms being significant at conventional levels is consistent with mild curvature, especially if the sample size is large.

2. A quadratic has a turning point somewhere, here evidently a maximum, but that turning point may lie outside the range of the data.

3. An approximately linear relationship between predicted probability and a predictor itself implies that the probability will escape [0, 1] at some points, but as in #2 perhaps what is going on within the range of the data is acceptable. Use of an appropriate link function (e.g. logit) would make everything fine.

[Nick Cox]

Let me add a further standard point. Linear and squared terms are usually highly correlated -- those thinking that the relationship is one of squaring by definition are right, but may still be surprised at how high the correlation can be. Either way, the point is that linear and squared terms need to be taken together. There isn't a useful substantive interpretation of linear and squared effects as separate effects even though coefficients and their uncertainties are stated separately.