Interpretation for simple slope analysis for curvilinear regression with interaction effects

When regressing income ($Y$) on age ($X$) moderated by gender ($Z$), I not only find significant effects for age ($X$), age squared ($X^2$), gender ($Z$), the interaction of age and gender ($XZ$), and the interaction of squared age and gender ($X^2Z$). Could anyone help me with how to interpret these results? Specifically, how can I calculate simple slopes for the interaction effects of age squared and gender in SPSS?

• The site shows your profile flair so there is no need to sign your posts. – user88 Sep 26 '12 at 10:35

I struggled for a long time with this exact problem. If it's longitudinal data, you might take the first derivative, $(Y_{t}-Y_{t-1})\over(X_{t}-X_{t-1})$. This will hopefully linearize the data and you can interpret the main effects in the model as being about initial rates of income change (make sure to center your X variable so that 0 falls on an age that is interpretable-- maybe earliest age in the sample, maybe the mean age in the sample) and the slopes as being about the rate at which this change accelerates or decelerates with age. The Z variable has the same interpretation as it normally would except like everything else, it's affect the rate of change rather than the actual income.