# 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?

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## 2 Answers

There really aren't simple slopes for interactions, at least, not in the sense of 'simple' that I think of.

The best way to look at complex models like this, in my experience, is to graph them. Make scatter plots of the actual and/or predicted values from the regression; put income on the Y-axis, age on the X-axis, plot dots for the actual data at various ages and make lines for the predicted values. Color the dots and lines (say, pink for women and blue for men, to be traditional).

Or, you could look at alternatives to quadratic curves. Your data almost certainly isn't generated by a parabolic process, it just happens to fit a parabola better than a straight line. But, what if you do a likelihood ratio test and find that a cubic or quartic curve fits even better? You can keep going and end up with a high order polynomial that fits your data really really well. And then you add or remove a few data points, and the fitted model is now a polynomial with completely different coefficients that fits that data really really well.

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.

If all the measurements are from different individuals, however, I'm not sure what to do. Maybe find a nonlinear model appropriate to the data? Easier said than done, though.