I've looked more into quadratic/U-shaped relationships, particularly searching for references. Before I proceed further in this thread, though, I think we need to clarify how this all got started. As some of the references I have found point out, and as is the motivation for the turning-point location approach that I have been recommending, quadratic is not synonymous with U-shaped, and often quadratic is just (mis)used as a proxy for any type of U-shaped, or even any type of curvilinear relationship. So maybe we need to back up and figure out just what we're getting at here. What motivated you to include the quadratic terms in the regression in the first place? You've already said that theory is silent on the matter. Did you have a hunch that previous modeling in this area that was just linear was missing something? Were you specifically interested in searching for a quadratic model, or were you interested more generally in non-linearity? If the latter, did you have anything specific in mind? I think we should answer these questions so we can pursue an approach that is tailored to the question we are actually facing.
Yes, there could be models that appear to be quadratic but are in fact some other form of curvilinearity. The turning point method is what I have always relied on to make this distinction, but with the reading I have done since yesterday, there may be better alternatives. But, as in my previous paragraph, let's first get clear exactly what the question is before we select a way to answer it.
The approach with two linear regressions on opposite sides of the data was pioneered b Simonsohn, and I can provide you a reference to it. But until we are sure that is what we need, I'd rather we hold off on that. In any case, if we need it, his reference has a link to code for this in R. I have not found any existing Stata code for it. The approach is conceptually fairly simple, and I don't doubt I could code it in Stata myself for particular models, although it would be difficult to write a fully general program that could be applied to any regression at all.
So does this mean that they may be several models in my work that would seem to be the quadratic model but are not? I will try to calculate all the quadratic models that I have to see their turning points and make sure of where their values fall.
Please, who has run such analysis and how do the commands look like so that I can try them? Please, if there is an alternative means of analyzing my data, I will be open to attempting them.
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