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I would have thought there'd be an answer to this question on here already, but I've been unable to find it, so I apologize if this is a repost.

If I have a regression model of the form Y ~ X + X^2, and both the X and X^2 terms come back significant, can I safely interpret both of these results? For example, let's say the coefficients for both X and X^2 are significantly positive, would it be appropriate to write something like this: "Y is significantly increasing as X increases (interpreting the linear X term) at a significantly increasing rate (interpreting the quadratic X^2 term)"?

Bajcz
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    Consider them literally as a double act. They act together. There is no sense in which one can be held constant without the other changing. In any case, why resort to a qualitative verbal summary when you can show the curve? Sometimes the square term is significant yet merely acts to tweak the curvature slightly. Some times the presence of a marked turning point is the main message. You need to plot the curve over the range of the data to see the combined effect; that is immensely more informative than a verbal paraphrase. – Nick Cox Jan 31 '18 at 19:23
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    Related: https://stats.stackexchange.com/questions/304831, https://stats.stackexchange.com/questions/34488, and https://stats.stackexchange.com/questions/28730. Their answers indicate that $x$ and $x^2$ are usually not interpreted separately and that the interpretation depends on how $x$ is measured. Consider, then, the reason you are using $x$ and $x^2$, what it means for your model, and what scientific question you are trying to answer. – whuber Jan 31 '18 at 19:26
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    While I completely agree with @NickCox, it may be that you are forced by circumstance to be verbal. In that case, you could write "Y can be modeled as a quadratic function of x", or something similar. – jbowman Jan 31 '18 at 19:27
  • Great comments all. I believe that I was already inclined to write about them in concert, as you recommend @NickCox. However, the reasoning behind my model is that I am looking for differences in the distributions of responds to a Likert-type survey question between two groups of people (X is binary & categorical) and I have reasons to expect both a rightward shift (linear shift) between the distributions AND a peakedness (quadratic) shift between them as well. My question really is--when can I say I am seeing both? – Bajcz Jan 31 '18 at 20:56
  • I'd consider using e..g. Legendre polynomials instead if it fits. – Nick Cox Jan 31 '18 at 21:40

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