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I seem to remember from my graduate statistics course that if higher order variables (i.e., X^2, X^3, etc) are significant in a polynomial regression analysis such as our quadratic regression, then the relationship between the DV and IVs is considered to be the highest order variable.

In other words, when I do a regression in the format of X + X^2, and both the linear (X) and quadratic (X^2) components of the analysis are significant, we report the relationship as quadratic?

Both the X and X^2 predictors are significant in the model but X is more significant is it still considered to be a quadratic relationship? Note also that a simple linear regression has a lower R^2 than the quadratic regression.

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    $\begingroup$ It's not entirely clear what you mean. If the relationship was quadratic then why would you try and model the two variables using a regression that doesn't include the quadratic term (i.e. a linear regression)? $\endgroup$ – Ian_Fin Aug 25 '16 at 10:59
  • $\begingroup$ I've edited the question to be more clear. My confusion is because the X relationship is more significant then X^2 $\endgroup$ – dorien Aug 25 '16 at 11:12
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    $\begingroup$ What do you mean "more significant"? Do you mean that the p value is bigger for the linear term than the quadratic term? if so, you may want to contemplate the phrase "the difference between significant and not significant is not itself significant" $\endgroup$ – Ian_Fin Aug 25 '16 at 11:13
  • $\begingroup$ Furthermore, the coefficient estimate & p-value for the lower-order effect will depend on the origin used for the scale on which $x$ is measured: it makes no sense to consider it in isolation as a "main effect". $\endgroup$ – Scortchi - Reinstate Monica Aug 25 '16 at 11:47
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If this is primarily a linguistic question 'What do I call it?' then I think you use the highest term. So if when you plot it the appearance is almost straight but with a slight curve it is still quadratic.

Some of the other issues about inclusion of terms of various orders have been dealt with extensively on this site, for instance here

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From my personal experience, I would choose one or another depending on two things.

First, depending on the application, the error between the regression line and the data that can be accepted may be different. If choosing just linear regression meets your error requirements, why not keeping it simple. It is usually a tradeoff between the precision of fit and the robustness (the ability to applicate it to other sets of same kind of data).

Second, sometimes we have some a priori knowledge on the data. For example, we know that the relationship between y and x is expected to be linear, or that it might be strongly non linear, etc. This may help also to choose between the regression models.

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