Quadratic effect in OLS regression I am currently trying to do an OLS regression using data of online product reviews and I have two questions:


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*Do I have to use both, the linear and the quadratic effect in the model or is it also okay to only keep the squared variable in the model? I read that I have to use both, but I do not really understand why. So why should this be like this?

*I am doing a regression with a helpfulness score of online product reviews as the dependent variable and the star rating of the reviews (integers between 1 and 5) as an independent variable. I would like to incorporate a squared effect, because I hypothesize that 1 and 5 star ratings are more helpful than moderate reviews (e.g. 3 stars). When I just square the star rating I get 0, 1, 4, 9, or 25 as possible values for the squared variable. However, to me it makes much more sense to first subtract 3 from the rating and then square the variables, because this better reflects the hypothesis that the extremer a rating the higher its helpfulness score. Now, I get 4, 1, 0, 1, or 4 as possible values for the squared variable. Would it make sense to do this?
Thanks for your answers!
 A: In my experience, I would say yes you would always adjust for lower level terms when fitting polynomial trends. This is the approach that is advocated in most of the biostatistics textbooks I've encountered. The reason for this is that the terms are guaranteed to have the correct interpretation. For instance, if you omit a linear (first order) term when fitting a quadratic effect, it is not guaranteed that the interpretation of the coefficient is the slope of the quadratic trend curve. You are constraining the fit so that the slope of the fit through the origin is 0. But when on Earth are we so confident that this is the case? Small measurement calibrations that bias or offset measurements by a single unit can catastrophically attenuate a quadratic slope. Relative to the small amount of power we spend to estimate the linear term, I advocate for using it always.
With respect to your second point, this argues all the more for including the linear term. I agree centering the value on 3 is nice because it gives a 0 and 5 star rating the same influence by symmetry (but the hypothesis that a certain rating is more "helpful" depends on what you estimate. You might estimate a negative quadratic trend on 3 star suggesting that moderate reviews are the most helpful... that's the point about being agnostic with statistics).
A: You don't have to use a linear term to use a quadratic, but it's usually a good idea. The only situation I wouldn't use it is when your theory tells you that you have a quadratic process. For instance, if you somehow are measuring kinetic energy as a function of speed, then there's no linear term in theory:
$$e=m\frac{v^2}{2}$$
These are rare cases when there's no need for linear terms. Otherwise, it's better to keep them. If you don't have a solid theory, keep them.
