Differing significance of linear and quadratic terms

I'm surprised this question has yet to be asked; hopefully it is an embarrassingly simple one.

I am fitting a negative binomial regression with 12 total covariates (6 linear variables and 6 corresponding quadratic variables). I understand that one must also include a linear term if a quadratic term is included in a model ("Does it make sense to add a quadratic term but not the linear term to a model"). However, what is the best approach when a linear term is significant while the quadratic is not..or vice versa? In the first case, would one include the linear term without the quadratic term in the final model? Conversely, if the quadratic term is significant while the linear was not - include both terms?

I can provide specific model output if necessary.

• Whether you remove an insignificant highest order term depends partly on what the purpose of the analysis is. When a lower order term is not significant it might only very rarely (if ever) make any sense to remove it. – Glen_b Apr 29 '15 at 0:52

Yes, that's exactly how this goes. If the quadratic term is not significant, there is no point in keeping it. But if the linear term is not significant, you should only exclude it if you have pretty strong theoretical reasons to believe that the relationship must be strictly quadratic. (Something like $E=mc^2$ comes to mind...)