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I read on this website what to do when the linear term is significant and the quadratic term is not, and vice versa. But what if they are both significant when added independently to the model, but not when added together? Which one should you choose?

M. Westerneng
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    See [Does it make sense to add a quadratic term but not the linear term to a model?](http://stats.stackexchange.com/a/28741/17230) & [Venables (1998), "Exegeses on linear models", S-Plus Users' Conference, Washington DC](http://www.stats.ox.ac.uk/pub/MASS3/Exegeses.pdf). – Scortchi - Reinstate Monica Aug 07 '15 at 11:29

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If adding a quadratic term to a linear term in a model does not lead to a significant improvement, the simple linear model is often used. Often this is the more plausible model in the applications.

Alternatively, you could make the choice based on which model has the lower residual sum of squares. As both models have the same number of parameters, this is equivalent to choosing the model that minimizes a certain information criterion (e.g., AIC or BIC).

Furthermore, there are formal tests for comparing such non-nested models that are popular especially in econometrics. For example, the encompassing test, the Cox test and J test.

Finally, you could also phrase the question differently: You don't know which functional form your regression takes. So you might investigate what kind of functional form a nonlinear spline term takes, e.g., by using a GAM.

Achim Zeileis
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