Quadratic model term without a linear term

[qrobinson]

Hi everyone,

I have a question regarding incorporating a quadratic term into models without a linear term.

We are considering the possibility that one of our environmental covariates may have a non-linear relationship to our response variable (occupancy). Specifically, we're hypothesizing that ambient temperature will positively influence our target species' habitat selection until a critical upper temperature is reached; at temperatures above this critical point, we're predicting that the species will become less likely to occupy the area.

We'd like to know if this non-linear relationship could be modeled with a single covariate representing the square of temperature, without including a term representing the un-squared covariate (i.e y = Bo + x^2 vs. y = Bo + x + x^2). The following stackexchange thread suggests that this can be done if the squared covariate's distribution is centered around zero. We're wondering if we could therefore normalize the squared covariate (thus centering its mean around zero), and then include only the squared covariate term in models?

http://stats.stackexchange.com/question ... to-a-model

As always any suggestions and guidance would be much appreciated.

Thanks,

Quinn

[Morten Frederiksen]

Hi Quinn,

I would very strongly advise against leaving out the linear term. By doing this, you would force the response variable (in this case occupancy) to have its maximum (or minimum) at a covariate value of zero. This is equivalent to assuming that the 'critical upper temperature' you mention is equal to the mean.

I imagine what you want to do is firstly establish whether a quadratic function provides a good description of the relationship between temperature and occupancy, and secondly estimate the 'critical upper temperature'. Inclusion of the linear term will provide the latter (e.g. through differentiation of the fitted curve).

Morten