Nabin - While it seems like a mechanistic model for a non-linear response is always preferable, it doesn't seem like there is an indication in the literature that a quadratic term should not be used when such a model is not available. (I looked through Sokal and Rolf's "Biometry", Legendre's "Numerical Ecology," and Gotelli's "A Primer of ecological statistics," and found nothing of note that would invalidate the use of an x^2 term). In fact, it seems to me that it is entirely valid to hypothesize that there is a non-linear mechanism in your system and to test that hypothesis with a statistical model; the only problem is if you draw conclusions about what is generating the non-linearity based on a statistically significant quadratic term. Quadratic terms are very common in evolutionary studies of non-linear selection gradients (Cf Endler 1986).
Regarding Nabin's particular questions: > 2. Is it necessary to have a significant linear term too? From what I understand, No, it does not matter if the linear term is significant. I think statisticians treat it as a nuisance parameter that should always be included in a model if there are higher-order terms but is not relevant for inference. Overall, you can focus just on the significance (and biological relevance) of the quadratic term. > How you interpret the result when both the linear and quadratic terms are > significant compared > to when the quadratic term is significant while the linear term is not? I'm not positive about this, but I think you can ignore the p-value of linear term either way. The statistical significance of the linear term will likely depend on the range over which you have data, or more importantly, over which it is biologically realistic. If you're data indicates a gentling response, this can be approximated with a straight line and you will perhaps get a significant linear term. If you're data has an actual curve or hump in it, you can't approximate this with a straight line and so your linear term will be non-significant. Any function with an x^2 term will have a hump in it - but the data may not. If this doesn't make sense, play around with graphing a quadratic function in excel and imagine how a model would be fit to different portions of the curve if that's the portion encompassed by your data. Your first question > 1. Is it necessary to have a positive linear term and a negative quadratic > (squared term) to support the curvilinear relationship? and third question > 3. How do you interpret when both the linear and quadratic terms have > negative beta coefficients? I think are answered with the same line of thinking. The sign of the linear term will depend on which portion of the hump-shaped x^2 function your data is covering. Question 1) is a bit confusing - Again, if you play around with a quadratic function in excel it may help you visualize the math behind what your statistical package is fitting - math that, as pointed out in a previous post, is detached from the biology of your situation. Here is where its important to remember the biology of what you're working on, something which R or SAS or excel knows nothing about. Does the sign of your quadratic term make sense? Does it fit your hypothesis? What is the biological reason for expecting a non-linear term? Should you transform the data instead of fitting a quadratic term? Does a quadratic term improve the fit of your model in terms of AIC? (Don't look at R^2 - including an R^2 will always improve your R^2 value) Two books that are excellent references for regression and linear models are Applied Linear Statistical Models by Michael Kutner et al and Applied Regression Analysis: a research tool by Rawlings though both require a fair bit of matrix algebra, the general regression hints and tips are often less math intensive Good luck! Nathan On Dec 18, 2010, at 3:17 PM, University of Maryland LISTSERV Server (14.5) wrote: > > From: Nabin Baral <[email protected]> > Date: December 17, 2010 6:27:05 AM PST > Subject: Interpreting quadratic terms in regression > Reply-To: Nabin Baral <[email protected]> > > > Hello Listserv: I am requesting your help in interpreting the results of > quadratic terms in multiple regression. > > I've hypothesized a curvilinear relationship (inverted U) between a > dependent and an independent variable. To test this hypothesis, I centered > the independent variable, squared it and entered it along with the linear > term in the regression model. I got different results with independent > variables and I have tried to summarize them in the following questions: > > 1. Is it necessary to have a positive linear term and a negative quadratic > (squared term) to support the curvilinear relationship? > > 2. Is it necessary to have a significant linear term too? How you interpret > the result when both the linear and quadratic terms are significant compared > to when the quadratic term is significant while the linear term is not? > > > > I would greatly appreciate your time and help. > > Happy holidays. > > Thanking you. > > Nabin Baral
