Statistical models usually employ a linear time variable and a squared time variable. I mean why the models do not include triple-squared time variables ? What are any reasons applying squared time variables instead of triple or higher power time variables?
Such model allow to fit a curved (i.e. polynomial) relationship between the predictor (e.g. time) and the response (e.g. crop yield). Using polynomials of 2nd order (quadratic, i.e. containing x² and x) is the simplest way to model a relationship with an optimum or pessimum. Higher order polynomials (cubic etc) can also be used, and they should be used when there is a sensible physical model (theroy) behind the modeled relationship. If there is no particular theory and some fitting is all what should be done, higher order polynomials can easily cause over-fitting (the model implies a too high precision, and conclusions are not reliable), so this has to be taken with care. If the relationship between response and predictor is complicated, one should consider using splines to describe the reliationship.
I agree Jochen's answer. However,there are many simple models the reason why old researchers have poor statistical power. Muhanmad, you try your mentioned model and may breakthrough your area.
The big danger in applying simpe polynomials (squares, cubes, roots) is that they form curves that may fit the data badly at its extremes. This is not such a problem when you are modelling a biological paramter whose extremes are rare (such as, say, height) so the resulting poor predictions are of little consequence - people don't want to extrapolate beyond the data space (why would we predict the blood pressure of a 2·5 metre person?).
However, modelling climate is quite different. Here, we are vitally interested in predicting beyond the data space. And it is here that simple polynomials can rapidly generate predictions that are very wrong. You might look at the use of fractional polynomials instead. These models fit curves using best-fit orthogonal transformations. The resulting curves have much better properties when extrapolated. See the work of Pat Royston, who has pioneered the use of these models.
His introduction to the area is an excellent read: http://www.stata-journal.com/sjpdf.html?articlenum=st0120 (Stata Journal articles are free to read after three years, and it's a superb resource!)
Abstract . Spline functions provide a useful and flexible basis for modeling relationships with continuous predictors. However, to limit instability and provide sensible regression models in the multivariable setting, a principled approach to model selection and function estimation is important. Here the multivariable fractional polynomials approach to model building is transferred to regression splines. The essential features are specifying a maximum acceptable complexity for each continuous function and applying a closed-test approach to each continuous predictor to simplify the model where possible. Important adjuncts are an initial choice of scale for continuous predictors (linear or logarithmic), which often helps one to generate realistic, parsimonious final models; a goodness-of-fit test for a parametric function of a predictor; and a preliminary predictor transformation to improve robustness.
firstly you draw scatter diagram and see what type of curve are there and then you decide what type of regression model is best fit.
thank you
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