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Usefulness of polynomial models

There are two situations where you might want to choose a polynomial model:

Your scientific model is described by a polynomial equation. This is rare in biology. Few chemical or pharmacological models are described by polynomial equations.

You don't have a scientific model, but want to fit a curve to interpolate unknown values. With this goal, you often don't care much about the details of the model. Instead, you care only about finding a model that goes near the data points. Polynomial models often work well.

Which polynomial model?

The order of a polynomial model expresses how many terms it has. Prism offers up to a sixth order equation (and it would be easier to enter higher order equations). The higher order equations have more inflection points.

Choosing the best polynomial model is often a matter of trial and error. If the curve doesn't follow the trend of your data, pick a higher order equation. If it wiggles too much, pick a lower order equation.

How are polynomial models special?

To a mathematician, polynomial models are very special. Strictly speaking, polynomial models are not 'nonlinear'. Even though a graph of X vs. Y is curved (in all but some special cases), the derivative of Y with respect to the parameters is linear.

Because polynomial models are not nonlinear, it is possible (but not with Prism) to fit polynomial models without fussing with initial values. And the fit can be in one step, rather than the iterative approach used for nonlinear models.

Since Prism treats polynomial models the same way it treats nonlinear models, it does require initial values (it chooses 1.0 for each parameter automatically). It doesn't matter what values are used -- polynomial regression cannot encounter false minima.

Why you should choose a centered polynomial equation

There are two problems with polynomial fits, often solved by centering:

When the X values are large, and start well above zero (for example, when  X is a calendar year), taking the very large X values to large powers can lead to math errors.

Even when the X values are not large, the parameters of the model are intertwined, so have high covariance and dependency. This results in large standard errors, wide confidence intervals, and huge confidence or prediction bands.  

Both these problems are solved by using a centered polynomial model. The idea of centering is simple. Subtract the mean of all X values from each X value, and use those differences instead of X in the model.

Fitting the centered model leads to exactly the same curve (unless the regular  approach led to math errors). Accordingly, the sum-of-squares are R2 are the same, as are results of model comparisons.However, the  parameters have different meanings, so have different best-fit values (except the first parameter which is the same), different standard errors and confidence intervals, smaller covariances and dependencies, and tighter confidence/prediction bands.

Read more about centered polynomial models.

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