Polynomial Regression Under General Linear Model
January 15, 2008 at 2:43 pm #49106
I’m curious why polynomial regression is under the General Linear Model. The General Linear Model assumes (1) the residuals follow a multivariate normal distribution and (2) the mean response of the dependent variable is identical to the the linear equation on the predictor side of the equation, aka “identity function.” However, polynomial regression does not necessarily assume a linear relationship between two variables. The response I always get is that if the beta coefficients themselves are not higher order even if you have a higher order variable (ex. X(squared)), it is linear regression, and consequently under the General Linear Model. More technically, the function(x) is linear as a function of the unknown paramters.
Do you know of a situation where a beta coefficient is higher ordered?
I recently saw an example of age and health, where it was determined that age and health do not have a linear relationship. The difference in health between a 30-yr old and a 40-yr old is not nearly as dramatic as the difference between a 60-yr old and a 70-yr old. The link between age and health is most likely nonlinear. It seems to me that you could either transform age (let’s assume the transformation is correct) or develop a model under “Generalized” Linear Modeling, where you can specify the type of exponential distribution of the DV and the appropriate link function.
Under polynomoial regression, by transforming age (assuming we make the correct transformation) are we in a sense making the predictor identical to the expected mean of the DV (“identity function”).
Any help would be greatly appreciated.
Ryan0January 15, 2008 at 3:25 pm #167224
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The problem is the definition of “linear equation”. A linear equation is one that is linear in the coefficients or one that can be made to fit this definition by transforms of either the Y or the X’s or both. Thus polynomials are under GLM. The book Fitting Equations to Data – Daniel and Wood discusses this issue at length.
As for an example where the beta coefficient is higher ordered
y = 1/(a +b**(-cx)) which is the basic equation for the S curve.
If the S curve asymptotes to 0 on the lower end and 1/a on the upper end then the equation is linearizable.
Its form becomes y = 1(a + be**(-x)) and the linear form is
1/y = a + be**(-x)0January 15, 2008 at 3:36 pm #167227
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