## Curve Fitting with Parametric Regression Analysis

### Uses

- Is a straight line the model that is
**best suited** to
describe our data? Or is there something to be gained by including
**curves **in
our equation? An additional curve is added to the model when an
additional polynomial term of the predictor variable is included in the
equation. Y modeled on X produces a straight line model, Y modeled on X
+ X^{2} will test for a curved model, Y modeled on X + X^{2} and X^{3} gives a model with two curves. These Curves are represented by adding a higher order polynomial term to the equation.
- The fit of the model (i.e., the SS
_{M}) will certainly increase
with each additional polynomial term, however, it is not clear whether the new equation is **significantly** better? Remember we loose **one df** for each term added to
our equation, so our MS_{M} model term in the numerator for the F-statistic may or may not increase.

### How is this done

- Build a
**linear model** of the data by regressing Y on X
**Center the independent variable** X by subtracting the mean for X from each value in it
-> Xc. The main purpose of this is to reduce collinearity between the independent variables.**Create the polynomial terms** by multiplying each value
in Xc with itself one time (quadratic term), 2 times (cubic term), etc.
- Build regression
**models of increasing complexity** by including additonal polynomials terms as predictor variables
- Test whether a higher degree model significantly improves the model's fit
- calculate F = (SS
_{M} for higher degree
model - SS_{M} for lower degree model) / (MS_{E} for higher degree model)
- compare to F-Tables with numerator df = 1 and denominator
df = residual df of the higher degree model

#### Things to consider

- you can always run this analysis with raw data, standardized
(i.e, z-transform), centered (i.e., subtract mean) or on ranked data to
make sure they give you essentially similar results.

last modified: 2/1/08