## Advanced Statistics - Biology 6030 |

## Bowling Green State University, Fall 2017 |

- 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.

- 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

- calculate F = (SS

- 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.

**Compare models** with different degrees of freedom - F statistic: (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

last modified: 2/1/08

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