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notes13 - Chapter 8 Timothy Hanson Department of Statistics...

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Chapter 8 Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1 / 23

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8.1 Polynomial regression Used when the relationship between Y and the predictor(s) is curvilinear. Example : we might add a quadratic term to a simple linear model to get a parabolic mean Y i = β 0 + β 1 x i 1 + β 11 x 2 i 1 + i . We can no longer interpret β 1 and β 11 “as usual.” Cannot hold x 1 constant and increase x 2 1 by one unit, or vice-versa! Adding higher order terms in PROC REG is a pain; it needs to be done in the DATA step. For PROC GLM, you can specify a model such as model outcome=age chol age*age age*chol; directly. 2 / 23
Higher degree polynomials The degree of a polynomial is the largest power the predictor is raised to. The previous model is a 2nd degree polynomial giving a quadratic-shaped mean function. Here is a third-order (cubic) in one predictor: Y i = β 0 + β 1 x i 1 + β 11 x 2 i 1 + β 111 x 3 i 1 i . A polynomial f ( x ) = β 0 + β 1 x + β 2 x 2 + · · · + β k x k can have up to k - 1 “turning points” or extrema. (p. 296) A k - 1th-order polynomial can go through ( x 1 , Y 1 ) , . . . , ( x k , Y k ) exactly ! 3 / 23

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General notes on fitting (p. 295) Predictors can be first centered by subtracting off the sample mean from each predictor, i.e. x * ij = x ij - ¯ x j is used as a predictor instead of x ij where ¯ x j = n - 1 n i =1 x ij . This reduces multicollinearity among, for example, x i 1 , x 2 i 1 , x 3 i 1 , etc. This isn’t always necessary. Polynomials of degree 4 (quartic) and higher should rarely be used; cubic and lower is okay. High-degree polynomials have unwieldy behavior and can provide extremely poor out of sample prediction. Extrapolation particularly dangerous (p. 294). A better option is to fit an “additive model” (discussed later); the degrees of freedom on the smoothers can mimic third, fourth degree polynomials while being better behaved. 4 / 23
Polynomial regression: more than one predictor In the case of multiple predictors with quadratic terms, cross-product terms should also be included, at least initially. Example : Quadratic regression, two predictors: Y i = β 0 + β 1 x i 1 + β 2 x i 2 | {z } 1st order + β 11 x 2 i 1 + β 22 x 2 i 2 + β 12 x i 1 x i 2 | {z } 2nd order + i . This is an example of a response surface , or parabolic surface (Chapter 30!) “Hierarchical model building,” (p. 299) stipulates that a model containing a particular term should also contain all terms of lower order including the cross-product terms. Degree of cross-product term is obtained by summing power for each predictor. e.g. the degree of β 1123 x 2 i 1 x i 2 x i 3 is 2 + 1 + 1 = 4. 5 / 23

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Hierarchical model building “When using a polynomial regression model as an approximation to the true regression function, statisticians will often fit a second-order or third-order model and then explore whether a lower-order model is adequate...With the hierarchical approach, if a polynomial term of a given order is retained, then all related terms of lower order are also retained in the model. Thus, one would not drop the quadratic term of a predictor variable but retain the cubic term in the model. Since the quadratic term is of lower order, it is viewed as providing more basic information
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notes13 - Chapter 8 Timothy Hanson Department of Statistics...

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