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Methods and Applications of Longitudinal Data Analysis by Xian Liu

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Appendix A

Orthogonal polynomials

Let the dependent variable Y be associated with an independent variable x in the form of a polynomial, given by

Yx=β0+β1x+β2x2+...+βKxK,

image(A.1)
where K is the highest order of polynomials specified in the model.

Equation (A.1) is a convenient curvilinear expression of y as a function of x and is fitted to observed pairs of associated values yT and xT where T = 1, …, n. Suppose that the variable xT progresses by constant intervals. It is then convenient to standardize the x-scale, written as

xT=Tn+12T=1,2,...,n,

and to fit Y(x) in terms of a weighted sum of orthogonal polynomial:

Yx=B0φ~0x+B1φ~1x+B2φ~2x+...+

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