Methods and Applications of Longitudinal Data Analysis

Appendix A

Orthogonal polynomials

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

$Y\left(x\right)={\beta }_0+{\beta }_{1}x+{\beta }_2{x}^2+\mathrm{...}+{\beta }_K{x}^K\text{,}$

(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 y_T and x_T where T = 1, …, n. Suppose that the variable x_T progresses by constant intervals. It is then convenient to standardize the x-scale, written as

$x_T=T-\frac{\left(n+1\right)}{2}T=\mathrm{1,2,...},n\text{,}$

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

$Y\left(x\right)=B_0{\tilde{\phi }}_0\left(x\right)+B_1{\tilde{\phi }}_1\left(x\right)+B_2{\tilde{\phi }}_2\left(x\right)+\mathrm{...}+$