Statistical Methods in Medical Research 4ed

12 Further regression models for a continuous
response
12.1 Polynomial regression
Reference was made in §11.6 to the possibility of creating new predictor variables
defined as the squares of existing variables, to cope with non-linear or curvilinear
relationships. This is an important idea, and is most easily studied in situations in
which there is originally only one predictor variable x. Instead of the linear
regression equation
introduced in §7.2, we consider the polynomial model
E…y† ˆa ‡ bx …12:1†
E…y† ˆa ‡ b 1x ‡ b 2x 2 ‡ ...‡ b px p : …12:2†
The highest power of x, denoted here by p, is called the degree of the
polynomial. Some typical shapes of low-degree polynomial curves are shown in
Fig. 12.1. The curve for p ˆ 2, when the term in x 2 is added, is called quadratic;
that for p ˆ 3 cubic, and that for p ˆ 4 quartic. Clearly, a wide variety of curves
can be represented by polynomials. The quadratic curve has one peak or trough;
the cubic has at most two peaks or troughs; and so on. A particular set of data
may be fitted well by a portion of a low-degree polynomial even though no peaks
or troughs are present. In particular, data showing a moderate amount of
curvature can often be fitted adequately by a quadratic curve.
The general principle of polynomial regression analysis is to regard the
successive powers of x as separate predictor variables. Thus, to fit the p-degree
polynomial (12.2), we could define x1 ˆ x, x2 ˆ x 2 , ...xp ˆ x p and apply the
standard methods of §11.6. It will often be uncertain which degree of polynomial
is required. Considerations of simplicity suggest that as low an order as possible
should be used; for example, we should normally use linear regression unless
there is any particular reason to use a higher-degree polynomial. The usual
approach is to use a slightly higher degree than one supposes to be necessary.
The highest-degree terms can then be dropped successively so long as they
contribute, separately or together, increments to the sum of squares (SSq)
which are non-significant when compared with the Residual SSq. Some problems
arising from this approach are illustrated in Example 12.1, p. 379.
378