Statistical Methods in Medical Research 4ed

polynomials can be used for generalized linear models (§4.1) and, indeed, wherever
a linear predictor is used to summarize the information from covariates,
such as a Cox regression (§17.8).
Polynomials, even when they are extended to include fractional polynomials,
have limitations as a source of smooth curves for the description and interpretation
of data. Their fitting is global rather than local, so changes in coefficient
values to accommodate the response in one part of the data can have consequences
(often unwanted) in other parts. Moreover, their explanatory power is
low: applications where there is background theory to guide the choice of curve
will rarely point to the use of polynomials. Methods for smoothing that avoid
many of the problems associated with polynomials are discussed in the next
section and more general aspects are considered in §12.4.
12.2 Smoothing and non-parametric regression
It is often useful to think of data as having been generated by a process that
could be represented by the equation:
Data ˆ Signal ‡ Noise:
The statistician's task is often to prise apart these two components of the data:
quantification of the `Signal' may be of primary interest but measurement of the
amount of noise is usually a necessary adjunct in an analysis. Indeed, the usual
multiple regression model (11.1) can be thought of in these terms, with `Signal'
associated with x1b 1 ‡ x2b 2 ‡ ...‡ xpb p and `Noise' with the random term e.
However, simple linear regression techniques, even with the extensions described
in §12.1, are not always the most suitable method. Even if an approach based on
§12.1 is ultimately adopted, it is useful to be able to employ a number of
approaches to an analysis, to find the most appropriate method and to assess
the robustness of any inferences being made.
There has been considerable progress in recent years in the theoretical and
practical development of smoothing techniques and much fuller treatments can
be found in, for example, Hastie and Tibshirani (1990), Green and Silverman
(1994) and Bowman and Azzalini (1997).
Moving averages, lines and kernel smoothing
12.2 Smoothing and non-parametric regression 387
Suppose that data of the form …xi, yi†, i ˆ 1, ..., n have been observed and it is
desired to produce some estimate of the signal, which might be thought of in this
case to be the relationship between y and x. Suppose the signal at the ith data
point is Si and the noise is ei, so
yi ˆ Si ‡ ei,