Statistical Methods in Medical Research 4ed

396 Further regression models
and the fractional polynomial are virtually indistinguishable. The fractional polynomial
has been fitted by the method outlined in §12.1. The close agreement with the entirely
independent fitting of a smoothing spline may give the statistician considerable encouragement.
It also illustrates how a non-parametric approach can often be reproduced with
a parametric alternative and the former may act as a useful guide to the selection of the
latter.
General remarks
While it will often be adequate to describe the relationship between two or more
variables using the methods based on a straight line that were described in Chapters
7 and 11, there are many circumstances when more elaborate models are
needed. When the aim is to produce a smooth representation of a relationship,
perhaps for interpolation or prediction, or to summarize a relationship succinctly,
the methods outlined in this and the previous section are of value. If a
parametric model based on polynomials fits the data adequately, then it has
the advantages that it can be described by a simple equation and predictions at
new x values can calculated simply. Moreover, the statistical properties will
essentially be those of linear models, which are well understood to be generally
good, although the effect of the way the model was selected may be less than
transparent. However, the family of polynomials gives a limited repertory of
curves and, although this is usefully extended by fractional polynomials,
it is valuable to have access to further methods. More general parametric
models, discussed in §12.4, can be useful, as can the smoothing methods discussed
above.
Techniques for smoothing data are a large and expanding part of modern
statistical methodology and the foregoing description has mentioned only a few
of the more widespread methods. One of the disadvantages of the methodology
is that the fitted curve is not easily described in a way that others could use. Most
of the software that will fit a non-parametric or locally weighted regression will
compute s(x) for values of x specified by the user that are not in the data set.
However, publishing s(x) in an accessible form is currently less straightforward.
In this respect, parametric methods retain an advantage. Careful combination
of parametric and non-parametric approaches may prove the most effective way
forward.
Many of the more sophisticated methods for smoothing, such as those using
basis functions, have not been mentioned. Nor have extensions to the methods
that were discussed. Examples of these include the addition of variability bands
to give some indication of the uncertainty that attends a given curve, allowing the
smoothing parameter to vary with the data and weighted non-parametric regression.
For more information on these and other topics, the reader is referred
to the texts by Silverman (1986), Hastie and Tibshirani (1990), Green and
Silverman (1994) and Bowman and Azzalini (1997).