Statistical Methods in Medical Research 4ed

392 Further regression models
SS ˆ Pn
‰ yi s…xi†Š 2
iˆ1
…12:9†
over functions defined by a few parameters, such as s…x† ˆa ‡ bx, or a higher
polynomial as in §12.1. An alternative approach to smoothing starts from
attempting to minimize (12.9) for a general smooth function s(x) (i.e. a curve
with a continuous slope). If the xis are distinct, it is clear that there will be many
smooth functions that pass through all the points, thereby giving SS ˆ 0. The
nth-degree polynomial fitted by regression is an example of a function that
interpolates the data in this way. If the data points are not distinct, then SS
will be minimized by any of the curves that interpolate the means of the yis for
each distinct xi. However, it is clear that such curves, while smooth, will generally
have to oscillate rapidly in order to pass through each point. Such curves are of
limited interest to the statistician, not only because such oscillatory responses are
intrinsically implausible as a measure of a signal in the data but also because any
measure of error in the data, which would naturally be estimated by a quantity
proportional to the minimized value of (12.9), is necessarily zero (or, for nondistinct
abscissae, a value unaffected by the fitted curve).
As noted above, curves which interpolate the data will generally have to
oscillate rapidly, in other words, as x increases, s…x† will have to change rapidly
from sloping upwards to sloping downwards and back again many times. The
rate at which the slope of a curve changes is the rate of change of the rate of
change of the curve. The rate of change of the curve is the derivative of s…x†, so
the rate of change of the rate of change is the derivative of this, or the second
derivative of s…x†, which is denoted by s 00 …x†. Consequently, curves that oscillate
rapidly will have large numerical values for s 00 …x†, whether positive or negative. If
this is true for a large portion of the curve, R ‰s 00 …u†Š 2 du will also be large; functions
for which this is true are often described as rough. It may help to note that
in this area `smoothness' is often used in two subtly different ways and care is
sometimes needed. Smoothness can refer to the slope of s…x† changing smoothly,
that is, without any abrupt changes or discontinuities: this is a common mathematical
use of the term. On the other hand, it can refer to functions that change
smoothly in the sense of not oscillating rapidly: here smooth is being used in its
colloquial sense of the opposite of rough, i.e. to describe a function with low
R ‰s 00 …u†Š 2 du. Functions that are smooth in the mathematical sense can be very
rough, and the sense in which smooth is being used often needs to be judged from
the context.
An approach to smoothing that is intermediate between methods based on
minimizing (12.9), subject to s…x† being determined by a few adjustable parameters
and the locally based methods described above, is to combine the measures
of fit to the data, SS, and of roughness. Suppose a function s…x† is chosen
which minimizes