Statistical Methods in Medical Research 4ed

Conduction velocity
30
25
20
15
10
5
–1 . 0
0 0 . 0 1 . 0 2 . 0 3 . 0 4 . 0
log(age + 0 . (a) 75)
(b)
Conduction velocity
25
20
15
10
5
Conduction velocity
Conduction velocity
0
–1
(c) (d)
. 0 0 . 0 1 . 0 2 . 0 3 . 0 4 . 0
log(age + 0 . 75)
25
20
15
10
5
–1 . 0
0
25
20
15
10
5
–1 . 0
0
0 . 0 1 . 0 2 . 0 3 . 0
log(age + 0 . 75)
0 . 0 1 . 0 2 . 0 3 . 0
log(age + 0 . 75)
Fig. 12.4 Nerve conduction velocity: (a) top left, raw data, vertical lines at full term, age 1, 10, 30 and 50
years; (b) top right, running mean (- - -), running line ( ), lowess (ÐÐ); (c) bottom left, Gaussian
kernel smoothing with smoothing parameter 0 06 (± ± ±) and 0 37 (ÐÐ); (d) bottom right, penalized
least squares a ˆ 1 0 (continuous), GCV ( ) and fractional polynomial degree 2, p ˆ (1, 3) (± ± ±).
weight, so the effect on s(x) is far less abrupt. This explains why the lowess smoother is the
smoothest of the three estimates of s(x) in this figure.
Figure 12.4(c) shows the effect of fitting two Gaussian kernel smoothers to the data.
The parameter h in (12 8) is set at 0 06 and 0 37. It seems implausible that mean conduction
velocities should go up and down with age in the rather `wiggly' way predicted by the
estimate of s…x† for the smaller h, suggesting that a larger value of h, such as in the other
estimate shown in Fig. 12.4(c), is to be preferred.
Penalized least squares and smoothing splines
12.2 Smoothing and non-parametric regression 391
The local nature of the smoothing methods just described is achieved quite
directlyÐthe methods deliberately construct estimates of s(x) that rely only, or
largely, on those observations that are within a certain distance of x. Regression
lines, on the other hand, are obtained by optimizing a suitable criterion that
expresses a desirable property for the estimator. So, for example, in the case of
normal errors, regression lines are obtained by minimizing
4.0
4 . 0