Statistical Methods in Medical Research 4ed

y
y
(a) Linear
p = 1
(c) Cubic
p = 3
Fig. 12.1 Illustrations of polynomials of up to the fourth degree.
x
x
y
y
(b) Quadratic
p = 2
(d) Quartic
p = 4
Note that, with n observations, all with different values of x, a polynomial of
degree p ˆ n 1 would have n p 1 ˆ n …n 1† 1 ˆ 0 DF. It is
always possible to fit a polynomial of degree n 1, so as to pass through n
points with different values of x, just as a straight line (p ˆ 1) can be drawn
through any two points. The Residual SSq is, therefore, also zero, and no
significance tests are possible. To provide a test of the adequacy of the model,
the degree of the polynomial should be considerably lower than the number of
observations.
Example 12.1
12.1 Polynomial regression 379
Table 12.1 gives the population size of England and Wales (in millions) as recorded at
decennial censuses between 1801 and 1971; there is a gap in the series at 1941, as no census
was taken in that year. It is of some interest to fit a smooth curve to the trend in
population size, first, to provide estimates for intermediate years and, secondly, for
projection beyond the end of the series, although demographers would in practice use
more sophisticated methods of projection. The figure for 1981 is given at the foot of the
main series to provide a comparison with estimates obtained by extrapolation from
the main series.
x
x