Statistical Methods in Medical Research 4ed

There is only one conventional polynomial of a given degree but the presence
of the vector p means that there is an infinite number of fractional polynomials
of degree m. On the one hand, this enriches the family of curves available to the
analyst but, on the other, it complicates the fitting of a fractional polynomial. As
pointed out above, the coefficients bj can be found using linear regression
techniques, but this is not the case unless the elements of p are effectively taken
to be fixed. Treating p as another collection of parameters to be estimated from
the data removes the simplicity from the fitting procedure that is one of the most
attractive features of fractional polynomials. Royston and Altman (1994) propose
the following simple, heuristic approach to fitting fractional polynomials.
The elements of p are assumed to belong to a small discrete set of possible
powers, which is usually taken to be P ˆf 2, 1, 1
2
12.1 Polynomial regression 385
,0, 1
2
,1,2, ..., Mg
where M could be 3 or possibly max {3, m}. Each m-tuple taken from P with
replacement defines a fractional polynomial and the m-tuple which gives rise to
the smallest residual mean square defines the best-fitting fractional polynomial
of degree m. It is also suggested that fractional polynomials of degree m and
m ‡ 1 can be compared using the usual F-ratio statistic for the ratio of residual
mean squares, but it must be remembered that the fractional polynomial of
higher degree has two extra parameters, a power and a coefficient, so the
numerator DF for the F statistic must be 2.
Figure 12.3 shows the population trend data with the cubic polynomial
previously fitted and the best-fitting fractional polynomial of degree 3. The
fractional polynomial has p ˆ…0, 0, 1
2 †Ðthat is, a quadratic in log(year) together
with a term p year. The residual mean square for the fractional polynomial is
0 300 compared with 0 365 for the cubic. The improvement in fit for a model
which at least nominally has the same number of terms may be very worthwhile
for some purposes. Interpolation and smoothing within the range of the data
may be two such purposes. However, some caution is also advisable.
The fractional polynomial shown in Fig. 12.3 has been found as a result of
considering 120 fractional polynomials of degree 3 and a further 44 of degree 2 or
1. The number of fractional polynomials considered obviously depends on the
size of the set P. IfP has P elements then there are P fractional polynomials of
degree 1, P…P ‡ 1†=2 of degree 2 and P…P ‡ 1†…P ‡ 2†=6 of degree 3. Some of the
statistical features of a fractional polynomial fitted using the method outlined,
such as the standard errors of the coefficients bj that are reported by the
regression program, will be unreliable because they will not take account of the
number of models that have been considered. This is, of course, true of conventional
polynomials, but the number of alternatives that are available from
consideration here is so much smaller that its effect is likely to be much less
important. For some purposes, the standard errors are unimportant; in other
cases it may be sensible to check the reliability of standard errors using techniques
such as the bootstrap (see §10.7). When so many models are considered, it