Statistical Methods in Medical Research 4ed

384 Further regression models
where m ˆ 3 and …p1, p2, p3† ˆ…1, 1 2 ,0†. The parentheses around the powers of
x are because x …0† is interpreted as log x, as in the Box±Cox transformation (see
§10.8).
The form (12.5), with powers p1 < p2 < ... pm taking real, rather than just
non-negative, integer values, provides a wider selection of curves than conventional
polynomials. However, the family can be widened further by allowing
equality between some of the powers. If pj ˆ pj‡1 then at first sight the
term
reduces to
b jx …pj† ‡ bj‡1x …pj†
…b j ‡ b j‡1†x …pj† ;
in other words, the expression really has degree m 1. However, if pj‡1 ˆ pj ‡ d,
then
b jx …pj† ‡ bj‡1x …pj‡d†
ˆ b j x …pj† ‡ bj‡1 x …pj† …x d
1†=d ! b j x …pj† ‡ bj‡1 x …pj† log x
as d ! 0 for suitable b j‡1 , b j . This rather heuristic argument suggests that the
family can be extended to include products of the form
x …p† log x
and also (see Royston & Altman, 1994) to include
x …p† …log x† j , j ˆ 2, ..., m 1:
The full definition of a fractional polynomial of degree m, f m…x; b, p†, depends
on a real-valued m-dimensional vector p with elements p1 p2 ... pm:
where
fm…x; b, p† ˆa ‡ Pm
bjHj…x† Hj…x† ˆx …pj† if pj 6ˆ pj 1 and Hj…x† ˆHj 1…x† log x, ifpj ˆ pj 1:
Since the coefficients bj enter fractional polynomials linearly they can be fitted in
exactly the same manner as conventional polynomialsÐnamely, using the
methods of §11.7. The appearance of terms in log x and the possibility of terms
such as x
p mean that some fractional polynomials cannot be used directly if
some x are negative or zero. A simple way out of this difficulty is to work with
x ‡ z rather than x, where z is such that x ‡ z is positive for all cases. However,
the choice of z can be a delicate matter.
jˆ1