Statistical Methods in Medical Research 4ed

410 Further regression models
suppose that the closeness of the above approximations depends on the departure
of f …x, b† from linearity.
Much work has been done on the issue of the degree of non-linearity or
curvature in non-linear regression; important contributions include Beale (1960)
and Bates and Watts (1980). Important aspects of non-linearity can be described
by a measure of curvature introduced by the latter authors. The measurement of
curvature involves considering what happens to f …x, b† as b moves through its
allowed values. The measure has two components, namely the intrinsic curvature
and the parameter-effects curvature. These two components are illustrated by the
model f …x, b† ˆ1 e bx . The same response curve results from g…x, g† ˆ1 g x
if g ˆ e b , so in a clear sense these response curves have the same curvature.
However, when fitting this curve to data, the statistician searches through the
parameter space to find a best-fitting value and it is clear, with one parameter
logarithmically related to the other, that the rate at which fitted values change
throughout this search will be highly dependent on the parameterization
adopted. Thus, while the intrinsic curvatures are similar, their parameter-effects
values are quite different.
Indeed, many features of a non-linear regression depend importantly on the
parameterization adopted. Difficulties in applying numerical methods to find ^ b
can often be alleviated by altering the parameterization used. Also, the shape of
likelihood surfaces and the performance of likelihood ratio tests can be sensitive
to the parameterization. More information on these matters can be found in the
encyclopaedic work by Seber and Wild (1989).
The foregoing discussion has assumed that a value ^ b which minimizes (12.22)
is available. As pointed out previously, there is no general closed-form solution
and numerical methods must be used. Equation (12.24) suggests an iterative
approach, with F evaluated at the current estimate being applied as in this
equation to give a new estimate of b. Indeed, this approach is closely related
to the Gauss±Newton method for obtaining numerical estimates. However, the
numerical analysis of non-linear regression can be surprisingly awkward,
with various subtle problems giving rise to unstable solutions. The simple
Gauss±Newton is not to be recommended and more sophisticated methods are
widely available in the form of suites of programs or, more conveniently, in
major statistical packages and these should be the first choice. Much more
on this problem can be found in Chapters 3, 13 and 14 of Seber and Wild
(1989).
Uses of non-linear regression
The methods described in Chapter 7 and §§12.1±12.2 are used, in a descriptive
way, to assess associations between variables and summarize relationships
between variables. Non-linear methods can be used in this context, but they