Statistical Methods in Medical Research 4ed

method achieves this by using maximum likelihood, penalized by roughness
penalties for each of the curves, as described in §12.2.
Although the approach of Cole and Green is attractive inasmuch as some
arbitrary features of the original LMS method have been removed or made more
systematic, the resulting curves are more complicated to describe in a succinct
mathematical form and this may be a limitation in some applications. Alternative
methods using a similar approach, namely transforming data to normality
and smoothing the dependence on age, are emerging. Royston and Wright (1998)
describe a method which makes use of fractional polynomials.
Distribution-free methods
The LMS method and other methods mentioned hitherto determine percentiles
on the basis that the data, perhaps after transformation, have a given distributional
form, usually the normal distribution. The percentiles are then computed
by substituting estimates of parameters into the expression for the percentiles
derived from the form of the distribution. If the data do not have this distribution
and if the transformation fails to make them have this distribution, then the
derived percentiles may be wrong. In view of this, some statisticians may prefer
an approach which does not make such assumptions.
One class of methods is based around the observation that the value of u
which minimizes the mean deviation
1
n
P n
iˆ1
jxi uj
over a sample x1, ..., xn is the sample median. This result can be extended; the
100pth percentile can be found by minimizing
1
n
P n
iˆ1
‰…1 p†jxi uj‡…2p 1†…xi u† ‡ Š,
12.3 Reference ranges 403
where z ‡ ˆ maxfz,0g. The method can be used to determine age-related percentiles
by allowing u to be a function of age. Details are in Koenker and Bassett
(1978), with further work in Newey and Powell (1987). This method seems to
have been little used in medical applications.
Another approach altogether is described by Healy et al. (1988) and it will be
convenient to refer to this method as the HRY (Healy, Rasbash and Yang)
method. In this model the median is described by a pth-order polynomial in age,
where p is chosen by the statistician on the basis of the data. For illustrative purposes,
suppose p ˆ 1 and the median at age t is a0 ‡ a1t. If the data actually follow
a normal distribution, then this will also be the mean. Furthermore, if the standard
deviation, s, does not change with age, then the ith percentile can be written as: