This paper is about Guttman's method for assigning optimal numerical scores
to categorical variables, and related methods, which underlie Healy's TW1
and TW2 bone-age standards. In giving the methodological underpinning, Heal
y and Goldstein noted that the scores they obtained could depend critically
on how essential constraints were specified, thus questioning the whole of
optimal score theory. This paper is concerned with resolving this difficul
ty; the resolution is surprisingly intricate, involving:
(i) the relationship between those optimality criteria expressed as ratios
and those not;
(ii) the distinction between weak constraints (identification constraints)
and strong constraints;
(iii) the relationship between working in terms of deviations from the mean
scores and so-called 'uninteresting solutions';
(iv) the use of short-cut algorithms that yield admissible solutions only w
hen the correct strong constraints are applied;
(v) generalizations that lead to reformulations of classical multivariate m
ethods with algorithmic as well as statistical consequences. (C) 1998 John
Wiley & Sons, Ltd.