The role of constraints in determining optimal scores

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, Healy 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 difficulty; 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 when the correct strong constraints are applied; (v) generalizations that lead to reformulations of classical multivariate methods with algorithmic as well as statistical consequences.