The algorithm is applicable to structures such as are obtained from additive conjoint measurement designs, unfolding theory, general Fechnerian scaling, some special types of multidimensional scaling, and ordinal multiple regression. A description is obtained of the space containing all possible numerical representations which can satisfy the structure, the size and shape of which is informative. The Abelson-Tukey maximin r2 solution is provided.

Signal detection theory (SDT) was developed to analyze the behavior of a single judge but also can be used to analyze decisions made by organizations or other social systems. SDT quantifies the ability to distinguish between signal and noise by separating accuracy of the detection system from response bias—the propensity to over-warn (too many false positives) or under-warn (too many misses). We apply SDT techniques to national and state-level data sets to analyze the ability of the child welfare services systems to detect instances of child maltreatment. Blacks have higher rates of referral and the system is less accurate for them than for Whites or Hispanics. The incidence of false positives—referrals leading to unsubstantiated findings—is higher for Blacks than for other groups, as is the incidence of false negatives—children for whom no referral was made but who are in fact neglected or abused. The rate of true positives–children for whom a referral was made and for whom the allegation was substantiated–is higher for Blacks. Values of d′ (signal strength) are roughly the same for Whites, Blacks, and Hispanics but there are pronounced group differences in C (a measure of the location of the decision threshold). Analyses show that the child welfare services system treats Blacks differently from Hispanics and Whites in ways that cannot be justified readily in terms of objective measures of group differences. This study illustrates the potential for JDM techniques such as SDT to contribute to understanding of system-level decision making processes.