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U-Scores for Multivariate Data in Sports


  • Wittkowski Knut M.

    (The Rockefeller University)

  • Song Tingting

    (The Rockefeller University)

  • Anderson Kent

    (University of California at Davis, Clinical and Translational Science Center)

  • Daniels John E.

    (Central Michigan University, Department of Mathematics)


In many sport competitions athletes, teams, or countries are evaluated based on several variables. The strong assumptions underlying traditional 'linear weight' scoring systems (that the relative importance, interactions and linearizing transformations of the variables are known) can often not be justified on theoretical grounds, and empirical `validation' of weights, interactions and transformations, is problematic when a `gold standard' is lacking. With µ-scores (u-scores for multivariate data) one can integrate information even if the variables have different scales and unknown interactions or if the events counted are not directly comparable, as long as the variables have an `orientation'. Using baseball as an example, we discuss how measures based on µ-scores can complement the existing measures for `performance' (which may depend on the situation) by providing the first multivariate measures for `ability' (which should be independent of the situation). Recently, µ-scores have been extended to situations where count variables are graded by importance or relevance, such as medals in the Olympics (Wittkowski 2003) or Tour-de-France jerseys (Cherchye and Vermeulen 2006, 2007). Here, we present extensions to `censored' variables (life-time achievements of active athletes), penalties (counting a win more than two ties) and hierarchically structured variables (Nordic, alpine, outdoor, and indoor Olympic events). The methods presented are not restricted to sports. Other applications of the method include medicine (adverse events), finance (risk analysis), social choice theory (voting), and economy (long-term profit).

Suggested Citation

  • Wittkowski Knut M. & Song Tingting & Anderson Kent & Daniels John E., 2008. "U-Scores for Multivariate Data in Sports," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 4(3), pages 1-24, July.
  • Handle: RePEc:bpj:jqsprt:v:4:y:2008:i:3:n:7

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    References listed on IDEAS

    1. Wittkowski, Knut M., 2003. "Novel Methods for Multivariate Ordinal Data applied to Genetic Diplotypes, Genomic Pathways, Risk Profiles, and Pattern Similarity," MPRA Paper 4570, University Library of Munich, Germany.
    2. Morales José F. & Song Tingting & Auerbach Arleen D. & Wittkowski Knut M., 2008. "Phenotyping Genetic Diseases Using an Extension of µ-Scores for Multivariate Data," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 7(1), pages 1-20, June.
    3. Callander, Steven & Wilson, Catherine H., 2006. "Context-dependent Voting," Quarterly Journal of Political Science, now publishers, vol. 1(3), pages 227-254, July.
    4. John C. Harsanyi, 1953. "Cardinal Utility in Welfare Economics and in the Theory of Risk-taking," Journal of Political Economy, University of Chicago Press, vol. 61, pages 434-434.
    5. Kenneth J. Arrow, 1950. "A Difficulty in the Concept of Social Welfare," Journal of Political Economy, University of Chicago Press, vol. 58, pages 328-328.
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