Goodman and Kruskal's λ: A new look at an old measure of association

Robert W. Makuch, Philip S. Rosenberg, Gwendolyn Scott

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We examine Goodman and Kruskal's λ using Efron's approach to regression and analysis of variance (ANOVA) for zero-one outcome data. For a binary response cross-classified by a single nominal predictor, we present a computationally simple ANOVA table in which λ is analogous to Pearson's R-square. We characterize the relationship between λ and the commonly used apparent error rate in logistic regression, and show that λ is based implicitly on a prediction rule for a saturated model with classification level 0.5. This relationship suggests that we can correct the apparent error rate for chance by defining a natural generalization of λ that we call PRE, the proportional reduction in error. We illustrate the use of λ and PRE in an analysis of prognostic factors for one-year survival in children with the acquired immunodeficiency syndrome (AIDS).

Original languageEnglish (US)
Pages (from-to)619-631
Number of pages13
JournalStatistics in Medicine
Issue number5
StatePublished - May 1989


  • AIDS
  • Apparent error rate
  • Goodman and Kruskal's λ
  • Logistic regression

ASJC Scopus subject areas

  • Epidemiology
  • Statistics and Probability


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