Goodman and Kruskal's lambda: a new look at an old measure of association

We examine Goodman and Kruskal's lambda 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 lambda is analogous to Pearson's R-square. We characterize the relationship between lambda and the commonly used apparent error rate in logistic regression, and show that lambda 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 lambda that we call PRE, the proportional reduction in error. We illustrate the use of lambda and PRE in an analysis of prognostic factors for one-year survival in children with the acquired immunodeficiency syndrome (AIDS).