Calculate Goodman-Kruskal lambda — proportional reduction in error
Try It FreeGoodman-Kruskal lambda (λ) is a measure of nominal association based on proportional reduction in error (PRE). It answers the question: "By what proportion does knowledge of one variable reduce errors in predicting the other?" Lambda ranges from 0 (no improvement in prediction) to 1 (perfect prediction).
CrossTabs.com calculates all three:
Lambda compares prediction errors with and without the independent variable:
Where E₁ = errors predicting Y without X (using the mode), E₂ = errors predicting Y with X (using conditional modes)
Lambda has a well-known limitation: it can equal 0 even when variables are strongly associated, as long as the modal category of Y is the same in every row. This happens because lambda only considers prediction via the mode. For this reason, always report lambda alongside other measures like the uncertainty coefficient or Cramér's V.
Lambda can be 0 whenever the modal category is the same across all rows (or columns). This is a known limitation. The variables may still be strongly associated — the distribution shapes differ, but the modes don't. Use the uncertainty coefficient or Cramér's V as alternatives.
Cramér's V is symmetric and measures the overall strength of association. Lambda is asymmetric (directional) and measures predictive improvement specifically. V is based on chi-square; lambda is based on prediction errors using the mode.
PRE measures are a family of association measures that quantify how much knowing one variable reduces prediction errors for another. Lambda is a PRE measure for nominal data; Somers' d and tau-b are PRE-like measures for ordinal data.