Association measure (nominal)
Goodman & Kruskal's Lambda Calculator (λ)
Goodman and Kruskal's lambda (λ) is a proportional-reduction-in-error measure: it tells you how much knowing one categorical variable reduces the errors you make predicting the other's most common category, on a scale from 0 (no help) to 1 (perfect prediction).
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Open the workspace →What is Goodman and Kruskal's lambda?
Lambda is a proportional-reduction-in-error (PRE)measure for nominal variables. The idea is simple: if you had to guess a case's category with no other information, your best bet is the most common category (the mode), and you would be wrong some of the time. Lambda asks how much that error shrinks once you know the value of a second variable.
It comes in an asymmetric form — predicting one specified variable from the other — and a symmetric form that averages both directions. Because it is built around the modal category, lambda has a concrete, prediction-based interpretation that many other association measures lack.
Formula
Definition
λ(row|col) = ( Σ_columns (max cell count in that column) − max row total ) / ( N − max row total )
- Σ_columns (max cell)
- = for each column, take the largest cell count, then sum those maxima across all columns
- max row total
- = the largest of the row marginal totals (the overall mode of the row variable)
- N
- = the grand total of all counts in the table
- symmetric λ
- = averages the two asymmetric lambdas (row|col and col|row)
Worked example
Worked example
Take the 3×3 table:
[[20, 10, 5], [10, 20, 10], [5, 10, 20]]
Row totals are 35, 40, 35; column totals are 35, 40, 35; N = 110.
To predict the row from the column, take the largest count in each column: 20, 20, and 20, which sum to 60. The largest row total is 40.
λ(row|col) = (60 − 40) / (110 − 40) = 20 / 70 = 0.286.
The symmetric lambda for this table is also 0.286. So λ = 0.286 — knowing the column reduces prediction error for the row by about 29%.
When to use it
Use it when
- Both variables are nominal (unordered categories).
- You want a prediction-oriented (PRE) interpretation of association strength.
- You can specify which variable predicts which, or report the symmetric average.
Not the right tool when
- The data are ordinal and order matters — use gamma, Kendall's tau, or Somers' d instead.
- You expect a small but real association concentrated in non-modal categories — lambda can miss it.
- Note: lambda can be 0 even when a chi-square test is significant, if the modal category is the same across groups.
How to interpret it
Rule of thumb
λ = 0 means the predictor gives no help; λ = 1 means perfect prediction. For example, λ = 0.286 means a 29% reduction in prediction error.
Frequently asked questions
- What does a lambda of 0 mean?
- A lambda of 0 means the predictor variable gives you no help: knowing it does not reduce the errors you make when guessing the other variable's most common category. You would predict the same modal category whether or not you knew the predictor.
- Why can lambda be 0 when chi-square is significant?
- Lambda only rewards a predictor when it changes which category is the mode. If the same category is the most common one across every group, your best guess never changes, so lambda is 0 — even though a chi-square test can still detect a significant difference in the rest of the distribution.
- Is lambda symmetric or asymmetric?
- Both. There are two asymmetric lambdas, one for predicting each variable from the other, and a symmetric lambda that averages the two directions. Use the asymmetric form when one variable is clearly the predictor; use the symmetric form when neither is.
References & further reading
- Goodman, L. A., & Kruskal, W. H. (1954). Measures of Association for Cross Classifications. Journal of the American Statistical Association.
- Goodman and Kruskal's lambda — Wikipedia
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