Proportional Reduction in Error Measures (Lambda and Uncertainty Coefficient)

Proportional reduction in error measures, Goodman-Kruskal lambda and Theil's uncertainty coefficient, quantify how much knowing one nominal variable reduces errors or entropy in predicting another.

오차 감소 비율 측도인 Goodman-Kruskal 람다와 Theil 불확실성 계수는 한 명목 변수를 알 때 다른 명목 변수를 예측하는 오차나 엔트로피가 얼마나 줄어드는지를 나타냅니다.

How much does knowing a customer's region help predict their preferred product category?

고객 지역을 알면 선호 제품군을 예측하는 데 얼마나 도움이 되는지 어떤 지표로 볼 수 있나요?

Lambda measures the proportional reduction in errors when you predict the modal category of the outcome within each level of the predictor instead of the overall mode, and the uncertainty coefficient measures the proportional reduction in entropy. Both range from 0 to 1, have asymmetric and symmetric versions, and come with asymptotic standard errors; independence itself is tested with the likelihood-ratio G-squared.

When to use Proportional Reduction in Error Measures (Lambda and Uncertainty Coefficient)

• Both variables are nominal and you want a directional measure of association.
• You want an interpretable 'percent of prediction error removed' effect size for a contingency table.
• You need an entropy-based association measure to compare tables of different sizes.

When to use something else

• Both variables are ordered; use gamma, Somers' d or Kendall's tau.
• You only need a symmetric strength of association for a chi-square test; Cramér's V is the usual choice.

Assumptions and requirements

• Observations are independent and each falls into one cell.
• Asymptotic standard errors need a reasonably large sample.
• Lambda's standard error assumes a unique modal cell in each row or column.

What the analysis reports

• Lambda (outcome given group, group given outcome, symmetric) with ASE and confidence interval
• Uncertainty coefficient (both directions and symmetric) with ASE and confidence interval
• Entropies and mutual information
• Likelihood-ratio G-squared test of independence

How to report it

Knowing [predictor] reduced errors in predicting [outcome] by lambda = [value] (ASE = [value]), and the uncertainty coefficient was U = [value] (ASE = [value]); G2([df], N = [value]) = [value], p = [value].

Related methods

• Cramér's V and Phi Association
• Chi-Square Test of Independence
• Ordinal Association Measures

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