Ordinal Association Measures (Gamma, Somers' d, Kendall's Tau-b and Tau-c)

Ordinal association measures (gamma, Somers' d, Kendall's tau-b and tau-c) summarize the association between two ordered categorical variables from the counts of concordant and discordant pairs.

순서형 연관 측도(감마, Somers d, Kendall tau-b·tau-c)는 순서가 있는 두 범주 변수 사이의 연관을 일치 쌍과 불일치 쌍의 수로 요약하는 지표입니다.

Is satisfaction level associated with education level when both are ordered categories?

만족도 수준과 학력 수준이 둘 다 순서형 범주일 때 둘 사이에 연관이 있는지 어떻게 보나요?

Each measure starts from the number of concordant pairs C and discordant pairs D in the cross-table: gamma ignores tied pairs, Somers' d is asymmetric and divides by pairs untied on the predictor, tau-b corrects for ties on both variables and tau-c adjusts for non-square tables. Confidence intervals use asymptotic standard errors and z tests use the standard error under independence.

When to use Ordinal Association Measures (Gamma, Somers' d, Kendall's Tau-b and Tau-c)

• Both variables are ordinal, such as Likert ratings or ordered income bands.
• You need a directional measure (Somers' d) of how well one ordered variable predicts another.
• The table is not square and you need tau-c.

When to use something else

• One or both variables are nominal; use Cramér's V, lambda or the uncertainty coefficient.
• Both variables are continuous with many distinct values; use Kendall's tau or Spearman's correlation directly.

Assumptions and requirements

• Observations are independent.
• The categories of both variables have a meaningful order.
• The sample is large enough for the normal approximation of the asymptotic standard errors.

What the analysis reports

• Concordant and discordant pair counts
• Gamma, Somers' d in both directions, tau-b and tau-c
• Asymptotic standard errors and confidence intervals
• z tests and p values under independence

How to report it

The ordinal association between [variable 1] and [variable 2] was gamma = [value], Somers' d = [value], and tau-b = [value], 95% CI [[lower], [upper]], z = [value], p = [value].

Related methods

• Kendall's Tau
• Spearman Rank Correlation
• Ordinal Logistic Regression

Run Ordinal Association Measures (Gamma, Somers' d, Kendall's Tau-b and Tau-c) on Quorant