Cramér's V and Phi Association

Cramér's V measures the strength of association between two nominal variables from the chi-square statistic, scaled to range from 0 to 1, and reduces to the phi coefficient for 2×2 tables.

Cramér의 V는 카이제곱 통계량을 바탕으로 두 명목 변수의 연관 강도를 0~1로 나타낸 지수이며, 2×2 표에서는 파이 계수와 같습니다.

The chi-square test says region and preferred brand are related; how strong is that association?

카이제곱 검정에서 지역과 선호 브랜드가 관련 있다고 나왔는데, 그 연관은 얼마나 강한가요?

Cramér's V divides chi-square by the sample size times the smaller of rows minus one and columns minus one and takes the square root, giving an effect size that does not grow with sample size. The bias-corrected version reduces the upward bias that occurs in small samples and large tables.

When to use Cramér's V and Phi Association

• Reporting an effect size alongside a chi-square test of independence
• Comparing association strength across tables of different sizes
• Both variables are nominal

When to use something else

• Both variables are ordinal, where Spearman correlation or Kendall's tau-b uses the ordering
• Paired nominal data such as before–after responses, where McNemar's test applies

Assumptions and requirements

• Independent observations
• Each observation falls in exactly one cell
• Expected counts large enough for the chi-square approximation

What the analysis reports

• Cramér's V and bias-corrected V
• Phi (signed for 2×2 tables)
• Chi-square, df and p value
• Bootstrap 95% CI for V
• Contingency table

How to report it

The association between [X] and [Y] was Cramér's V = [value], 95% CI [[value], [value]], χ²([df], N = [value]) = [value], p = [value].

Related methods

• Chi Square Test
• Fisher's Exact Test
• Entropy and Mutual Information Analysis

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