Cramer’s
V. A correlation coefficient that may be used with
nominal data. It is often included with chi-square test reports.
More details about Cramer's V.
Cramér's V is a measure of association for categorical data, particularly useful when examining the strength of the relationship between two nominal variables. It provides a value between 0 (no association) and 1 (perfect association), making it an excellent follow-up measure after conducting a Chi-Square test. Here's the formula:
V = √(χ² / (n × min(k - 1, r - 1)))
Where:
χ² is the Chi-Square statistic.
n is the total number of observations.
k is the number of categories in one variable.
r is the number of categories in the other variable.
Example: Preferences for Therapy Types and Age Groups
Imagine a psychology researcher is studying whether preferences for therapy (e.g., Cognitive Behavioral Therapy (CBT), Group Therapy, Art Therapy) are associated with age groups (Youth, Adults, Elderly). The observed data is:
| Age Group | CBT | Group Therapy | Art Therapy | Total |
|---|---|---|---|---|
| Youth | 20 | 15 | 10 | 45 |
| Adults | 30 | 25 | 20 | 75 |
| Elderly | 10 | 15 | 5 | 30 |
| Total | 60 | 55 | 35 | 150 |
Suppose the Chi-Square statistic (χ²) for this data is 10.5.
Step 1: Plug Values Into the Formula
We know:
χ² = 10.5 (calculated from the Chi-Square test).
n = 150 (total number of observations).
k = 3 (categories of therapy types).
r = 3 (categories of age groups).
The smaller of k - 1 and r - 1 is 2 (min(3 - 1, 3 - 1)).
Now substitute these values: V = √(χ² / (n × min(k - 1, r - 1))) V = √(10.5 / (150 × 2)) V = √(10.5 / 300) V = √0.035 V ≈ 0.187
Step 2: Interpret the Result
The Cramér's V value is 0.187, which indicates a weak association between therapy preferences and age groups. While there is some relationship, it is not strong based on this analysis.
Cramér's V is particularly useful because it adjusts for the number of categories, unlike the Chi-Square value alone, providing a standardized measure of association.
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