Related Concepts of Cramer’s V
- Chi-Squared Test: Before calculating Cramer’s V, a chi-squared test is frequently used to evaluate whether there is a significant relationship between the categorical variables. This test determines whether the observed frequency distribution differs considerably from the expected frequency distribution, given that the variables are independent.
- Contingency Table: Also known as a cross-tabulation or crosstab, this table shows the frequency distribution of two or more categorical variables. Each cell in the table represents the number of observations that belong to a specific set of categories.
- Degrees of freedom: as used in chi-squared tests and Cramer’s V calculation, describe the number of independent bits of information available when certain constraints are placed. For a contingency table, degrees of freedom are calculated.
- Nominal Data: Categorical variables without a natural ranking or order within them. Gender, racial or and categorical variables that indicate groups are a few examples.
- Ordinal data: are categorical variables with a built-in ranking or ordering of the categories. Examples include comments on the Likert scale (strongly disagree, agree, neutral, disagree, disagree strongly) or education level (high school, college, graduate school, etc.).
Cramer’s V is determined using the chi-squared statistic and the dimensions of the contingency table. It is defined as the square root of the chi-squared statistic divided by the total number of observations multiplied by the smallest of the number of rows minus one and columns minus one. It can be mathematically represented as follows:
[Tex]V = \sqrt{\frac{\chi^2}{n \times \min(c – 1, r – 1)}} [/Tex]
Where,
- V represents Cramer’s V
- X2 is the chi-squared statistic
- n is the total number of observations
- r is the number of rows in the contingency table
- c is the number of columns in the contingency table
Assume we have 500 students’ survey responses, who divided their top three topics into three categories: math, science, and literature. We also gather information about their gender, which is divided into Male and Female categories. We aim to find out if there is a correlation between a person’s favorite subjects and gender.
# Load the rcompanion package
library(rcompanion)
# Create the contingency table
subject_gender <- matrix(c(50, 60, 40, 70, 90, 80), nrow = 3, byrow = TRUE,
dimnames = list(c("Math", "Science", "Literature"),
c("Male", "Female")))
# Calculate Cramer's V
cramers_v <- cramerV(subject_gender)
# Print the result
print(cramers_v)
Output:
Cramer V
0.1379
A Cramer’s V of 0.1379 indicates a relatively weak association between the two categorical variables. It suggests that while there may be some relationship between the variables, it is not particularly strong.
Calculate Cramer’s V for market research
Let’s say a business want to examine the correlation between product categories and customer satisfaction levels. They gather survey information from three hundred clients, classifying product categories as home appliances, apparel, and electronics and assigning satisfaction ratings of Low, Medium, or High.
# Create the contingency table
satisfaction_product <- matrix(c(30, 40, 20, 50, 30, 10, 20, 40, 20), nrow = 3,
byrow = TRUE, dimnames = list(c("Low", "Medium", "High"),
c("Electronics", "Clothing", "Home Appliances")))
# Calculate Cramer's V
cramers_v <- cramerV(satisfaction_product)
# Print the result
print(cramers_v)
Output:
Cramer V
0.1914
How to Calculate Cramer’s V in R
Cramer’s V is a measure of the relationship between two categorical variables, similar to the Pearson correlation coefficient for continuous variables. It goes from 0 to 1, with 0 representing no relationship and 1 indicating perfect relationship. You may calculate Cramer’s V in R by calling the assocstats() function from the vcd package in R Programming Language.
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