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.

Related Concepts of Cramer’s V

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

R

# 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.

R

# 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

Applications of Cramer’s V

Cramer’s V is a measure of the relationship between two categorical variables that has applications in many fields where knowing the link or relationship among variables is important. Here are some applications for Cramer’s V solutions :

1. Social Science: In sociology, psychology, and other social sciences, Cramer’s V can be used to evaluate survey data to determine the correlations between various demographic parameters (e.g., gender, age, education level) and attitudes, behaviors, or preferences.
2. Market Research: Cramer’s V can assist in the analysis of customer survey data to detect connections between demographic factors (such as age, income, and geography) and consumer preferences, purchasing behaviors, or brand loyalty.
3. Medical: In medical research, Cramer’s V can be used to assess categorical data from studies looking into the relationship between risk factors (e.g., smoking, diet) and health outcomes.
4. Educational Research: In educational research, Cramer’s V can be used to investigate the association between student demographics (for example, socioeconomic status, parental education) and academic performance, attendance rates, or behavioral consequences.
5. Quality Control: In the manufacturing or service industries, Cramer’s V can aid in determining the relationship between categorical variables such as product quality (e.g., defective vs. non-defective) and process factors.

Benefits of using Cramer’s V

1. Cramer’s V is a standardized measure of association between categorical variables, allowing for comparisons across datasets or studies. Because it runs from 0 to 1, with 0 indicating no linkage and 1 showing perfect correlation, it provides a straightforward and understandable statistic.
2. Applicability to Contingency Tables of Any Size: Cramer’s V is adaptable to contingency tables of any size, making it useful for examining relationships between several categorical variables with varying levels.
3. Interpretability: Cramer’s V is simple to understand. A value near to zero implies a weak or no relationship between variables, whereas a value closer to one indicates a strong correlation. This makes it accessible to researchers and practitioners from a variety of disciplines.
4. Robustness: Cramer’s V is resistant to changes in sample size, making it appropriate for assessing data from research with various sample sizes. It delivers accurate estimates of association even with tiny sample numbers.
5. Non-parametric measure: Cramer’s V is a nonparametric measure, which means it makes no assumptions about the data’s distribution. This quality makes it appropriate for assessing categorical data that may not satisfy the assumptions of parametric tests.

Conclusion

Cramer’s V is an effective technique for investigating the relationship between category variables over multiple fields. Its mathematical underpinning, combined with practical applications and benefits, make it an invaluable tool for academics looking for deeper insights into categorical data relationships. Researchers can use Cramer’s V to discover important connections, guide decision-making processes, and drive breakthroughs in a variety of disciplines.