How To Compare Categorical Variables: Comprehensive Guide

Comparing categorical variables effectively is crucial for data analysis and decision-making. compare.edu.vn provides a detailed guide on utilizing various statistical methods and tools to compare categorical data, enabling informed choices. Discover advanced techniques and practical examples for categorical data comparison, improving your analytical skills.

1. Understanding Categorical Variables and Their Importance

Categorical variables, also known as qualitative variables, represent data that can be divided into groups or categories. These variables are distinct from numerical variables, which represent quantities that can be measured. Categorical variables can be nominal, meaning they have no inherent order (e.g., colors, types of fruit), or ordinal, meaning they have a natural order (e.g., education level, satisfaction ratings).

Understanding categorical variables is essential because they frequently appear in various fields, including:

• Market Research: Analyzing customer preferences for different product features.
• Healthcare: Comparing treatment outcomes across different patient groups.
• Social Sciences: Investigating the relationship between demographic factors and social behaviors.
• Education: Assessing student performance based on different teaching methods.
• Business Analytics: Evaluating the success of different marketing campaigns.

The ability to effectively compare categorical variables allows you to identify patterns, relationships, and differences that can inform strategic decisions and improve outcomes.

2. Key Challenges in Comparing Categorical Variables

Comparing categorical variables presents unique challenges compared to comparing numerical data. These challenges include:

• Non-Numerical Nature: Categorical data does not have numerical values, making it impossible to use standard arithmetic operations like addition or subtraction.
• Data Representation: Categorical data is often represented as frequencies or counts within each category, requiring specific statistical techniques to analyze.
• Interpretation Complexity: The meaning of differences between categories can be subjective and context-dependent, making interpretation more complex than with numerical data.
• Sample Size Sensitivity: Some statistical tests for categorical data are sensitive to small sample sizes, potentially leading to inaccurate or misleading results.
• Multidimensionality: Analyzing the relationship between multiple categorical variables can be challenging due to the increased complexity of the data.

Overcoming these challenges requires a solid understanding of appropriate statistical methods and tools designed for categorical data analysis.

3. Statistical Tests for Comparing Categorical Variables

Several statistical tests are specifically designed to compare categorical variables. These tests help determine whether there are significant differences or associations between the categories. Here’s an overview of the most commonly used tests:

3.1 Chi-Square Test

The Chi-Square test is one of the most widely used methods for assessing the relationship between two categorical variables. It examines whether the observed frequencies in a contingency table differ significantly from the expected frequencies under the assumption of independence.

When to Use:

• You want to determine if there is a statistically significant association between two categorical variables.
• The variables are nominal or ordinal.
• The sample size is sufficiently large (generally, expected frequencies in each cell should be at least 5).

How it Works:

1. Formulate Hypotheses:
   • Null Hypothesis (H0): There is no association between the two variables.
   • Alternative Hypothesis (H1): There is an association between the two variables.
2. Create a Contingency Table: Organize the data into a table showing the frequencies for each combination of categories.
3. Calculate Expected Frequencies: For each cell in the table, calculate the expected frequency using the formula:
   E = (Row Total * Column Total) / Grand Total
4. Calculate the Chi-Square Statistic: Use the formula:
   χ² = Σ [(Observed Frequency - Expected Frequency)² / Expected Frequency]
5. Determine Degrees of Freedom: Calculate the degrees of freedom using the formula:
   df = (Number of Rows - 1) * (Number of Columns - 1)
6. Compare to Critical Value: Compare the calculated χ² value to a critical value from the Chi-Square distribution table based on the degrees of freedom and chosen significance level (e.g., 0.05).
7. Interpret Results:
   • If the calculated χ² value is greater than the critical value, reject the null hypothesis and conclude that there is a significant association between the variables.
   • If the calculated χ² value is less than the critical value, fail to reject the null hypothesis and conclude that there is no significant association between the variables.

Example:

Suppose you want to determine if there is an association between smoking status (smoker, non-smoker) and lung cancer (yes, no). You collect data from 500 individuals and create a contingency table:

	Lung Cancer (Yes)	Lung Cancer (No)	Total
Smoker	60	140	200
Non-Smoker	30	270	300
Total	90	410	500

Calculate the expected frequencies, χ² statistic, and compare it to the critical value. If the χ² value exceeds the critical value, you can conclude that there is a significant association between smoking and lung cancer.

3.2 Fisher’s Exact Test

Fisher’s Exact Test is a non-parametric test used to determine if there is a significant association between two categorical variables in a 2×2 contingency table. Unlike the Chi-Square test, Fisher’s Exact Test is particularly useful when dealing with small sample sizes or when the expected frequencies in any cell are less than 5.

When to Use:

• You want to determine if there is a statistically significant association between two categorical variables in a 2×2 contingency table.
• The sample size is small, or the expected frequencies in any cell are less than 5.
• The data consists of independent observations.

How it Works:

1. Formulate Hypotheses:

   • Null Hypothesis (H0): There is no association between the two variables.
   • Alternative Hypothesis (H1): There is an association between the two variables.

2. Create a 2×2 Contingency Table: Organize the data into a table showing the frequencies for each combination of categories.

   	Group 1	Group 2	Total
   Category A	a	b	a+b
   Category B	c	d	c+d
   Total	a+c	b+d	N

3. Calculate the p-value: The p-value is calculated using the hypergeometric distribution:
   p = [(a+b)! * (c+d)! * (a+c)! * (b+d)!] / [a! * b! * c! * d! * N!]

4. Interpret Results:

   • If the p-value is less than or equal to the chosen significance level (e.g., 0.05), reject the null hypothesis and conclude that there is a significant association between the variables.
   • If the p-value is greater than the significance level, fail to reject the null hypothesis and conclude that there is no significant association between the variables.

Example:

Suppose you want to determine if there is an association between a new drug (treatment) and patient improvement (yes, no). You collect data from 20 patients and create a 2×2 contingency table:

	Improved (Yes)	Improved (No)	Total
Treatment	7	3	10
Control	1	9	10
Total	8	12	20

Using Fisher’s Exact Test, you calculate the p-value. If the p-value is less than 0.05, you can conclude that there is a significant association between the treatment and patient improvement.

3.3 McNemar’s Test

McNemar’s Test is a statistical test used to determine if there is a significant difference between two related categorical variables, especially in a before-and-after study design. It is particularly useful when analyzing paired or matched data, where each subject is measured under two different conditions.

When to Use:

• You want to determine if there is a statistically significant change or difference between two related categorical variables.
• The data is paired or matched, meaning each subject is measured under two different conditions (e.g., before and after an intervention).
• The variables are binary (dichotomous).

How it Works:

1. Formulate Hypotheses:

   • Null Hypothesis (H0): There is no difference between the two related variables (i.e., the proportion of subjects changing from one category to another is the same in both directions).
   • Alternative Hypothesis (H1): There is a difference between the two related variables.

2. Create a 2×2 Contingency Table: Organize the data into a table showing the frequencies of changes between the two conditions.

   	Condition 2 (Positive)	Condition 2 (Negative)	Total
   Condition 1 (Positive)	a	b	a+b
   Condition 1 (Negative)	c	d	c+d
   Total	a+c	b+d	N

   • a: Number of subjects positive in both conditions
   • b: Number of subjects positive in Condition 1 but negative in Condition 2
   • c: Number of subjects negative in Condition 1 but positive in Condition 2
   • d: Number of subjects negative in both conditions

3. Calculate the McNemar’s Test Statistic: The McNemar’s test statistic is calculated as:
   χ² = (b - c)² / (b + c)

4. Determine Degrees of Freedom: The degrees of freedom (df) for McNemar’s Test is always 1.

5. Compare to Critical Value: Compare the calculated χ² value to a critical value from the Chi-Square distribution table based on df = 1 and the chosen significance level (e.g., 0.05).

6. Interpret Results:

   • If the calculated χ² value is greater than the critical value, reject the null hypothesis and conclude that there is a significant difference between the two related variables.
   • If the calculated χ² value is less than the critical value, fail to reject the null hypothesis and conclude that there is no significant difference between the two related variables.

Example:

Suppose you want to evaluate the effectiveness of an advertising campaign on brand awareness. You survey 100 customers before and after the campaign, asking if they are aware of the brand (yes/no). The results are as follows:

	After Campaign (Aware)	After Campaign (Not Aware)	Total
Before Campaign (Aware)	40	10	50
Before Campaign (Not Aware)	20	30	50
Total	60	40	100

• a = 40 (Aware before and after)
• b = 10 (Aware before, not after)
• c = 20 (Not aware before, aware after)
• d = 30 (Not aware before or after)

Calculate the McNemar’s test statistic:
χ² = (10 - 20)² / (10 + 20) = 100 / 30 = 3.33

Compare this value to the critical value from the Chi-Square distribution table with df = 1 and α = 0.05 (critical value ≈ 3.84). Since 3.33 < 3.84, you fail to reject the null hypothesis and conclude that there is no significant difference in brand awareness before and after the campaign.

3.4 Cochran’s Q Test

Cochran’s Q test is a statistical test used to determine if there is a significant difference between three or more related categorical variables, especially in a repeated measures or matched study design. It is an extension of McNemar’s test for multiple related groups.

When to Use:

• You want to determine if there is a statistically significant difference between three or more related categorical variables.
• The data is paired or matched, meaning each subject is measured under multiple conditions.
• The variables are binary (dichotomous).

How it Works:

1. Formulate Hypotheses:

   • Null Hypothesis (H0): There is no difference between the related variables (i.e., the proportion of subjects with a positive outcome is the same across all conditions).
   • Alternative Hypothesis (H1): There is a difference between the related variables.

2. Organize the Data: Arrange the data in a table where each row represents a subject, and each column represents a condition. The values in the table are binary (0 or 1), indicating the presence or absence of the outcome in each condition.

3. Calculate the Test Statistic: Cochran’s Q test statistic is calculated as:
   Q = [(k - 1) * (k * ΣCj² - (ΣCj)²)] / [k * ΣRi - ΣRi²]

   • k: Number of conditions
   • Cj: Column total for condition j
   • Ri: Row total for subject i
   • ΣCj²: Sum of the squares of the column totals
   • (ΣCj)²: Square of the sum of the column totals
   • ΣRi: Sum of the row totals
   • ΣRi²: Sum of the squares of the row totals

4. Determine Degrees of Freedom: The degrees of freedom (df) for Cochran’s Q test is k – 1, where k is the number of conditions.

5. Compare to Critical Value: Compare the calculated Q value to a critical value from the Chi-Square distribution table based on df = k – 1 and the chosen significance level (e.g., 0.05).

6. Interpret Results:

   • If the calculated Q value is greater than the critical value, reject the null hypothesis and conclude that there is a significant difference between the related variables.
   • If the calculated Q value is less than the critical value, fail to reject the null hypothesis and conclude that there is no significant difference between the related variables.

Example:

Suppose you want to evaluate the effectiveness of three different treatments for a skin condition. You recruit 25 patients and measure whether their condition improves (1 = yes, 0 = no) after each treatment. The data is organized as follows:

Subject	Treatment 1	Treatment 2	Treatment 3
1	1	0	1
2	0	1	0
…	…	…	…
25	1	1	1

After calculating the column totals (Cj) and row totals (Ri), you compute the Cochran’s Q test statistic. Compare the Q value to the critical value from the Chi-Square distribution table with df = 2 and α = 0.05. If Q is greater than the critical value, you can conclude that there is a significant difference between the effectiveness of the three treatments.

3.5 Chi-Square Test for Trend

The Chi-Square Test for Trend is a statistical test used to determine if there is a significant trend or pattern between an ordinal categorical variable and a binary categorical variable. It is particularly useful when assessing whether the proportions of one category increase or decrease systematically across ordered categories of another variable.

When to Use:

• You want to determine if there is a statistically significant trend or pattern between an ordinal categorical variable and a binary categorical variable.
• One variable has two categories (binary), and the other variable has multiple mutually exclusive but ordered categories (ordinal).
• You want to assess whether the proportions in the two groups show a consistent increase or decrease across the ordered categories.

How it Works:

1. Formulate Hypotheses:

   • Null Hypothesis (H0): There is no trend between the two variables (i.e., the proportions of the binary variable are the same across all ordered categories of the ordinal variable).
   • Alternative Hypothesis (H1): There is a trend between the two variables.

2. Organize the Data: Arrange the data in a contingency table where the rows represent the binary variable categories, and the columns represent the ordered categories of the ordinal variable.

3. Assign Scores to Ordinal Categories: Assign numerical scores (e.g., 1, 2, 3, …) to the ordered categories of the ordinal variable. The scores should reflect the order of the categories.

4. Calculate the Test Statistic: The Chi-Square Test for Trend statistic is calculated as:
   χ²trend = [N * (Σ(xi * Oi) - (Σxi * ΣOi) / N)²] / [(Σxi² - (Σxi)² / N) * (ΣOi - (ΣOi)² / N)]

   • N: Total sample size
   • xi: Score for category i of the ordinal variable
   • Oi: Observed frequency in the category of the binary variable for category i of the ordinal variable
   • Σ(xi * Oi): Sum of the product of the scores and observed frequencies
   • Σxi: Sum of the scores
   • ΣOi: Sum of the observed frequencies
   • Σxi²: Sum of the squares of the scores

5. Determine Degrees of Freedom: The degrees of freedom (df) for the Chi-Square Test for Trend is always 1.

6. Compare to Critical Value: Compare the calculated χ²trend value to a critical value from the Chi-Square distribution table based on df = 1 and the chosen significance level (e.g., 0.05).

7. Interpret Results:

   • If the calculated χ²trend value is greater than the critical value, reject the null hypothesis and conclude that there is a significant trend between the two variables.
   • If the calculated χ²trend value is less than the critical value, fail to reject the null hypothesis and conclude that there is no significant trend between the two variables.

Example:

Suppose you want to assess whether there is a trend between age group and the proportion of individuals with a certain disease. You collect data from 400 individuals and organize the data as follows:

Age Group	Disease (Yes)	Disease (No)	Total
20-30	15	85	100
31-40	25	75	100
41-50	35	65	100
51-60	45	55	100
Total	120	280	400

Assign scores to the age groups (e.g., 1 for 20-30, 2 for 31-40, 3 for 41-50, 4 for 51-60). Calculate the Chi-Square Test for Trend statistic and compare it to the critical value from the Chi-Square distribution table with df = 1 and α = 0.05. If the calculated χ²trend value exceeds the critical value, you can conclude that there is a significant trend between age group and the presence of the disease.

3.6 Choosing the Right Test

Selecting the appropriate statistical test depends on several factors:

• Type of Variables: Are the variables nominal, ordinal, or a combination?
• Number of Groups: How many groups are being compared?
• Sample Size: Is the sample size large enough to meet the assumptions of the test?
• Data Structure: Is the data independent or paired/matched?

Here’s a summary table to help you choose the right test:

Test	Variable Type	Number of Groups	Sample Size	Data Structure
Chi-Square Test	Nominal or Ordinal	Two or More	Large	Independent
Fisher’s Exact Test	Nominal	Two	Small	Independent
McNemar’s Test	Binary	Two	Any	Paired/Matched
Cochran’s Q Test	Binary	Three or More	Any	Paired/Matched
Chi-Square Test for Trend	Ordinal (with scores) and Binary	N/A	Any	Independent

By carefully considering these factors, you can select the most appropriate test to accurately compare your categorical variables.

4. Measures of Association for Categorical Variables

While statistical tests help determine if a significant association exists between categorical variables, measures of association quantify the strength and direction of that association. These measures provide additional insights into the nature of the relationship.

4.1 Odds Ratio (OR)

The Odds Ratio (OR) is a measure of association between two binary categorical variables. It quantifies the odds of an event occurring in one group relative to the odds of it occurring in another group.

When to Use:

• You want to quantify the association between two binary categorical variables.
• The data is organized in a 2×2 contingency table.
• The study design is cross-sectional, case-control, or cohort.

How to Calculate:

Given a 2×2 contingency table:

	Outcome (Yes)	Outcome (No)
Group 1	a	b
Group 2	c	d

The Odds Ratio is calculated as:
OR = (a/b) / (c/d) = (a*d) / (b*c)

Interpretation:

• OR > 1: The odds of the outcome are higher in Group 1 compared to Group 2.
• OR < 1: The odds of the outcome are lower in Group 1 compared to Group 2.
• OR = 1: There is no association between the group and the outcome.

Example:

Suppose you want to assess the association between a risk factor and a disease. You collect data and create the following contingency table:

	Disease (Yes)	Disease (No)
Exposed	60	40
Not Exposed	30	70

The Odds Ratio is:
OR = (60*70) / (40*30) = 4200 / 1200 = 3.5

Interpretation: The odds of having the disease are 3.5 times higher for individuals who are exposed to the risk factor compared to those who are not exposed.

4.2 Relative Risk (RR)

The Relative Risk (RR), also known as the risk ratio, is a measure of association between two binary categorical variables in cohort studies or clinical trials. It quantifies the risk of an event occurring in one group relative to the risk of it occurring in another group.

When to Use:

• You want to quantify the association between two binary categorical variables in a cohort study or clinical trial.
• You can directly calculate the incidence rates (risks) of the outcome in each group.

How to Calculate:

Given a 2×2 contingency table:

	Outcome (Yes)	Outcome (No)	Total
Group 1	a	b	a+b
Group 2	c	d	c+d

The Relative Risk is calculated as:
RR = (a / (a+b)) / (c / (c+d))

Interpretation:

• RR > 1: The risk of the outcome is higher in Group 1 compared to Group 2.
• RR < 1: The risk of the outcome is lower in Group 1 compared to Group 2.
• RR = 1: There is no association between the group and the outcome.

Example:

Suppose you conduct a clinical trial to assess the effectiveness of a new drug in preventing a disease. The results are as follows:

	Disease (Yes)	Disease (No)	Total
Treatment	10	90	100
Control	20	80	100

The Relative Risk is:
RR = (10 / 100) / (20 / 100) = 0.1 / 0.2 = 0.5

Interpretation: The risk of developing the disease is 0.5 times (or 50%) lower in the treatment group compared to the control group.

4.3 Phi Coefficient (φ)

The Phi Coefficient (φ) is a measure of association between two binary categorical variables. It is essentially a Pearson correlation coefficient applied to binary data.

When to Use:

• You want to quantify the association between two binary categorical variables.
• The data is organized in a 2×2 contingency table.

How to Calculate:

Given a 2×2 contingency table:

	Variable 2 (Yes)	Variable 2 (No)
Variable 1 (Yes)	a	b
Variable 1 (No)	c	d

The Phi Coefficient is calculated as:
φ = (ad - bc) / sqrt((a+b)(c+d)(a+c)(b+d))

Interpretation:

• φ ranges from -1 to +1.
• φ = +1: Perfect positive association.
• φ = -1: Perfect negative association.
• φ = 0: No association.

Example:

Suppose you want to assess the association between gender and smoking status. You collect data and create the following contingency table:

	Smoker (Yes)	Smoker (No)
Male	40	60
Female	20	80

The Phi Coefficient is:
φ = (40*80 - 60*20) / sqrt((40+60)(20+80)(40+20)(60+80)) = (3200 - 1200) / sqrt(100*100*60*140) = 2000 / sqrt(84000000) ≈ 0.218

Interpretation: There is a weak positive association between being male and being a smoker.

4.4 Cramer’s V

Cramer’s V is a measure of association between two nominal categorical variables. It is used when at least one of the variables has more than two categories.

When to Use:

• You want to quantify the association between two nominal categorical variables.
• At least one of the variables has more than two categories.

How to Calculate:

1. Calculate the Chi-Square Statistic (χ²): Use the Chi-Square test to determine if there is a significant association between the two variables.

2. Calculate Cramer’s V:
   V = sqrt(χ² / (N * (min(r-1, c-1))))

   • χ²: Chi-Square statistic
   • N: Total sample size
   • r: Number of rows in the contingency table
   • c: Number of columns in the contingency table
   • min(r-1, c-1): The smaller of (r-1) and (c-1)

Interpretation:

• V ranges from 0 to +1.
• V = 0: No association.
• V = +1: Perfect association.

Example:

Suppose you want to assess the association between education level and job satisfaction. You collect data and create the following contingency table:

	Very Satisfied	Somewhat Satisfied	Not Satisfied
High School	20	30	50
Bachelor’s	40	50	10
Master’s	60	30	10

First, calculate the Chi-Square statistic (χ²). Then, calculate Cramer’s V:
V = sqrt(χ² / (N * (min(r-1, c-1))))
Assuming χ² = 50 and N = 300:
V = sqrt(50 / (300 * (min(3-1, 3-1)))) = sqrt(50 / (300 * 2)) = sqrt(50 / 600) ≈ 0.289

Interpretation: There is a moderate association between education level and job satisfaction.

4.5 Choosing the Right Measure of Association

Selecting the appropriate measure of association depends on the nature of the variables and the study design:

Measure	Variable Type	Study Design
Odds Ratio	Binary	Cross-Sectional, Case-Control, Cohort
Relative Risk	Binary	Cohort, Clinical Trial
Phi Coefficient	Binary	Any
Cramer’s V	Nominal (at least one with more than two categories)	Any

By understanding the characteristics of each measure, you can choose the most appropriate one to quantify the strength and direction of the association between your categorical variables.

5. Practical Tools and Software for Comparing Categorical Variables

Several software packages and tools can facilitate the comparison of categorical variables. These tools offer features for data organization, statistical analysis, and visualization, making the process more efficient and accurate.

5.1 SPSS

SPSS (Statistical Package for the Social Sciences) is a widely used statistical software package that offers a comprehensive suite of tools for analyzing categorical data.

Key Features:

• Contingency Table Analysis: Easily create and analyze contingency tables for multiple categorical variables.
• Chi-Square Test: Perform Chi-Square tests with automatic calculation of expected frequencies, degrees of freedom, and p-values.
• Fisher’s Exact Test: Conduct Fisher’s Exact Test for 2×2 tables with small sample sizes.
• McNemar’s Test: Analyze paired categorical data using McNemar’s Test.
• Cochran’s Q Test: Compare three or more related categorical variables with Cochran’s Q Test.
• Measures of Association: Calculate Odds Ratios, Relative Risks, Phi Coefficient, and Cramer’s V.
• Visualization: Create bar charts, pie charts, and other visualizations to explore and present categorical data.

Benefits:

• User-friendly interface.
• Extensive documentation and support.
• Suitable for both beginners and advanced users.

Limitations:

• Commercial software, requiring a license.
• Can be expensive for individual users.

5.2 R

R is a free and open-source statistical programming language that provides a highly flexible and powerful environment for analyzing categorical data.

Key Features:

• Extensive Package Library: Access a vast collection of packages for statistical analysis, including stats, vcd, and DescTools.
• Contingency Table Analysis: Create and manipulate contingency tables with functions like table() and xtabs().
• Chi-Square Test: Perform Chi-Square tests using the chisq.test() function.
• Fisher’s Exact Test: Conduct Fisher’s Exact Test with the fisher.test() function.
• McNemar’s Test: Analyze paired categorical data using the mcnemar.test() function.
• Cochran’s Q Test: Implement Cochran’s Q Test with custom code or specialized packages.
• Measures of Association: Calculate Odds Ratios, Relative Risks, Phi Coefficient, and Cramer’s V using various functions and packages.
• Visualization: Create publication-quality graphics with packages like ggplot2 and lattice.

Benefits:

• Free and open-source.
• Highly customizable and extensible.
• Active and supportive community.

Limitations:

• Steeper learning curve compared to SPSS.
• Requires programming knowledge.

5.3 SAS

SAS (Statistical Analysis System) is a comprehensive statistical software suite used extensively in business, healthcare, and research settings.

Key Features:

• Contingency Table Analysis: Generate and analyze contingency tables using procedures like PROC FREQ.
• Chi-Square Test: Perform Chi-Square tests with options for Yates’ correction and other adjustments.
• Fisher’s Exact Test: Conduct Fisher’s Exact Test for 2×2 tables.
• McNemar’s Test: Analyze paired categorical data using the AGREE option in PROC FREQ.
• Cochran’s Q Test: Implement Cochran’s Q Test with custom code or specialized procedures.
• Measures of Association: Calculate Odds Ratios, Relative Risks, Phi Coefficient, and Cramer’s V with various options in PROC FREQ.
• Visualization: Create a variety of charts and graphs with SAS/GRAPH.

Benefits:

• Powerful and reliable.
• Suitable for large datasets and complex analyses.
• Widely used in industry and academia.

Limitations:

• Commercial software, requiring a license.
• Can be expensive.
• Steeper learning curve.

5.4 Python

Python, with its rich ecosystem of libraries, is increasingly popular for statistical analysis and data science. Libraries like SciPy, Statsmodels, and scikit-learn provide powerful tools for comparing categorical variables.

Key Features:

• Contingency Table Analysis: Create and manipulate contingency tables using pandas DataFrames and the crosstab() function.
• Chi-Square Test: Perform Chi-Square tests using the chi2_contingency() function from scipy.stats.
• Fisher’s Exact Test: Conduct Fisher’s Exact Test with the fisher_exact() function from scipy.stats.
• McNemar’s Test: Analyze paired categorical data using the mcnemar() function from statsmodels.stats.contingency_tables.
• Cochran’s Q Test: Implement Cochran’s Q Test with custom code or specialized packages.
• Measures of Association: Calculate Odds Ratios, Relative Risks, Phi Coefficient, and Cramer’s V using custom functions or specialized packages.
• Visualization: Create informative plots with libraries like matplotlib and seaborn.

Benefits:

• Free and