Categorical Data Analysis

 

Module 2: Categorical Data Analysis

Our first series provided an overview of categorical variables, distinguishing them from numerical data. It emphasized the unique analytical and statistical considerations for categorical variables due to differences in measurement scales, analysis techniques, and data encoding. It highlighted two categorical variable types: nominal (no inherent order) and ordinal (with inherent order). Subtypes like binary, multi-category, hierarchical, and label-set nominal variables were explored, along with equidistant/unequally spaced ordinal categories. Unequal intervals in ordinal categories necessitate qualitative comparison and non-parametric tests for analysis.

In today's episode, we will look at how to deal with missingness in categorical data.

Missing Categorical Data

Missing data refers to the absence or lack of values for one or more variables in a dataset. It can occur for reasons such as non-response in a survey, data entry errors, or system issues. Understanding the types of missing data is crucial for appropriate handling and analysis. Here are some common missing data types:

1.      Missing Completely at Random (MCAR):

• MCAR implies that the missingness of a variable is unrelated to both observed and unobserved data. In other words, the missingness occurs randomly across the dataset, regardless of other variables or patterns. Dealing with MCAR can be straightforward.

How to deal with MCAR

1. Complete Case Analysis:
• In this method, rows with missing categorical variables are simply excluded from the analysis. It is suitable when the missing values are minimal and randomly distributed across the dataset. However, this approach may lead to a loss of information if the missing data is not 'missing completely at random'       (MCAR).

Example

In this example, Table 1 contains missing values denoted by NaN. Complete case analysis involves removing rows with missing values to focus on cases with complete information. We have the complete case in Table 2.

b.      Mode Imputation:

·  Mode imputation involves replacing the missing categorical values with the mode, which is the most frequent category in the variable. This method is suitable when the missing values are assumed to be like the most common category. However, it may lead to an overrepresentation of the mode if missingness is related to other variables.

Example 

In Table 1, you can see that two data points are missing in the "Category" column, represented by "Missing." These missing values will be imputed using the mode of the available data.

After applying mode imputation, Table 2 shows the complete case with the missing values filled in using the mode value of the "Category" column, which is "Category B" in this case.

2.  Missing at Random (MAR):

MAR indicates that the probability of missingness depends only on observed variables and can be predicted by them. In this scenario, the absence of data is connected in a structured manner to other variables present in the dataset, despite lacking a direct link to the variable that is absent.

It's important to note that the "missing at random" assumption is often difficult to verify completely, as it requires understanding the underlying mechanisms causing missingness. Researchers often rely on domain knowledge and statistical techniques to assess whether the assumption is plausible for their specific dataset.

How to deal with MAR:

·    Various imputation methods can be employed for MAR, including predictive imputation techniques such as regression imputation, propensity score matching, or multiple imputation. These methods use observed variables to predict the missing values.

Here are some techniques tailored to handling MAR in categorical data:

1. Mode Imputation: For each categorical variable, impute missing values with the mode (most frequent category) of that variable. This approach assumes that missing values are likely to belong to the most common category.
2. Create an "Unknown" Category: Introduce a new category (e.g., "Unknown" or "Missing") for each categorical variable to explicitly represent missing data. This maintains the distinction between missing values and observed categories.
3. Conditional Imputation: Use conditional probabilities or frequency distributions to impute missing values. This involves estimating the probability of a certain category given the values of other observed variables, and then randomly selecting a category based on these probabilities.
4. Hot Deck Imputation: Replace missing values with values from similar cases that have complete data. This method retains the characteristics of the observed cases and helps to preserve the original distribution of the categorical variable.
5. Multiple Imputation: Perform multiple imputations by generating multiple datasets with imputed values for the missing entries. These datasets are then analyzed separately, and the results are combined to account for the uncertainty introduced by imputation.
6. Logistic Regression Imputation: If you suspect relationships between missingness and other observed variables, perform logistic regression to predict the probability of being missing for each category based on those variables. Then impute missing values using these predicted probabilities.
7. Propensity Score Matching: For categorical variables, you can calculate propensity scores that represent the likelihood of a certain category given the values of other observed variables. Match cases with missing data to cases with complete data based on similar propensity scores and observed categories.
8. Cluster Analysis: Use cluster analysis to group observations with similar missing data patterns. Analyze each cluster separately, taking into account the specific missingness characteristics within each group.
9. Specialized Software: Some statistical software packages provide built-in functions for imputing missing categorical data, such as the MICE package in R or PROC MI in SAS.
10. Validation: Assess the impact of imputation methods on your analysis results. Compare outcomes with and without imputation and consider the sensitivity of your conclusions to different imputation approaches.

Remember that the choice of method should align with the assumptions about the nature of missingness in your data. Exploring the patterns of missing data and understanding the underlying mechanisms can guide the selection of an appropriate technique to treat missing at random in your categorical dataset.

 Using Chi-square to analyze MAR

To test if missingness in a dataset is Missing at Random (MAR), you can use statistical techniques such as chi-square analyses.

The purpose is to assess whether there is a significant association between the missingness of data and the categories. If the p-value obtained from the chi-squared test is small (typically below a chosen significance level, often 0.05), it suggests that there is a statistically significant relationship between the variables, which might indicate that the missing data is not Missing at Random (MAR).

However, if the p-value is large (greater than the chosen significance level), it suggests that there is no significant relationship between the missingness and the categories. In this case, it provides some evidence in favor of the assumption that the missing data is Missing at Random (MAR), as the missingness is not systematically related to the categories.

Keep in mind that the chi-squared test alone might not definitively determine whether the data is MAR. It provides statistical evidence, but the final conclusion should also take into account domain knowledge, understanding of the data collection process, and potentially other analyses to validate the assumption.

Example:

We create some made-up data using Python for our analyses. You can copy the code and work along.

import numpy as np

import pandas as pd

from scipy.stats import chi2_contingency, ttest_ind

 # Create a mock dataset with missing values

np.random.seed(42)

categories = ['A', 'B', 'C', 'D', None]

data = {'Category': np.random.choice(categories,

                                     p=[0.2, 0.2, 0.2, 0.2, 0.2], size=100)}

# Convert the dictionary to a pandas DataFrame df = pd.DataFrame(data) # Create a contingency table contingency_table = pd.crosstab(index=df['Category'], columns='count') # Perform Chi-squared test chi2, p, dof, expected = chi2_contingency(contingency_table)

# Perform T-test

missing_values = df[df['Category'].isnull()]

non_missing_values = df[df['Category'].notnull()]

t_statistic, t_p_value = ttest_ind(missing_values.index, non_missing_values.index)

print("Contingency Table:")

print(contingency_table)

print("\nChi-squared Test:")

print(f"Chi-squared value: {chi2}")

print(f"P-value: {p}")

print("\nT-test:")

print(f"T-statistic: {t_statistic}")

print(f"P-value: {t_p_value}")

# Output: Not part of the code

Contingency Table: col_0 count Category A 28 B 18 C 17 D 19 Chi-squared Test: Chi-squared value: 0.0 P-value: 1.0 T-test: T-statistic: -0.5901964523195302 P-value: 0.556417432493813

# Conclusion: Not part of the code

Let's interpret the outcomes of the chi-squared test and the t-test to determine whether the missing values are at random or not:

1. Chi-squared Test:
   • Chi-squared value: 0.0
   • P-value: 1.0

The chi-squared value of 0.0 indicates that there is no difference between the observed and expected frequencies in the contingency table. This could imply that the distribution of categories (including missing values) does not deviate from what would be expected if the missing values were randomly distributed among the categories. The p-value of 1.0 suggests that the test result is not statistically significant, further indicating that there is no evidence to reject the null hypothesis that the missing values are distributed randomly.

2. T-test:
   • T-statistic: -0.5901964523195302
   • P-value: 0.556417432493813

The t-test compares the indices of rows with missing values and non-missing values. The t-statistic of -0.5901964523195302 indicates that there isn't a significant difference in the means of these indices. The p-value of 0.556417432493813 is greater than the common significance levels (e.g., 0.05), which suggests that there is not enough evidence to reject the null hypothesis that the missing values are randomly distributed.

Overall, both the chi-squared test and the t-test results suggest that there is no strong evidence to conclude that the missing values are not random