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Bivariate Analysis 101

Cross-tabs, Correlations & Statistical Tests for Development Data
ImpactMojo Workshop Series • Understanding Relationships Between Variables
75-90 Minutes

Workshop 1: Measuring Associations & Statistical Testing

Target Audience: Researchers, data analysts, program evaluators, and development practitioners who need to understand relationships between variables

Prerequisites: Basic statistics knowledge, familiarity with descriptive statistics

Materials Needed: Computers with statistical software (R/Python/Stata/Excel), calculators, sample datasets

Learning Objectives

By the end of this workshop, participants will be able to:

Part 1: Understanding Variable Types and Analysis Methods

20 minutes

The Education-Income Question: Choosing the Right Analysis Approach

Research Question: Is there a relationship between education and household income in Indian households?

The Data Decisions:

  • Education variable: Years of schooling (0-20) OR Categories (None, Primary, Secondary, Higher)?
  • Income variable: Continuous rupees OR Income quartiles OR Above/below poverty line?
  • Analysis choice: This determines which statistical method to use!

Three Different Approaches:

  1. Correlation analysis: Years of education × Continuous income (r = 0.47)
  2. Cross-tabulation: Education categories × Income quartiles (χ² test)
  3. Group comparison: Mean income by education level (ANOVA)

Same underlying relationship, different analytical lenses - each reveals different aspects of the association.

Variable Types and Their Analytical Implications

Categorical Variables

Nominal: Religion, occupation, program participation

Ordinal: Education levels, income quintiles, satisfaction ratings

Analysis Methods: Frequencies, proportions, cross-tabs, chi-square tests

Continuous Variables

Interval: Temperature, test scores

Ratio: Income, age, household size, expenditure

Analysis Methods: Means, correlations, t-tests, regression

Mixed Types

Common in Development: Many variables can be treated as either categorical or continuous

Examples: Age (years vs. age groups), Income (continuous vs. quintiles)

Decision Rule: Consider research question and statistical power

The Bivariate Analysis Decision Matrix

Variable 1 Type
Variable 2 Type
Categorical
Continuous
Categorical

Cross-tabulation

Chi-square test

Cramer's V

Fisher's exact test

Group Comparison

t-test (2 groups)

ANOVA (3+ groups)

Box plots

Continuous

Group Comparison

t-test (2 groups)

ANOVA (3+ groups)

Box plots

Correlation

Pearson's r

Spearman's ρ

Scatter plots

Part 2: Cross-Tabulation and Chi-Square Analysis

25 minutes

Building and Interpreting Cross-Tabulations

Example: Education Level × Contraceptive Use (NFHS-5 Data)

Education Level Using Contraception Not Using Total % Using
No Education 8,234 12,566 20,800 39.6%
Primary 4,127 4,573 8,700 47.4%
Secondary 15,678 12,322 28,000 56.0%
Higher 7,891 4,109 12,000 65.8%
Total 35,930 33,570 69,500 51.7%

Key Observations:

  • Clear positive association: Higher education → Higher contraceptive use
  • 26 percentage point difference between no education (39.6%) and higher education (65.8%)
  • Monotonic relationship: Each education level shows higher use than the previous

Cross-Tabulation Best Practices:

  • Row vs. Column percentages: Choose based on which variable is the "predictor"
  • Sample size concerns: Avoid cells with <5 expected observations
  • Missing data: Decide whether to include "Unknown" as a category
  • Survey weights: Use weighted frequencies for population estimates

Problem Set: Chi-Square Analysis (15 minutes)

Scenario: Analyzing the relationship between wealth quintile and child immunization status using NFHS data.

Dataset: Child Immunization by Wealth Quintile Sample: 45,000 children aged 12-23 months Observed Data: Fully Partially Not Total Immunized Immunized Immunized Poorest (Q1) 2,400 3,200 3,400 9,000 Poorer (Q2) 3,100 2,800 3,100 9,000 Middle (Q3) 3,800 2,700 2,500 9,000 Richer (Q4) 4,200 2,400 2,400 9,000 Richest (Q5) 4,500 2,100 2,400 9,000 ------ ------ ------ ------ Total 18,000 13,200 13,800 45,000
Problem 1: Descriptive Analysis (5 minutes)

Calculate percentages and identify patterns:

Wealth Quintile % Fully Immunized % Not Immunized Gap (Fully - Not)
Poorest (Q1) _____% _____% _____ pp
Poorer (Q2) _____% _____% _____ pp
Richest (Q5) _____% _____% _____ pp

Interpretation Questions:

  • What is the difference in full immunization rates between richest and poorest quintiles?
  • Is there a clear monotonic relationship between wealth and immunization?
  • Which immunization category shows the strongest wealth gradient?
Problem 2: Chi-Square Test (10 minutes)
# Chi-square test calculation steps: # Step 1: Calculate expected frequencies # Expected = (Row Total × Column Total) / Grand Total # Step 2: Calculate chi-square statistic # χ² = Σ [(Observed - Expected)² / Expected] # Step 3: Determine degrees of freedom # df = (rows - 1) × (columns - 1) = 4 × 2 = 8 # Step 4: Compare to critical value or use p-value # Critical value at α = 0.05, df = 8: 15.507

Expected Frequencies for Selected Cells:

Cell Observed Expected (O-E)²/E
Q1, Fully Immunized 2,400 _____ (18,000×9,000/45,000) _____
Q5, Not Immunized 2,400 _____ (13,800×9,000/45,000) _____

Results Interpretation:

  • Calculate total χ² value: _____
  • Is this significant at α = 0.05? (Compare to 15.507): Yes/No
  • What does this tell us about the wealth-immunization relationship?
  • What is the practical significance of this finding?

Part 3: Correlation Analysis

20 minutes

Understanding Correlation Coefficients

Correlation Type When to Use Range Interpretation
Pearson's r Both variables continuous, linear relationship -1 to +1 Linear association strength
Spearman's ρ Ordinal variables or non-linear relationships -1 to +1 Monotonic association strength
Kendall's τ Small samples, many tied ranks -1 to +1 Rank-based association
Point-biserial One binary, one continuous variable -1 to +1 Effect size for group differences

Correlation Interpretation Guidelines:

  • |r| < 0.1: Negligible association
  • 0.1 ≤ |r| < 0.3: Small association
  • 0.3 ≤ |r| < 0.5: Medium association
  • |r| ≥ 0.5: Large association
  • Statistical vs. Practical Significance: Large samples can yield significant but tiny correlations

Problem Set: Correlation Analysis with Development Data (12 minutes)

Dataset: Household-level data from National Sample Survey (NSS) Consumption Expenditure

Variables for Analysis: • household_size: Number of members (1-15) • per_capita_exp: Monthly per capita expenditure (₹500-25,000) • education_years: Average education of adults (0-20 years) • asset_index: Household asset score (0-100) • land_owned: Agricultural land in acres (0-50) Sample correlations (n = 25,000 households): • household_size ↔ per_capita_exp: r = -0.18* • education_years ↔ per_capita_exp: r = 0.52* • asset_index ↔ per_capita_exp: r = 0.73* • land_owned ↔ per_capita_exp: r = 0.31* • education_years ↔ asset_index: r = 0.45* *All correlations significant at p < 0.001
Problem 1: Correlation Interpretation (6 minutes)

Analyze each correlation:

Variable Pair r Value Strength Direction Practical Meaning
Household size ↔ Per capita expenditure -0.18 _____ _____ _____
Education ↔ Per capita expenditure 0.52 _____ _____ _____
Assets ↔ Per capita expenditure 0.73 _____ _____ _____

Interpretation Questions:

  • Which variable has the strongest association with per capita expenditure?
  • Why might household size be negatively correlated with per capita expenditure?
  • Is the education-expenditure correlation practically significant for policy?
Problem 2: Statistical Significance vs. Effect Size (6 minutes)

Scenario: With n = 25,000, even correlations as small as |r| = 0.02 are statistically significant (p < 0.05).

Critical Thinking Questions:

  1. If r = 0.02 between gender and expenditure (p < 0.001), is this meaningful?
  2. How would you decide which correlations deserve further investigation?
  3. What additional analysis would you do for the education-expenditure relationship?
  4. How might survey design (clustering, weights) affect these correlations?

Part 4: Group Comparisons - t-tests and ANOVA

20 minutes

Choosing the Right Test

Scenario Test Assumptions Output
2 groups, continuous outcome Independent t-test Normality, equal variances t-statistic, p-value, mean difference
2 groups, non-normal data Mann-Whitney U Independent observations U-statistic, p-value, median difference
3+ groups, continuous outcome One-way ANOVA Normality, equal variances F-statistic, p-value, group means
3+ groups, non-normal data Kruskal-Wallis Independent observations H-statistic, p-value, median ranks

Problem Set: ANOVA Analysis (15 minutes)

Research Question: Does mean household expenditure differ significantly across education levels of household head?

Sample Data: Monthly Household Expenditure by Education Level (Sample sizes and means from NSS data) Education Groups: • No Education (n = 8,500): Mean = ₹12,450, SD = ₹6,200 • Primary (n = 6,200): Mean = ₹15,680, SD = ₹7,100 • Secondary (n = 7,800): Mean = ₹19,230, SD = ₹8,900 • Higher (n = 4,500): Mean = ₹28,560, SD = ₹12,400 Overall: N = 27,000, Grand Mean = ₹17,890
Problem 1: Descriptive Analysis (5 minutes)

Calculate differences and effect sizes:

Comparison Mean Difference (₹) % Difference Effect Size (Cohen's d)
Higher vs. No Education _____ (28,560 - 12,450) _____% _____ (use pooled SD)
Secondary vs. Primary _____ _____% _____

Interpretation:

  • Which education gap shows the largest absolute difference?
  • Which gap shows the largest percentage difference?
  • Are these differences practically meaningful for household welfare?
Problem 2: ANOVA Interpretation (10 minutes)
ANOVA Results Summary: F-statistic: 485.7 df: 3, 26,996 p-value: < 0.001 R-squared: 0.052 (5.2% of variance explained) Post-hoc tests (Tukey HSD): All pairwise comparisons significant at p < 0.001

Critical Analysis Questions:

  1. Statistical Significance: The F-test is highly significant. What does this tell us?
  2. Effect Size: Only 5.2% of expenditure variance is explained by education. Is this meaningful?
  3. Practical Significance: A ₹16,000 difference between highest and lowest groups - is this substantial?
  4. Alternative Explanations: What other factors might explain expenditure differences?
  5. Policy Implications: How would you use these findings in education policy arguments?

Key Insight: With large samples, statistical significance is almost guaranteed. Focus on effect sizes and practical significance. A 129% difference in expenditure between education groups, while explaining only 5.2% of variance, suggests both meaningful group differences AND substantial within-group variation.

Reflection & Integration

10 minutes

Bivariate Analysis Planning

Individual Reflection (7 minutes):

Think about a research question from your work:

  1. Your research question: _____
  2. Variable 1 type (categorical/continuous): _____
  3. Variable 2 type (categorical/continuous): _____
  4. Appropriate analysis method: _____
  5. Expected result (positive/negative/no association): _____
  6. What would constitute practical significance: _____

Quick Share (3 minutes): Exchange analysis planning strategies with a partner

Key Takeaway

Bivariate analysis is about understanding relationships, not just finding statistical significance. Always consider practical significance, effect sizes, and alternative explanations. The goal is insight that informs action, not just p-values.

Resources for Statistical Analysis

Statistical Software:

Reference Guides:

Effect Size Calculators:

Next Steps in ImpactMojo:

This handout is part of the ImpactMojo 101 Knowledge Series
Licensed under CC BY-NC-ND 4.0 • Free to use with attribution • www.impactmojo.in

For statistical calculation templates, effect size guides, and interpretation frameworks, visit the ImpactMojo platform.