Multivariate Analysis Quick Reference Guide

Essential formulas, interpretations, and decision rules for South Asian development research

Method Selection Decision Tree

Research Question	Method	Data Requirements	Example
"Are X and Y related?"	Correlation	2 continuous variables, n ≥ 30	Education & income relationship
"Do groups differ?"	ANOVA	Continuous outcome, categorical groups	Comparing training programs
"What predicts outcome?"	Regression	Continuous outcome, mixed predictors	Factors affecting malnutrition
"Can we predict values?"	Regression	Historical data, stable relationships	Forecasting crop yields

CORRELATION ANALYSIS

Purpose

Measure strength and direction of relationships between variables

Correlation Coefficient (r)
Range: -1 to +1
r = 0: No relationship
r = ±1: Perfect relationship

Types

Pearson's r: Linear, continuous data
Spearman's ρ: Ranked/ordinal data
Point-biserial: Continuous × binary

Interpretation Guide

\|r\| Value	Strength	Interpretation
0.0 - 0.3	Weak	Little practical value
0.3 - 0.7	Moderate	Meaningful relationship
0.7 - 1.0	Strong	Very important relationship

South Asian Examples:
• Maternal education ↔ child malnutrition: r = -0.74
• Rainfall ↔ crop yield: r = 0.62
• Distance to school ↔ enrollment: r = -0.68

Key Limitations

Correlation ≠ causation - Strong relationships don't prove cause
Outlier sensitivity - Extreme values distort results
Linear assumption - May miss curved relationships
Confounding variables - Third factors may explain relationships

Statistical Significance: For n=120, critical value = ±0.18 (α = 0.05)
Practical Significance: Consider both r-value and real-world importance

ANOVA (ANALYSIS OF VARIANCE)

Purpose

Compare means across multiple groups to identify significant differences

F-statistic
F = Between-group variance / Within-group variance
Higher F = More likely groups differ

Types

One-way: One factor (treatment types)
Two-way: Two factors (treatment × gender)
Repeated measures: Same subjects over time

Key Assumptions

Independence: Observations are independent
Normality: Data is normally distributed
Homogeneity: Equal variances across groups
Random sampling: Representative samples

Bangladesh Agriculture Example:
• Control: 2,340 kg/hectare
• Seeds only: 2,780 kg/hectare
• Training only: 2,650 kg/hectare
• Combined: 3,120 kg/hectare
F = 24.87, p < 0.001

Interpretation Steps

Check F-statistic: F > critical value → significant differences exist
Check p-value: p < 0.05 → reject null hypothesis (groups are equal)
Post-hoc tests: Determine which specific groups differ
Effect size: Calculate η² to assess practical significance

Post-hoc Tests:
• Tukey's HSD: All pairwise comparisons, moderate power
• Bonferroni: Conservative, controls Type I error
• Scheffé: Most conservative, best for unequal sample sizes

REGRESSION ANALYSIS

Purpose

Model relationships, make predictions, quantify effects of multiple variables

Simple Linear: Y = β₀ + β₁X + ε
Multiple: Y = β₀ + β₁X₁ + β₂X₂ + ... + ε

Key Components

β₀ (Intercept): Y when all X = 0
β₁, β₂ (Slopes): Change in Y per unit X
R²: Variance explained (0-1)
p-values: Statistical significance

Interpretation Guide

Component	Interpretation
β coefficient	Effect size (practical significance)
p-value < 0.05	Statistically significant
R² = 0.68	68% of variance explained
95% CI	Range of plausible values

Indian Child Health Example:
Malnutrition = 52.34 + 1.24(Distance) - 0.31(Maternal_Ed)
R² = 0.682 (68% variance explained)

Model Diagnostics

Residual analysis: Check for patterns in errors
Multicollinearity: VIF > 5 indicates problems
Influential points: Cook's distance > 1 is concerning
Homoscedasticity: Equal variance of residuals

Sample Size Rule: Minimum 10-15 observations per predictor variable
Variable Selection: Include theoretically important variables, avoid overfitting

South Asian Context Considerations

Cultural & Contextual Factors

Data Patterns

Seasonal variations: Monsoon affects agriculture data
Social hierarchies: Caste/class affect responses
Gender dynamics: Women's responses vary by context
Regional differences: State-level variations significant

Missing Data Patterns

Not random: Often reflects social exclusion
Seasonal migration: Affects data collection
Literacy barriers: Survey completion issues
Cultural sensitivity: Some topics avoided

Common South Asian Development Variables

Statistical Significance vs. Practical Importance

Domain	Typical Variables	Expected Relationships	Common Issues
Education	Enrollment, test scores, dropout rates	Income (+), Distance (-), Gender gaps	Seasonal attendance, quality measures
Health	Malnutrition, vaccination, mortality	Education (+), Income (+), Access (+)	Reporting accuracy, cultural practices
Agriculture	Yield, adoption rates, income	Rainfall (+), Extension (+), Credit (+)	Weather variability, market access
Infrastructure	Water access, electricity, roads	Investment (+), Governance (+)	Equity issues, maintenance quality

Statistical Significance

p < 0.05: Result unlikely due to chance
Confidence intervals: Don't include null value
Sample size matters: Large n can make small effects significant

Practical Importance

Effect size: Is the change meaningful?
Cost-benefit: Worth the investment?
Policy relevance: Can we act on this?

Example: A 0.5% improvement in test scores might be statistically significant with n=10,000 students, but may not justify expensive interventions.

Reporting Results for Different Audiences

Quick Diagnostic Checklist

Audience	Focus On	Language	Visual Tools
Policymakers	Practical significance, cost implications	Plain language, avoid jargon	Simple charts, infographics
Field Teams	Implementation insights, local patterns	Operational terms	Maps, bar charts
Researchers	Statistical details, assumptions, limitations	Technical terminology	Detailed tables, diagnostic plots
Donors	Impact evidence, scalability	Results-focused	Before/after comparisons

Before Analysis

□ Sample size adequate? (n ≥ 30 for correlation, 20+ per group for ANOVA)
□ Data quality checked? (outliers, missing data patterns)
□ Assumptions met? (normality, independence, homogeneity)
□ Variables appropriate? (measurement level, theoretical basis)

During Interpretation

□ Statistical significance assessed? (p-values, confidence intervals)
□ Practical significance considered? (effect sizes, real-world meaning)
□ Cultural context applied? (South Asian development realities)
□ Limitations acknowledged? (assumptions, causation, generalizability)