Multivariate Analysis Problem Sets

Scenario

You are analyzing factors affecting child health outcomes in rural Indian villages. You have collected data from 120 villages across three states (Uttar Pradesh, Bihar, Rajasthan) on various health, economic, and infrastructure indicators.

Sample Data Summary (n=120 villages)

Variable	Mean	Std Dev	Range
Child malnutrition rate (%)	32.4	12.8	8.5 - 58.2
Distance to health center (km)	8.7	5.2	1.2 - 24.8
Maternal literacy rate (%)	45.3	18.6	12.0 - 82.0
Household income (₹/month)	8,450	3,200	2,100 - 18,900
Access to clean water (%)	67.8	22.4	15.0 - 95.0

Part A: Correlation Analysis

Calculate and interpret correlations between child malnutrition rate and each predictor variable.

Given correlation coefficients:

Malnutrition vs. Distance to health center: r = 0.68
Malnutrition vs. Maternal literacy: r = -0.74
Malnutrition vs. Household income: r = -0.52
Malnutrition vs. Clean water access: r = -0.61

Questions:

Which variable has the strongest relationship with malnutrition? Interpret this correlation.
Which correlations are statistically significant at α = 0.05? (Critical value for n=120: r = ±0.18)
What do the negative correlations tell us about these relationships?
Can we conclude that improving maternal literacy will directly reduce malnutrition? Why or why not?

Your Analysis for Part A:

1. Strongest relationship:

2. Statistical significance:

3. Negative correlations interpretation:

4. Causation discussion:

Part B: ANOVA Analysis

Compare malnutrition rates across the three states using ANOVA.

State-wise malnutrition data:

State	n (villages)	Mean malnutrition (%)	Std Dev
Uttar Pradesh	45	38.2	11.4
Bihar	42	35.8	13.2
Rajasthan	33	25.1	10.8

ANOVA Results:

F-statistic: 18.74
p-value: < 0.001
Critical F-value (α = 0.05): 3.07

Questions:

State the null and alternative hypotheses for this ANOVA.
What is your conclusion based on the F-statistic and p-value?
Which state has the highest malnutrition rate? What might explain this difference?
What post-hoc test would you recommend to determine which states differ significantly?

Your Analysis for Part B:

1. Hypotheses:

H₀:

H₁:

2. Conclusion:

3. State comparison:

4. Post-hoc test recommendation:

Part C: Regression Analysis

Build a multiple regression model to predict malnutrition rates.

Regression Results:

Model: Malnutrition = β₀ + β₁(Distance) + β₂(Maternal_Lit) + β₃(Income) + β₄(Clean_Water)

Variable	Coefficient	Std Error	t-value	p-value
Intercept	52.34	4.28	12.23	< 0.001
Distance to health center	1.24	0.18	6.89	< 0.001
Maternal literacy	-0.31	0.06	-5.17	< 0.001
Household income (₹1000s)	-0.89	0.24	-3.71	< 0.001
Clean water access	-0.15	0.04	-3.75	< 0.001

Model Statistics: R² = 0.682, Adjusted R² = 0.671, F = 61.8, p < 0.001

Questions:

Interpret the coefficient for maternal literacy. What does -0.31 mean in practical terms?
What percentage of variance in malnutrition is explained by this model?
Predict the malnutrition rate for a village with: Distance = 5km, Maternal literacy = 60%, Income = ₹10,000, Clean water = 80%
Which variable has the largest impact on malnutrition? Justify your answer.

Your Analysis for Part C:

1. Maternal literacy coefficient interpretation:

2. Variance explained:

3. Prediction calculation:

Malnutrition = 52.34 + 1.24(5) + (-0.31)(60) + (-0.89)(10) + (-0.15)(80)

= 52.34 + ____ + ____ + ____ + ____

= _____%

4. Largest impact variable:

Scenario

You are evaluating the effectiveness of different agricultural interventions in Bangladesh. A randomized controlled trial was conducted with 300 farmers across 4 treatment groups over 2 growing seasons. Your task is to analyze which interventions are most effective and identify factors that predict adoption success.

Treatment Groups and Results

Treatment Group	n	Mean Yield (kg/hectare)	Std Dev	Adoption Rate (%)
Control (Traditional methods)	75	2,340	420	-
Improved seeds only	75	2,780	380	68
Extension training only	75	2,650	450	72
Seeds + Training combination	75	3,120	390	84

Part A: Method Selection and Design

Before conducting analysis, you need to determine the most appropriate statistical methods.

Questions:

What statistical method would you use to compare yields across the four treatment groups? Justify your choice.
List three assumptions you need to check before conducting this analysis.
If you wanted to examine the relationship between farmer education (years) and adoption rate, what method would you use?
Design a regression model to predict crop yield. What variables would you include and why?

Your Method Selection Analysis:

1. Method for comparing yields:

2. Statistical assumptions to check:

3. Method for education-adoption relationship:

4. Regression model design:

Part B: ANOVA and Interpretation

Analyze the yield differences across treatment groups.

ANOVA Results:

F-statistic: 24.87
p-value: < 0.001
Effect size (η²): 0.201

Tukey's HSD Post-hoc Results (p-values):

Comparison	Mean Difference	p-value
Control vs. Seeds only	440	< 0.001
Control vs. Training only	310	0.002
Control vs. Seeds + Training	780	< 0.001
Seeds only vs. Training only	130	0.421
Seeds only vs. Seeds + Training	340	0.001
Training only vs. Seeds + Training	470	< 0.001

Questions:

Interpret the F-statistic and what it tells us about treatment effectiveness.
Which treatment groups differ significantly from the control? What are the practical implications?
Is there a significant difference between "Seeds only" and "Training only"? What does this suggest?
Calculate the effect size of the combined treatment compared to control. Is this a meaningful difference?

Your Interpretation:

1. F-statistic interpretation:

2. Significant differences from control:

3. Seeds vs. Training comparison:

4. Combined treatment effect size:

Effect size = (3120 - 2340) / pooled SD = _____ / _____ = _____

Part C: Multiple Factor Analysis

You have additional data on farmer characteristics and environmental factors.

Additional Variables:

Farmer education (years): Mean = 6.2, SD = 3.4
Farm size (hectares): Mean = 1.8, SD = 0.9
Rainfall during season (mm): Mean = 1,245, SD = 180
Soil quality index (1-10): Mean = 6.8, SD = 1.2

Correlation Matrix with Yield:

Variable	Correlation with Yield	p-value
Farmer education	0.45	< 0.001
Farm size	0.23	0.008
Rainfall	0.67	< 0.001
Soil quality	0.52	< 0.001

Questions:

Which environmental factor has the strongest relationship with yield? Is this relationship practically meaningful?
If you were to build a comprehensive regression model, what would be your top 3 predictor variables?
How might you test for interactions between treatment type and farmer education?
What are the policy implications of these correlations for agricultural extension programs?

Your Analysis:

1. Strongest environmental factor:

2. Top 3 predictors for regression model:

3. Testing treatment × education interaction:

4. Policy implications:

Scenario

You are conducting a comprehensive study of water access inequality in Pakistani cities. You have data from 85 urban areas with information on infrastructure investment, governance quality, population demographics, and water service delivery outcomes. Your goal is to identify the key drivers of water access equity.

Part A: Research Design Challenge

You need to design an appropriate analytical strategy for this complex policy question.

Available Variables:

Dependent variables: Overall water access (%), Water access for low-income households (%)
Infrastructure: Per capita water investment (PKR), Infrastructure age (years)
Governance: Transparency index (1-10), Corruption perception (1-10)
Demographics: Population size, Population density, Urban poverty rate (%)
Geographic: Province (Punjab, Sindh, KPK, Balochistan), Rainfall (mm/year)

Your Task:

Design a complete analytical strategy using all three methods (correlation, ANOVA, regression) to answer: "What factors best explain water access inequality in Pakistani cities?"

Your Research Design:

Step 1 - Correlation Analysis Plan:

Step 2 - ANOVA Analysis Plan:

Step 3 - Regression Analysis Plan:

Expected Outcomes & Policy Implications:

Part B: Data Interpretation Challenge

You receive the following results from your analysis. Interpret the findings and make policy recommendations.

Key Findings:

Strongest correlation: Transparency index vs. overall water access (r = 0.78)
ANOVA result: Significant differences in water access across provinces (F = 12.4, p < 0.001)
Regression R²: Model explains 69% of variance in water access equity
Top predictors: Transparency (β = 4.2), Infrastructure investment (β = 0.003), Urban poverty (β = -0.8)

Provincial Means:

Province	Overall Access (%)	Low-income Access (%)	Equity Gap
Punjab	78	65	13
Sindh	72	54	18
KPK	69	58	11
Balochistan	45	28	17

Critical Thinking Questions:

What story do these results tell about water access in Pakistan?
Why might transparency be more important than infrastructure investment?
How would you explain the provincial differences to a policy maker?
What are three specific, evidence-based policy recommendations?
What additional data would strengthen your analysis?

Your Policy Analysis:

1. Overall story:

2. Transparency vs. infrastructure:

3. Provincial differences explanation:

4. Three policy recommendations:

a)

b)

c)

5. Additional data needed:

Instructions for All Problem Sets

Problem Set 1: Rural Health Outcomes in India

Scenario

Sample Data Summary (n=120 villages)

Part A: Correlation Analysis

Given correlation coefficients:

Questions:

Your Analysis for Part A:

Part B: ANOVA Analysis

State-wise malnutrition data:

ANOVA Results:

Questions:

Your Analysis for Part B:

Part C: Regression Analysis

Regression Results:

Questions:

Your Analysis for Part C:

Problem Set 2: Agricultural Productivity in Bangladesh

Scenario

Treatment Groups and Results

Part A: Method Selection and Design

Questions:

Your Method Selection Analysis:

Part B: ANOVA and Interpretation

ANOVA Results:

Tukey's HSD Post-hoc Results (p-values):

Questions:

Your Interpretation:

Part C: Multiple Factor Analysis

Additional Variables:

Correlation Matrix with Yield:

Questions:

Your Analysis:

Problem Set 3: Urban Water Access in Pakistan

Scenario

Part A: Research Design Challenge

Available Variables:

Your Task:

Your Research Design:

Part B: Data Interpretation Challenge

Key Findings:

Provincial Means:

Critical Thinking Questions:

Your Policy Analysis:

Answer Key & Hints

Problem Set 1 Hints:

Problem Set 2 Hints:

Problem Set 3 Hints:

Assessment Rubric