R Notebook

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Facilitator: CDAM Experts

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Session 7: ANOVA and Non-Parametric Tests

By the end of this session, you will be able to:

• Understand when and how to use Analysis of Variance (ANOVA) .

• Perform one-way and two-way ANOVA , including interpretation of results.

• Conduct post-hoc tests like Tukey’s HSD to identify group differences.

• Apply non-parametric alternatives (e.g., Kruskal-Wallis, Wilcoxon rank-sum) when assumptions of ANOVA are not met.

• Use these statistical tools to answer questions like:

  • Are there significant differences in salary across departments?

  • Is there an interaction between gender and job level on performance?

Introduction to ANOVA

What is ANOVA?

• ANOVA stands for Analysis of Variance — a statistical method used to compare means across more than two groups .

It helps answer questions like: “Is there a statistically significant difference in average salary among employees in different departments?”

Key Concepts:

• Null Hypothesis (H₀): All group means are equal

• Alternative Hypothesis (H₁): At least one group mean is different

• F-statistic: Ratio of variance between groups to variance within groups

• p-value: Probability that observed differences occurred by chance

1. One-Way ANOVA

Used when comparing means of a continuous outcome variable across one categorical independent variable with more than two levels .

Example:Does average salary differ significantly by department?

data <- data.frame(
  group = factor(c("A", "A", "A", "B", "B", "B", "C", "C", "C")),
  value = c(5, 6, 7, 8, 9, 10, 11, 12, 13))

#View(data)

# Perform one-way ANOVA
anova_result = aov(value ~ group, data = data)

# Print summary of ANOVA
summary(anova_result)

            Df Sum Sq Mean Sq F value Pr(>F)    
group        2     54      27      27  0.001 ***
Residuals    6      6       1                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

2. Post-hoc Test (Tukey’s HSD): Identify which groups differ significantly after a significant ANOVA result.

# Example: Tukey's HSD post-hoc test
tukey_result <- TukeyHSD(anova_result)

# Print results
print(tukey_result)

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = value ~ group, data = data)

$group
    diff       lwr      upr     p adj
B-A    3 0.4947644 5.505236 0.0242291
C-A    6 3.4947644 8.505236 0.0007942
C-B    3 0.4947644 5.505236 0.0242291

3. Two-Way ANOVA: Examine the effects of two factors and their interaction.

# Example: Two-way ANOVA
data1 <- data.frame(
  factor1 = factor(c("X", "X", "X", "Y", "Y", "Y")),
  factor2 = factor(c("P", "Q", "R", "P", "Q", "R")),
  value = c(5, 6, 7, 8, 9, 10))

head(data1)

tail(data1)

# Perform two-way ANOVA
anova_result1 <- aov(value ~ factor1 + factor2, data = data1)

# Print summary of ANOVA
summary(anova_result1)

            Df Sum Sq Mean Sq   F value Pr(>F)    
factor1      1   13.5    13.5 4.213e+30 <2e-16 ***
factor2      2    4.0     2.0 6.241e+29 <2e-16 ***
Residuals    2    0.0     0.0                     
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

4. Wilcoxon Rank-Sum Test (Mann-Whitney U Test): Compare medians of two independent samples.

# Example: Wilcoxon rank-sum test
group1 <- c(5, 6, 7)
group2 <- c(8, 9, 10)

# Perform Wilcoxon rank-sum test
wilcoxon_result <- wilcox.test(group1, group2)

# Print results
print(wilcoxon_result)

    Wilcoxon rank sum exact test

data:  group1 and group2
W = 0, p-value = 0.1
alternative hypothesis: true location shift is not equal to 0

5. Kruskal-Wallis Test: Compare medians across more than two groups.

# Example: Kruskal-Wallis test
data <- data.frame(
  group = factor(c("A", "A", "A", "B", "B", "B", "C", "C", "C")),
  value = c(5, 6, 7, 8, 9, 10, 11, 12, 13))

# Perform Kruskal-Wallis test
kruskal_result <- kruskal.test(value ~ group, data = data)
krus

G1;H1;Errorh: object 'krus' not found
gG1;H1;Error during wrapuph: not that many frames on the stack
Error: no more error handlers available (recursive errors?); invoking 'abort' restart
g

# Print results
print(kruskal_result)

    Kruskal-Wallis rank sum test

data:  value by group
Kruskal-Wallis chi-squared = 7.2, df = 2, p-value = 0.02732

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Homework

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Practice ANOVA and Non-Parametric Tests :

• Use the aov() function for one-way and two-way ANOVA.

• Use TukeyHSD() for post-hoc analysis.

• Use wilcox.test() for the Wilcoxon rank-sum test.

• Use kruskal.test() for the Kruskal-Wallis test.

Homework Assignment

Exercise A: Import a Dataset

• Download a public dataset (customerData)

• Load it into R using import() or read.csv()

• Print first few rows and check structure

Exercise B: One-Way ANOVA

• Identify a numeric outcome variable and a categorical grouping variable

• Run a one-way ANOVA

• Interpret the result

Exercise C: Post-Hoc Test

• If ANOVA is significant, run Tukey’s HSD

• Report which groups differ significantly

Exercise D: Two-Way ANOVA

• Include a second categorical variable

• Interpret both main effects and interaction

Exercise E: Non-Parametric Tests

• Check assumptions of ANOVA (normality, homogeneity)

• If violated, run Kruskal-Wallis or Wilcoxon tests instead

• Interpret the result

Exercise F: Write a Summary

• Summarize your findings clearly in plain language

• Include:

  • What question you were answering

  • Which variables were tested

  • Whether the result was significant

  • What conclusion can be drawn