Lab Overview

Time: ~30 minutes

Goal: Practice one-way ANOVA analysis from start to finish using real public health data

Learning Objectives:

Understand when and why to use ANOVA instead of multiple t-tests
Set up hypotheses for ANOVA
Conduct and interpret the F-test
Perform post-hoc tests when appropriate
Check ANOVA assumptions
Calculate and interpret effect size (η²)

Structure:

Part A: Guided Example (follow along)
Part B: Your Turn (independent practice)

Submission: Upload your completed .Rmd file and published to Brightspace by the end of class.

PART A: GUIDED EXAMPLE

Example: Blood Pressure and BMI Categories

Research Question: Is there a difference in mean systolic blood pressure (SBP) across three BMI categories (Normal weight, Overweight, Obese)?

Why ANOVA? We have one continuous outcome (SBP) and one categorical predictor with THREE groups (BMI category). Using multiple t-tests would inflate our Type I error rate.

Step 1: Setup and Data Preparation

# Load necessary libraries
library(tidyverse)   # For data manipulation and visualization
library(knitr)       # For nice tables
library(car)         # For Levene's test
library(NHANES)      # NHANES dataset

# Load the NHANES data
data(NHANES)

Create analysis dataset:

# Set seed for reproducibility
set.seed(553)

# Create BMI categories and prepare data
bp_bmi_data <- NHANES %>%
  filter(Age >= 18 & Age <= 65) %>%  # Adults 18-65
  filter(!is.na(BPSysAve) & !is.na(BMI)) %>%
  mutate(
    bmi_category = case_when(
      BMI < 25 ~ "Normal",
      BMI >= 25 & BMI < 30 ~ "Overweight",
      BMI >= 30 ~ "Obese",
      TRUE ~ NA_character_
    ),
    bmi_category = factor(bmi_category, 
                         levels = c("Normal", "Overweight", "Obese"))
  ) %>%
  filter(!is.na(bmi_category)) %>%
  select(ID, Age, Gender, BPSysAve, BMI, bmi_category)

# Display first few rows
head(bp_bmi_data) %>% 
  kable(caption = "Blood Pressure and BMI Dataset (first 6 rows)")

Blood Pressure and BMI Dataset (first 6 rows)
ID	Age	Gender	BPSysAve	BMI	bmi_category
51624	34	male	113	32.22	Obese
51624	34	male	113	32.22	Obese
51624	34	male	113	32.22	Obese
51630	49	female	112	30.57	Obese
51647	45	female	118	27.24	Overweight
51647	45	female	118	27.24	Overweight

# Check sample sizes
table(bp_bmi_data$bmi_category)

## 
##     Normal Overweight      Obese 
##       1939       1937       2150

Interpretation: We have 6026 adults with complete BP and BMI data across three BMI categories.

Step 2: Descriptive Statistics

# Calculate summary statistics by BMI category
summary_stats <- bp_bmi_data %>%
  group_by(bmi_category) %>%
  summarise(
    n = n(),
    Mean = mean(BPSysAve),
    SD = sd(BPSysAve),
    Median = median(BPSysAve),
    Min = min(BPSysAve),
    Max = max(BPSysAve)
  )

summary_stats %>% 
  kable(digits = 2, 
        caption = "Descriptive Statistics: Systolic BP by BMI Category")

Descriptive Statistics: Systolic BP by BMI Category
bmi_category	n	Mean	SD	Median	Min	Max
Normal	1939	114.23	15.01	113	78	221
Overweight	1937	118.74	13.86	117	83	186
Obese	2150	121.62	15.27	120	82	226

Observation: The mean SBP appears to increase from Normal (114.2) to Overweight (118.7) to Obese (121.6).

But is this difference statistically significant?

Step 3: Visualize the Data

# Create boxplots with individual points
ggplot(bp_bmi_data, 
  aes(x = bmi_category, y = BPSysAve, fill = bmi_category)) +
  geom_boxplot(alpha = 0.7, outlier.shape = NA) +
  geom_jitter(width = 0.2, alpha = 0.1, size = 0.5) +
  scale_fill_brewer(palette = "Set2") +
  labs(
    title = "Systolic Blood Pressure by BMI Category",
    subtitle = "NHANES Data, Adults aged 18-65",
    x = "BMI Category",
    y = "Systolic Blood Pressure (mmHg)",
    fill = "BMI Category"
  ) +
  theme_minimal(base_size = 12) +
  theme(legend.position = "none")

What the plot tells us:

There appears to be a trend: higher BMI categories have higher median SBP
The boxes overlap, but the obese group appears shifted upward
Variability (box heights) looks similar across groups

Step 4: Set Up Hypotheses

Null Hypothesis (H₀): μ_Normal = μ_Overweight = μ_Obese
(All three population means are equal)

Alternative Hypothesis (H₁): At least one population mean differs from the others

Significance level: α = 0.05

Step 5: Fit the ANOVA Model

# Fit the one-way ANOVA model
anova_model <- aov(BPSysAve ~ bmi_category, data = bp_bmi_data)

# Display the ANOVA table
summary(anova_model)

##                Df  Sum Sq Mean Sq F value Pr(>F)    
## bmi_category    2   56212   28106   129.2 <2e-16 ***
## Residuals    6023 1309859     217                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interpretation:

F-statistic: 129.24
Degrees of freedom: df₁ = 2 (k-1 groups), df₂ = 6023 (n-k)
p-value: < 2e-16 (very small)
Decision: Since p < 0.05, we reject H₀
Conclusion: There is statistically significant evidence that mean systolic BP differs across at least two BMI categories.

Step 6: Post-Hoc Tests (Tukey HSD)

Why do we need this? The F-test tells us that groups differ, but not which groups differ. Tukey’s Honest Significant Difference controls the family-wise error rate for multiple pairwise comparisons.

# Conduct Tukey HSD test
tukey_results <- TukeyHSD(anova_model)
print(tukey_results)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = BPSysAve ~ bmi_category, data = bp_bmi_data)
## 
## $bmi_category
##                       diff      lwr      upr p adj
## Overweight-Normal 4.507724 3.397134 5.618314     0
## Obese-Normal      7.391744 6.309024 8.474464     0
## Obese-Overweight  2.884019 1.801006 3.967033     0

# Visualize the confidence intervals
plot(tukey_results, las = 0)

Interpretation:

Comparison	Mean Diff	95% CI	p-value	Significant?
Overweight - Normal	4.51	[3.4, 5.62]	1.98e-13	Yes
Obese - Normal	7.39	[6.31, 8.47]	< 0.001	Yes
Obese - Overweight	2.88	[1.8, 3.97]	1.38e-09	Yes

Conclusion: All three pairwise comparisons are statistically significant. Obese adults have higher SBP than overweight adults, who in turn have higher SBP than normal-weight adults.

Step 7: Calculate Effect Size

# Extract sum of squares from ANOVA table
anova_summary <- summary(anova_model)[[1]]

ss_treatment <- anova_summary$`Sum Sq`[1]
ss_total <- sum(anova_summary$`Sum Sq`)

# Calculate eta-squared
eta_squared <- ss_treatment / ss_total

cat("Eta-squared (η²):", round(eta_squared, 4), "\n")

## Eta-squared (η²): 0.0411

cat("Percentage of variance explained:", round(eta_squared * 100, 2), "%")

## Percentage of variance explained: 4.11 %

Interpretation: BMI category explains 4.11% of the variance in systolic BP.

Effect size guidelines: Small (0.01), Medium (0.06), Large (0.14)
Our effect: Small

While statistically significant, the practical effect is modest—BMI category alone doesn’t explain most of the variation in blood pressure.

Step 8: Check Assumptions

ANOVA Assumptions:

Independence: Observations are independent (assumed based on study design)
Normality: Residuals are approximately normally distributed
Homogeneity of variance: Equal variances across groups

# Create diagnostic plots
par(mfrow = c(2, 2))
plot(anova_model)

par(mfrow = c(1, 1))

Diagnostic Plot Interpretation:

Residuals vs Fitted: Points show random scatter around zero with no clear pattern → Good!
Q-Q Plot: Points follow the diagonal line reasonably well → Normality assumption is reasonable
Scale-Location: Red line is relatively flat → Equal variance assumption is reasonable
Residuals vs Leverage: No points beyond Cook’s distance lines → No highly influential outliers

# Levene's test for homogeneity of variance
levene_test <- leveneTest(BPSysAve ~ bmi_category, data = bp_bmi_data)
print(levene_test)

## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value  Pr(>F)  
## group    2  2.7615 0.06328 .
##       6023                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Levene’s Test Interpretation:

p-value: 0.0633
If p < 0.05, we would reject equal variances
Here: Equal variance assumption is met

Overall Assessment: With n > 2000, ANOVA is robust to minor violations. Our assumptions are reasonably satisfied.

Step 9: Report Results

Example Results Section:

We conducted a one-way ANOVA to examine whether mean systolic blood pressure (SBP) differs across BMI categories (Normal, Overweight, Obese) among 6,026 adults aged 18-65 from NHANES. Descriptive statistics showed mean SBP of 114.2 mmHg (SD = 15) for normal weight, 118.7 mmHg (SD = 13.9) for overweight, and 121.6 mmHg (SD = 15.3) for obese individuals.

The ANOVA revealed a statistically significant difference in mean SBP across BMI categories, F(2, 6023) = 129.24, p < 0.001. Tukey’s HSD post-hoc tests indicated that all pairwise comparisons were significant (p < 0.05): obese adults had on average 7.4 mmHg higher SBP than normal-weight adults, and 2.9 mmHg higher than overweight adults.

The effect size (η² = 0.041) indicates that BMI category explains 4.1% of the variance in systolic blood pressure, representing a small practical effect. These findings support the well-established relationship between higher BMI and elevated blood pressure, though other factors account for most of the variation in SBP.

PART B: YOUR TURN - INDEPENDENT PRACTICE

Practice Problem: Physical Activity and Depression

Research Question: Is there a difference in the number of days with poor mental health across three physical activity levels (None, Moderate, Vigorous)?

Your Task: Complete the same 9-step analysis workflow you just practiced, but now on a different outcome and predictor.

Step 1: Data Preparation

# Prepare the dataset
set.seed(553)

mental_health_data <- NHANES %>%
  filter(Age >= 18) %>%
  filter(!is.na(DaysMentHlthBad) & !is.na(PhysActive)) %>%
  mutate(
    activity_level = case_when(
      PhysActive == "No" ~ "None",
      PhysActive == "Yes" & !is.na(PhysActiveDays) & PhysActiveDays < 3 ~ "Moderate",
      PhysActive == "Yes" & !is.na(PhysActiveDays) & PhysActiveDays >= 3 ~ "Vigorous",
      TRUE ~ NA_character_
    ),
    activity_level = factor(activity_level, 
                           levels = c("None", "Moderate", "Vigorous"))
  ) %>%
  filter(!is.na(activity_level)) %>%
  select(ID, Age, Gender, DaysMentHlthBad, PhysActive, activity_level)

# YOUR TURN: Display the first 6 rows and check sample sizes
# View first 6 rows
head(mental_health_data)

## # A tibble: 6 × 6
##      ID   Age Gender DaysMentHlthBad PhysActive activity_level
##   <int> <int> <fct>            <int> <fct>      <fct>         
## 1 51624    34 male                15 No         None          
## 2 51624    34 male                15 No         None          
## 3 51624    34 male                15 No         None          
## 4 51630    49 female              10 No         None          
## 5 51647    45 female               3 Yes        Vigorous      
## 6 51647    45 female               3 Yes        Vigorous

table(mental_health_data$activity_level)

## 
##     None Moderate Vigorous 
##     3139      768     1850

YOUR TURN - Answer these questions:

How many people are in each physical activity group?
- None: 3139
- Moderate: 768
- Vigorous: 1850

Step 2: Descriptive Statistics

# YOUR TURN: Calculate summary statistics by activity level
# Hint: Follow the same structure as the guided example
# Variables to summarize: n, Mean, SD, Median, Min, Max

mental_health_summary <- mental_health_data %>%
  group_by(activity_level) %>%
  summarise(
    N = n(),
    Mean = mean(DaysMentHlthBad),
    SD = sd(DaysMentHlthBad),
    Median = median(DaysMentHlthBad),
    Min = min(DaysMentHlthBad),
    Max = max(DaysMentHlthBad),
    .groups = "drop"
  )

mental_health_summary

## # A tibble: 3 × 7
##   activity_level     N  Mean    SD Median   Min   Max
##   <fct>          <int> <dbl> <dbl>  <dbl> <int> <int>
## 1 None            3139  5.08  9.01      0     0    30
## 2 Moderate         768  3.81  6.87      0     0    30
## 3 Vigorous        1850  3.54  7.17      0     0    30

YOUR TURN - Interpret:

Which group has the highest mean number of bad mental health days? Ans : None
Which group has the lowest? Ans: Vigorous —

Step 3: Visualization

# YOUR TURN: Create boxplots comparing DaysMentHlthBad across activity levels
# Hint: Use the same ggplot code structure as the example
# Change variable names and labels appropriately
ggplot(mental_health_data,
       aes(x = activity_level, y = DaysMentHlthBad)) +
  geom_boxplot(fill = "lightblue", alpha = 0.7) +
  labs(
    title = "Poor Mental Health Days by Physical Activity Level",
    x = "Physical Activity Level",
    y = "Days of Poor Mental Health (Past 30 Days)"
  ) +
  theme_minimal()

YOUR TURN - Describe what you see:

Do the groups appear to differ?
Are the variances similar across groups? Ans: Groups do differ, with higher medians among inactive individuals Variability is larger in the None group —

Step 4: Set Up Hypotheses

YOUR TURN - Write the hypotheses:

Null Hypothesis (H₀): Mean number of poor mental health days is equal across all physical activity levels.

Alternative Hypothesis (H₁): At least one physical activity level has a different mean number of poor mental health days.

Significance level: α = 0.05

Step 5: Fit the ANOVA Model

# YOUR TURN: Fit the ANOVA model
# Outcome: DaysMentHlthBad
# Predictor: activity_level
anova_model <- aov(DaysMentHlthBad ~ activity_level,
                   data = mental_health_data)

summary(anova_model)

##                  Df Sum Sq Mean Sq F value   Pr(>F)    
## activity_level    2   3109  1554.6   23.17 9.52e-11 ***
## Residuals      5754 386089    67.1                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

YOUR TURN - Extract and interpret the results:

F-statistic: 23.2
Degrees of freedom: 2,5754
p-value: 9.5 × 10⁻¹¹
Decision (reject or fail to reject H₀): Since p < 0.05 → Reject H₀
Statistical conclusion in words:There is a statistically significant difference in mean poor mental health days across physical activity levels.

Step 6: Post-Hoc Tests

# YOUR TURN: Conduct Tukey HSD test
# Only if your ANOVA p-value < 0.05
TukeyHSD(anova_model)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = DaysMentHlthBad ~ activity_level, data = mental_health_data)
## 
## $activity_level
##                         diff       lwr        upr     p adj
## Moderate-None     -1.2725867 -2.045657 -0.4995169 0.0003386
## Vigorous-None     -1.5464873 -2.109345 -0.9836298 0.0000000
## Vigorous-Moderate -0.2739006 -1.098213  0.5504114 0.7159887

YOUR TURN - Complete the table:

Comparison	Mean Difference	95% CI Lower	95% CI Upper	p-value	Significant?
Moderate - None	-1.2726	-2.046	-0.4995	0.0003
Vigorous - None	-1.5465	-2.109		-0.9836	0.0000
Vigorous - Moderate	-0.2739	-1.098	-0.9836	0.7160

Interpretation:

Which specific groups differ significantly?

Ans: while any activity is better than none, increasing intensity from moderate to vigorous did not result in a statistically significant further reduction in poor mental health days for this sample.

Step 7: Calculate Effect Size

# YOUR TURN: Calculate eta-squared
# Hint: Extract Sum Sq from the ANOVA summary
anova_table <- summary(anova_model)[[1]]
eta_sq <- anova_table["activity_level", "Sum Sq"] /
          sum(anova_table[, "Sum Sq"])

eta_sq

## [1] 0.007988564

YOUR TURN - Interpret:

η² = 0.008
Percentage of variance explained: 0.007989×100= 0.8%
Effect size classification (small/medium/large): 0.01/0.06/0.14
What does this mean practically? Ans: While there is a statistically significant difference in poor mental health days across activity levels, physical activity explains only a tiny proportion of the variation. Other factors likely play a bigger role in mental health.

Step 8: Check Assumptions

# YOUR TURN: Create diagnostic plots
par(mfrow = c(2,2))
plot(anova_model)

par(mfrow = c(1,1))

YOUR TURN - Evaluate each plot:

Residuals vs Fitted: Residuals are scattered fairly. No obvious pattern reasonibily met.
Q-Q Plot: Residuals are approximately normally distributed
Scale-Location: Points mostly flat along horizontal line, slight spread at higher fitted values
Residuals vs Leverage: No single observation is highly influential, so ANOVA results are robust.

# YOUR TURN: Conduct Levene's test
library(car)

leveneTest(DaysMentHlthBad ~ activity_level,
           data = mental_health_data)

## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value    Pr(>F)    
## group    2  23.168 9.517e-11 ***
##       5754                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

YOUR TURN - Overall assessment:

Are assumptions reasonably met?
Do any violations threaten your conclusions?

ASns: The p-value is 9.5 × 10⁻¹¹, which is far below 0.05. Interpretation: This indicates that the assumption of homogeneity of variance is violated—the variances across your groups are not equal. —

Step 9: Write Up Results

YOUR TURN - Write a complete 2-3 paragraph results section:

Include: 1. Sample description and descriptive statistics 2. F-test results 3. Post-hoc comparisons (if applicable) 4. Effect size interpretation 5. Public health significance

Your Results Section:

Ans: A one-way ANOVA was conducted to evaluate the relationship between physical activity levels (None, Moderate, and Vigorous) and the number of poor mental health days reported by adults in the NHANES dataset (N = 5,757). Descriptive statistics revealed that individuals with no physical activity reported the highest average of poor mental health days (M = 5.08, SD = 9.01), followed by those with moderate activity (M = 3.81, SD = 6.87) and vigorous activity $M = 3.54, SD = 7.17). While the descriptive data suggests a trend of better mental health with increased activity, diagnostic checks—specifically Levene’s Test (F(2, 5754) = 23.2, p < .001) and the Q-Q plot—indicated a violation of the assumption of homogeneity of variance and normality, likely due to the high frequency of “0” days reported in the dataset.Despite these distributional challenges, the ANOVA was statistically significant, F(2, 5754) = 23.2, p < .001. Post-hoc comparisons using the Tukey HSD test indicated that individuals in the “None” category had significantly more poor mental health days than both the “Moderate” group (diff = -1.27, p < .001) and the “Vigorous” group (diff = -1.55, p < .001). However, there was no statistically significant difference between the Moderate and Vigorous groups (p = .716), suggesting that the primary benefit is associated with engaging in any consistent activity rather than the intensity of that activity. The calculated effect size, eta-squared ( = 0.008), indicates that activity level explains approximately 0.8% of the variance in mental health days. While this effect size is small by traditional standards, the results hold notable public health significance. Even a small reduction in poor mental health days across a large population can translate to substantial improvements in quality of life and economic productivity. These findings support public health initiatives that encourage sedentary individuals to adopt at least moderate physical activity to help manage mental well-being.

Reflection Questions

1. How does the effect size help you understand the practical vs. statistical significance?

Ans: Effect size quantifies the magnitude of a difference or relationship, independent of sample size. While a p-value tells you whether an effect is statistically significant, it doesn’t indicate whether the difference is large enough to matter in practice.

2. Why is it important to check ANOVA assumptions? What might happen if they’re violated?

Ans: Checking assumptions ensures that your conclusions are valid and reliable, and if assumptions are violated, alternative methods (e.g., Welch ANOVA or nonparametric tests) may be necessary.

3. In public health practice, when might you choose to use ANOVA?

Ans: ANOVA helps determine whether group differences are statistically significant, guiding decisions about where to focus interventions, allocate resources, or further investigate health disparities.

4. What was the most challenging part of this lab activity? The step 8 and 9 description. —

Submission Checklist

Before submitting, verify you have:

Completed all code chunks in Part B
Filled in all interpretation questions
Completed the Tukey HSD comparison table
Written a complete results section
Answered all reflection questions
Rendered your .Rmd file to HTML without errors
Checked that all outputs display correctly

To submit: Upload both your .Rmd file and the HTML output to Brightspace.

Lab completed on: February 05, 2026

GRADING RUBRIC (For TA Use)

Total Points: 15

Category	Criteria	Points	Notes
Code Execution	All code chunks run without errors	4	- Deduct 1 pt per major error - Deduct 0.5 pt per minor warning
Completion	All “YOUR TURN” sections attempted	4	- Part B Steps 1-9 completed - All fill-in-the-blank answered - Tukey table filled in
Interpretation	Correct statistical interpretation	4	- Hypotheses correctly stated (1 pt) - ANOVA results interpreted (1 pt) - Post-hoc results interpreted (1 pt) - Assumptions evaluated (1 pt)
Results Section	Professional, complete write-up	3	- Includes descriptive stats (1 pt) - Reports F-test & post-hoc (1 pt) - Effect size & significance (1 pt)

Detailed Grading Guidelines

Code Execution (4 points):

4 pts: All code runs perfectly, produces correct output
3 pts: Minor issues (1-2 small errors or warnings)
2 pts: Several errors but demonstrates understanding
1 pt: Major errors, incomplete code
0 pts: Code does not run at all

Completion (4 points):

4 pts: All sections attempted thoughtfully
3 pts: 1-2 sections incomplete or minimal effort
2 pts: Several sections missing
1 pt: Only partial completion
0 pts: Little to no work completed

Interpretation (4 points):

4 pts: All interpretations correct and well-explained
3 pts: Minor errors in interpretation
2 pts: Several interpretation errors
1 pt: Significant misunderstanding of concepts
0 pts: No interpretation provided

Results Section (3 points):

3 pts: Publication-quality, complete results section
2 pts: Good but missing some elements
1 pt: Incomplete or poorly written
0 pts: No results section written

Common Deductions

-0.5 pts: Missing sample sizes in write-up
-0.5 pts: Not reporting confidence intervals
-1 pt: Incorrect hypothesis statements
-1 pt: Misinterpreting p-values
-1 pt: Not checking assumptions
-0.5 pts: Poor formatting (no tables, unclear output)

In-Class Lab Activity: Analysis of Variance (ANOVA)

EPI 553: Principles of Statistical Inference II

Muntasir Masum

2026-02-05

Lab Overview

PART A: GUIDED EXAMPLE

Example: Blood Pressure and BMI Categories

Research Question: Is there a difference in mean systolic blood pressure (SBP) across three BMI categories (Normal weight, Overweight, Obese)?

Step 1: Setup and Data Preparation

Step 2: Descriptive Statistics

But is this difference statistically significant?

Step 3: Visualize the Data

Step 4: Set Up Hypotheses

Step 5: Fit the ANOVA Model

Step 6: Post-Hoc Tests (Tukey HSD)

Step 7: Calculate Effect Size

Step 8: Check Assumptions

Step 9: Report Results

PART B: YOUR TURN - INDEPENDENT PRACTICE

Practice Problem: Physical Activity and Depression

Research Question: Is there a difference in the number of days with poor mental health across three physical activity levels (None, Moderate, Vigorous)?

Step 1: Data Preparation

Step 2: Descriptive Statistics

Step 3: Visualization

Step 4: Set Up Hypotheses

Step 5: Fit the ANOVA Model

Step 6: Post-Hoc Tests

Step 7: Calculate Effect Size

Step 8: Check Assumptions

Step 9: Write Up Results

Reflection Questions

Submission Checklist

GRADING RUBRIC (For TA Use)

Detailed Grading Guidelines

Common Deductions