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
library(dplyr)

# 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

head(mental_health_data) %>% 
  kable(caption = "Poor Mental Health and Physical Activity Dataset (first 6 rows)")

Poor Mental Health and Physical Activity Dataset (first 6 rows)
ID	Age	Gender	DaysMentHlthBad	PhysActive	activity_level
51624	34	male	15	No	None
51624	34	male	15	No	None
51624	34	male	15	No	None
51630	49	female	10	No	None
51647	45	female	3	Yes	Vigorous
51647	45	female	3	Yes	Vigorous

# Check sample sizes
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
summary_stats <- 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)
  )
summary_stats %>% 
  kable(digits = 2, 
        caption = "Descriptive Statistics: Poor Mental Health by Physical Activity level")

Descriptive Statistics: Poor Mental Health by Physical Activity level
activity_level	n	Mean	SD	Max
None	3139	5.08	9.01	30
Moderate	768	3.81	6.87	30
Vigorous	1850	3.54	7.17	30

YOUR TURN - Interpret:

Which group has the highest mean number of bad mental health days? Answer: Group which had None physical activity
Which group has the lowest? Answer: Vigorous group

Step 3: Visualization

# YOUR TURN: Create boxplots comparing DaysMentHlthBad across activity levels

ggplot(mental_health_data, 
  aes(x = activity_level, y = DaysMentHlthBad, fill = activity_level)) +
  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 = "Poor Mental Health Days by Physical Activity Level",
    subtitle = "NHANES Data, Adults aged 18-65",
    x = "Physical Activity Level",
    y = "Poor Mental Health Days",
    fill = "Activity level"
  ) +
  theme_minimal(base_size = 12) +
  theme(legend.position = "none")

# Hint: Use the same ggplot code structure as the example
# Change variable names and labels appropriately

YOUR TURN - Describe what you see:

Do the groups appear to differ?

All three groups have the same median: 0 poor mental health days.That means most people do not have mental health issues.

However, Q3 is highest among those with none physical activity, lower in the moderate activity level groups and lowest among the vigorous activity levels. This means those with vigorous activity level tend to have a fewer poor-mental-helath days.

Are the variances similar across groups?

The boxplots indicate greater variability in poor mental-health days among adults reporting “None” physical activity, as evidenced by a larger interquartile range and wider spread of values compared to the “Moderate” and “Vigorous” groups.

All groups showed long upper tails extending to 30 days, reflecting right-skewed distributions with occasional extreme values.

Step 4: Set Up Hypotheses

Null Hypothesis (H₀): μ_none = μ_moderate = μ_vigirous
(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(DaysMentHlthBad ~ activity_level, data = mental_health_data)

# Display the ANOVA table
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

Interpretation:

F-statistic: 23.17
Degrees of freedom: df₁ = 2 (k-1 groups), df₂ = 5754 (n-k)
p-value: < 9.52e-11 (very small)
Decision: Since p < 0.05, we reject H₀
Conclusion: There is statistically significant evidence that mean poor mental health days differs across at least two activity level groups.

YOUR TURN - Extract and interpret the results:

F-statistic: 23.17
Degrees of freedom: 2
p-value: 9.52e-11
Decision (reject or fail to reject H₀): ** Since p < 0.05, we reject H₀
Statistical conclusion in words:There is statistically significant evidence that mean poor mental health days differs across at least two activity level groups.

Step 6: Post-Hoc Tests

# 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 = 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

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

YOUR TURN - Complete the table:

Comparison	Mean Difference	95% CI Lower	95% CI Upper	p-value	Significant?
Moderate - None	-1.2725867	-2.045657	-0.4995169	0.0003386	Yes
Vigorous - None	-1.5464873	-2.109345	-0.9836298	0.0000000	Yes
Vigorous - Moderate	-0.2739006	-1.098213	0.5504114	0.7159887	No

Interpretation: Answer: Based on the Tukey-adjusted 95% confidence intervals, both the moderate- and vigorous- activity groups differed significantly from the no-activity group because their intervals excluded zero, whereas the confidence interval for the vigorous-moderate comparison included zero, indicating no significant difference.

Step 7: Calculate Effect Size

# YOUR TURN: Calculate eta-squared
# Hint: Extract Sum Sq from the ANOVA summary

# 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): 0.008

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

## Percentage of variance explained: 0.8 %

YOUR TURN - Interpret:

η² = 0.008
Percentage of variance explained: 0.08%
Effect size classification (small/medium/large): small
What does this mean practically?

Only 0.8% of the variance in poor mental health days is explained by physical activity level 99.2% of the variance is due to other factors (genetics, stress, socioeconomic status, sleep, diet, social support, etc.)

Effective size indicate a small portion of the variance.

Individual Level: Physical activity is a very small piece of the mental health puzzle. You can’t predict someone’s mental health well just by knowing their activity level.

Population Level: Even small effects matter when applied to millions of people. A difference of 1.5 fewer poor mental health days per month across a population is still meaningful for public health.

Multifactorial Nature: Mental health is complex with many contributing factors - physical activity is just one small (but significant) piece.

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: Points are clustered at a few fitted values with no obvious curved pattern and the smoothing line is roughly flat, suggesting the linearity assumption is generally reasonable.
Q-Q Plot: Plot revealed pronounced departures from the reference line, particularly in the tails of the distribution, indicating that the residuals deviated substantially from normality.
Scale-Location: Plot displayed relatively constant variability of standardized residuals across fitted values, with no clear increasing or decreasing trend, suggesting that the assumption of constant variance is mostly satisfied.
Residuals vs Leverage: Plot indicated that all observations fall within the Cook’s distance boundaries and show low leverage, meaning no single data point is unduly influencing the model.

# YOUR TURN: Conduct Levene's test
levene_test <- leveneTest(DaysMentHlthBad ~ activity_level, data = mental_health_data)
print(levene_test)

## 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?

Independence of Observations: LIKELY MET Each participant should be independent but data shows duplicate IDs
Normality:LIKELY VIOLATED The outcome measurement(DaysMentHlthBad) is highly right-skewed All groups have Median = 0, suggesting most people report 0 days
Homogeneity of Variance: VIOLATED Levene’s Test: F(2, 5754) = 23.17, p < .001 Variances differ significantly across groups

Do any violations threaten your conclusions? YES - The independence violation is a problem:

Duplicate IDs are the biggest threat Standard ANOVA assumes each observation is independent Repeated measures from the same person violate this Could lead to underestimated standard errors and inflated significance
Homogeneity violation - Minor threat With n = 5,757, the F-test is robust Unlikely to substantially affect conclusions Could consider Welch’s ANOVA as sensitivity check
Non-normality - Minimal threat Large sample size provides protection Could consider non-parametric alternative (Kruskal-Wallis) as sensitivity check —

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:

Reflection Questions

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

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

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

4. What was the most challenging part of this lab activity?

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 04, 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)

Lab 02 Practice Assignment: Analysis of Variance (ANOVA)

EPI 553: Principles of Statistical Inference II

Marina Frimpong

February 4, 2026

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

All groups showed long upper tails extending to 30 days, reflecting right-skewed distributions with occasional extreme values.

Step 4: Set Up Hypotheses

Step 5: Fit the ANOVA Model

Step 6: Post-Hoc Tests

Step 7: Calculate Effect Size

Multifactorial Nature: Mental health is complex with many contributing factors - physical activity is just one small (but significant) piece.

Step 8: Check Assumptions

Step 9: Write Up Results

Reflection Questions

Submission Checklist

GRADING RUBRIC (For TA Use)

Detailed Grading Guidelines

Common Deductions