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

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 = 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, hence skewed to the right.
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:

The study analyzed 5,757 adults with complete data on physical activity and mental health outcomes. The sample included three groups based on physical activity level: None (n = 3,139, 54.5%), Moderate (n = 768, 13.3%), and Vigorous (n = 1,850, 32.1%). Participants with no physical activity reported an average of 5.08 days (SD = 9.01) of poor mental health in the past 30 days, compared to 3.81 days (SD = 6.87) for moderate activity and 3.54 days (SD = 7.17) for vigorous activity. Although Levene’s test indicated unequal variances across groups (F(2, 5754) = 23.17, p < .001), the ANOVA was conducted given the robustness of the F-test to this violation with large sample sizes.

The one-way ANOVA revealed a statistically significant effect of physical activity level on poor mental health days, F(2, 5754) = 23.17, p < .001, η² = 0.008. Tukey’s HSD post-hoc tests showed that adults with no physical activity reported significantly more poor mental health days than those with moderate activity (mean difference = 1.27 days, 95% CI [0.50, 2.05], p < .001) and vigorous activity (mean difference = 1.55 days, 95% CI [0.98, 2.11], p < .001). However, no significant difference emerged between moderate and vigorous activity levels (mean difference = 0.27 days, 95% CI [-0.55, 1.10], p = .72). The effect size was small, with physical activity level accounting for less than 1% of the variance in poor mental health days. Despite the modest effect size, the absolute difference of approximately 1.5 fewer poor mental health days per month among physically active adults represents a meaningful public health benefit at the population level, suggesting that promoting any level of physical activity may contribute to improved mental health outcomes.

Reflection Questions

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

Effect size tells me whether a statistically significant finding actually matters in the real world. In my analysis, I found p < .001 (statistically significant), but η² = 0.008 means physical activity only explains 0.8% of mental health variance. This shows me that while the effect is real, it’s very small - physical activity is just one tiny piece of the mental health puzzle. Without effect size, I might have overinterpreted my significant p-value and thought exercise was way more important than it actually is for individual mental health outcomes.

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

Checking assumptions ensures the validity of ANOVA results. Violated independence increases Type I error rates. Violated homogeneity of variance affects accuracy of p-values and confidence intervals. Violated normality impacts the F-test, though large samples provide robustness. In this analysis, the large sample size (n = 5,757) mitigated concerns about heterogeneity and non-normality, but duplicate IDs posed a more serious threat to independence.

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

I would use ANOVA when comparing a continuous health outcome across three or more groups. For example: comparing blood pressure across different diet interventions (Mediterranean, low-carb, standard), testing if disease rates differ across multiple geographic regions, evaluating if health education programs at different intensity levels affect knowledge scores differently, or like in my study, comparing mental health days across physical activity levels. ANOVA is particularly useful in public health because we often want to compare multiple intervention strategies or exposure levels at once, rather than just two groups.

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

Interpreting the discrepancy between statistical significance (p < .001) and practical significance (η² = 0.008) required careful consideration of both individual-level and population-level implications. Additionally, determining which assumption violations were acceptable given the large sample size versus which threatened validity required applying statistical judgment rather than following fixed rules.

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

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)

Untitled

Marina Frimpong

2026-02-05