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)

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

# 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, n=6)
## # 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
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 = 3, 
        caption = "Descriptive statistics for Mental Health by physical activity level")
Descriptive statistics for Mental Health by physical activity level
activity_level n Mean SD Median Min Max
None 3139 5.084 9.010 0 0 30
Moderate 768 3.811 6.873 0 0 30
Vigorous 1850 3.537 7.171 0 0 30

YOUR TURN - Interpret:

  • Which group has the highest mean number of bad mental health days? The group with the highest mean number of bad mental health days was participants who did no physical activity at all with a mean of 5.084.
  • Which group has the lowest? The group with the lowest mean number of bad mental health days was participants who did vigorous physical activity with a mean of 3.537.

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, fill = activity_level)) +
  geom_boxplot(alpha = 0.7) +
  geom_jitter(width = 0.2, alpha = 0.3, size = 0.5) +
  scale_fill_brewer(palette = "Set2") +
  labs(
    title = "Days of Bad Mental Health compared to Physical Activity Levels",
    subtitle = "NHANES 2017-2018, Adults aged 18+",
    x = "Physical Activity Level",
    y = "Bad Mental Health Days (g/cm²)",
    fill = "Activity Level"
  ) +
  theme_minimal(base_size = 12) +
  theme(legend.position = "none")

YOUR TURN - Describe what you see:

  • Do the groups appear to differ? Yes the groups appear to differ. As physical activity increases, bad mental health days decrease’s. No physical activity group has the highest median days of bad mental health while the vigorous group has the lowest median of bad mental health days.
  • Are the variances similar across groups? No, the variances are not similar across the groups. The no physical activity group shows the widest spread, with a large variability. The moderate group has a slightly smaller variability. The vigorous group has the smallest variance. —

Step 4: Set Up Hypotheses

YOUR TURN - Write the hypotheses:

Null Hypothesis (H₀):

There is no difference in the number of bad mental health days across physical activity level (none, moderate, vigorous).

Alternative Hypothesis (H₁):

There is a difference in the number of bad mental health days across physical activity levels (none, moderate, vigorous).

Significance level: α = 0.05

Step 5: Fit the ANOVA Model

# YOUR TURN: Fit the ANOVA model
# Outcome: DaysMentHlthBad
# Predictor: activity_level

# Fit one-way ANOVA
anova_model <- aov(DaysMentHlthBad ~ activity_level, data = mental_health_data)

# Display ANOVA table
anova_table <- summary(anova_model)
print(anova_table)
##                  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: 23.17
  • Degrees of freedom: 2
  • p-value: 9.52e-11
  • Decision (reject or fail to reject H₀): reject H₀
  • Statistical conclusion in words: The p-value of 9.52 X 10^-11 is smaller than the significance level of 0.05, we can reject the null hypothesis. This means that there is a statistical significance between the number of bad mental health days and physical activity levels.

Step 6: Post-Hoc Tests

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

# 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)
## 
## $activity_level
##                         diff       lwr        upr     p adj
## Moderate-None     -1.2725867 -2.045657 -0.4995169 0.0003386
## Vigorous-None     -1.5464873 -2.109345 -0.9836298 0.0000000
## Vigorous-Moderate -0.2739006 -1.098213  0.5504114 0.7159887
# Extract and format results
tukey_summary <- as.data.frame(tukey_results$activity_level)
tukey_summary$Comparison <- rownames(tukey_summary)
tukey_summary <- tukey_summary[, c("Comparison", "diff", "lwr", "upr", "p adj")]

# Add interpretation columns
tukey_summary$Significant <- ifelse(tukey_summary$`p adj` < 0.05, "Yes", "No")
tukey_summary$Direction <- ifelse(
  tukey_summary$Significant == "Yes",
  ifelse(tukey_summary$diff > 0, "First group higher", "Second group higher"),
  "No difference"
)

kable(tukey_summary, digits = 4,
      caption = "Tukey HSD post-hoc comparisons with 95% confidence intervals")
Tukey HSD post-hoc comparisons with 95% confidence intervals
Comparison diff lwr upr p adj Significant Direction
Moderate-None Moderate-None -1.2726 -2.0457 -0.4995 0.0003 Yes Second group higher
Vigorous-None Vigorous-None -1.5465 -2.1093 -0.9836 0.0000 Yes Second group higher
Vigorous-Moderate Vigorous-Moderate -0.2739 -1.0982 0.5504 0.7160 No No difference

YOUR TURN - Complete the table:

Comparison Mean Difference 95% CI Lower 95% CI Upper p-value Significant?
Moderate - None
Vigorous - None
Vigorous - Moderate

Interpretation:

Which specific groups differ significantly?

Moderate vs. None: The none group has significantly more bad mental health days then the moderate group.

Vigorous vs. None: The none group has significantly more bad mental health days then the vigorous group.

Vigorous vs. Moderate: There is no significant difference.

Step 7: Calculate Effect Size

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

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: 0.8%
  • Effect size classification (small/medium/large): medium
  • What does this mean practically?

An η² = 0.008 value indicates that physical activity level is only 0.8% of the variation of bad mental health days. This indicates that while physical activity is statistically significantly associated with mental health days, it only accounts for a small portion of mental health days.

Step 8: Check Assumptions

# YOUR TURN: Create diagnostic plots

par(mfrow = c(2, 2))
plot(anova_model, which = 1:4)

YOUR TURN - Evaluate each plot:

  1. Residuals vs Fitted:

The residuals are roughly centered around zero with no visible pattern. This suggests that independence assumption holds and the linearity assumption for this model is reasonably met.

  1. Q-Q Plot:

The residuals deviate from the diagonal line, especially in the upper tail. This indicates that normality is violated.

  1. Scale-Location:

The residuals are not constant across the fitted values and might indicate some non-constant variance but may not have a severe affect.

  1. Residuals vs Leverage: There are no identifiable outliers that influence the conclusion.
# YOUR TURN: Conduct Levene's test
library (car)
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?

The assumptions are reasonably met.

  • Do any violations threaten your conclusions?

The assumptions show mild violations but are not severe enough to change our overall conclusion.

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 included 5757 participants that were divided up into 3 groups based on physical activity level (none-3139, moderate-768, vigorous-1850). A one way ANOVA was conducted to examine the effect of physical activity on bad mental health days. The results indicated a statistically significant difference between the groups with an F-statistic of 23.17 and a p-value of 9.52e-11. The moderate vs. None group showed that the none group has significantly more bad mental health days then the moderate group. The vigorous vs. None group showed that the none group has significantly more bad mental health days then the vigorous group. The vigorous vs. Moderate group showed that there was no significant difference. The calculated eta-squared pf 0.008 shows that only 0.8% of bad mental health days is explained by physical activity. This represents a medium effect, suggesting that physical activity has a moderate impact on bad mental health days beyond the statistics. The results suggest that physical activity levels can have an influence on bad mental health days. From a public health perspective, creating interventions that show a positive correlation between physical activity and bad mental health days could help improve the overall populations mental health. By creating more interventions, we can help improve long term health outcomes and help reduce mental health disease at a individual level and a community based level.

Reflection Questions

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

Effect size allows us to see the real world terms and not just if they are statistically significant or not. In this example, η² = 0.008 value shows a small portion of mental health days is affected by physical activity levels. This shows the practical relvance.

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

ANOVA assumes normality and if the assumptions are violated it can inflate Type 1 and Type 2 error rates which affects your conclusion.

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

ANOVA is used when you are comparing 3 or more groups.

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

The most challenging part of this lab was interpreting the diagnostic plots and making sure that I was reading them correclty.