PART A: GUIDED EXAMPLES

Setup: Load Packages and Data

# Load NHANES data
data(NHANES)

# Select adult participants with complete data
nhanes_adult <- NHANES %>%
  filter(Age >= 18, Age <= 80) %>%
  select(Age, Weight, Height, BMI, BPSysAve, BPDiaAve, 
         Pulse, PhysActive, SleepHrsNight) %>%
  na.omit()

# Display sample
# Display sample size
data.frame(
  Metric = "Sample Size",
  Value = paste(nrow(nhanes_adult), "adults")
) %>%
  kable()
Metric Value
Sample Size 7133 adults 
head(nhanes_adult, 8) %>%
  kable(digits = 1, caption = "NHANES Adult Data Sample")
NHANES Adult Data Sample
Age Weight Height BMI BPSysAve BPDiaAve Pulse PhysActive SleepHrsNight
34 87.4 164.7 32.2 113 85 70 No 4
34 87.4 164.7 32.2 113 85 70 No 4
34 87.4 164.7 32.2 113 85 70 No 4
49 86.7 168.4 30.6 112 75 86 No 8
45 75.7 166.7 27.2 118 64 62 Yes 8
45 75.7 166.7 27.2 118 64 62 Yes 8
45 75.7 166.7 27.2 118 64 62 Yes 8
66 68.0 169.5 23.7 111 63 60 Yes 7

Dataset Description:

  • Age: Age in years
  • Weight: Weight in kg
  • BMI: Body Mass Index (kg/m²)
  • BPSysAve: Average systolic blood pressure (mmHg)
  • BPDiaAve: Average diastolic blood pressure (mmHg)
  • Pulse: 60 second pulse rate
  • SleepHrsNight: Hours of sleep per night

PART B: YOUR TURN - Practice Problems

Now it’s your turn to practice! Use the same NHANES dataset and follow the examples above.

Total Points: 25 points

Problem 1: Weight and Height (10 points)

Research Question: Is there a correlation between weight and height among US adults?

Your tasks:

  1. Create a scatterplot with a fitted line (2 points)
  2. Calculate Pearson correlation using cor.test() and display with tidy() (3 points)
  3. Test for statistical significance and state your conclusion (2 points)
  4. Calculate r² and interpret in 2-3 sentences (3 points)
# a. Scatterplot
ggplot(nhanes_adult, aes(x = Height, y = Weight)) +
  geom_point(alpha = 0.3, color = "steelblue") +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  labs(
    title = "Height vs Weight",
    subtitle = "NHANES Data, Adults 18-80 years",
    x = "Height (cm)",
    y = "Weight (kgs)"
  ) +
  theme_minimal()

# b. Correlation test with tidy() display
# Calculate Pearson correlation
cor_height_weight <- cor.test(nhanes_adult$Height, nhanes_adult$Weight)

# Display results in clean table
tidy(cor_height_weight) %>%
  select(estimate, statistic, p.value, conf.low, conf.high) %>%
  kable(
    digits = 3,
    col.names = c("r", "t-statistic", "p-value", "95% CI Lower", "95% CI Upper"),
    caption = "Pearson Correlation: Height and Weight"
  )
Pearson Correlation: Height and Weight
r t-statistic p-value 95% CI Lower 95% CI Upper
0.451 42.618 0 0.432 0.469 
# Calculate r-squared
r_squared_height_weight <- cor_height_weight$estimate^2

data.frame(
  Measure = c("r²", "Variance Explained"),
  Value = c(
    round(r_squared_height_weight, 4),
    paste0(round(r_squared_height_weight * 100, 2), "%")
  )
) %>%
  kable(caption = "Effect Size")
Effect Size
Measure Value
cor r² 0.203
Variance Explained 20.3% 
# c. Statistical significance
# p<0.001 suggests statistical significance

# d. r² and interpretation (write as comment)
# R-squared value is 0.203. This means that 20.3% of the variation in weight can be explained by height.
  
# Hypothesis Test:
# H₀: ρ = 0 (no correlation between height and weight in population)
# H₁: ρ ≠ 0 (correlation exists)
# α = 0.05

# Results:
# r = 0.451: Moderate positive correlation
# p < 0.001: Statistically significant (reject H₀)
# 95% CI [0.432, 0.469]: Doesn’t contain zero (confirms significance)

Problem 2: Correlation Matrix Analysis (10 points)

Research Question: What are the relationships among BMI, weight, and height?

Your tasks:

  1. Create a correlation matrix for: Weight, Height, BMI (3 points)
  2. Visualize the matrix using corrplot (3 points)
  3. Identify which pair has the strongest correlation (2 points)
  4. Explain why that correlation makes sense biologically/mathematically (2 points)
# a. Correlation matrix
# Select cardiovascular variables
BMI_vars <- nhanes_adult %>%
  select(Weight, Height, BMI)

# Calculate correlation matrix
BMI_matrix <- cor(BMI_vars, use = "complete.obs")

# Display as table
BMI_matrix %>%
  kable(digits = 3, caption = "BMI Correlation Matrix")
BMI Correlation Matrix
Weight Height BMI
Weight 1.000 0.451 0.880
Height 0.451 1.000 -0.012
BMI 0.880 -0.012 1.000 
# b. Visualize with corrplot

# Create correlation plot
corrplot(BMI_matrix, 
         method = "circle",
         type = "lower",
         tl.col = "black",
         tl.srt = 45,
         addCoef.col = "black",
         number.cex = 0.7,
         col = colorRampPalette(c("#3498db", "white", "#e74c3c"))(200),
         title = "BMI Correlations",
         mar = c(0,0,2,0))

# c. Strongest correlation:
# Weight and BMI 

# d. Explanation (write as comment)
# Weight and BMI have a correlation of r = 0.88, which is a strongly positive correlation. The r-squared value of 0.774 suggests that weight accounts for 77.4% of the variation for BMI. This makes sense as BMI is a calculation of weight and height.

Problem 3: Sleep and Age (5 points)

Research Question: Is there a relationship between hours of sleep and age?

Your tasks:

  1. Create a scatterplot (1 point)
  2. Calculate Pearson correlation and display with tidy() (2 points)
  3. Interpret whether the relationship is statistically significant (2 points)
# a. Scatterplot
ggplot(nhanes_adult, aes(x = Age, y = SleepHrsNight)) +
  geom_point(alpha = 0.3, color = "steelblue") +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  labs(
    title = "Age vs Sleep",
    subtitle = "NHANES Data, Adults 18-80 years",
    x = "Age (years)",
    y = "Sleep (hours per night)"
  ) +
  theme_minimal()

# b. Correlation with tidy()
# Calculate Pearson correlation
cor_age_sleep <- cor.test(nhanes_adult$Age, nhanes_adult$SleepHrsNight)

# Display results in clean table
tidy(cor_age_sleep) %>%
  select(estimate, statistic, p.value, conf.low, conf.high) %>%
  kable(
    digits = 3,
    col.names = c("r", "t-statistic", "p-value", "95% CI Lower", "95% CI Upper"),
    caption = "Pearson Correlation: Age and Sleep"
  )
Pearson Correlation: Age and Sleep
r t-statistic p-value 95% CI Lower 95% CI Upper
0.023 1.904 0.057 -0.001 0.046 
# Calculate r-squared
r_squared_age_sleep <- cor_age_sleep$estimate^2

data.frame(
  Measure = c("r²", "Variance Explained"),
  Value = c(
    round(r_squared_age_sleep, 4),
    paste0(round(r_squared_age_sleep * 100, 2), "%")
  )
) %>%
  kable(caption = "Effect Size")
Effect Size
Measure Value
cor r² 5e-04
Variance Explained 0.05% 
# c. Interpretation (write as comment)
# There is not a statistical significance between age and sleep (p>0.05). While there is a weak positive correlation (r=0.023), age only accounts for 0.05% of the variation in sleep for the adults in this study.