# 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

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)
# YOUR CODE HERE

# 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 = "Weight vs. Height",
    subtitle = "NHANES Data, Adults 18-80 years",
    x = "Height (cm)",
    y = "Weight (kg)"
  ) +
  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 
# c. Statistical significance
#The Pearson Correlation between height and weight is statistically significant, because the p-value is 0 and the 95% confidence interval [0.432,0.469] does not include 0. The r value is 0.451, meaning height and weight are moderately positively correlated. As height increases, weight also tends to increase. 

# d. r² and interpretation (write as comment)

# Calculate r-squared
r_squared <- cor_height_weight$estimate^2

data.frame(
  Measure = c("Correlation (r)", "Coefficient of Determination (r²)", 
              "Variance Explained"),
  Value = c(
    round(cor_height_weight$estimate, 3),
    round(r_squared, 3),
    paste0(round(r_squared * 100, 1), "%")
  )
) %>%
  kable(caption = "Summary of Correlation Strength")
Summary of Correlation Strength
Measure Value
Correlation (r) 0.451
Coefficient of Determination (r²) 0.203
Variance Explained 20.3% 
#R squared is 0.203, meaning that height only  explains about 20.3% of the variation in weight, suggesting other factors also play important roles.

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)
# YOUR CODE HERE

# a. Correlation matrix
# Select cardiovascular variables
bmi_vars <- nhanes_adult %>%
  select(Weight, Height, BMI)

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

# Display as table
cor_matrix %>%
  kable(digits = 3, caption = "Antropometric Health Correlation Matrix")
Antropometric Health 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(cor_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 = "Antropometric Health Correlations",
         mar = c(0,0,2,0))

# c. Strongest correlation:
data.frame(
  Relationship = c(
    "Weight & BMI",
    "Height & Weight",
    "Height & BMI"
  ),
  Correlation = c(
    round(cor_matrix["Weight", "BMI"], 3),
    round(cor_matrix["Height", "Weight"], 3),
    round(cor_matrix["Height", "BMI"], 3)
    
  ),
  Strength = c("Strong","Moderate","Weak")
) %>%
  kable(caption = "Notable Correlations Summary")
Notable Correlations Summary
Relationship Correlation Strength
Weight & BMI 0.880 Strong
Height & Weight 0.451 Moderate
Height & BMI -0.012 Weak 
# d. Explanation (write as comment)
#  Weight and BMI have the strongest positive correlation at r = 0.880, height and weight have a moderate positive correlation at r = 0.451, and height and BMI have a weak negative correlation at r = -0.012. The strong correlation between weight and BMI makes sense, as weight plays a large role in a person's BMI. Height has a weak correlation with BMI, suggesting it is impacted by other factors.

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)
# YOUR CODE HERE

# 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 = "Hours of Sleep at Night vs. Age",
    subtitle = "NHANES Data, Adults 18-80 years",
    x = "Age (yrs)",
    y = "Hours of Sleep at 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 Hours of Sleep at Night"
  )
Pearson Correlation: Age and Hours of Sleep at Night
r t-statistic p-value 95% CI Lower 95% CI Upper
0.023 1.904 0.057 -0.001 0.046 
# c. Interpretation (write as comment)
# The Pearson Correlation between Age and Hours slept at night is not statistically signficant, because the p-value is 0.057 (>0.05) and the 95% confidence interval [-0.001,0.046] includes 0. The r value is 0.023, meaning age and hours slept at night are very weakly positively correlated, if correlated at all.

Bonus (Optional, 5 extra points)

Challenge: Investigate the relationship between two variables of your choice from the NHANES dataset. Include:

  • Scatterplot
  • Correlation test with clean display
  • Assumption checks
  • Thoughtful interpretation Research Question: Is there a relationship between weight and pulse rate?
# YOUR CODE HERE
# Create scatterplot
ggplot(nhanes_adult, aes(x = Weight, y = Pulse)) +
  geom_point(alpha = 0.3, color = "steelblue") +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  labs(
    title = "Weight vs Pulse",
    subtitle = "NHANES Data, Adults 18-80 years",
    x = "Weight (kg)",
    y = "Pulse Rate (measured in 60s)"
  ) +
  theme_minimal()

# Calculate Pearson correlation
cor_weight_pulse <- cor.test(nhanes_adult$Weight, nhanes_adult$Pulse)

# Display results in clean table
tidy(cor_weight_pulse) %>%
  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: Weight and Pulse"
  )
Pearson Correlation: Weight and Pulse
r t-statistic p-value 95% CI Lower 95% CI Upper
0.063 5.295 0 0.039 0.086 
# Calculate r-squared
r_squared <- cor_weight_pulse$estimate^2

data.frame(
  Measure = c("Correlation (r)", "Coefficient of Determination (r²)", 
              "Variance Explained"),
  Value = c(
    round(cor_weight_pulse$estimate, 3),
    round(r_squared, 3),
    paste0(round(r_squared * 100, 1), "%")
  )
) %>%
  kable(caption = "Summary of Correlation Strength")
Summary of Correlation Strength
Measure Value
Correlation (r) 0.063
Coefficient of Determination (r²) 0.004
Variance Explained 0.4% 
# Q-Q plots for normality
par(mfrow = c(1, 2))

qqnorm(nhanes_adult$Weight, main = "Q-Q Plot: Weight")
qqline(nhanes_adult$Weight, col = "red")

qqnorm(nhanes_adult$Pulse, main = "Q-Q Plot: Pulse")
qqline(nhanes_adult$Pulse, col = "red")

par(mfrow = c(1, 1))

#Interpretation:

#There is a statistically significant weak positive correlation between weight and pulse (r = 0.063, 95% confidence interval [0.039,0.086]). As weight increases, pulse rate tends to increase. However, weight explains only about 0.4% of the variation in pulse, suggesting other factors also play important roles. The assumptions are all met, as the scatter plot shows linearity and the Q-Q plot shows normality, as the points follow the red line.

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