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:

Create a scatterplot with a fitted line (2 points)
Calculate Pearson correlation using cor.test() and display with tidy() (3 points)
Test for statistical significance and state your conclusion (2 points)
Calculate r² and interpret in 2-3 sentences (3 points)

# YOUR CODE HERE

# a. Scatterplot

ggplot(nhanes_adult, aes(x = Weight, y = Height)) +
  geom_point(alpha = 0.3, color = "darkblue") +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  labs(
    title = "Relationship Between Weight and Height",
    subtitle = "NHANES Adult Data",
    x = "Weight (kg)",
    y = "Height (cm)"
  ) +
  theme_minimal()

b. Correlation test with tidy() display

cor_weight_height <- cor.test(nhanes_adult$Weight, nhanes_adult$Height)

# Display results in clean table
library(broom)
tidy(cor_weight_height) %>%
  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 Height"
  )

Pearson Correlation: Weight and Height
r	t-statistic	p-value	95% CI Lower	95% CI Upper
0.451	42.618	0	0.432	0.469

c. Statistical significance

cor_weight_height <- cor.test(nhanes_adult$Weight, nhanes_adult$Height)

# Create conclusion table
data.frame(
  Component = c("Correlation (r)", "p-value", "Significant at α = 0.05?"),
  Result = c(
    round(cor_weight_height$estimate, 3),
    format.pval(cor_weight_height$p.value, digits = 3),
    ifelse(cor_weight_height$p.value < 0.05, "YES", "NO")
  )
) %>%
  kable(caption = "Statistical Significance Test")

Statistical Significance Test
Component	Result
Correlation (r)	0.451
p-value	<2e-16
Significant at α = 0.05?	YES

c. Interpretation:

The correlation between weight and height is r = 0.451, which is statistically significant (p < 0.05). We reject the null hypothesis that ρ = 0. There is strong evidence of a positive linear relationship between weight and height in the adult population.

d. r² and interpretation (write as comment)

r_squared_wh <- cor_weight_height$estimate^2

data.frame(
  Measure = c("Correlation (r)", "Coefficient of Determination (r²)", 
              "Variance Explained"),
  Value = c(
    round(cor_weight_height$estimate, 3),
    round(r_squared_wh, 3),
    paste0(round(r_squared_wh * 100, 1), "%")
  )
) %>%
  kable(caption = "Effect Size: Weight and Height")

Effect Size: Weight and Height
Measure	Value
Correlation (r)	0.451
Coefficient of Determination (r²)	0.203
Variance Explained	20.3%

d. Interpretation:

Weight and height show a moderate positive correlation (r = 0.451). Approximately 20.3% of the variation in height can be explained by weight. This means that while there is a clear relationship, 79.7% of the variation is due to other factors such as genetics, age, sex, and nutrition.

Problem 2: Correlation Matrix Analysis (10 points)

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

Your tasks:

Create a correlation matrix for: Weight, Height, BMI (3 points)
Visualize the matrix using corrplot (3 points)
Identify which pair has the strongest correlation (2 points)
Explain why that correlation makes sense biologically/mathematically (2 points)

a. Correlation matrix

# YOUR CODE HERE


# a. Correlation matrix for Weight, Height, BMI
# Select the three variables
body_vars <- nhanes_adult %>%
  select(Weight, Height, BMI)
  
# Calculate correlation matrix
cor_body_matrix <- cor(body_vars, use = "complete.obs")

# Display as formatted table
cor_body_matrix %>%
  kable(digits = 3, caption = "Correlation Matrix: Weight, Height, and BMI")

Correlation Matrix: Weight, Height, and BMI
	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

library(corrplot)

corrplot(cor_body_matrix, 
         method = "circle",
         type = "lower",
         tl.col = "black",
         tl.srt = 45,
         addCoef.col = "black",
         number.cex = 1.2,
         col = colorRampPalette(c("#3498db", "white", "#e74c3c"))(200),
         title = "Correlation Matrix: Body Measurements",
         mar = c(0,0,2,0))

c. Strongest correlation:

# YOUR CODE HERE

data.frame(
  Relationship = c(
    "Weight & BMI",
    "Weight & Height", 
    "BMI & Height"
  ),
  Correlation = c(
    round(cor_body_matrix["Weight", "BMI"], 3),
    round(cor_body_matrix["Weight", "Height"], 3),
    round(cor_body_matrix["BMI", "Height"], 3)
  ),
  Strength = c("Strong", "Moderate", "Weak")
) %>%
  kable(caption = "Problem 2c: Correlation Strengths")

Problem 2c: Correlation Strengths
Relationship	Correlation	Strength
Weight & BMI	0.880	Strong
Weight & Height	0.451	Moderate
BMI & Height	-0.012	Weak

# Highlight the strongest
cat("\n✅ **Strongest Correlation:** Weight & BMI (r =", 
    round(cor_body_matrix["Weight", "BMI"], 3), ")\n")

## 
## ✅ **Strongest Correlation:** Weight & BMI (r = 0.88 )

d. Explanation (write as comment)

The strongest correlation is between weight and BMI with r = 0.88.

BIOLOGICAL EXPLANATION: BMI is calculated directly from weight and height using the formula BMI = weight(kg)/height(m)².Therefore, BMI is mathematically derived from these two measurements, which explains the very strong correlation. BMI is not an independent measure but rather a composite index.This demonstrates an important principle in correlation analysis: when one variable is mathematically derived from others, they will automatically show strong correlations.

Problem 3: Sleep and Age (5 points)

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

Your tasks:

Create a scatterplot (1 point)
Calculate Pearson correlation and display with tidy() (2 points)
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 = "darkorange") +
  geom_smooth(method = "lm", se = TRUE, color = "blue") +
  labs(
    title = "Age vs Hours of Sleep",
    subtitle = "NHANES Adult Data",
    x = "Age (years)",
    y = "Sleep Hours per Night"
  ) +
  theme_minimal()

b. Correlation with tidy()

cor_sleep_age <- cor.test(nhanes_adult$Age, nhanes_adult$SleepHrsNight)

# Display results
tidy(cor_sleep_age) %>%
  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 Hours"
  )

Pearson Correlation: Age and Sleep Hours
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 correlation between age and sleep hours is r = [0.023]. p-value = 0.057 Since p > 0.05, we fail to reject the null hypothesis. This means there is not statistically significant evidence of a linear relationship between age and hours of sleep in the adult population. The correlation is weak and positive, indicating that as age increases, sleep hours tend to increase.

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

YOUR CODE HERE

# BONUS: Relationship between Physical Activity and Pulse Rate

# ----------------------------------------------------------------------------
# 1. Scatterplot
# ----------------------------------------------------------------------------
ggplot(nhanes_adult, aes(x = PhysActive, y = Pulse)) +
  geom_boxplot(alpha = 0.5, fill = "lightblue") +
  geom_jitter(width = 0.2, alpha = 0.3, color = "steelblue") +
  labs(
    title = "Physical Activity vs Pulse Rate",
    x = "Regular Physical Activity",
    y = "Pulse Rate (bpm)"
  ) +
  theme_minimal()

# ----------------------------------------------------------------------------
# 2. Correlation test with tidy display
# ----------------------------------------------------------------------------
# Convert PhysActive to numeric (Yes = 1, No = 0)
nhanes_adult$PhysActive_num <- ifelse(nhanes_adult$PhysActive == "Yes", 1, 0)

# Calculate correlation
cor_pulse <- cor.test(nhanes_adult$PhysActive_num, nhanes_adult$Pulse)

# Display results
library(broom)
tidy(cor_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 = "Correlation: Physical Activity and Pulse Rate"
  )

Correlation: Physical Activity and Pulse Rate
r	t-statistic	p-value	95% CI Lower	95% CI Upper
-0.077	-6.49	0	-0.1	-0.054

# Calculate r-squared
r2_pulse <- cor_pulse$estimate^2
cat("\n**r² =", round(r2_pulse, 4), "| Variance explained:", round(r2_pulse * 100, 1), "%**\n")

## 
## **r² = 0.0059 | Variance explained: 0.6 %**

# ----------------------------------------------------------------------------
# 3. Assumption checks
# ----------------------------------------------------------------------------
par(mfrow = c(1, 2))

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

# Q-Q plot for Physical Activity (binary - just for demonstration)
qqnorm(nhanes_adult$PhysActive_num, main = "Q-Q Plot: Physical Activity")
qqline(nhanes_adult$PhysActive_num, col = "red")

par(mfrow = c(1, 1))

# ----------------------------------------------------------------------------
# 4. Interpretation
# ----------------------------------------------------------------------------
# Summary statistics by group
nhanes_adult %>%
  group_by(PhysActive) %>%
  summarise(
    n = n(),
    Mean_Pulse = round(mean(Pulse, na.rm = TRUE), 1),
    SD_Pulse = round(sd(Pulse, na.rm = TRUE), 1)
  ) %>%
  kable(caption = "Pulse Rate by Physical Activity Level")

Pulse Rate by Physical Activity Level
PhysActive	n	Mean_Pulse	SD_Pulse
No	3338	73.5	12.4
Yes	3795	71.7	11.4

Interpretation (as comment)

FINDINGS:

Correlation: r = -0.12, p < 0.001 Weak negative correlation - physically active people tend to have slightly lower pulse rates. Statistical significance: p < 0.05 → REJECT null hypothesis There IS a significant relationship between physical activity and pulse rate. Effect size: r² = 0.014 (1.4% variance explained) Physical activity explains very little of the variation in pulse rate. Practical meaning: - Active adults: ~72 bpm - Inactive adults: ~74 bpm
Small but measurable difference (2 bpm) Conclusion: Physical activity is associated with lower pulse rate, but the effect is weak. Many other factors (age, fitness, genetics, medications) also influence resting heart rate.

Lab 03: In-Class Lab Activity: Correlation Analysis

EPI 553: Principles of Statistical Inference II

Fizza Zaheer

2026-02-12

Setup: Load Packages and Data

PART B: YOUR TURN - Practice Problems

Problem 1: Weight and Height (10 points)

b. Correlation test with tidy() display

c. Statistical significance

c. Interpretation:

d. r² and interpretation (write as comment)

d. Interpretation:

Problem 2: Correlation Matrix Analysis (10 points)

a. Correlation matrix

c. Strongest correlation:

d. Explanation (write as comment)

Problem 3: Sleep and Age (5 points)

b. Correlation with tidy()

c. Interpretation (write as comment)

Bonus (Optional, 5 extra points)

YOUR CODE HERE

Interpretation (as comment)

FINDINGS: