knitr::opts_chunk$set(
  echo = TRUE,
  warning = FALSE,
  message = FALSE,
  fig.align = "center",
  fig.width = 8,
  fig.height = 6
)

# Load required packages
library(tidyverse)

## Warning: package 'tidyverse' was built under R version 4.5.2

## Warning: package 'ggplot2' was built under R version 4.5.2

## Warning: package 'tibble' was built under R version 4.5.2

## Warning: package 'tidyr' was built under R version 4.5.2

## Warning: package 'readr' was built under R version 4.5.2

## Warning: package 'purrr' was built under R version 4.5.2

## Warning: package 'dplyr' was built under R version 4.5.2

## Warning: package 'stringr' was built under R version 4.5.2

## Warning: package 'forcats' was built under R version 4.5.2

## Warning: package 'lubridate' was built under R version 4.5.2

## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr     1.1.4     ✔ readr     2.1.6
## ✔ forcats   1.0.1     ✔ stringr   1.6.0
## ✔ ggplot2   4.0.1     ✔ tibble    3.3.1
## ✔ lubridate 1.9.4     ✔ tidyr     1.3.2
## ✔ purrr     1.2.1     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

library(ggplot2)
library(knitr)

## Warning: package 'knitr' was built under R version 4.5.2

library(broom)      # For clean statistical output

## Warning: package 'broom' was built under R version 4.5.2

library(corrplot)

## Warning: package 'corrplot' was built under R version 4.5.2

## corrplot 0.95 loaded

library(NHANES)

## Warning: package 'NHANES' was built under R version 4.5.2

# Set seed
set.seed(553)

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:

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 = 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",
    y = "Weight (kg)",
    x = "Height (cm)"
  ) +
  theme_minimal()

# b. Correlation test with tidy() display
cor_wt_ht <- cor.test(nhanes_adult$Height, nhanes_adult$Weight)

tidy(cor_wt_ht) %>%
  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

#Hypothesis Test

#H₀: ρ = 0 (no correlation between height and weight in population)
#H₁: ρ ≠ 0 (correlation exists)
#α = 0.05

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

#Conclusion: Reject H₀, there is a positive correlation between height and weight in the population.


# d. r² and interpretation (write as comment)

r_squared_wtht <- cor_wt_ht$estimate^2

data.frame(
  Measure = c("r²", "Variance Explained"),
  Value = c(
    round(r_squared_wtht, 4),
    paste0(round(r_squared_wtht * 100, 2), "%")
  )
) %>%
  kable(caption = "Effect Size")

Effect Size
	Measure	Value
cor	r²	0.203
	Variance Explained	20.3%

#r² = 0.203, meaning height explains 20.3% of the variance in weight. This indicates that height is an important contributor to weight, but there are likely other important factors that also contribute to weight.

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)

# YOUR CODE HERE

# a. Correlation matrix
body_vars <- nhanes_adult %>%
  select(Weight, Height, BMI)

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

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

Weight, Height, and 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
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 = "Weight, Height, BMI Correlations",
         mar = c(0,0,2,0))

# c. Strongest correlation:

#Weight and BMI have the strongest correlation with r = 0.880.

# d. Explanation (write as comment)

#This correlation makes sense because BMI is calculated as BMI = weight (kg) / height (m) ^2. Thus, as weight increases, BMI also increases. 
#

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 = "steelblue") +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  labs(
    title = "Age Vs. Hours of Sleep",
    subtitle = "NHANES Data, Adults 18-80 years",
    y = "Sleep per night (hours)",
    x = "Age (years)"
  ) +
  theme_minimal()

# b. Correlation with tidy()

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

tidy(cor_age_slp) %>%
  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"
  )

Pearson Correlation: Age and Hours of Sleep
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)

# This relationship is not statistically significant as p > 0.05. Thus, we do not have evidence to conclude that age and hours of sleep are correlated.

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

ggplot(nhanes_adult, aes(x = Age, y = Pulse)) +
  geom_point(alpha = 0.3, color = "steelblue") +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  labs(
    title = "Age Vs. Pulse (BPM)",
    subtitle = "NHANES Data, Adults 18-80 years",
    y = "Pulse (BPM)",
    x = "Age (years)"
  ) +
  theme_minimal()

# b. Correlation with tidy()

cor_age_pul <- cor.test(nhanes_adult$Age, nhanes_adult$Pulse)

tidy(cor_age_pul) %>%
  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 Pulse"
  )

Pearson Correlation: Age and Pulse
r	t-statistic	p-value	95% CI Lower	95% CI Upper
-0.153	-13.116	0	-0.176	-0.131

#r-squared

r_squared_age_pul <- cor_age_pul$estimate^2

data.frame(
  Measure = c("r²", "Variance Explained"),
  Value = c(
    round(r_squared_age_pul, 4),
    paste0(round(r_squared_age_pul * 100, 2), "%")
  )
) %>%
  kable(caption = "Effect Size")

Effect Size
	Measure	Value
cor	r²	0.0236
	Variance Explained	2.36%

#Assumption tests

par(mfrow = c(1, 2))

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

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

par(mfrow = c(1, 1))

#Assumption 1: Linearity

#The assumption of linearity has been met, as indicated by the scatterplot. 

#Assumption 2: Normality

#Both Age and Pulse are approximately normally distributed; the points in the Q-Q plots tend to follow the red line. Although points on the tails deviate from the normality assumption, the sample size is large (n = 7133), so the test should still be valid.

#Assumption 3: No extreme outliers 

#There are a few points that could be considered extreme outliers in the scatterplot, but there are not many and given the large sample size, the effect of these should be minimal. 

#Conclusion

#Since p<0.05, we reject our null hypothesis and conclude there is a negative correlation between Age and Pulse. That is, as age increases, pulse decreases. This is confirmed by the 95% CI [-0.176, -0.131], which does not intersect with 0. r² = 0.0236, indicating only 2.36% of the variance in pulse is explained by age. This suggests that despite the statistically significant correlation, age is not a major contributor to pulse and that other variables are likely more important in affecting pulse.

Grading Rubric

Problem 1: Weight and Height (10 points)

1. Scatterplot properly formatted with labels: 2 points
1. Correct correlation with clean display: 3 points
1. Significance test correctly interpreted: 2 points
1. r² calculated and interpreted: 3 points

Problem 2: Correlation Matrix (10 points)

1. Correct matrix calculated: 3 points
1. Well-formatted correlation plot: 3 points
1. Strongest correlation identified: 2 points
1. Biological/mathematical explanation: 2 points

Problem 3: Sleep and Age (5 points)

1. Scatterplot: 1 point
1. Correlation calculated and displayed: 2 points
1. Interpretation of significance: 2 points

Submission Instructions

Save your work with your name: Correlation_Lab_YourName.Rmd
Knit to HTML to create your report
Publish to RPubs:
- Click the Publish button (blue icon) in the HTML preview window
- Choose RPubs from the options
- Follow the prompts to publish (create account if needed)
- Copy your RPubs URL
Submit to Brightspace:
- Upload your .Rmd file
- Paste your RPubs link in the assignment comments or submission text box
Due: End of class today

Grading: This lab is worth 15% of your in-class lab grade. The lowest 2 lab grades are dropped.

Additional Resources

R Functions Used Today

cor.test() - Calculate correlation and test significance
tidy() - Clean display of statistical test results
cor() - Calculate correlation matrix
corrplot() - Visualize correlation matrix
ggplot() + geom_point() - Scatterplots
geom_smooth(method="lm") - Add fitted regression line
qqnorm() / qqline() - Check normality

For More Help

Textbook: Chapter 6 - Correlation Analysis
Office Hours: See syllabus
TA Help: See syllabus
R Documentation: Type ?cor.test in console

Remember:

✓ Correlation measures LINEAR relationships only
✓ Always visualize your data first
✓ Correlation ≠ Causation
✓ Check your assumptions
✓ Consider confounding and alternative explanations

This lab activity was created for EPI 553: Principles of Statistical Inference II
University at Albany, College of Integrated Health Sciences
Spring 2026

Lab 03: In-Class Lab Activity: Correlation Analysis

EPI 553: Principles of Statistical Inference II

Josh Macera

2026-02-10

PART A: GUIDED EXAMPLES

Setup: Load Packages and Data

PART B: YOUR TURN - Practice Problems

Problem 1: Weight and Height (10 points)

Problem 2: Correlation Matrix Analysis (10 points)

Problem 3: Sleep and Age (5 points)

Bonus (Optional, 5 extra points)

Grading Rubric

Problem 1: Weight and Height (10 points)

Problem 2: Correlation Matrix (10 points)

Problem 3: Sleep and Age (5 points)

Submission Instructions

Additional Resources

R Functions Used Today

For More Help