# 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, TotChol) %>%
  na.omit()

# Display sample
# Display sample size
data.frame(
  Metric = "Sample Size",
  Value = paste(nrow(nhanes_adult), "adults")
) %>%
  kable()

Metric	Value
Sample Size	6778 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	TotChol
34	87.4	164.7	32.2	113	85	70	No	4	3.5
34	87.4	164.7	32.2	113	85	70	No	4	3.5
34	87.4	164.7	32.2	113	85	70	No	4	3.5
49	86.7	168.4	30.6	112	75	86	No	8	6.7
45	75.7	166.7	27.2	118	64	62	Yes	8	5.8
45	75.7	166.7	27.2	118	64	62	Yes	8	5.8
45	75.7	166.7	27.2	118	64	62	Yes	8	5.8
66	68.0	169.5	23.7	111	63	60	Yes	7	5.0

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 = "Height vs Weight",
    subtitle = "NHANES Data, Adults 18-80 years",
    x = "Height (cm)",
    y = "Weight (kg)"
  ) +
  theme_minimal()

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

#Results in clean table
tidy(cor_ht_wt) %>%
  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
0.453	41.836	0	0.434	0.472

# c. Statistical significance
 
##**r = 0.451 moderate positive correlation
##**p < 0.001 statistically significant
##**95% CI [0.432, 0.469] significant


# d. r² and interpretation (write as comment)
r_squared <- cor_ht_wt$estimate^2

data.frame(
  Measure = c("Correlation (r)", "Coefficient of Determination (r²)", 
              "Variance Explained"),
  Value = c(
    round(cor_ht_wt$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)	0.453
Coefficient of Determination (r²)	0.205
Variance Explained	20.5%

Interpretation There is a statistically significant moderate positive correlation between height and weight. As height increases, weight usually increases as well. Height only explains 20.3% of variation in weight, indicating other factors play a role. —

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
# Selecting variables
cardio_vars <- nhanes_adult %>%
  select (BMI, Weight, Height)

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

cor_matrix %>%
  kable(digits = 3, caption = "BMI, Height, & Weight Correlation Matrix")

BMI, Height, & Weight Correlation Matrix
	BMI	Weight	Height
BMI	1.000	0.881	-0.009
Weight	0.881	1.000	0.453
Height	-0.009	0.453	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 = "BMI, Height, & Weight Correlations",
         mar = c(0,0,2,0))

# c. Strongest correlation:
data.frame(
  Relationship = c(
    "Weight & BMI",
    "Height & Weight"
  ),
  Correlation = c(
    round(cor_matrix["Weight", "BMI"], 3),
    round(cor_matrix["Height", "Weight"], 3)
  ),
  Strength = c("Strong", "Moderate")
) %>%
  kable(caption = "Notable Correlations Summary")

Notable Correlations Summary
Relationship	Correlation	Strength
Weight & BMI	0.881	Strong
Height & Weight	0.453	Moderate

# d. Explanation (write as comment)
# Weight and BMI show the strongest correlation where r=0.880, which reflects the biological connection between them. Height and weight are not as strongly correlated, but still a moderate positive trend, showing that other factors influence weight, rather than height alone.

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 = "darkgreen") +
  geom_smooth(method = "lm", se = TRUE, color = "red", fill = "pink") +
  labs(
    title = "Age vs Hours of Sleep",
    x = "Age (years)",
    y = "Hours of Sleep (hrs)"
  ) +
  theme_minimal()

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

# Display results
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"
  )

Pearson Correlation: Age and Hours of Sleep
r	t-statistic	p-value	95% CI Lower	95% CI Upper
0.024	1.989	0.047	0	0.048

r_squared_age <- cor_age_sleep$estimate^2

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

Effect Size
	Measure	Value
cor	r²	6e-04
	Variance Explained	0.06%

# c. Interpretation (write as comment)
# The correlation between age and hours of sleep was r= 0.023, indicating a weak positive correlation. The association is not statistically significant as the p value is 0.057. The 95% CI of [-0.001, 0.046] includes zero confirming the correlation is not significant. Age only explains 0.05% of variation in hours of sleep, illustrating that other factors play a more significant role in how many hours people sleep.

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
# Scatterplot investigating the relationship between age and total cholesterol
ggplot(nhanes_adult, aes(x = Age, y = TotChol)) +
  geom_point(alpha = 0.3, color = "darkgreen") +
  geom_smooth(method = "lm", se = TRUE, color = "red", fill = "pink") +
  labs(
    title = "Age vs Total Cholesterol",
    x = "Age (years)",
    y = "Total Cholesterol"
  ) +
  theme_minimal()

# Correlation with tidy()
cor_age_chol <- cor.test(nhanes_adult$Age, nhanes_adult$TotChol)

# Display results
tidy(cor_age_chol) %>%
  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 Total Cholesterol"
  )

Pearson Correlation: Age and Total Cholesterol
r	t-statistic	p-value	95% CI Lower	95% CI Upper
0.16	13.329	0	0.137	0.183

r_squared_age <- cor_age_chol$estimate^2

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

Effect Size
	Measure	Value
cor	r²	0.0255
	Variance Explained	2.55%

# Check assumptions
##**Assumption 1: Linearity** (already checked with scatterplot ✓)

##**Assumption 2: Bivariate Normality**

# Q-Q plots for normality
par(mfrow = c(1, 2))

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

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

par(mfrow = c(1, 1))

Assessment: Both age and total cholesterol are approximately normally distributed. Some deviation in the tails, but with large sample size, the correlation test is robust to minor violations.

Assumption 3: No Extreme Outliers (scatterplot shows no extreme outliers ✓)

Interpretation The correlation between age and total cholesterol is r = 0.16, indicating a weak positive correlation. The p-value of <0.0001 shows statistical significance. The 95% CI, [0.137, 0.183], further confirms the statistical significance as it doesn’t include zero. Age explains only 2.55% of variation showing that other variables play a more substantial role to total cholesterol levels. ```

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

Set seed

set.seed(553) ```

Correlation_Lab_Sarah_Andres

Sarah Andres

2026-02-10

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

Set seed