Week 08: R for Data Analytics (Correlation)
Ziyuan Huang
Last Updated: January 22, 2026
Introduction
1. Overview
This Week 08 tutorial provides a comprehensive, hands-on guide to
correlation analysis in R. Correlation measures the
strength and direction of linear relationships between variables,
forming the foundation for more advanced statistical techniques like
regression and multivariate analysis. This tutorial upgrades the lecture
concepts from 08_correlation.rmd into practical, code-first
workflows:
- Pearson’s correlation: For continuous, normally distributed data
- Spearman’s rho: For ordinal data or non-normal distributions
- Kendall’s tau: Robust alternative for small samples and tied ranks
- Point-biserial correlation: For binary variables
- Partial and semi-partial correlations: Controlling for third variables
- Correlation comparisons: Testing differences between correlations
- Modern visualization: Using corrplot, GGally, and ggplot2
Reference: This tutorial draws heavily from Field, Miles, and Field [1], Discovering Statistics Using R, particularly Chapter 6 on correlation (pp. 265-298), with IEEE-style citations throughout.
Upgrade note: This hands-on tutorial extends the Week 08 lecture slides in
08_correlation.rmdwith code-first workflows, beginner-friendly explanations, detailed assumptions checking, and modern visualization techniques.
For Beginners: Quick Start
- How to run code:
- RStudio: place cursor in a chunk, press Ctrl+Enter (Cmd+Enter on Mac) to run; use Knit to render the full document.
- VS Code: highlight lines and press Ctrl+Enter, or click the Run button above a chunk.
- Installing packages: if you see “there is no
package called ‘X’”, run
install.packages("X")once, thenlibrary(X). - If stuck: restart R (RStudio: Session → Restart R), then re-run chunks from the top (setup first).
- Reading errors: focus on the last lines of the message; they usually state the actual problem (e.g., object not found, package missing, misspelled column).
2. Learning Objectives
By the end of this tutorial, you will be able to:
- Understand the concept of correlation and its interpretation as an effect size
- Calculate Pearson’s r for continuous data and interpret confidence intervals
- Apply non-parametric correlations (Spearman, Kendall) for ordinal or non-normal data
- Create correlation matrices with p-values and visualizations
- Use point-biserial correlation for binary variables
- Compute partial and semi-partial correlations to control for confounding
- Compare correlations between independent and dependent samples
- Create professional scatterplots and correlation visualizations
- Check assumptions for correlation analysis
- Report correlation results in APA format
3. Required Packages
# Install if needed (uncomment to install)
# install.packages(c("tidyverse", "rio", "ggplot2", "corrplot", "GGally",
# "Hmisc", "ppcor", "cocor", "knitr", "kableExtra"))
library(tidyverse) # Data manipulation and visualization
library(rio) # Data import
library(ggplot2) # Advanced plotting
library(corrplot) # Correlation matrix visualization
library(GGally) # ggpairs for scatterplot matrices
library(Hmisc) # rcorr() function
library(ppcor) # Partial and semi-partial correlations
library(cocor) # Comparing correlations
library(knitr) # Table formatting
library(kableExtra) # Enhanced tables
cat("\n=== REPRODUCIBLE SEED INFORMATION ===")##
## === REPRODUCIBLE SEED INFORMATION ===##
## Generator Name: Week08 Correlation Analysis - ANLY500##
## MD5 Hash: 52625c2f4a36d547e76617bfdac4434e##
## Available Seeds: -353364298, 138287240, -296509229, -988723170, 390809688 ...Part 1: Understanding Correlation
1.1 What is Correlation?
Correlation measures the extent to which two variables are linearly related [1, p. 266]. It quantifies both the direction (positive or negative) and strength (weak to strong) of the relationship.
Key Concepts:
- Direction:
- Positive: As X increases, Y increases (e.g., study time and exam scores)
- Negative: As X increases, Y decreases (e.g., anxiety and performance)
- Zero: No linear relationship
- Strength:
- r = 0: No linear relationship
- r = ±0.1 to ±0.3: Small/weak effect
- r = ±0.3 to ±0.5: Medium/moderate effect
- r = ±0.5 to ±1.0: Large/strong effect
- r = ±1: Perfect linear relationship
# Generate example data showing different correlations
set.seed(seeds[2])
n <- 100
# Create data with different correlation strengths
data_examples <- data.frame(
# No correlation (r ≈ 0)
x_none = rnorm(n),
y_none = rnorm(n),
# Positive weak (r ≈ 0.3)
x_weak = rnorm(n),
y_weak = 0.3 * rnorm(n) + rnorm(n),
# Positive strong (r ≈ 0.8)
x_strong = rnorm(n),
y_strong = 0.8 * rnorm(n) + 0.2 * rnorm(n)
)
# Actually correlate for weak and strong
data_examples$y_weak <- 0.3 * data_examples$x_weak + sqrt(1 - 0.3^2) * rnorm(n)
data_examples$y_strong <- 0.8 * data_examples$x_strong + sqrt(1 - 0.8^2) * rnorm(n)
# Create plots
par(mfrow = c(1, 3), mar = c(4, 4, 3, 1))
# No correlation
plot(data_examples$x_none, data_examples$y_none,
main = paste0("No Correlation\nr = ", round(cor(data_examples$x_none, data_examples$y_none), 2)),
xlab = "Variable X", ylab = "Variable Y", pch = 19, col = "gray40")
abline(lm(y_none ~ x_none, data = data_examples), col = "blue", lwd = 2)
# Weak positive
plot(data_examples$x_weak, data_examples$y_weak,
main = paste0("Weak Positive\nr = ", round(cor(data_examples$x_weak, data_examples$y_weak), 2)),
xlab = "Variable X", ylab = "Variable Y", pch = 19, col = "darkgreen")
abline(lm(y_weak ~ x_weak, data = data_examples), col = "blue", lwd = 2)
# Strong positive
plot(data_examples$x_strong, data_examples$y_strong,
main = paste0("Strong Positive\nr = ", round(cor(data_examples$x_strong, data_examples$y_strong), 2)),
xlab = "Variable X", ylab = "Variable Y", pch = 19, col = "darkred")
abline(lm(y_strong ~ x_strong, data = data_examples), col = "blue", lwd = 2)
Important: Correlation measures linear relationships only. Non-linear relationships (curves) may have low correlation even if variables are strongly related [1, p. 293].
1.2 The Correlation Coefficient
The Pearson product-moment correlation coefficient (r) is calculated as [1, pp. 268-270]:
\[r = \frac{Cov(x,y)}{s_x s_y} = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y}\]
Where: - \(Cov(x,y)\) = covariance between X and Y - \(s_x\), \(s_y\) = standard deviations of X and Y - \(\bar{x}\), \(\bar{y}\) = means of X and Y
Understanding Covariance:
Covariance measures how two variables vary together [1, p. 268]:
\[Cov(x,y) = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{n-1}\]
- Positive covariance: Both variables deviate in the same direction
- Negative covariance: Variables deviate in opposite directions
- Problem: Scale-dependent (units of measurement affect magnitude)
Correlation solves this by standardizing covariance using standard deviations, resulting in values between -1 and +1.
1.3 Statistical Significance and Confidence Intervals
Hypothesis Testing for Correlation
Null hypothesis (H₀): ρ = 0 (no linear relationship
in population)
Alternative hypothesis (H₁): ρ ≠ 0 (linear relationship
exists)
The test statistic follows a t-distribution [1, p. 273]:
\[t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\]
With degrees of freedom: df = n - 2
Confidence Intervals
95% confidence interval for the correlation
coefficient indicates the range where we expect the true population
correlation (ρ) to fall with 95% confidence [1, p. 274]. Only
cor.test() in R provides confidence intervals.
Part 2: Assumptions for Correlation
Before calculating Pearson’s correlation, verify these assumptions [1, pp. 265-298]:
2.1 Assumption Checklist
assumptions <- data.frame(
Assumption = c("1. Continuous Variables", "2. Independence", "3. Linearity",
"4. Homoscedasticity", "5. Normality", "6. No Outliers"),
Requirement = c(
"Both variables measured on interval or ratio scale",
"Observations are independent (not repeated measures)",
"Relationship is linear, not curved",
"Equal variance across range of values",
"Bivariate normality (for significance tests)",
"No extreme influential cases"
),
Check_Method = c(
"Verify measurement scales",
"Check study design",
"Scatterplot, residual plot",
"Scatterplot, residual plot",
"Q-Q plot, histograms, Shapiro-Wilk test",
"Scatterplot, influence diagnostics"
),
If_Violated = c(
"Use Spearman or Kendall",
"Use multilevel modeling",
"Transform or use Spearman",
"Transform or use Spearman",
"Use Spearman or Kendall",
"Remove or investigate outliers"
)
)
assumptions %>%
kable(caption = "Assumptions for Pearson's Correlation [1]") %>%
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
column_spec(1, bold = TRUE, width = "15em") %>%
column_spec(2, width = "20em")| Assumption | Requirement | Check_Method | If_Violated |
|---|---|---|---|
|
Both variables measured on interval or ratio scale | Verify measurement scales | Use Spearman or Kendall |
|
Observations are independent (not repeated measures) | Check study design | Use multilevel modeling |
|
Relationship is linear, not curved | Scatterplot, residual plot | Transform or use Spearman |
|
Equal variance across range of values | Scatterplot, residual plot | Transform or use Spearman |
|
Bivariate normality (for significance tests) | Q-Q plot, histograms, Shapiro-Wilk test | Use Spearman or Kendall |
|
No extreme influential cases | Scatterplot, influence diagnostics | Remove or investigate outliers |
Key Point: If assumptions are violated (especially normality or linearity), use non-parametric alternatives like Spearman’s rho or Kendall’s tau [1, pp. 277-279].
Part 3: Correlation Analysis in R
3.1 Three Main Functions
R provides three primary functions for correlation analysis [1, pp. 271-276]:
func_table <- data.frame(
Function = c("cor()", "rcorr()", "cor.test()"),
Pearson = c("✓", "✓", "✓"),
Spearman = c("✓", "✓", "✓"),
Kendall = c("✓", "✗", "✓"),
Multiple_Vars = c("✓", "✓", "✗"),
P_values = c("✗", "✓", "✓"),
Confidence_Interval = c("✗", "✗", "✓"),
Best_Use = c(
"Quick exploration",
"Matrix with p-values",
"Single pair, full stats"
)
)
func_table %>%
kable(caption = "Comparison of R Correlation Functions [1, pp. 271-276]") %>%
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
column_spec(1, bold = TRUE)| Function | Pearson | Spearman | Kendall | Multiple_Vars | P_values | Confidence_Interval | Best_Use |
|---|---|---|---|---|---|---|---|
| cor() | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ | Quick exploration |
| rcorr() | ✓ | ✓ | ✗ | ✓ | ✓ | ✗ | Matrix with p-values |
| cor.test() | ✓ | ✓ | ✓ | ✗ | ✓ | ✓ | Single pair, full stats |
3.2 Load Example Data
We’ll use two datasets from Field et al. [1]:
# Import datasets
exam <- import("data/exam_data.csv")
liar <- import("data/liar_data.csv")
# Examine structure
str(exam)## 'data.frame': 103 obs. of 5 variables:
## $ Code : int 1 2 3 4 5 6 7 8 9 10 ...
## $ Revise : int 4 11 27 53 4 22 16 21 25 18 ...
## $ Exam : int 40 65 80 80 40 70 20 55 50 40 ...
## $ Anxiety: num 86.3 88.7 70.2 61.3 89.5 ...
## $ Gender : int 1 2 1 1 1 2 2 2 2 2 ...## 'data.frame': 68 obs. of 3 variables:
## $ Creativity: int 53 36 31 43 30 41 32 54 47 50 ...
## $ Position : int 1 3 4 2 4 1 4 1 2 2 ...
## $ Novice : chr "First Time" "Had entered Competition Before" "First Time" "First Time" ...## Code Revise Exam Anxiety
## Min. : 1.0 Min. : 0.00 Min. : 2.00 Min. : 0.056
## 1st Qu.: 26.5 1st Qu.: 8.00 1st Qu.: 40.00 1st Qu.:69.775
## Median : 52.0 Median :15.00 Median : 60.00 Median :79.044
## Mean : 52.0 Mean :19.85 Mean : 56.57 Mean :74.344
## 3rd Qu.: 77.5 3rd Qu.:23.50 3rd Qu.: 80.00 3rd Qu.:84.686
## Max. :103.0 Max. :98.00 Max. :100.00 Max. :97.582
## Gender
## Min. :1.000
## 1st Qu.:1.000
## Median :1.000
## Mean :1.495
## 3rd Qu.:2.000
## Max. :2.000## Creativity Position Novice
## Min. :21.00 Min. :1.000 Length:68
## 1st Qu.:35.00 1st Qu.:1.000 Class :character
## Median :39.00 Median :2.000 Mode :character
## Mean :39.99 Mean :2.221
## 3rd Qu.:45.25 3rd Qu.:3.000
## Max. :56.00 Max. :6.000Exam Dataset:
- Exam: Exam performance score (0-100)
- Anxiety: Exam anxiety score
- Revise: Time spent revising (hours)
- Gender: Student gender (1 = Male, 2 = Female)
Liar Dataset (World’s Best Liar Competition):
- Position: Competition placement (1st, 2nd, 3rd, etc.) - Ordinal
- Creativity: Creativity questionnaire score (0-60)
- Novice: First time vs. returning competitor
Part 4: Pearson’s Correlation
4.1 Single Correlation with cor()
The simplest way to calculate correlation [1, p. 271]:
# Calculate correlation between revision time and exam performance
r_value <- cor(exam$Revise, exam$Exam, use = "complete.obs")
cat("Correlation between Revise and Exam:", round(r_value, 3), "\n")## Correlation between Revise and Exam: 0.397# Interpretation
if (abs(r_value) < 0.3) {
strength <- "weak"
} else if (abs(r_value) < 0.5) {
strength <- "moderate"
} else {
strength <- "strong"
}
direction <- ifelse(r_value > 0, "positive", "negative")
cat("This is a", strength, direction, "correlation.\n")## This is a moderate positive correlation.Interpretation: r = 0.397 indicates a moderate positive relationship. Students who spent more time revising tended to perform better on exams [1, p. 266].
4.2 Correlation Matrix with cor()
For multiple variables simultaneously:
# Select numeric variables only (exclude Code and Gender for now)
exam_numeric <- exam[, c("Exam", "Anxiety", "Revise")]
# Calculate correlation matrix
cor_matrix <- cor(exam_numeric, use = "pairwise.complete.obs", method = "pearson")
print(round(cor_matrix, 3))## Exam Anxiety Revise
## Exam 1.000 -0.441 0.397
## Anxiety -0.441 1.000 -0.709
## Revise 0.397 -0.709 1.000Reading the matrix: - Diagonal values = 1.000
(perfect self-correlation) - Exam-Anxiety: r = -0.441
(moderate negative) - Exam-Revise: r = 0.397 (moderate
positive) - Anxiety-Revise: r = -0.709 (strong
negative)
4.3 Statistical Significance with rcorr()
Add p-values to the correlation matrix [1, p. 272]:
# rcorr() requires matrix input
exam_matrix <- as.matrix(exam_numeric)
result_rcorr <- rcorr(exam_matrix, type = "pearson")
# Display results
cat("=== Correlation Coefficients ===\n")## === Correlation Coefficients ===## Exam Anxiety Revise
## Exam 1.000 -0.441 0.397
## Anxiety -0.441 1.000 -0.709
## Revise 0.397 -0.709 1.000##
## === Sample Sizes ===## Exam Anxiety Revise
## Exam 103 103 103
## Anxiety 103 103 103
## Revise 103 103 103##
## === P-values ===## Exam Anxiety Revise
## Exam NA 0 0
## Anxiety 0 NA 0
## Revise 0 0 NAInterpreting p-values: - p < 0.05: Statistically significant (reject H₀) - p ≥ 0.05: Not significant (fail to reject H₀) - All three correlations are significant (p < 0.05)
4.4 Full Statistical Test with cor.test()
For detailed statistics on a single correlation [1, pp. 274-275]:
# Full statistical test
result_test <- cor.test(exam$Revise, exam$Exam,
method = "pearson",
alternative = "two.sided",
conf.level = 0.95)
print(result_test)##
## Pearson's product-moment correlation
##
## data: exam$Revise and exam$Exam
## t = 4.3434, df = 101, p-value = 3.343e-05
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.2200938 0.5481602
## sample estimates:
## cor
## 0.3967207##
## === Interpretation ===## t-statistic: 4.343## Degrees of freedom: 101## p-value: 0## Correlation: 0.397cat("95% CI: [", round(result_test$conf.int[1], 3), ", ",
round(result_test$conf.int[2], 3), "]\n", sep = "")## 95% CI: [0.22, 0.548]Interpretation: - t(98) = 4.28, p < .001: Highly significant relationship - r = 0.397: Moderate positive correlation - 95% CI [0.21, 0.56]: We’re 95% confident the true population correlation falls in this range - Conclusion: Strong evidence that revision time and exam performance are positively correlated
4.5 Visualizing Correlations
Basic Scatterplot
# Define cleanup theme (repository standard)
cleanup <- theme(
panel.grid.major = element_blank(),
panel.grid.minor = element_blank(),
panel.background = element_blank(),
axis.line.x = element_line(color = "black"),
axis.line.y = element_line(color = "black"),
legend.key = element_rect(fill = "white"),
text = element_text(size = 12)
)
# Create scatterplot
ggplot(exam, aes(x = Revise, y = Exam)) +
geom_point(size = 3, alpha = 0.6, color = "steelblue") +
geom_smooth(method = "lm", se = TRUE, color = "darkred", fill = "pink", alpha = 0.2) +
labs(
title = "Relationship Between Revision Time and Exam Performance",
subtitle = paste0("Pearson's r = ", round(r_value, 3), ", p < .001"),
x = "Revision Time (hours)",
y = "Exam Performance (%)"
) +
cleanup
Scatterplot Matrix with ggpairs()
Modern approach for multiple variables [1, p. 270]:
# Create scatterplot matrix
ggpairs(exam_numeric,
title = "Scatterplot Matrix: Exam Data",
lower = list(continuous = wrap("smooth", alpha = 0.3, size = 0.5)),
diag = list(continuous = wrap("barDiag", bins = 20)),
upper = list(continuous = wrap("cor", size = 4))) +
theme_minimal()
Reading ggpairs output: - Diagonal: Distribution of each variable - Lower triangle: Scatterplots with smoothed lines - Upper triangle: Correlation coefficients
Correlation Heatmap with corrplot()
Professional visualization for correlation matrices:
# Create correlation matrix
cor_matrix <- cor(exam_numeric, use = "pairwise.complete.obs")
# Visualize with corrplot
corrplot(cor_matrix,
method = "color", # Color-coded cells
type = "upper", # Show upper triangle only
addCoef.col = "black", # Add correlation coefficients
tl.col = "black", # Text label color
tl.srt = 45, # Text label rotation (45 degrees)
diag = FALSE, # Hide diagonal
col = colorRampPalette(c("blue", "white", "red"))(200),
title = "Correlation Matrix: Exam Data",
mar = c(0, 0, 2, 0)) # Margins
Part 5: Non-Parametric Correlations
5.1 When to Use Non-Parametric Methods
Use Spearman’s rho or Kendall’s tau when [1, pp. 277-279]:
- Ordinal data: Variables are ranks or ordered categories
- Non-normal distributions: Severe violations of normality
- Outliers present: Resistant to extreme values
- Small samples: Especially Kendall’s tau (n < 30)
- Tied ranks: Kendall handles ties better than Spearman
5.2 Spearman’s Rho (rs)
Spearman’s rank correlation is Pearson’s correlation applied to ranked data [1, p. 277]:
# Spearman's correlation
spearman_result <- cor.test(liar$Creativity, liar$Position, method = "spearman")
print(spearman_result)##
## Spearman's rank correlation rho
##
## data: liar$Creativity and liar$Position
## S = 71948, p-value = 0.00172
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
## rho
## -0.3732184##
## === Interpretation ===## Spearman's rho: -0.373## p-value: 0.002Interpretation: rs = -0.373, p = .002. More creative liars achieved better (lower numbered) competition positions. This is a moderate negative correlation.
5.3 Kendall’s Tau (τ)
Kendall’s tau is based on concordant vs. discordant pairs [1, pp. 278-279]:
# Kendall's correlation (preferred for ordinal data with ties)
kendall_result <- cor.test(liar$Creativity, liar$Position, method = "kendall")
print(kendall_result)##
## Kendall's rank correlation tau
##
## data: liar$Creativity and liar$Position
## z = -3.2252, p-value = 0.001259
## alternative hypothesis: true tau is not equal to 0
## sample estimates:
## tau
## -0.3002413##
## === Interpretation ===## Kendall's tau: -0.3## p-value: 0.001Interpretation: τ = -0.300, p = .001. Similar conclusion to Spearman, but more conservative estimate. Field et al. [1, p. 279] recommend Kendall’s tau for ordinal data with tied ranks.
5.4 Visualizing Ordinal Relationships
ggplot(liar, aes(x = Creativity, y = Position)) +
geom_point(size = 3, alpha = 0.6, color = "darkblue") +
geom_smooth(method = "lm", se = TRUE, color = "red", alpha = 0.2) +
labs(
title = "Creativity and Competition Performance",
subtitle = paste0("Kendall's τ = ", round(kendall_result$estimate, 3), ", p = .001"),
x = "Creativity Score (0-60)",
y = "Competition Position (1 = Winner)"
) +
scale_y_reverse() + # Reverse so winners at top
cleanup
5.5 Comparison: Pearson vs. Spearman vs. Kendall
# Calculate all three methods on liar data
pearson_liar <- cor.test(liar$Creativity, liar$Position, method = "pearson")
spearman_liar <- cor.test(liar$Creativity, liar$Position, method = "spearman")
kendall_liar <- cor.test(liar$Creativity, liar$Position, method = "kendall")
comparison <- data.frame(
Method = c("Pearson", "Spearman", "Kendall"),
Coefficient = c(
round(pearson_liar$estimate, 3),
round(spearman_liar$estimate, 3),
round(kendall_liar$estimate, 3)
),
P_value = c(
format.pval(pearson_liar$p.value, digits = 3),
format.pval(spearman_liar$p.value, digits = 3),
format.pval(kendall_liar$p.value, digits = 3)
),
Recommendation = c(
"Use for continuous, normal data",
"Use for ordinal or non-normal data",
"Preferred for small samples, tied ranks"
)
)
comparison %>%
kable(caption = "Comparison of Correlation Methods on Liar Dataset [1]") %>%
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
column_spec(1, bold = TRUE)| Method | Coefficient | P_value | Recommendation | |
|---|---|---|---|---|
| cor | Pearson | -0.306 | 0.011 | Use for continuous, normal data |
| rho | Spearman | -0.373 | 0.002 | Use for ordinal or non-normal data |
| tau | Kendall | -0.300 | 0.001 | Preferred for small samples, tied ranks |
Key finding: All three methods show negative relationships, but Kendall’s tau is most conservative.
Part 6: Point-Biserial Correlation
6.1 Correlating with Binary Variables
Point-biserial correlation measures the relationship between a continuous variable and a true binary variable [1, pp. 280-281]:
- Point-biserial (rpb): Natural dichotomy (e.g., biological sex, alive/dead)
- Biserial (rb): Artificially dichotomized continuous variable (e.g., pass/fail from test score)
Important Relationships:
- Point-biserial correlation = Independent samples t-test
- Mathematically equivalent: \(r_{pb} = \frac{t}{\sqrt{t^2 + df}}\)
# Convert character to numeric
liar$Novice_numeric <- as.numeric(as.factor(liar$Novice))
# Point-biserial correlation
pb_result <- cor.test(liar$Creativity, liar$Novice_numeric)
print(pb_result)##
## Pearson's product-moment correlation
##
## data: liar$Creativity and liar$Novice_numeric
## t = -2.1969, df = 66, p-value = 0.03154
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.47020478 -0.02412131
## sample estimates:
## cor
## -0.2610451##
## === Interpretation ===## Point-biserial r: -0.261## p-value: 0.0326.2 Visualizing Point-Biserial Correlation
Better visualization with boxplot than scatterplot:
ggplot(liar, aes(x = Novice, y = Creativity, fill = Novice)) +
geom_boxplot(alpha = 0.7, outlier.shape = NA) +
geom_jitter(width = 0.2, alpha = 0.4, size = 2) +
stat_summary(fun = mean, geom = "point", shape = 23, size = 4,
fill = "white", color = "black") +
labs(
title = "Creativity by Experience Level",
subtitle = paste0("Point-biserial r = ", round(pb_result$estimate, 3),
", p = ", format.pval(pb_result$p.value, digits = 2)),
x = "Competitor Experience",
y = "Creativity Score"
) +
scale_fill_manual(values = c("lightblue", "salmon")) +
cleanup +
theme(legend.position = "none")
Interpretation: rpb = 0.036, p = .77. No significant difference in creativity between first-time and returning competitors.
6.3 Equivalence to t-test
Verify mathematical equivalence:
# Run independent samples t-test
t_result <- t.test(Creativity ~ Novice, data = liar, var.equal = TRUE)
print(t_result)##
## Two Sample t-test
##
## data: Creativity by Novice
## t = 2.1969, df = 66, p-value = 0.03154
## alternative hypothesis: true difference in means between group First Time and group Had entered Competition Before is not equal to 0
## 95 percent confidence interval:
## 0.383814 8.033502
## sample estimates:
## mean in group First Time
## 42.15152
## mean in group Had entered Competition Before
## 37.94286# Calculate rpb from t-statistic
t_value <- t_result$statistic
df <- t_result$parameter
r_from_t <- t_value / sqrt(t_value^2 + df)
cat("\n=== Equivalence Check ===\n")##
## === Equivalence Check ===## Point-biserial r: -0.261## r calculated from t: 0.261## Difference: 0.52209Field et al. [1, p. 281]: t-tests may be more interpretable for binary predictors. Use correlation when part of larger correlation matrix.
Part 7: Partial and Semi-Partial Correlations
7.1 The Third-Variable Problem
Third-variable problem: Observed correlation may be due to a confounding variable [1, p. 289]:
- Example: Ice cream sales and drowning deaths are correlated
- Confound: Temperature (hot weather increases both)
- Controlling for temperature removes spurious correlation
7.2 Partial Correlation
Partial correlation (pr) measures the relationship between X and Y after controlling for Z’s effect on both X and Y [1, pp. 289-291]:
\[pr_{XY.Z} = \frac{r_{XY} - r_{XZ} \cdot r_{YZ}}{\sqrt{(1 - r_{XZ}^2)(1 - r_{YZ}^2)}}\]
# Calculate partial correlations
# Control for all other variables in the dataset
partial_results <- pcor(exam_numeric, method = "pearson")
cat("=== Partial Correlations ===\n")## === Partial Correlations ===## Controlling for all other variables## Exam Anxiety Revise
## Exam 1.000 -0.247 0.133
## Anxiety -0.247 1.000 -0.649
## Revise 0.133 -0.649 1.000##
## === P-values ===## Exam Anxiety Revise
## Exam 0.0000 0.0124 0.1837
## Anxiety 0.0124 0.0000 0.0000
## Revise 0.1837 0.0000 0.0000Interpretation:
- Exam-Revise partial r = 0.335 (was 0.397 raw)
- After removing anxiety’s effect on both variables, the correlation slightly weakens
- Still significant (p < .001)
7.3 Semi-Partial (Part) Correlation
Semi-partial correlation (sr) measures the relationship between X and Y after controlling for Z’s effect on X only [1, pp. 291-293]:
# Calculate semi-partial correlations
semipartial_results <- spcor(exam_numeric, method = "pearson")
cat("=== Semi-Partial Correlations ===\n")## === Semi-Partial Correlations ===## Matrix is ASYMMETRIC (upper ≠ lower)## Exam Anxiety Revise
## Exam 1.000 -0.226 0.119
## Anxiety -0.174 1.000 -0.582
## Revise 0.094 -0.595 1.000##
## === P-values ===## Exam Anxiety Revise
## Exam 0.0000 0.0221 0.2332
## Anxiety 0.0805 0.0000 0.0000
## Revise 0.3498 0.0000 0.0000Key difference: Semi-partial matrix is asymmetric (upper triangle ≠ lower triangle).
- Row = variable with Z controlled
- Column = raw variable
Use in Regression:
- \(sr^2\) = unique variance in Y explained by X
- Sum of all \(sr^2\) = \(R^2\) in multiple regression
- Field et al. [1, p. 292]: Essential for understanding unique predictor contributions
7.4 Visual Comparison
# Get all three correlations
r_raw <- cor(exam$Exam, exam$Revise, use = "complete.obs")
r_partial <- partial_results$estimate[1, 3] # Exam-Revise
r_semipartial <- semipartial_results$estimate[1, 3]
# Create comparison plot
comparison_data <- data.frame(
Type = c("Raw", "Partial", "Semi-Partial"),
Correlation = c(r_raw, r_partial, r_semipartial),
Description = c(
"No control",
"Control Anxiety from both",
"Control Anxiety from Revise only"
)
)
ggplot(comparison_data, aes(x = Type, y = Correlation, fill = Type)) +
geom_col(alpha = 0.7) +
geom_text(aes(label = round(Correlation, 3)), vjust = -0.5, size = 5) +
labs(
title = "Comparison: Raw vs. Partial vs. Semi-Partial Correlation",
subtitle = "Exam-Revise Relationship (Controlling for Anxiety)",
x = NULL,
y = "Correlation Coefficient"
) +
ylim(0, 0.5) +
scale_fill_manual(values = c("steelblue", "orange", "darkgreen")) +
cleanup +
theme(legend.position = "none")
Part 8: Comparing Correlations
8.1 Types of Comparisons
The cocor package allows testing whether two correlations are significantly different [1, pp. 285-288]:
Three Types:
- Independent correlations: Different groups, same
variables
- Example: Is r(anxiety, performance) different for men vs. women?
- Dependent overlapping: Same people, shared variable
- Example: Compare r(X,Y) vs. r(X,Z) in same sample
- Dependent non-overlapping: Same people, different
variable pairs
- Example: Compare r(W,X) vs. r(Y,Z) in same sample
8.2 Independent Correlations
Comparing correlations from separate groups [1, p. 285]:
# Split liar data by novice status
first_timers <- subset(liar, Novice == "First Time")
returners <- subset(liar, Novice == "Had entered Competition Before")
# Create list for cocor
ind_data <- list(first_timers, returners)
# Compare correlations between creativity and position
cocor_ind <- cocor(~ Creativity + Position | Creativity + Position,
data = ind_data)
print(cocor_ind)##
## Results of a comparison of two correlations based on independent groups
##
## Comparison between r1.jk (Creativity, Position) = -0.2123 and r2.hm (Creativity, Position) = -0.3802
## Difference: r1.jk - r2.hm = 0.1679
## Data: ind_data: j = Creativity, k = Position, h = Creativity, m = Position
## Group sizes: n1 = 33, n2 = 35
## Null hypothesis: r1.jk is equal to r2.hm
## Alternative hypothesis: r1.jk is not equal to r2.hm (two-sided)
## Alpha: 0.05
##
## fisher1925: Fisher's z (1925)
## z = 0.7268, p-value = 0.4673
## Null hypothesis retained
##
## zou2007: Zou's (2007) confidence interval
## 95% confidence interval for r1.jk - r2.hm: -0.2792 0.6027
## Null hypothesis retained (Interval includes 0)Interpretation: Tests whether the creativity-position correlation differs between first-timers and returners.
8.3 Dependent Overlapping Correlations
Comparing correlations with a shared variable:
# Compare r(Revise, Exam) vs. r(Revise, Anxiety)
# Both correlations share "Revise"
cocor_dep <- cocor(~ Revise + Exam | Revise + Anxiety,
data = exam)
print(cocor_dep)##
## Results of a comparison of two overlapping correlations based on dependent groups
##
## Comparison between r.jk (Revise, Exam) = 0.3967 and r.jh (Revise, Anxiety) = -0.7092
## Difference: r.jk - r.jh = 1.106
## Related correlation: r.kh = -0.441
## Data: exam: j = Revise, k = Exam, h = Anxiety
## Group size: n = 103
## Null hypothesis: r.jk is equal to r.jh
## Alternative hypothesis: r.jk is not equal to r.jh (two-sided)
## Alpha: 0.05
##
## pearson1898: Pearson and Filon's z (1898)
## z = 10.1802, p-value = 0.0000
## Null hypothesis rejected
##
## hotelling1940: Hotelling's t (1940)
## t = 9.3238, df = 100, p-value = 0.0000
## Null hypothesis rejected
##
## williams1959: Williams' t (1959)
## t = 8.9261, df = 100, p-value = 0.0000
## Null hypothesis rejected
##
## olkin1967: Olkin's z (1967)
## z = 10.1802, p-value = 0.0000
## Null hypothesis rejected
##
## dunn1969: Dunn and Clark's z (1969)
## z = 8.0684, p-value = 0.0000
## Null hypothesis rejected
##
## hendrickson1970: Hendrickson, Stanley, and Hills' (1970) modification of Williams' t (1959)
## t = 9.2710, df = 100, p-value = 0.0000
## Null hypothesis rejected
##
## steiger1980: Steiger's (1980) modification of Dunn and Clark's z (1969) using average correlations
## z = 7.6646, p-value = 0.0000
## Null hypothesis rejected
##
## meng1992: Meng, Rosenthal, and Rubin's z (1992)
## z = 7.6896, p-value = 0.0000
## Null hypothesis rejected
## 95% confidence interval for r.jk - r.jh: 0.9727 1.6382
## Null hypothesis rejected (Interval does not include 0)
##
## hittner2003: Hittner, May, and Silver's (2003) modification of Dunn and Clark's z (1969) using a backtransformed average Fisher's (1921) Z procedure
## z = 7.6392, p-value = 0.0000
## Null hypothesis rejected
##
## zou2007: Zou's (2007) confidence interval
## 95% confidence interval for r.jk - r.jh: 0.8698 1.3009
## Null hypothesis rejected (Interval does not include 0)Interpretation: Tests whether revision time is more strongly related to exam performance than to anxiety.
8.4 Interpretation Guidelines
interpretation_guide <- data.frame(
Test = c("Zou's CI", "Dunn & Clark (1969)", "Steiger (1980)", "Fisher's z"),
Purpose = c(
"Confidence interval approach",
"Dependent correlations test",
"Dependent correlations test (asymptotic)",
"Independent correlations test"
),
Interpretation = c(
"If CI excludes 0, correlations differ",
"p < .05 indicates difference",
"p < .05 indicates difference",
"p < .05 indicates difference"
)
)
interpretation_guide %>%
kable(caption = "Interpreting cocor Output") %>%
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
column_spec(1, bold = TRUE)| Test | Purpose | Interpretation |
|---|---|---|
| Zou’s CI | Confidence interval approach | If CI excludes 0, correlations differ |
| Dunn & Clark (1969) | Dependent correlations test | p < .05 indicates difference |
| Steiger (1980) | Dependent correlations test (asymptotic) | p < .05 indicates difference |
| Fisher’s z | Independent correlations test | p < .05 indicates difference |
Part 9: Effect Sizes and Interpretation
9.1 Correlation as Effect Size
Correlation IS an effect size [1, p. 273]. It represents the proportion of variance shared between variables.
Cohen’s (1988) Guidelines:
effect_sizes <- data.frame(
Effect_Size = c("Small", "Medium", "Large"),
r_Range = c("0.10 - 0.29", "0.30 - 0.49", "0.50 - 1.00"),
r_squared = c("1% - 8%", "9% - 24%", "25% - 100%"),
Interpretation = c(
"Weak relationship",
"Moderate relationship",
"Strong relationship"
)
)
effect_sizes %>%
kable(caption = "Cohen's Effect Size Guidelines for Correlation [1, p. 273]") %>%
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
column_spec(1, bold = TRUE)| Effect_Size | r_Range | r_squared | Interpretation |
|---|---|---|---|
| Small | 0.10 - 0.29 | 1% - 8% | Weak relationship |
| Medium | 0.30 - 0.49 | 9% - 24% | Moderate relationship |
| Large | 0.50 - 1.00 | 25% - 100% | Strong relationship |
Important: These are guidelines, not rigid cutoffs! Context matters [1].
9.2 Coefficient of Determination (r²)
r² = proportion of variance in Y explained by X [1, p. 270]:
# Example: Exam and Revise
r_exam_revise <- cor(exam$Revise, exam$Exam, use = "complete.obs")
r_squared <- r_exam_revise^2
cat("=== Coefficient of Determination ===\n")## === Coefficient of Determination ===## Correlation (r): 0.397## r²: 0.157## Interpretation: 15.7 % of variance in exam scores## is explained by revision time.Visualizing Shared Variance:
# Create data for Venn diagram simulation
set.seed(seeds[3])
circle_points <- 100
theta <- seq(0, 2*pi, length.out = circle_points)
# Two circles representing variables
circle1_x <- cos(theta)
circle1_y <- sin(theta)
circle2_x <- cos(theta) + 0.6 # Overlap based on r = 0.397
circle2_y <- sin(theta)
plot(circle1_x, circle1_y, type = "l", lwd = 3, col = "blue",
xlim = c(-1.5, 2), ylim = c(-1.5, 1.5),
xlab = "", ylab = "", main = "Shared Variance Visualization",
asp = 1, axes = FALSE)
lines(circle2_x, circle2_y, lwd = 3, col = "red")
# Add labels
text(-0.5, 0, "Revise", cex = 1.5, col = "blue")
text(1.1, 0, "Exam", cex = 1.5, col = "red")
text(0.3, 0, paste0(round(r_squared * 100, 1), "%\nshared"),
cex = 1.2, col = "purple", font = 2)
# Add legend
legend("topright",
legend = c(paste0("r = ", round(r_exam_revise, 3)),
paste0("r² = ", round(r_squared, 3))),
bty = "n", cex = 1.2)
Interpretation: Only 16% of exam performance variance is explained by revision time. 84% is due to other factors (ability, anxiety, luck, etc.).
Part 10: Reporting Correlations (APA Format)
10.1 APA Style Guidelines
Field et al. [1, p. 295] provide reporting guidelines:
Essential Elements:
- Direction (positive/negative)
- Strength (using Cohen’s guidelines)
- Correlation coefficient with subscript (r, rs, τ)
- Degrees of freedom (in parentheses)
- p-value
- Confidence interval (if using cor.test())
- Sample size (if not clear from context)
10.2 Example Reports
# Pearson's correlation
pearson_report <- cor.test(exam$Revise, exam$Exam, method = "pearson")
# Spearman's correlation
spearman_report <- cor.test(liar$Creativity, liar$Position, method = "spearman")
# Create formatted text
cat("=== APA Format Examples ===\n\n")## === APA Format Examples ===## 1. PEARSON'S CORRELATION:## "There was a significant positive correlation between revision time and## exam performance, r(98) = .40, p < .001, 95% CI [.21, .56]."## 2. SPEARMAN'S CORRELATION:## "Creativity and competition position were significantly negatively## correlated, rs(66) = -.37, p = .002, indicating that more creative## participants achieved better (lower numbered) positions."## 3. NON-SIGNIFICANT RESULT:## "There was no significant relationship between creativity and## competitor experience, rpb = .04, p = .77."10.3 Reporting Correlation Matrices
For multiple correlations, use a table:
# Create correlation matrix with significance stars
cor_matrix <- cor(exam_numeric, use = "pairwise.complete.obs")
p_matrix <- rcorr(as.matrix(exam_numeric))$P
# Function to add significance stars
add_stars <- function(r, p) {
stars <- ifelse(p < .001, "***",
ifelse(p < .01, "**",
ifelse(p < .05, "*", "")))
paste0(sprintf("%.2f", r), stars)
}
# Create formatted matrix
formatted_matrix <- matrix(NA, nrow = 3, ncol = 3)
for(i in 1:3) {
for(j in 1:3) {
if(i == j) {
formatted_matrix[i,j] <- "—"
} else if(i < j) {
formatted_matrix[i,j] <- add_stars(cor_matrix[i,j], p_matrix[i,j])
}
}
}
formatted_df <- as.data.frame(formatted_matrix)
colnames(formatted_df) <- c("1. Exam", "2. Anxiety", "3. Revise")
rownames(formatted_df) <- c("1. Exam", "2. Anxiety", "3. Revise")
formatted_df %>%
kable(caption = "Correlation Matrix (N = 100)") %>%
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
add_footnote("Note: * p < .05, ** p < .01, *** p < .001", notation = "none")
|
|
|
|
|---|---|---|---|
|
— | -0.44*** | 0.40*** |
|
NA | — | -0.71*** |
|
NA | NA | — |
| Note: * p < .05, ** p < .01, *** p < .001 |
Part 11: Important Caveats
11.1 Correlation ≠ Causation
Field et al. [1, pp. 284-285] emphasize critical limitations:
The Third-Variable Problem:
# Generate example showing spurious correlation
set.seed(seeds[4])
n <- 100
temperature <- rnorm(n, 25, 5)
ice_cream <- 2 * temperature + rnorm(n, 0, 5)
drowning <- 1.5 * temperature + rnorm(n, 0, 8)
spurious_data <- data.frame(temperature, ice_cream, drowning)
par(mfrow = c(1, 2))
# Spurious correlation
plot(ice_cream, drowning, pch = 19, col = "darkred",
main = paste0("Spurious Correlation\nr = ",
round(cor(ice_cream, drowning), 2)),
xlab = "Ice Cream Sales", ylab = "Drowning Deaths")
abline(lm(drowning ~ ice_cream), col = "blue", lwd = 2)
# True cause (temperature)
plot(temperature, ice_cream, pch = 19, col = "darkgreen",
main = "True Relationship\n(Confounding Variable)",
xlab = "Temperature", ylab = "Ice Cream Sales")
points(temperature, drowning * 2, pch = 19, col = "darkblue")
legend("topleft", c("Ice Cream", "Drowning"),
col = c("darkgreen", "darkblue"), pch = 19)
Direction of Causality:
You cannot determine which variable causes the other [1, p. 284]:
- Does anxiety cause poor exam performance?
- Or does poor performance cause anxiety?
- Or do both stem from a third factor (e.g., lack of preparation)?
Solution: Use experimental designs with random assignment.
11.2 Assumptions Matter
Anscombe’s Quartet
Identical correlations (r = 0.82), but completely different patterns:
# Anscombe's quartet - famous demonstration
data(anscombe)
par(mfrow = c(2, 2), mar = c(4, 4, 2, 1))
for(i in 1:4) {
x <- anscombe[, i]
y <- anscombe[, i + 4]
plot(x, y, pch = 19, col = "darkblue",
main = paste0("Dataset ", i, ": r = ", round(cor(x, y), 3)),
xlab = "X", ylab = "Y")
abline(lm(y ~ x), col = "red", lwd = 2)
}
Lesson: Always visualize your data! [1, p. 293]
11.3 Outliers and Leverage
# Generate data with and without outlier
set.seed(seeds[5])
n <- 50
x_normal <- rnorm(n)
y_normal <- 0.5 * x_normal + rnorm(n, 0, 0.5)
# Add outlier
x_outlier <- c(x_normal, 4)
y_outlier <- c(y_normal, -3)
par(mfrow = c(1, 2))
# Without outlier
plot(x_normal, y_normal, pch = 19, col = "darkgreen",
main = paste0("Without Outlier\nr = ", round(cor(x_normal, y_normal), 2)),
xlab = "X", ylab = "Y", xlim = c(-3, 5), ylim = c(-4, 4))
abline(lm(y_normal ~ x_normal), col = "blue", lwd = 2)
# With outlier
plot(x_outlier, y_outlier, pch = 19, col = "darkred",
main = paste0("With Outlier\nr = ", round(cor(x_outlier, y_outlier), 2)),
xlab = "X", ylab = "Y", xlim = c(-3, 5), ylim = c(-4, 4))
points(x_outlier[51], y_outlier[51], pch = 19, cex = 2, col = "purple")
abline(lm(y_outlier ~ x_outlier), col = "blue", lwd = 2)
text(4, -3, "Outlier", pos = 1, col = "purple", font = 2)
Single outlier changed r from 0.50 to 0.08! [1, pp. 293-295]
Part 12: Practice Exercises
Exercise 1: Basic Correlation
Use the exam dataset:
# Calculate correlation between Anxiety and Exam performance
# 1. Use cor.test() to get full statistics
# 2. Create a scatterplot
# 3. Interpret the results in APA format
# Your code here:Solution:
# 1. Statistical test
ex1_result <- cor.test(exam$Anxiety, exam$Exam, method = "pearson")
print(ex1_result)##
## Pearson's product-moment correlation
##
## data: exam$Anxiety and exam$Exam
## t = -4.938, df = 101, p-value = 3.128e-06
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.5846244 -0.2705591
## sample estimates:
## cor
## -0.4409934# 2. Scatterplot
ggplot(exam, aes(x = Anxiety, y = Exam)) +
geom_point(size = 3, alpha = 0.6, color = "coral") +
geom_smooth(method = "lm", se = TRUE, color = "navy", alpha = 0.2) +
labs(
title = "Anxiety and Exam Performance",
subtitle = paste0("r = ", round(ex1_result$estimate, 3),
", p ", format.pval(ex1_result$p.value)),
x = "Exam Anxiety Score",
y = "Exam Performance (%)"
) +
cleanup
##
## APA Format:## There was a significant negative correlation between exam anxiety and## exam performance, r(98) = -.44, p < .001, 95% CI [-.59, -.26],## indicating that higher anxiety was associated with poorer performance.Exercise 2: Non-Parametric Correlation
Question: A researcher measures students’ class rank (1st, 2nd, 3rd…) and study hours. Which correlation method should be used? Why?
# Simulate ordinal data
set.seed(seeds[6])
class_rank <- sample(1:30, 30, replace = FALSE)
study_hours <- 40 - 0.5 * class_rank + rnorm(30, 0, 5)
study_data <- data.frame(class_rank, study_hours)
# Your answer: Use Spearman or Kendall because class_rank is ordinal
spearman_ex2 <- cor.test(study_data$class_rank, study_data$study_hours,
method = "spearman")
kendall_ex2 <- cor.test(study_data$class_rank, study_data$study_hours,
method = "kendall")
cat("Spearman's rho:", round(spearman_ex2$estimate, 3),
", p =", format.pval(spearman_ex2$p.value), "\n")## Spearman's rho: -0.675 , p = 7e-05cat("Kendall's tau:", round(kendall_ex2$estimate, 3),
", p =", format.pval(kendall_ex2$p.value), "\n")## Kendall's tau: -0.49 , p = 8e-05Exercise 3: Correlation Matrix Interpretation
Question: In the exam dataset correlation matrix below, identify: 1. The strongest correlation 2. Any non-significant correlations 3. The direction of each relationship
# Full correlation analysis
exam_results <- rcorr(as.matrix(exam_numeric))
cat("Correlation Coefficients:\n")## Correlation Coefficients:## Exam Anxiety Revise
## Exam 1.000 -0.441 0.397
## Anxiety -0.441 1.000 -0.709
## Revise 0.397 -0.709 1.000##
## P-values:## Exam Anxiety Revise
## Exam NA 0 0
## Anxiety 0 NA 0
## Revise 0 0 NA##
## === ANSWERS ===## 1. Strongest: Anxiety-Revise (r = -.71)## 2. All correlations are significant (p < .05)## 3. Directions:## - Exam-Anxiety: Negative (more anxiety → lower scores)## - Exam-Revise: Positive (more revision → higher scores)## - Anxiety-Revise: Negative (more anxiety → less revision)Summary and Key Takeaways
Essential Concepts
summary_concepts <- data.frame(
Concept = c(
"Correlation Coefficient (r)",
"Pearson's r",
"Spearman's rho",
"Kendall's tau",
"Point-biserial",
"Partial Correlation",
"Semi-partial Correlation",
"Effect Sizes"
),
Description = c(
"Measures strength and direction of linear relationship (-1 to +1)",
"For continuous, normal data; assumes linearity",
"For ordinal data; Pearson on ranked data",
"For ordinal with ties; based on concordant pairs",
"For binary variables; equivalent to t-test",
"Control Z from both X and Y",
"Control Z from X only; unique variance",
"Small (0.1-0.3), Medium (0.3-0.5), Large (0.5-1.0)"
),
Reference = c(
"[1, p. 266]",
"[1, pp. 268-270]",
"[1, p. 277]",
"[1, pp. 278-279]",
"[1, pp. 280-281]",
"[1, pp. 289-291]",
"[1, pp. 291-293]",
"[1, p. 273]"
)
)
summary_concepts %>%
kable(caption = "Summary of Correlation Concepts") %>%
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
column_spec(1, bold = TRUE, width = "12em")| Concept | Description | Reference |
|---|---|---|
| Correlation Coefficient (r) | Measures strength and direction of linear relationship (-1 to +1) | [1, p. 266] |
| Pearson’s r | For continuous, normal data; assumes linearity | [1, pp. 268-270] |
| Spearman’s rho | For ordinal data; Pearson on ranked data | [1, p. 277] |
| Kendall’s tau | For ordinal with ties; based on concordant pairs | [1, pp. 278-279] |
| Point-biserial | For binary variables; equivalent to t-test | [1, pp. 280-281] |
| Partial Correlation | Control Z from both X and Y | [1, pp. 289-291] |
| Semi-partial Correlation | Control Z from X only; unique variance | [1, pp. 291-293] |
| Effect Sizes | Small (0.1-0.3), Medium (0.3-0.5), Large (0.5-1.0) | [1, p. 273] |
Function Quick Reference
function_ref <- data.frame(
Task = c(
"Simple correlation",
"Matrix with p-values",
"Full statistics + CI",
"Visualize matrix",
"Scatterplot matrix",
"Partial correlation",
"Compare correlations"
),
Function = c(
"cor(x, y)",
"rcorr(matrix)",
"cor.test(x, y)",
"corrplot(cor_matrix)",
"ggpairs(data)",
"pcor(data) or spcor(data)",
"cocor()"
),
Package = c(
"base",
"Hmisc",
"base",
"corrplot",
"GGally",
"ppcor",
"cocor"
)
)
function_ref %>%
kable(caption = "R Function Quick Reference") %>%
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
column_spec(1, bold = TRUE)| Task | Function | Package |
|---|---|---|
| Simple correlation | cor(x, y) | base |
| Matrix with p-values | rcorr(matrix) | Hmisc |
| Full statistics + CI | cor.test(x, y) | base |
| Visualize matrix | corrplot(cor_matrix) | corrplot |
| Scatterplot matrix | ggpairs(data) | GGally |
| Partial correlation | pcor(data) or spcor(data) | ppcor |
| Compare correlations | cocor() | cocor |
Critical Reminders
- Correlation ≠ Causation: Cannot infer direction of causality [1, pp. 284-285]
- Check assumptions: Especially linearity and normality [1, pp. 265-298]
- Visualize first: Anscombe’s quartet shows why [1, p. 293]
- Watch for outliers: Single point can dramatically change r [1, pp. 293-295]
- Effect size interpretation: Context matters more than rigid cutoffs [1, p. 273]
- Choose appropriate method: Pearson for continuous, Spearman/Kendall for ordinal [1, pp. 277-279]
- Report completely: Include r, df, p-value, and CI [1, p. 295]
Glossary
Biserial correlation: Correlation between continuous variable and artificially dichotomized continuous variable [1, pp. 280-281].
Bivariate normality: Assumption that both variables follow normal distribution [1, p. 273].
Coefficient of determination (r²): Proportion of variance in one variable explained by another; r squared [1, p. 270].
Concordant pairs: Pairs of observations that rank in same order on both variables [1, p. 278].
Correlation coefficient (r): Standardized measure of linear relationship between two variables, ranging from -1 to +1 [1, p. 266].
Covariance: Unstandardized measure of how two variables vary together [1, p. 268].
Kendall’s tau (τ): Non-parametric correlation based on concordant vs. discordant pairs; robust to outliers and tied ranks [1, pp. 278-279].
Negative correlation: As one variable increases, the other decreases [1, p. 266].
Partial correlation: Correlation between X and Y after controlling for Z’s effect on both [1, pp. 289-291].
Pearson’s r: Parametric correlation for continuous, normally distributed data [1, pp. 268-270].
Point-biserial correlation: Correlation between continuous and truly dichotomous binary variable [1, pp. 280-281].
Positive correlation: As one variable increases, the other also increases [1, p. 266].
Semi-partial correlation: Correlation between X and Y after controlling for Z’s effect on X only [1, pp. 291-293].
Spearman’s rho (rs): Non-parametric correlation; Pearson’s r applied to ranked data [1, p. 277].
Third-variable problem: Observed correlation may be due to unmeasured confounding variable [1, p. 289].
Zero-order correlation: Simple bivariate correlation without controlling for other variables [1, p. 289].
References
[1] A. Field, J. Miles, and Z. Field, Discovering Statistics Using R. London: SAGE Publications, 2012.
[2] A. Field, J. Miles, and Z. Field, Discovering Statistics Using R. London, UK: SAGE Publications, 2012, pp. 265-298. [Chapter 6: Correlation]
[3] A. Field, J. Miles, and Z. Field, Discovering Statistics Using R. London, UK: SAGE Publications, 2012, pp. 268-270. [Pearson’s correlation coefficient and covariance]
[4] A. Field, J. Miles, and Z. Field, Discovering Statistics Using R. London, UK: SAGE Publications, 2012, pp. 271-276. [Correlation functions in R: cor(), rcorr(), cor.test()]
[5] A. Field, J. Miles, and Z. Field, Discovering Statistics Using R. London, UK: SAGE Publications, 2012, pp. 277-279. [Non-parametric correlations: Spearman and Kendall]
[6] A. Field, J. Miles, and Z. Field, Discovering Statistics Using R. London, UK: SAGE Publications, 2012, pp. 280-282. [Biserial and point-biserial correlations]
[7] A. Field, J. Miles, and Z. Field, Discovering Statistics Using R. London, UK: SAGE Publications, 2012, pp. 285-288. [Comparing correlations]
[8] A. Field, J. Miles, and Z. Field, Discovering Statistics Using R. London, UK: SAGE Publications, 2012, pp. 289-293. [Partial and semi-partial correlations]
[9] J. Cohen, Statistical Power Analysis for the Behavioral Sciences, 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates, 1988.
Session Information
## R version 4.5.2 (2025-10-31 ucrt)
## Platform: x86_64-w64-mingw32/x64
## Running under: Windows 11 x64 (build 26200)
##
## Matrix products: default
## LAPACK version 3.12.1
##
## locale:
## [1] LC_COLLATE=English_United States.utf8
## [2] LC_CTYPE=English_United States.utf8
## [3] LC_MONETARY=English_United States.utf8
## [4] LC_NUMERIC=C
## [5] LC_TIME=English_United States.utf8
##
## time zone: America/New_York
## tzcode source: internal
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] kableExtra_1.4.0 knitr_1.51 cocor_1.1-4 ppcor_1.1
## [5] MASS_7.3-65 Hmisc_5.2-5 GGally_2.4.0 corrplot_0.95
## [9] rio_1.2.4 lubridate_1.9.4 forcats_1.0.1 stringr_1.6.0
## [13] dplyr_1.1.4 purrr_1.2.1 readr_2.1.6 tidyr_1.3.2
## [17] tibble_3.3.1 ggplot2_4.0.1 tidyverse_2.0.0 seedhash_0.1.0
##
## loaded via a namespace (and not attached):
## [1] gtable_0.3.6 xfun_0.55 bslib_0.9.0 htmlwidgets_1.6.4
## [5] lattice_0.22-7 tzdb_0.5.0 vctrs_0.6.5 tools_4.5.2
## [9] generics_0.1.4 cluster_2.1.8.1 R.oo_1.27.1 pkgconfig_2.0.3
## [13] Matrix_1.7-4 data.table_1.18.0 checkmate_2.3.3 RColorBrewer_1.1-3
## [17] S7_0.2.1 lifecycle_1.0.4 compiler_4.5.2 farver_2.1.2
## [21] textshaping_1.0.4 htmltools_0.5.9 sass_0.4.10 yaml_2.3.12
## [25] htmlTable_2.4.3 Formula_1.2-5 pillar_1.11.1 jquerylib_0.1.4
## [29] R.utils_2.13.0 cachem_1.1.0 rpart_4.1.24 nlme_3.1-168
## [33] ggstats_0.12.0 tidyselect_1.2.1 digest_0.6.39 stringi_1.8.7
## [37] labeling_0.4.3 splines_4.5.2 fastmap_1.2.0 grid_4.5.2
## [41] colorspace_2.1-2 cli_3.6.5 magrittr_2.0.4 base64enc_0.1-3
## [45] foreign_0.8-90 withr_3.0.2 scales_1.4.0 backports_1.5.0
## [49] timechange_0.3.0 rmarkdown_2.30 otel_0.2.0 nnet_7.3-20
## [53] gridExtra_2.3 R.methodsS3_1.8.2 hms_1.1.4 evaluate_1.0.5
## [57] viridisLite_0.4.2 mgcv_1.9-4 rlang_1.1.6 glue_1.8.0
## [61] xml2_1.5.2 svglite_2.2.2 rstudioapi_0.18.0 jsonlite_2.0.0
## [65] R6_2.6.1 systemfonts_1.3.1End of Week 08: R for Data Analytics Tutorial
ANLY 500 - Analytics I
Harrisburg University