Correlation and rank correlation in R

In many research projects, you want to find out whether and how strongly two variables are related – for example, height ↔ weight, study time ↔ exam grade, or CO₂ emissions ↔ income. Correlation analyses provide simple but important tools for this.

Pearson correlation (r)

Pearson’s product–moment correlationr measures the strength and direction of a linear relationship between two metric variables. Values range from -1 (perfectly negative), through 0 (no linear relationship), to +1 (perfectly positive).

The assumptions for conducting a Pearson correlation are:

1. Metric scale (interval/ratio).
2. Linear relationship – always check this using a scatterplot!
3. No major outliers; they can severely distort r.
4. Homoscedasticity and an approximately normal distribution are helpful, but not essential for simply estimating r; they are important for significance tests, however.

This is the formula (but you don’t necessarily need to memorize it):

$r=\frac{\sum_{i=1}^{n}(x_i-\bar x)\,(y_i-\bar y)}{\sqrt{\sum_{i=1}^{n}(x_i-\bar )^2}\,\sqrt{\sum_{i=1}^{n}(y_i-\bar y)^2}}$

In R, you can implement this as follows:

# Beispiel: Körpergröße vs. Gewicht
data <- data.frame(
  groesse = c(168,172,181,176,169,185,160,178,190,174),
  gewicht = c(62,  70,  88,  75,  65,  92,  55,  80,  95,  72)
)

# Visualisierung
plot(data$groesse, data$gewicht,
     xlab = "Größe (cm)", ylab = "Gewicht (kg)",
     main = "Scatterplot Größe vs. Gewicht")
abline(lm(gewicht ~ groesse, data), lty = 2)

# Pearson-Korrelation + Test
cor(data$groesse, data$gewicht, method = "pearson")
cor.test(data$groesse, data$gewicht, method = "pearson")

The function cor.test() returns a t-value, the 95% confidence interval, and the p-value in addition to r.

For the interpretation, you should focus mainly on these three values:

• Absolute value of r indicates strength:
  • 0 – 0.3: weak
  • 0.3 – 0.6: moderate
  • 0.6 – 1.0: strong
• Sign indicates direction.
• Significance (p-value) indicates whether the observed relationship could have occurred by chance. However: Significant ≠ practically relevant (keyword: effect size).

Beware the causality trap: “Correlation does not imply causation” (see causality)

Rank correlation (Spearman $\rho$)

Spearman’s $\rho$ (rho) measures the monotonic relationship between two variables by transforming the data into ranks and then applying Pearson’s $r$ to these ranks. It is more robust to outliers & non-normality and is suitable for ordinal data.

You use rank correlation (instead of Pearson correlation) when…

• The data are measured on an ordinal scale or show outliers / skewness.
• The relationship is monotonic, but not necessarily linear (e.g., a saturating effect).

In R, you implement this as follows:

# Beispiel: Stress-Rang vs. Prüfungsrang
set.seed(1)
stress <- sample(1:100, 15)  # 1 = wenig Stress
note   <- sample(1:100, 15)  # 1 = beste Note

cor.test(stress, note, method = "spearman",
         exact = FALSE)      # exact=FALSE für größere n

The result contains $/rho$ and a p-value.

For the interpretation, you could say (assuming $\rho=-0.72 (p < 0.01)$): A higher stress rank (more stress) is monotonically associated with poorer exam ranks.

Kendall’s tau ($\tau$)

Kendall’s $\tau$ is based on pairwise comparisons (concordant vs. discordant pairs). It is particularly robust with small samples and many ties. Values: $[-1,1]$.

cor.test(stress, note, method = "kendall")

When is Kendall’s tau better than Spearman’s?

• Many ties (identical values) in the ranks.
• Very small n (< 20).

Visualization

1. Basic plot plot(x, y); abline(lm(y ~ x))
2. ggplot2 with a smooth line library(ggplot2) ggplot(data, aes(groesse, gewicht)) + geom_point() + geom_smooth(method = "lm", se = FALSE)
3. Correlation­ matrix as a heatmap library(reshape2); library(ggplot2) m <- cor(data, method = "spearman") m2 <- melt(m) ggplot(m2, aes(Var1, Var2, fill = value)) + geom_tile() + scale_fill_gradient2(limits = c(-1,1)) + theme_minimal()

Comparison & decision aid

Criterion	Pearson $r$	Spearman $\rho$	Kendall $\tau$
Data level	Metric	Ordinal+	Ordinal+
Relationship	Linear	Monotonic	Monotonic
Robust to outliers	No	Moderate	High
Small n + ties	Moderate	Moderate	Good
Test statistic	t-distribution	t-approximation	z-approximation

---

Example study

You collect data from 60 students: mg of caffeine per day vs. reaction­time (ms).

set.seed(42)
koffein <- rpois(60, lambda = 200)
reaktion <- 300 - 0.15*koffein + rnorm(60, 0, 20)
df <- data.frame(koffein, reaktion)

Then let’s take a look at the whole thing visually and analytically:

plot(df$koffein, df$reaktion)    # Streuplot
cor.test(df$koffein, df$reaktion)

Result: $r\approx -0.65, p < 0.001$ → higher caffeine consumption is (linearly) associated with shorter reaction time.

Because outliers (> 450 mg) appear, you should also compare it with Spearman:

cor.test(df$koffein, df$reaktion, method="spearman")

Often, $\rho$ is slightly smaller (e.g., −0.60), but it confirms the monotonic trend.