R Correlation

Correlation measures the strength and direction of the linear relationship between two numeric variables. A correlation coefficient ranges from -1 (perfect negative) to +1 (perfect positive), with 0 meaning no linear relationship. R calculates correlation in one function call and visualizes it with a few more.

The Correlation Coefficient

Value range:  -1  ────── 0 ────── +1

  +1  Perfect positive: as X rises, Y rises proportionally
  +0.7  Strong positive relationship
  +0.3  Weak positive relationship
   0   No linear relationship
  -0.3  Weak negative relationship
  -0.7  Strong negative relationship
  -1  Perfect negative: as X rises, Y falls proportionally

Scatter plot shapes:
  r = +1        r = +0.7      r = 0         r = -0.7
  ·             · ·           · · · ·       · · ·
  ·            · ·          · · · · ·         · · ·
  ·           · ·         · · · · · ·           · ·
 ↗           ↗           → scattered           ↘

Pearson Correlation

study_hours <- c(1,2,3,4,5,6,7,8,9,10)
exam_score  <- c(45,52,58,65,70,74,80,85,88,95)

cor(study_hours, exam_score)
# [1] 0.9973  → very strong positive correlation

# With hypothesis test
cor.test(study_hours, exam_score)
# t = 41.7, df = 8, p-value = 1.8e-10
# 95% CI: [0.990, 0.999]
# cor = 0.997

Spearman Rank Correlation

# Use when data is not normally distributed or has outliers
cor(study_hours, exam_score, method="spearman")

# Kendall's tau — another non-parametric option
cor(study_hours, exam_score, method="kendall")

Method        Assumption         Best For
──────────────────────────────────────────────────────────────
Pearson       Linear, normal     Continuous, normal data
Spearman      Monotonic          Ordinal or non-normal data
Kendall       Monotonic          Small samples, many ties

Correlation Matrix

students <- data.frame(
  study_hrs = c(5,8,3,9,6,4,7,10,2,6),
  sleep_hrs = c(7,6,8,6,7,8,7,5,9,6),
  score     = c(72,88,58,95,78,62,84,96,45,74),
  stress    = c(4,6,3,7,5,3,5,8,2,4)
)

cor(students)
#           study_hrs sleep_hrs  score stress
# study_hrs     1.000    -0.652  0.982  0.905
# sleep_hrs    -0.652     1.000 -0.614 -0.790
# score         0.982    -0.614  1.000  0.876
# stress        0.905    -0.790  0.876  1.000

Visualizing a Correlation Matrix

library(ggplot2)
library(reshape2)

cor_matrix <- cor(students)
melted      <- melt(cor_matrix)

ggplot(melted, aes(Var1, Var2, fill=value)) +
  geom_tile(color="white") +
  geom_text(aes(label=round(value,2)), size=3) +
  scale_fill_gradient2(low="tomato", mid="white", high="steelblue",
                        midpoint=0, limits=c(-1,1)) +
  labs(title="Correlation Heatmap", x=NULL, y=NULL, fill="r") +
  theme_minimal() +
  theme(axis.text.x=element_text(angle=45, hjust=1))

corrplot Package — Dedicated Correlation Visualization

library(corrplot)
corrplot(cor(students),
         method  = "circle",     # "color","number","pie","shade"
         type    = "upper",      # show upper triangle only
         tl.col  = "black",
         tl.srt  = 45,
         addCoef.col = "white",  # add coefficients
         col     = COL2("RdBu"))

Correlation vs Causation

High correlation does NOT mean one variable causes the other.

Example: Ice cream sales and drowning rates correlate strongly.
Cause: Both increase in summer (confounding variable = temperature).

  Ice cream sales ↑   r = 0.85   Drowning rates ↑
          ↑                              ↑
          └──────── Summer heat ─────────┘
                   (true cause of both)

Always look for confounding variables before claiming causation.

Partial Correlation

library(ppcor)

# Correlation between study hours and score,
# controlling for stress level
pcor.test(students$study_hrs, students$score, students$stress)
# Partial correlation = 0.943, p = 0.0008

Correlation analysis is the starting point for understanding relationships in your data. Run a correlation matrix on all numeric variables at the beginning of any analysis — it reveals which variables move together, guides variable selection for modeling, and immediately flags suspicious patterns worth investigating further.