Correlation | R for Psychological Research

Set up

R code

library(tidyverse)
students <- read_csv("data/students.csv", show_col_types = FALSE)

What r means

Four scatter plots with r of minus 0.8, 0, 0.3 and 0.8, from a tight falling band to a shapeless cloud to a tight rising band

r goes from −1 to +1.
Sign = direction: + both go up together; − one goes up as the other goes down.
Size = strength. Rough guide: .10 small, .30 medium, .50 large.

Plot first

R code

ggplot(students, aes(x = StudyHours, y = ExamScore)) +
  geom_point(alpha = 0.5) +
  geom_smooth(method = "lm")

Output

`geom_smooth()` using formula = 'y ~ x'
Warning messages:
1: Removed 6 rows containing non-finite outside the scale range
(`stat_smooth()`).
2: Removed 6 rows containing missing values or values outside the scale
range (`geom_point()`).

Scatter plot of study hours against exam score with a rising line

A clear rising pattern. The plot also shows if the pattern is curved or driven by a few odd points, which r alone can't tell you.

Calculate and test

R code

cor(students$StudyHours, students$ExamScore, use = "complete.obs")

Output

[1] 0.6892234

use = "complete.obs" skips students with a missing value. To test it:

R code

cor.test(students$StudyHours, students$ExamScore)

Output

	Pearson's product-moment correlation

data:  students$StudyHours and students$ExamScore
t = 16.255, df = 292, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.6240309 0.7448873
sample estimates:
      cor 
0.6892234

cor = 0.69: a large positive correlation.
df = 292: students with both values (294) minus 2.
p-value < 2.2e-16: extremely small. Report as p < .001.
95% CI 0.62 to 0.74: where the true r probably lies.

Many at once

R code

students %>%
  select(StudyHours, Attendance, Sleep, Stress, ExamScore) %>%
  cor(use = "pairwise.complete.obs") %>%
  round(2)

Output

           StudyHours Attendance Sleep Stress ExamScore
StudyHours       1.00       0.14 -0.20   0.04      0.69
Attendance       0.14       1.00  0.12  -0.05      0.39
Sleep           -0.20       0.12  1.00  -0.30     -0.06
Stress           0.04      -0.05 -0.30   1.00     -0.01
ExamScore        0.69       0.39 -0.06  -0.01      1.00

Find a row and a column to read a pair. The diagonal is always 1.

Spearman's rho

For ratings (like stress, 1 to 10), skewed data or outliers, use Spearman's correlation. It uses ranks instead of raw values:

R code

cor.test(students$Stress, students$Sleep, method = "spearman")

Output

	Spearman's rank correlation rho

data:  students$Stress and students$Sleep
S = 5179207, p-value = 2.139e-06
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
-0.274165 
Warning message:
In cor.test.default(students$Stress, students$Sleep, method = "spearman") :
  Cannot compute exact p-value with ties

The warning about "ties" is normal for ratings. Report as r_s(288) = −.27, p < .001.

Correlation is not causation

A causes B?

More study → higher scores.

B causes A?

Doing well → enjoying study more.

C causes both?

Motivation → more study and higher scores.

A correlation can't tell these apart. Only an experiment can show cause and effect.

Exercises

Exercise 12.1 Dataset: students.csv

Attendance and exam

Plot Attendance against ExamScore, then test the correlation and report it.

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R code

ggplot(students, aes(x = Attendance, y = ExamScore)) +
  geom_point(alpha = 0.5) +
  geom_smooth(method = "lm")
cor.test(students$Attendance, students$ExamScore)

Output

`geom_smooth()` using formula = 'y ~ x'

	Pearson's product-moment correlation

data:  students$Attendance and students$ExamScore
t = 7.262, df = 298, p-value = 3.342e-12
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.2871311 0.4799306
sample estimates:
      cor 
0.3877638

Scatter plot of attendance against exam score with a rising line

Attendance and exam scores were positively correlated, r(298) = .39, 95% CI [.29, .48], p < .001: a medium relationship.

Exercise 12.2 Dataset: students.csv

Sleep and stress

Test the (Pearson) correlation between Sleep and Stress. Is it positive or negative? Write the result.

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R code

cor.test(students$Sleep, students$Stress)

Output

	Pearson's product-moment correlation

data:  students$Sleep and students$Stress
t = -5.3437, df = 288, p-value = 1.854e-07
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.4016298 -0.1917994
sample estimates:
      cor 
-0.300344

Sleep and stress were negatively correlated, r(288) = −.30, 95% CI [−.40, −.19], p < .001. Students who sleep more tend to feel less stressed.

Exercise 12.3 Dataset: students.csv

A third variable?

Someone says: "Studying more causes higher grades." Using the data, show that motivation could explain part of the link.

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Compare study hours and exam scores across motivation groups with group_by() and summarise().

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R code

students %>%
  group_by(Motivation) %>%
  summarise(mean_study = mean(StudyHours, na.rm = TRUE), mean_exam = mean(ExamScore))

Output

# A tibble: 3 × 3
  Motivation mean_study mean_exam
  <chr>           <dbl>     <dbl>
1 High             17.2      73.7
2 Low              11.6      58.7
3 Medium           13.3      67.7

Motivated students both study more and score higher, so motivation could drive both.

Key points

Plot first, then cor.test().
Report r(df), the CI and p.
method = "spearman" for ratings or skewed data.
Correlation is not causation.