30  Spearman Rank Correlation

30.1 Spearman Rank Correlation Scale Tests

The Spearman Rank Correlation Coefficient, often referred to as Spearman’s rho, is a non-parametric measure of rank correlation. It assesses how well the relationship between two variables can be described using a monotonic function. This test is ideal for cases where the variables may not meet the assumptions necessary for Pearson’s correlation coefficient, such as not having a normal distribution of data or a linear relationship.

30.1.1 Assumptions

The Spearman Rank Correlation test operates under the following assumptions:

  1. Monotonic Relationship: The relationship between the variables should be monotonic, either increasing or decreasing, but not necessarily at a constant rate.
  2. Ordinal Data: The test can be applied to ordinal data or to continuous data that do not meet the assumptions required for Pearson’s correlation.

30.1.2 Hypotheses

The hypotheses for the Spearman Rank Correlation test are:

  • Null Hypothesis (H₀): There is no association between the two variables (the correlation is zero).
  • Alternative Hypothesis (H₁): There is an association between the two variables (the correlation is not zero).

30.1.3 Formula

The Spearman’s rho ($ \() is calculated as follows:\)$ = 1 - $$ Where: - \(d_i\) is the difference between the ranks of corresponding variables. - \(n\) is the number of observations.

30.1.4 Calculation Steps

  1. Rank each variable separately. Assign average ranks in case of ties.
  2. Compute the difference (\(d\)) between the ranks of each pair of corresponding variables.
  3. Square each difference (\(d_i^2\)).
  4. Sum all squared differences.
  5. Substitute the summed value into the formula to find $ $.

30.1.5 Interpretation

The Spearman’s rho values range from -1 to +1: - A \(\rho\) of +1 indicates a perfect positive association. - A \(\rho\) of -1 indicates a perfect negative association. - A \(\rho\) of 0 suggests no association.

The significance of \(\rho\) can be tested using tables of critical values or computationally to determine if the observed correlation is unlikely under the null hypothesis.

30.1.6 Example Problem

Suppose a researcher wants to examine if there is a correlation between the ranks of employees based on their performance scores and peer ratings. Here are the data for 5 employees:

  • Performance Scores: 90, 85, 80, 95, 70
  • Peer Ratings: 88, 80, 85, 90, 75

Hypotheses:

  • Null Hypothesis (H₀): There is no correlation between performance scores and peer ratings.
  • Alternative Hypothesis (H₁): There is a correlation between performance scores and peer ratings.

30.1.7 Spearman Rank Correlation using Excel:

Download the Excel file link here

30.1.8 Spearman Rank Correlation using R:

R
# Data for performance scores and peer ratings
performance_scores <- c(90, 85, 80, 95, 70)
peer_ratings <- c(88, 80, 85, 90, 75)

# Perform Spearman Rank Correlation
spearman_result <- cor(performance_scores, peer_ratings, method = "spearman")

# Print the results
print(spearman_result)
[1] 0.9

30.1.9 Spearman Rank Correlation using Python:

Python
import scipy.stats as stats

# Data for performance scores and peer ratings
performance_scores = [90, 85, 80, 95, 70]
peer_ratings = [88, 80, 85, 90, 75]

# Perform Spearman Rank Correlation
rho, p_value = stats.spearmanr(performance_scores, peer_ratings)

# Print the results
print("Spearman's rho:", rho, "P-value:", p_value)
Spearman's rho: 0.8999999999999998 P-value: 0.03738607346849875

This test is particularly valuable in research areas where data are ordinal or do not meet the prerequisites for parametric tests, providing a robust method for correlation analysis under such conditions.