Tests of correlation: Spearman's and Pearson's

Question 1

Types of Correlation Coefficients

Answer 1

1. Pearson’s Correlation Coefficient (r):
   o Measures the strength and direction of linear relationships between two continuous variables.
   o Ranges from -1 to +1.
2. Spearman’s Rank Correlation Coefficient (ρ):
   o Measures the strength and direction of a monotonic relationship between two ranked variables.
   o Also ranges from -1 to +1 but does not require a linear relationship.

Question 2

Pearson’s Correlation Coefficient (r)

Answer 2

• When to Use:
  o Two continuous variables.
  o Data should be normally distributed.
  o Homoscedasticity (equal variance) is assumed.
• Formula:
  r=n(Σxy)−(Σx)(Σy)[nΣx2−(Σx)2][nΣy2−(Σy)2]r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}r=[nΣx2−(Σx)2][nΣy2−(Σy)2]n(Σxy)−(Σx)(Σy)
• Interpretation:
  o r > 0: Positive correlation (as one variable increases, the other does too).
  o r < 0: Negative correlation (as one variable increases, the other decreases).
  o r = 0: No correlation.

Question 3

Example of Pearson’s Correlation

Answer 3

• Scenario: Investigating the relationship between hours studied and exam scores.
• Data Collection: Collect data from students on hours studied and their corresponding scores.
• Analysis: Calculate the Pearson’s r to see if there’s a significant correlation between study hours and exam performance.

Question 4

Spearman’s Rank Correlation Coefficient (ρ)

Answer 4

• When to Use:
  o Two ordinal variables or when data is not normally distributed.
  o For monotonic relationships (one variable consistently increases or decreases with the other).
• Formula:
  ρ=1−6Σd2n(n2−1)\rho = 1 - \frac{6 \Sigma d^2}{n(n^2 - 1)}ρ=1−n(n2−1)6Σd2
  where ddd is the difference between the ranks of each pair of observations.
• Interpretation:
  o ρ > 0: Positive correlation.
  o ρ < 0: Negative correlation.
  o ρ = 0: No correlation.

Question 5

Example of Spearman’s Correlation

Answer 5

• Scenario: Assessing the relationship between students’ ranks in sports and their ranks in academics.
• Data Collection: Rank students based on sports performance and academic scores.
• Analysis: Calculate Spearman’s ρ to determine if there’s a significant correlation between the two rankings.

Question 6

Reporting Results

Answer 6

• Format: When reporting, include the correlation coefficient, p-value, and sample size.
  o Example for Pearson’s: “There was a strong positive correlation between hours studied and exam scores, r(28) = 0.85, p < 0.001.”
  o Example for Spearman’s: “Spearman’s rank correlation indicated a significant positive relationship between sports and academic ranks, ρ(20) = 0.72, p < 0.01.”

Question 7

Assumptions of Correlation Tests

Answer 7

• Pearson’s:
  o Linearity: Relationship should be linear.
  o Normality: Data should be approximately normally distributed.
  o Homoscedasticity: Variances should be equal.
• Spearman’s:
  o No assumptions about the distribution of data.
  o Data should be ordinal, interval, or ratio.

Question 8

Advantages and Limitations

Answer 8

• Advantages:
  o Provides insight into the strength and direction of relationships.
  o Easy to compute and interpret.
• Limitations:
  o Correlation does not imply causation.
  o Sensitive to outliers, especially Pearson’s correlation.