Tests of correlation: Spearman's and Pearson's Flashcards
1
Q
Types of Correlation Coefficients
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- Pearson’s Correlation Coefficient (r):
o Measures the strength and direction of linear relationships between two continuous variables.
o Ranges from -1 to +1. - 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.
2
Q
Pearson’s Correlation Coefficient (r)
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- 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.
3
Q
Example of Pearson’s Correlation
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- 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.
4
Q
Spearman’s Rank Correlation Coefficient (ρ)
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- 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.
5
Q
Example of Spearman’s Correlation
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- 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.
6
Q
Reporting Results
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- 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.”
7
Q
Assumptions of Correlation Tests
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- 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.
8
Q
Advantages and Limitations
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- 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.
9
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