Tests of correlation: Spearman's and Pearson's Flashcards

1
Q

Types of Correlation Coefficients

A
  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.
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2
Q

Pearson’s Correlation Coefficient (r)

A
  • 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.
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3
Q

Example of Pearson’s Correlation

A
  • 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.
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4
Q

Spearman’s Rank Correlation Coefficient (ρ)

A
  • 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.
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5
Q

Example of Spearman’s Correlation

A
  • 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.
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6
Q

Reporting Results

A
  • 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.”
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7
Q

Assumptions of Correlation Tests

A
  • 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.
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8
Q

Advantages and Limitations

A
  • 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.
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9
Q
A
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