Pearson's Product-Moment Correlation Coefficient & Spearman's Rank-Order Correlation Coefficient

Definition (#f7aeae)

Important (#edcae9)

Extra (#fffe9d)

Test of Normality for Correlation:

  • Correlation analysis assumes that the data for both variables are normally distributed.

  • If the data is not normally distributed, it can lead to inaccurate or misleading correlation results.

  • Tests for Normality:

    • Shapiro-Wilk Test:

      • Commonly used for testing the normality of a variable’s distribution.

      • Null hypothesis: Data is normally distributed.

      • If p-value is greater than 0.05, the data is normally distributed.

      • If p-value is less than 0.05, the data is not normally distributed.

    • Kolmogorov-Smirnov Test:

      • Tests how closely a sample matches a normal distribution.

      • Similar to Shapiro-Wilk, a p-value greater than 0.05 suggests normality.

Sample size less than 50: Shapiro-Wilk.

Sample size more than 50: Kolmogorov-Smirnov.

Analysis: Test of normality

  1. Analyse → Descriptive Statistics → Explore.

  2. Put all variables in “Dependent List”

  3. Click “Plots” → Tick “Normality plots with tests” → Click “continue”.

  4. Click “OK”

    Step 2 - 4

Interpretation:

  • Kolmogorov-Smirnov or Shapiro-Wilk test not significant (p > .05).

  • Data is normally distributed.

  • Use parametric test.

  • Kolmogorov-Smirnov or Shapiro-Wilk test significant (p < .05).

  • Data is not normally distributed.

  • Use non-parametric test.

Skewness:

  • Tells us about the symmetry of a distribution.

  • Shows whether the data is leaning more to one side.

  • If skewness = 0, the data is perfectly symmetrical (a normal distribution).

  • Types:

    • Postive skew:

      • Tail is longer on the right side.

      • More scores are concentrated on the left.

    • Negative skew:

      • Tail is longer on the left side.

      • More scores are concentrated on the right.

  • Importance in correlation:

    • Pearson’s correlation assumes both variables have a normal (symmetrical) distribution.

    • High skewness can affect the accuracy of the correlation.

    • If data is skewed, consider using Spearman’s correlation instead.

Rules:

  1. 0: Perfectly normal (symmetrical) distribution.

  2. -0.5 to 0.5: Considered approximately symmetric or normal.

  3. 0.5 to 1 (or -0.5 to -1): Moderate skew. It still can be acceptable depending on the context.

  4. Greater than 1 (or less than -1): High skew, meaning the data is clearly skewed and less normal.

Kurtosis:

  • Refers to the “peakedness” or “flatness” of a distribution.

  • If kurtosis = 0, it’s close to a normal distribution.

  • Positive kurtosis: The distribution has thicker tails and a sharper peak than normal, meaning there are more extreme values.

  • Negative kurtosis: It has thinner tails and a flatter peak.

Rules:

  1. Kurtosis between -1 and +1 is generally acceptable for most analyses that assume normality.

  2. Kurtosis values beyond +/-2 indicate a distribution that is likely not normal and may require transformation or other statistical adjustments.

  3. Like with skewness, Spearman’s correlation is preferred when normality is violated.

Handling outliers:

  • Use a robust measure of central tendency - Mean.

  • Winsorize the data, replace with the nearest values thata re not outliers.

  • Exclude the outlier. (With justification)

  • Use non parametric tests.

Cronbach’s alpha:

  • Measure of internal consistency.

  • Tells us how well a set of items measure the same underlying construct.

  • Values range from 0-1.

    • Above 0.9: Excellent.

    • 0.8 - 0.9: Good.

    • 0.7 0.8: Acceptable.

    • 0.6 - 0.7: Questionable.

    • Below 0.6: Poor.

Importance:

  • Ensures that the survey/scale items work together as a group.

  • Tells us whether the items are reliable or consistent.

  • Helps improve the quality of psychological instruments.

When to use:

  • After developing or using likert type scale.

  • Before doing analysis, to check if the items are reliable.

Pearson’s Product-Moment Correlation Coefficient:

Overview:

  • Measures the strength and direction of the linear relationship between 2 continuous variables.

  • Widely used statistics to explore associations between variables such as test scores, behaviors, and traits.

  • Symbol: r.

  • Values range from -1 to +1.

Assumptions:

  • Both variables should be continuous and approximately normally distributed.

  • A linear relationship exists between the variables.

  • No significant outliers.

  • Independence of observations.

Spearman's Rank-Order Correlation Coefficient:

Overview:

  • A non -parametric measure of the strength and direction of association between 2 ranked or ordinal variables.

  • Values range from -1 to +1.

  • Use when data is ordinal or not normally distributed.

    • Relationship is not linear, but still consistent in direction.

    • Ranks are used instead of raw scores.

    • Useful for small samples or non-parametric data.

Assumptions:

  • Variables should be at least ordinal.

  • Monotonic relationship (as one increases, the other tends to increase or decrease).

  • Doesn’t require normality.

  • Less sensitive to outliers.

Analysis: Correlation

  1. Analyze → Correlate → Bivariate.

  2. Put variables in “Variables”.

  3. Tick either “Pearson” or “Spearman” for “correlation coefficient” → Click “OK.

    Step 2-3
Output

Template:

  • What test you ran and why & what variables were plugged in.

  • Is there a significant relation between the variables.

    • Direction and strength.

  • Report the “equation” above with the values.

  • Interpretation:

    • If significant: When one variable increases, what happened to the other.

    • If not significant: No relation.