Correlation Analysis

What Correlation Measures

Correlation measures the strength and direction of the linear relationship between two continuous variables. The correlation coefficient (r or ρ) ranges from -1 to +1.

• ●r = +1: Perfect positive linear relationship (as X increases, Y increases proportionally)
• ●r = 0: No linear relationship
• ●r = -1: Perfect negative linear relationship (as X increases, Y decreases proportionally)
• ●r = 0.3: Weak positive, r = 0.5: Moderate positive, r = 0.7: Strong positive

Pearson vs. Spearman Correlation

There are two main types of correlation coefficient:

• ●Pearson r: Parametric. Measures linear relationship. Requires both variables to be continuous and approximately normally distributed. Sensitive to outliers.
• ●Spearman ρ (rho): Nonparametric. Measures monotonic (not necessarily linear) relationship using ranks. Works for ordinal data and when normality is violated. More robust to outliers.
• ●In practice: Use Pearson if data is normal and continuous; use Spearman if skewed, ordinal, or if outliers are present.

R² — Coefficient of Determination

R² (r squared) is the proportion of variance in one variable explained by the other. It is a more intuitive measure of effect size than r alone.

• ●r = 0.5 → R² = 0.25 → 25% of variance in Y is explained by X
• ●r = 0.7 → R² = 0.49 → 49% of variance explained
• ●r = 0.9 → R² = 0.81 → 81% of variance explained

Correlation ≠ Causation

A significant correlation does not prove that one variable causes the other. Always consider three explanations for any observed correlation:

1. 1.X causes Y (the causal interpretation you want)
2. 2.Y causes X (reverse causation)
3. 3.A third variable Z causes both X and Y (confounding)

Correlation Matrices

When you have multiple variables, a correlation matrix shows all pairwise correlations simultaneously. This is useful for exploring data and identifying multicollinearity before regression.

• ●Diagonal is always r = 1 (variable with itself)
• ●Matrix is symmetric (r of X,Y = r of Y,X)
• ●Asterisks (*) typically indicate p < 0.05, (**) p < 0.01, (***) p < 0.001
• ●High intercorrelations among predictors (r > 0.7) can cause problems in regression (multicollinearity)