Regression & Correlation

Correlation Analysis (Pearson, Spearman, Kendall)

Computes Pearson r, Spearman ρ and Kendall τ including p-values and confidence intervals.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > Correlation

Description

Complete correlation analysis for two variables: parametric Pearson r (linear relationship), rank-based Spearman ρ (monotone relationship) and Kendall τ (pair-based concordance measure). For each method the correlation coefficient, t- or z-statistic, two-sided p-value, r² and Fisher- or normal-approximated confidence interval are returned.

Formulas

Pearson

r = \frac{\sum _{i = 1}^{n} ( x _{i} - x ˉ ) ( y _{i} - y ˉ )}{\sum ( x _{i} - x ˉ ) ^{2} \cdot \sum ( y _{i} - y ˉ ) ^{2}}

Pearson correlation coefficient: measures linear dependence

Spearman

ρ = r_{Pearson} (rank (x), rank (y))

Spearman ρ: Pearson correlation of ranks (using average ranks for ties)

Kendall

τ = \frac{C - D}{( n _{0} - n _{x} ) ( n _{0} - n _{y} )}

Kendall τ_b: concordant minus discordant pairs, normalised by tie-corrected maximum

t-Statistik

t = r \frac{n - 2}{1 - r ^{2}} \sim t (n - 2)

t-statistic for r and ρ; when |r| = 1 we set t = ±∞ and p = 0 (perfect-correlation special case)

Assumptions

For exact Pearson p-values: bivariate normality
Spearman/Kendall: only a monotone relationship is required
Observations are independent

Limitations

Correlation does not imply causation
Pearson is sensitive to outliers and non-linear relationships
For perfect correlation (|r| = 1) the classical t-formula is singular — the implementation must handle this special case robustly

References

Hollander, M., Wolfe, D. A., Chicken, E. (2013), Nonparametric Statistical Methods, 3rd Ed., Wiley
NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.13