Statistical Analysis
2x2 Association Coefficients: Pearson Phi, Yule’s Q & Yule’s Y Colligation
Measure binary 2x2 contingency association on [-1, +1] with Pearson’s Phi coefficient and marginal-prevalence-invariant Yule’s Q and Yule’s Y formulas.
Why Phi depends on base-rate prevalence while Yule’s Q and Y stay invariant
On a 2x2 table [[a, b], [c, d]], Pearson’s Phi coefficient (ad - bc) / SQRT((a+b)*(c+d)*(a+c)*(b+d)) equals the exact Pearson correlation r between two 0/1 indicator variables. However, Phi can reach +/-1.0 only when row and column marginals match; oversampling cases in a case-control study changes Phi even when the underlying cross-product odds ratio ad / bc is unchanged.
Yule’s Q = (ad - bc) / (ad + bc) (the 2x2 special case of Goodman-Kruskal Gamma) and Yule’s Y = (SQRT(ad) - SQRT(bc)) / (SQRT(ad) + SQRT(bc)) normalize strictly through cross-products, making both measures 100% invariant to multiplying any row or column by a positive sampling factor.
| Measure | Formula on [[a,b],[c,d]] | Original [[36,12],[9,48]] | 2x Case Oversampled [[72,24],[9,48]] |
|---|---|---|---|
| Odds Ratio (OR) | (a * d) / (b * c) | 16.0000 | 16.0000 (Invariant) |
| Pearson Phi | (ad - bc) / SQRT(R1*R2*C1*C2) | 0.5960 | 0.5735 (Shifts with marginals) |
| Yule’s Q | (ad - bc) / (ad + bc) | 0.8824 | 0.8824 (Invariant) |
| Yule’s Y (Colligation) | (SQRT(ad) - SQRT(bc)) / (SQRT(ad) + SQRT(bc)) | 0.6000 | 0.6000 (Invariant) |
2x2 Pearson Phi Coefficient Formula
Compute row and column totals inside LET and guard zero marginal products so degenerate tables return 0 cleanly.
=LET(ad, B2 * C3, bc, C2 * B3, denom, SQRT((B2 + C2) * (B3 + C3) * (B2 + B3) * (C2 + C3)), IF(denom = 0, 0, ROUND((ad - bc) / denom, 4)))Yule’s Q & Yule’s Y Formulas
Express Yule’s Q and Y directly from ad and bc cross-products to preserve [-1, +1] scale invariance across unequal base rates.
=LET(ad, B2 * C3, bc, C2 * B3, IF(ad + bc = 0, 0, ROUND((SQRT(ad) - SQRT(bc)) / (SQRT(ad) + SQRT(bc)), 4)))Try the example
Use the original values or change them to test your understanding.