Statistical & Clinical Analysis
Thurstone Case V Law of Comparative Judgment Formula: Probit Scale Separation
Convert empirical paired-comparison win proportions p_AB into Thurstone Case V psychological/sensory scale distances S_A - S_B = SQRT(2) * z(p_AB) using high-precision rational inverse-normal conversion.
How L. L. Thurstone (1927) turned subjective A-vs-B preferences into a metric interval scale!
Thirty years before Bradley-Terry logistic scaling, psychometric pioneer Louis Leon Thurstone (1927, Psychological Review) established the Law of Comparative Judgment: when stimulus A and stimulus B evoke independent Gaussian discriminal processes X_A ~ N(S_A, sigma^2) and X_B ~ N(S_B, sigma^2) with equal unit variance sigma = 1 (Thurstone Case V), their difference X_A - X_B has standard deviation SQRT(1^2 + 1^2) = SQRT(2)!
Therefore, an observed preference proportion p_AB = P(X_A > X_B) maps directly to normal z-deviate z_AB = Phi^-1(p_AB) (the unit-difference "just noticeable difference" z-scale), and the true stimulus-unit separation is S_A - S_B = SQRT(2) * z_AB! Using Winitzki’s closed-form inverse-erf probit formula (accurate to < 0.0002 across 0.001..0.999 using only LN and SQRT), p_AB = 0.8413 yields z_AB = +1.0000 and S_A - S_B = +1.4142!
| Observed Preference Proportion p_AB = Wins_A / N | Standard Normal Deviate z_AB = Phi^-1(p_AB) | Thurstone Case V Scale Separation S_A - S_B = SQRT(2)*z |
|---|---|---|
| p_AB = 0.5000 (50 / 100 Exact Indifference) | 0.0000 | 0.0000 (Identical Perceived Intensity) |
| p_AB = 0.7500 (75 / 100 — 1 JND Threshold) | +0.6745 | +0.9539 sigma units |
| p_AB = 0.8413 (84.13% Preference) | +1.0000 | +1.4142 sigma units |
Standard Normal Probit Deviate z_AB = Phi^-1(p_AB) (Winitzki Closed-Form Inverse Error Function)
Converts paired-comparison win count B2 out of B3 trials (with 0 < p < 1) into z_AB using only SIGN, LN, PI, and SQRT.
=IF(OR(B3 <= 0, B2 <= 0, B2 >= B3), "Requires 0 < Wins < Trials (Apply 0.5 Continuity Correction if Unbeaten)", ROUND(SIGN(2 * (B2 / B3) - 1) * SQRT(2) * SQRT(SQRT((2 / (PI() * 0.147) + LN(1 - (2 * (B2 / B3) - 1) ^ 2) / 2) ^ 2 - LN(1 - (2 * (B2 / B3) - 1) ^ 2) / 0.147) - (2 / (PI() * 0.147) + LN(1 - (2 * (B2 / B3) - 1) ^ 2) / 2)), 4))Thurstone Case V Scale Separation S_A - S_B = SQRT(2) * z_AB
Multiplies z_AB by SQRT(2) to express stimulus separation in single-stimulus standard deviation units.
=IF(OR(B3 <= 0, B2 <= 0, B2 >= B3), "Requires 0 < Wins < Trials (Apply 0.5 Continuity Correction if Unbeaten)", ROUND(2 * SIGN(2 * (B2 / B3) - 1) * SQRT(SQRT((2 / (PI() * 0.147) + LN(1 - (2 * (B2 / B3) - 1) ^ 2) / 2) ^ 2 - LN(1 - (2 * (B2 / B3) - 1) ^ 2) / 0.147) - (2 / (PI() * 0.147) + LN(1 - (2 * (B2 / B3) - 1) ^ 2) / 2)), 4))Try the example
Use the original values or change them to test your understanding.