Statistical & Psychometrics
Guttman’s Split-Half Lambda-4 (λ₄) Reliability Formula (Constrained-Free Variances)
Calculate Louis Guttman’s (1945) split-half test reliability λ_4 = 2 * (1 - (s_A^2 + s_B^2) / s_X^2) without Spearman-Brown’s strict equal-half-variance assumption!
Why Spearman-Brown overestimates reliability when odd and even test halves have unequal variance!
The classical Spearman-Brown prophecy formula 2*r_AB / (1 + r_AB) assumes strictly parallel half-tests with equal true-score and error variances (s_A^2 = s_B^2). Whenever odd vs. even or Part-A vs. Part-B exam subtests differ in spread, Spearman-Brown inflates reliability! Louis Guttman (Psychometrika, 1945) derived the exact lower-bound split-half reliability coefficient (equivalent to Rulon 1939 and 2-part Cronbach’s α): λ_4 = 2 * (1 - (s_A^2 + s_B^2) / s_X^2), requiring ONLY Part-A variance s_A^2, Part-B variance s_B^2, and total-score variance s_X^2!
Audit a certification exam split into Part A (s_A^2 = 4.20) and Part B (s_B^2 = 5.40) with total-score variance s_X^2 = 16.00 (implied half-test correlation r_AB = (s_X^2 - s_A^2 - s_B^2) / (2 * SQRT(s_A^2 * s_B^2)) = 0.6719): Guttman’s λ_4 = 2 * (1 - 9.60 / 16.00) = 0.8000 (whereas Spearman-Brown slightly overestimates 0.8038 due to unequal half variances)!
| Exam Total Variance s_X² (s_A² = 4.20, s_B² = 5.40) | Implied Half Correlation r_AB | Guttman Split-Half λ₄ | Psychometric Grade |
|---|---|---|---|
| s_X² = 16.00 (Strong Inter-Half Covariance) | 0.6719 | 0.8000 | High-Stakes Licensing Standard (>= 0.80) |
| s_X² = 12.00 (Weaker Inter-Half Covariance) | 0.2520 | 0.4000 | Poor Internal Consistency (< 0.70) |
| Invalid s² <= 0 | Undefined | "Invalid Half-Test Variances" | Positivity Guard |
Guttman Split-Half Reliability Coefficient λ₄
Evaluates 2 * (1 - (s_A^2 + s_B^2) / s_X^2) across Part-A variance B2, Part-B variance B3, and total-test variance B4.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Half-Test Variances", ROUND(2 * (1 - ((B2 + B3) / B4)), 4))Implied Pearson Correlation r_AB Between Test Halves
Recovers r_AB = (s_X^2 - s_A^2 - s_B^2) / (2 * SQRT(s_A^2 * s_B^2)) from the variance identity Var(A + B) = Var(A) + Var(B) + 2*Cov(A, B).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Half-Test Variances", ROUND((B4 - B2 - B3) / (2 * SQRT(B2 * B3)), 4))Try the example
Use the original values or change them to test your understanding.