SpearmanBrown

Spearman-Brown, or the Spearman-Brown prophecy formula, is a result in classical test theory that describes how the reliability of a test changes when the length of the test is altered. It originated from the work of Charles Spearman and a collaborator in the early 20th century and remains a standard tool in psychometrics for planning and evaluating assessments.

The general form predicts the reliability after changing test length. If a test has observed reliability r and its length is increased by a factor k, the predicted reliability is r' = (k r) / [1 + (k - 1) r]. When estimating full-test reliability from a split-half correlation r_half, the Spearman-Brown correction is r_full = (2 r_half) / (1 + r_half). These formulas are connected: the split-half correction is a special application of the same idea.

Applications include planning test development and administration. It is used to estimate how many items or how much testing time is needed to reach a desired level of reliability, to compare different test forms, and to adjust split-half estimates to the reliability of the full test.

Assumptions and limitations: The formula relies on classical test theory assumptions such as tau-equivalence of items, parallel forms, and uncorrelated errors. It presumes additive scoring and linear changes in reliability with length. Reliability estimates can be biased if items differ markedly in quality or if the test is not close to parallel form. It provides quick, approximate projections rather than exact guarantees.

In practice, the Spearman-Brown prophecy formula remains a basic tool for reliability planning, best used alongside other estimates and with awareness of its underlying assumptions.