The Spearman–Brown prediction formula, also known as the Spearman–Brown prophecy formula, is a formula relating psychometric reliability to test length and used by psychometricians to predict the reliability of a test after changing the test length.
[1] The method was published independently by Spearman (1910) and Brown (1910).
, is estimated as: where n is the number of "tests" combined (see below) and
The formula predicts the reliability of a new test composed by replicating the current test n times (or, equivalently, creating a test with n parallel forms of the current exam).
Values of n less than one may be used to predict the effect of shortening a test.
[4][5] After splitting the whole item into arbitrary halves, the correlation between the split-halves can be converted into reliability by applying the Spearman-Brown formula.
Although the Spearman-Brown formula is rarely used as a split-half reliability coefficient after the development of tau-equivalent reliability, this method is still useful for two-item scales.
[6] Cho (2016)[7] suggests using systematic nomenclature and formula expressions, criticizing that reliability coefficients have been represented in a disorganized and inconsistent manner with historically inaccurate and uninformative names.
The systematic name proposed for the Spearman-Brown formula is split-half parallel reliability.
In addition, the following equivalent systematic formula has been proposed.
Split-half parallel reliability and split-half tau-equivalent reliability have the assumption that split-halves have the same length.
Split-half congeneric reliability mitigates this assumption.
Angoff (1953)[11] and Feldt (1975)[12] published the split-half congeneric reliability assuming that the length of each split-half was proportional to the sum of the variances and covariances.
[7] The name Spearman-Brown seems to imply a partnership, but the two authors were competitive.
This formula originates from two papers published simultaneously by Brown (1910) and Spearman (1910) in the British Journal of Psychology.
Charles Spearman had a hostile relationship with Karl Pearson who worked together in King's College London, and they exchanged papers that criticized and ridiculed each other.
[13] William Brown received his Ph.D. under Pearson's guidance.
An important part of Brown's doctoral dissertation[14] was devoted to criticizing Spearman's work on the rank correlation.
[16] For example, Spearman established the first theory of reliability[15] and is called "the father of classical reliability theory.
"[17] This is an example of Matthew Effect or Stigler's law of eponymy.
Brown (1910) explicitly presented this formula as a split-half reliability coefficient, but Spearman (1910) did not.
Second, the formal derivation of Brown (1910) is more concise and elegant than that of Spearman (1910).
Brown (1910) is based on his doctoral dissertation, which was already available at the time of publication.
Fourth, it is the APA style to list the authors in alphabetical order.
This formula is commonly used by psychometricians to predict the reliability of a test after changing the test length.
This relationship is particularly vital to the split-half and related methods of estimating reliability (where this method is sometimes known as the "Step Up" formula).
[2] The formula is also helpful in understanding the nonlinear relationship between test reliability and test length.
Test length must grow by increasingly larger values as the desired reliability approaches 1.0.
For example, if a highly reliable test was lengthened by adding many poor items then the achieved reliability will probably be much lower than that predicted by this formula.
For the reliability of a two-item test, the formula is more appropriate than Cronbach's alpha (used in this way, the Spearman-Brown formula is also called "standardized Cronbach's alpha", as it is the same as Cronbach's alpha computed using the average item intercorrelation and unit-item variance, rather than the average item covariance and average item variance).