Spearman–Brown prediction formula
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. The method was published independently by Spearman (1910) and Brown (1910). Calculation. Predicted reliability, formula_1, is estimated as: where "n" is the number of "tests" combined (see below) and formula_3 is the reliability of the current "test". 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). Thus "n" = 2 implies doubling the exam length by adding items with the same properties as those in the current exam. Values of "n" less than one may be used to predict the effect of shortening a test. Forecasting test length. The formula can also be rearranged to predict the number of replications required to achieve a degree of reliability: ,where formula_5 is the Pearson correlation between the split-halves. 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. Its relation to other split-half reliability coefficients. Split-half parallel reliability. Cho (2016) 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 assumption of the Spearman-Brown formula is that split-halves are parallel, which means that the variances of the split-halves are equal. The systematic name proposed for the Spearman-Brown formula is split-half parallel reliability. In addition, the following equivalent systematic formula has been proposed. formula_6 Split -half tau-equivalent reliability. Split-half tau-equivalent reliability is a reliability coefficient that can be used when the variances of split-halves are not equal. Flanagan-Rulon (formula_7, formula_8), Guttman (formula_9) suggested the following formula expressions: