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. It is also vital to the "step-up" phase of split-half and related methods of estimating reliability.
Calculation
Predicted reliability, <math>{\rho}^*_{xx'}</math>, is estimated as:
:<math>{\rho}^*_{xx'}=\frac{n{\rho}_{xx'{1+(n-1){\rho}_{xx'</math>
where n is the number of "tests" combined (see below) and <math>{\rho}_{xx'}</math> 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). If an 80-item test is reduced to a comparable set of 50 items, then n = 50/80 = 0.625. 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 (e.g., n = 0.5 implies halving the test length).
Forecasting test length
The formula can also be rearranged to predict the number of replications required to achieve a degree of reliability:
:<math>n=\frac</math>
,where <math>{\rho}_{12}</math> 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 Cronbach's alpha, this method is still useful for two-item scales.
Split-half reliability
When split halves can be assumed to be parallel, the Spearman–Brown formula can be used to "step-up" the correlation of the two halves (<math>{\rho}_{12}</math>):
<math>{\rho}_{XX'}=\frac{2{\rho}_{12{1+{\rho}_{12</math>
When the halves can be assumed to be essentially tau equivalent (and thus the variances of split-halves are not equal), Flanagan-Rulon suggested two possible estimates (<math> {\rho} _ {FR1} </math>, <math> {\rho} _ {FR2} </math>),
<math>{\rho}_{FR1}=\frac{4{\rho}_{12}{\sigma}_{1}{\sigma}_{2