# 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.[1] The method was published independently by Spearman (1910) and Brown (1910).[2][3]

## Calculation

Predicted reliability, ${\displaystyle {\rho }_{xx'}^{*}}$ , is estimated as:

${\displaystyle {\rho }_{xx'}^{*}={\frac {n{\rho }_{xx'}}{1+(n-1){\rho }_{xx'}}}}$

where n is the number of "tests" combined (see below) and ${\displaystyle {\rho }_{xx'}}$  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:

${\displaystyle n={\frac {{\rho }_{xx'}^{*}(1-{\rho }_{xx'})}{{\rho }_{xx'}(1-{\rho }_{xx'}^{*})}}}$

## Use and related topics

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).[4]

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.

If the longer/shorter test is not parallel to the current test, then the prediction will not be strictly accurate. 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).[5]

Item response theory item information provides a much more precise means of predicting changes in the quality of measurement by adding or removing individual items.[citation needed]

## Citations

1. ^ Allen, M.; Yen W. (1979). Introduction to Measurement Theory. Monterey, CA: Brooks/Cole. ISBN 0-8185-0283-5.
2. ^ Stanley, J. (1971). Reliability. In R. L. Thorndike (Ed.), Educational Measurement. Second edition. Washington, DC: American Council on Education
3. ^ Wainer, H., & Thissen, D. (2001). True score theory: The traditional method. In H. Wainer and D. Thissen, (Eds.), Test Scoring. Mahwah, NJ:Lawrence Erlbaum
4. ^ Stanley, J. (1971). Reliability. In R. L. Thorndike (Ed.), Educational Measurement. Second edition. Washington, DC: American Council on Education
5. ^ Eisinga, R.; Te Grotenhuis, M.; Pelzer, B. (2013). "The reliability of a two-item scale: Pearson, Cronbach or Spearman-Brown?". International Journal of Public Health. 58 (4): 637–642. doi:10.1007/s00038-012-0416-3.

## References

• Spearman, Charles, C. (1910). Correlation calculated from faulty data. British Journal of Psychology, 3, 271–295.
• Brown, W. (1910). Some experimental results in the correlation of mental abilities. British Journal of Psychology, 3, 296–322.