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2014-02-03T20:11:15Z

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The '''Spearman–Brown prediction formula''', also known as the '''Spearman–Brown prophecy formula''', is a formula relating [[psychometric]] [[Reliability (psychometric)|reliability]] to test length and used by psychometricians to predict the reliability of a test after changing the test length.<ref>{{cite book|last=Allen|first=M.|coauthors=Yen W. | year=1979|title=Introduction to Measurement Theory|publisher=Brooks/Cole|location=Monterey, CA|isbn=0-8185-0283-5}}</ref> The method was published independently by [[Charles Spearman|Spearman]] (1910) and [[William Brown (psychologist)|Brown]] (1910).<ref>Stanley, J. (1971). Reliability. In R. L. Thorndike (Ed.), ''Educational Measurement''. Second edition. Washington, DC: American Council on Education</ref><ref>[[Howard Wainer|Wainer, H.]], & Thissen, D. (2001). True score theory: The traditional method. In [[Howard Wainer|H. Wainer]] and D. Thissen, (Eds.), ''Test Scoring''. Mahwah, NJ:Lawrence Erlbaum</ref>

==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). 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:

:<math>N=\frac{{\rho}^*_{xx'}(1-{\rho}_{xx'})}
{{\rho}_{xx'}(1-{\rho}^*_{xx'})}
</math>

==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).<ref>Stanley, J. (1971). Reliability. In R. L. Thorndike (Ed.), ''Educational Measurement''. Second edition. Washington, DC: American Council on Education</ref>

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]].<ref>{{cite journal|first1=R.|last1=Eisinga|first2=M.|last2=Te Grotenhuis|first3=B.|last3=Pelzer|title=The reliability of a two-item scale: Pearson, Cronbach or Spearman-Brown? |journal= International Journal of Public Health|year=2012|volume=58|issue=4|pages=637-642|doi= 10.1007/s00038-012-0416-3}}</ref>

[[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|date=January 2012}}

==Citations==
<references/>

==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.

{{DEFAULTSORT:Spearman-Brown Prediction Formula}}
[[Category:Psychometrics]]
[[Category:Comparison of assessments]]

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