Spearman-Brown Prophecy Formula

Charles Spearman and William Brown separately derived a formula to predict the reliability of a test [Page 1403]when its length was altered by the addition or subtraction of parallel items. Each presented their formula in 1910 in Volume Three of The British Journal of Psychology. Because their articles were published at the same time, the formula is known as the Spearman—Brown prophecy formula.

The basic premise of classical test theory is that an observed test score consists of the sum of the following two components: true score and error score. For any individual examinee, the two cannot be separated, but for a group of examinees, the variance attributed to each source can be estimated. Test reliability, which is an important characteristic of test score quality, is defined as the ratio of true score variance to observed score variance.

By definition, error scores are random perturbations—they covary zero with each other as well as with true scores. It is well established within statistics that the variance of a sum is equal to the sum of the variances plus two times the covariances.

Consider a test to which items are added that measure the same construct equally as well as the items already in the test (parallel items). When an item is added, the true score variance of the test is increased by the true score variance of the item as well as two times the sum of true score covariances. The error variance of the test is increased by the item error score variance, but because error scores have zero covariance with everything else (by definition), the true score variance of a test increases more rapidly than the error score variance. Thus, adding parallel items increases the reliability of a test.

An early important use of the prophecy formula allowed the estimation of score reliability from a single test administration. For example, when using the split-half reliability calculation method, a single test is divided into two halves with equivalent properties and the reliability of the half tests is calculated by correlating the two. The Spearman—Brown formula allows the test developer to estimate the reliability of the full-length test from the reliability of the test halves. Additionally, the formula allows estimation of full-length test reliability when any number of individual items are added to or subtracted from a test.

The Spearman—Brown formula relies on the assumption of strictly parallel tests or test items. Parallel tests or items are those that have identical properties, measure the same constructs, and have the same level of difficulty. The Spearman—Brown prophecy formula will only produce a valid estimation if the change in the length of the test is derived from parallel components. If an increase in test length is accomplished by adding items of a lesser quality, the Spearman—Brown formula will overestimate the reliability of the longer test. If the new items are statistically better than the average items of the original test, then Spearman—Brown will underestimate the reliability.

The Spearman—Brown prophecy is expressed in the following equation:

where k is the factor by which the length of the test is changed or the ratio of the new test length to the old test length, rAB is the reliability of the original test, and

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