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Spearman-Brown Formula

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What is the Spearman-Brown Formula?

The Spearman-Brown Formula (also called the Spearman-Brown Prophecy Formula) is a measure of test reliability. It’s usually used when the length of a test is changed and you want to see if reliability has increased.
The formula is:

rkk = k(r11) / [1 + (k – 1)* r11]

Where:


  • rkk = reliability of a test “k” times as long as the original test,
  • r11 = reliability of the original test(e.g. Cronbach’s Alpha),
  • k = factor by which the length of the test is changed. To find k, divide the number of items on the original test by the number of items on the new test. If you had 10 items on the original and 20 on the new, k would be 20/10 = 2.

Example question: a test made up of 12 items has a reliability (R11) of .68. If the number of items is doubled to 24, will the reliability of the test improve?

Solution: Insert the given numbers into the formula and solve.
We are given:

  • r11 = .68.
  • k = 24/12 = 2.

So:
rkk = 2(.68) / [1 + (2 – 1)* .68] = .81.

Doubling the test increases the reliability from .68 to .81.

  • For the formula to work properly, the two tests must be equivalent in difficulty. If you double a test and add only easy/poor questions, the results from the Spearman-Brown Formula will be invalid.
  • Although increasing test items is one way to increase reliability, it’s not always possible to do so. For example, doubling the (already lengthy) GRE would lead to examinee fatigue.

Two Item Tests

It’s long been recognized that two-item tests are problematic to begin with. If you must use a two-item test, you’ll find there is further disagreement about how to test reliability. Many researchers use Cronbach’s for measuring two-item test reliability, but other researchers claim that use is meaningless and the Spearman-Brown or even the Pearson Correlation coefficient should be used instead.

In general, Cronbach’s will underestimate reliability, sometimes dramatically. The Spearman-Brown is always higher than Cronbach’s alpha, and is therefore a more appropriate measure of reliability for two-item tests.

References:
Brown, W. (1910). Some experimental results in the correlation of mental abilities. British Journal of Psychology, 3, 296–322.
Eisinga, R. 2012. The Reliability of a Two-Item Scale: Pearson, Cronbach, or Spearman-Brown. Int J Public Health. Available here.
Stanley, J. (1971). Reliability. In R. L. Thorndike (Ed.), Educational Measurement. Second edition. Washington, DC: American Council on Education

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Stephanie Glen. "Spearman-Brown Formula" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/spearman-brown/
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