Spearman–Brown prediction formula
The Spearman–Brown prediction formula, also known as the Spearman–Brown prophecy formula, is a formula relating
Calculation
Predicted reliability, , is estimated as:
where n is the number of "tests" combined (see below) and 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:
Split-half reliability
Until the development of
,where 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
Its relation to other split-half reliability coefficients
Split-half parallel reliability
Cho (2016)[6] suggests using systematic nomenclature and formula expressions, criticizing that reliability coefficients have been represented in a disorganized and inconsistent manner with historically inaccurate and uninformative names. The assumption of the Spearman-Brown formula is that split-halves are parallel, which means that the variances of the split-halves are equal. The systematic name proposed for the Spearman-Brown formula is split-half parallel reliability. In addition, the following equivalent systematic formula has been proposed.
Split-half tau-equivalent reliability
Split-half
Where , , , and is the variance of the first split-half, the second half, the sum of the two split-halves, and the difference of the two split-halves, respectively.
These formulas are all algebraically equivalent. The systematic formula [9] is as follows.
.
Split-half congeneric reliability
Split-half parallel reliability and split-half tau-equivalent reliability have the assumption that split-halves have the same length. Split-half
History
The name Spearman-Brown seems to imply a partnership, but the two authors were competitive. This formula originates from two papers published simultaneously by Brown (1910) and Spearman (1910) in the
This formula should be referred to as the Brown-Spearman formula for the following reasons:
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).[22]
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).[23]
Citations
- ISBN 0-8185-0283-5.
- ^ Stanley, J. (1971). Reliability. In R. L. Thorndike (Ed.), Educational Measurement. Second edition. Washington, DC: American Council on Education
- ^ Wainer, H., & Thissen, D. (2001). True score theory: The traditional method. In H. Wainer and D. Thissen, (Eds.), Test Scoring. Mahwah, NJ:Lawrence Erlbaum
- ^ Kelley, T. L. (1924). Note on the Reliability of a Test: A reply to Dr. Crum ’s criticism. Journal of Educational Psychology, 15, 193–204. doi: 10.1037 / h0072471.
Kuder, G. F., & Richardson, M. W. (1937). The theory of the estimation of test reliability. Psychometrika, 2, 151-160. doi: 10.1007 / BF02288391. - ^ 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
- ^ Cho, E. (2016). Making reliability reliable: A systematic approach to reliability coefficients. Organizational Research Methods, 19, 651-682. doi:10.1177/1094428116656239.
- ^ Flanagan, J. C. (1937). A proposed procedure for increasing the efficiency of objective tests. Journal of Educational Psychology, 28, 17-21. doi: 10.1037 / h0057430. Rulon, P. J. (1939). A simplified procedure for determining the reliability of a test by split-halves. Harvard Educational Review, 9, 99-103.
- ^ Guttman, L. (1945). A basis for analyzing test-retest reliability. Psychometrika, 10, 255-282. doi: 10.1007 / BF02288892.
- ^ Cho, E. (2016). Making reliability reliable: A systematic approach to reliability coefficients. Organizational Research Methods, 19, 651-682. doi: 10.1177 / 1094428116656239.
- ^ Raju, N. S. (1970). New formula for estimating total test reliability from parts ofunequal length. Proceedings of the 78th Annual Convention ofAPA, 5, 143-144.
- ^ Angoff, W. H. (1953). Test reliability and effective test length. Psychometrika, 18(1), 1-14.
- ^ Feldt, L. S. (1975). Estimation of the reliability of a test divided into two parts of unequal length. Psychometrika, 40(4), 557-561.
- ^ Cho, E. (2016). Making reliability reliable: A systematic approach to reliability coefficients. Organizational Research Methods, 19, 651-682. doi:10.1177/1094428116656239.
- ^ Cowles, M. (2005) Statistics in psychology: An historical perspective. New York: Psychology Press.
- ^ Later published as a book Brown, W. (1911). The essentials of mental measurement. London: Cambridge University Press.
- ^ Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, 15, 72-101.
- ^ Cho, E. & Chun, S. (2018). Fixing a broken clock: A historical review of the originators reliability coefficients including Cronbach's alpha. Survey Research, 19 (2), 23-54.
- ^ Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, 15, 72-101.
- ^ Cronbach, L. J., Rajaratnam, N., & Gleser, G. C. (1963). Theory of generalizability: A liberalization of reliability theory. British Journal of Statistical Psychology, 16, 137-163. doi: 10.1111 / j.2044-8317.1963.tb00206.x.
- ^ Cho, E. & Chun, S. (2018). Fixing a broken clock: A historical review of the originators reliability coefficients including Cronbach's alpha. Survey Research, 19 (2), 23-54.
- ^ Traub, R. E. (1997). Classical test theory in historical perspective. Educational Measurement: Issues and Practice, 16, 8-14. doi: 10.1111 / j.1745-3992.1997.tb00603.x.
- ^ Stanley, J. (1971). Reliability. In R. L. Thorndike (Ed.), Educational Measurement. Second edition. Washington, DC: American Council on Education
- PMID 23089674.
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.