 |
Visual Statistics Studio ♦ Outline of Visual Statistics ♦ Library |
Krus, D. J., & Helmstadter, G. C. (1987)
The relationship between correlational and internal consistency notions of test
reliability. Educational and Psychological Measurement, 47, 911-915.
The relationship
between correlational and internal consistency notions of test reliability
David J. Krus and
The role of the Spearman-Brown conceptualization of the coefficient of reliability was discussed within the context of the modern formulations of the internal-consistency reliability.
Discussions of the Spearman-Brown formula for correction of the “odd-even’’ or ‘‘split-half” reliability coefficients (Brown 1910; Spearman, 1910) often emphasize only its "prophecy’’ properties; i.e., its ability to estimate reliability that would result from changing the length of a given test. This particular interpretation neither leads to deliberations about the structural properties of test batteries, nor stresses the historical importance of the Spearman-Brown formula as a precursor of the modern formulations of internal-consistency reliability.
The purpose of this paper is to provide a new perspective on the classical theory of the true and error scores and to highlight its role in formulation of the modern concepts of the internal consistency reliability.
Reliability within the context of the classical test theory
Within the context of the classical test theory, where an observed score Xo is considered to be composed of the sum of the true and error components
|
(1) |
|
reliability has been defined as the ratio of the variance of the true test scores to the variance of the obtained test scores
|
(2) |
|
which can also be expressed as
|
(3) |
|
since and thus .
Within the theoretical framework of the Spearman-Brown coefficient of reliability which was subsequently elaborated into the theory of true and error scores, the obtained scores Xo are conceptualized as consisting of two halves, Xa and Xb. Thus
|
(4) |
|
and the variance of this composite can be computed as
|
(5) |
|
Assuming that the variances and are equal, Eq. 5 can be simplified
|
(6) |
|
and variance of the error component can be written as
|
(7) |
|
The Spearman-Brown coefficient of reliability thus can be conceptualized as
|
(8) |
|
and simplified to its classic form
|
(9) |
|
Structural Similarities of the Spearman-Brown and the Jackson-Hoyt coefficients of reliability.
To see how this early conceptualization of test reliability relates to the modern reliability estimates by the Jackson Hoyt formula, let us consider the correlational matrix R between the two halves of the variable X
|
(10) |
|
containing two eigenvalues and . These eigenvalues can he computed as roots of the equation
|
(11) |
|
as
|
(12) |
|
and
|
(13) |
|
Substituting to Eq. 8, the S-B coefficient of reliability can be conceptualized as
|
(14) |
|
The variance components p and of the first and second principal components and are proportional the mean squares for rows and the mean squares for interactions within the Jackson-Hoyt reliability formula
|
(15) |
|
which removes the assumption of the equality of variances and generalizes the solution from a single variable of the test scores to its adjacent data matrix, developed a generation later.
REFERENCES
Brown, W. (1910) Some experimental results in the correlation of mental abilities. British Journal of Psychology, 3, 296-322.
Hoyt. C. (1941) Test reliability estimated by analysis of variance. Psychometrika, 6, 153-160.
Spearman, C. (1910) Correlation calculated from faulty data. British Journal of Psychology, 3, 271-295.
