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On the Theory of a Set of Tests Which Differ Only in Length NICHD

Kristof, Walter
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Report Number:
ETS Research Bulletin
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Subject/Key Words:
National Institute for Child Health and Human Development (NICHD), Mathematical Formulas, Measurement Techniques, Models, Psychometrics, Sampling, Statistical Analysis, Test Theory


This paper presents a contribution to the sampling theory of a set of homogeneous tests which differ only in length, test length being regarded as an essential test parameter. Observed variance- covariance matrices of such measurements are taken to follow a Wishart distribution. The familiar true score-and-error concept of classical test theory is employed. Upon formulation of the basic model it is shown that in a combination of such tests forming a "total" test, the signal-to-noise ratio of the components is additive and that the inverse of the population variance-covariance matrix of the component measures has all of its off-diagonal elements equal, regardless of distributional assumptions. This fact facilitates the subsequent derivation of a statistical sampling theory, there being at most m + 1 free parameters when m is the number of component tests. In developing the theory, the cases of known and unknown test lengths are treated separately. For both cases maximum likelihood estimators of the relevant parameters are derived. It is argued that the resulting formulas will remain reasonable even if the distributional assumptions are too narrow. Under these assumptions, however, maximum-likelihood ratio tests of the validity of the model and of hypotheses concerning reliability and standard error of measurement of the total test are given. It is shown in each case that the maximum-likelihood equations possess precisely one acceptable solution under rather natural conditions. Application of the methods can be effected without the use of a computer. Two numerical examples are appended by way of illustration.

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