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Statistical Inferences About the Error Variance

Kristof, Walter
Publication Year:
Report Number:
ETS Research Bulletin
Document Type:
Page Count:
Subject/Key Words:
Error of Measurement, Hypothesis Testing, Maximum Likelihood Statistics, Sampling, Statistical Inference, Test Theory


This paper is a presentation of an essential part of the sampling theory of the error variance and the standard error of measurement. An experimental assumption is that several factorially equivalent tests with equal variances are available. These may be either final forms of the same test or obtained by dividing one test into several parts. The simple model of independent and normally distributed errors of measurement with zero mean is employed. No assumption is made about the form of the distributions of true and observed scores. This implies unrestricted freedom in defining the population. First maximum-likelihood estimators of the error variance and the standard error of measurement are obtained, their sampling distributions given, and their properties investigated. Then unbiased estimators are defined and their distributions derived. The precision of estimation is given special consideration from various points of view. Next, rigorous statistical tests are developed to test hypotheses about error variances on the basis of one and two samples. Also the construction of confidence intervals is treated. Finally, Bartlett's test of homogeneity of variances is used to provide a multi-sample test of equality of error variances.

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