Inequalities Among Lower Bounds to Reliability: With Applications to Test Construction and Factor Analysis

Peter M. Bentler · J. Arthur Woodward

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Psychometric Methodologies and Testing Educational Assessment and Pedagogy Reliability and Agreement in Measurement

Abstract

A chain of lower-bound inequalities leading to the greatest lower bound to reliability is established for the internal consistency of a composite of unit-weighted components. The chain includes the maximum split-half coefficient, the lowest coefficient consistent with nonimaginary common factors, and the lowest coefficient consistent with nonimaginary common and unique factors. Optimization theory is utilized to determine the conditions that are requisite for the inequalities. Convergence proofs demonstrate that the coefficients can be attained. Rapid algorithms obtain estimates of the coefficients with sample data. The theory yields methods for splitting items into maximally similar sets and for exploratory factor analysis based on a theoretical solution to the communality problem.

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References

1951Coefficient Alpha and the Internal Structure of Tests 1969Statistical Theories of Mental Test Scores 1960Modern factor analysis 1945A Basis for Analyzing Test-Retest Reliability 1974Analyzing psychological data by structural analysis of covariance matrices 1977Lower Bounds for the Reliability of the Total Score on a Test Composed of Non-Homogeneous Items: I: Algebraic Lower Bounds 1977Lower Bounds for the Reliability of the Total Score on a Test Composed of Non-Homogeneous Items: II: A Search Procedure to Locate the Greatest Lower Bound 1972A lower-bound method for the dimension-free measurement of internal consistency

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