Coefficient alpha under simultaneous violations of essential tau-equivalence and uncorrelated errors

Date of Award




Degree Name

Doctor of Philosophy (Ph.D.)


Educational Research

First Committee Member

Richard Williams, Committee Chair


Coefficient alpha was evaluated with respect to how well it estimates the reliability of composite tests or scales under simultaneous violations of two classical test theory assumptions: essential tau-equivalence and uncorrelated errors. It is known that under the violation of essential tau-equivalence, true score correlations are not equal to unity and alpha underestimates the classical reliability coefficient, or the proportion of true score variance to observed score variance (Novick & Lewis, 1967; Zimmerman, Zumbo and Lalond, 1993). It is also known that under the violation of uncorrelated errors, or error score correlations not equal to zero, alpha could overestimate the classical reliability coefficient (Zimmerman et al., 1993). However, no previous studies have examined the interactive effects of the simultaneous violations of both assumptions on alpha.To study the interactive effects, computer simulated true and error scores were generated. Correlations were systematically reduced from unity among true scores and increased from zero among error scores. It was found that as correlations among true scores decreased from unity, alpha also decreased. However, as the spread and magnitude of correlations among error scores were increased from zero, the reduction in alpha was attenuated to the point that alpha equaled or even overestimated the classical reliability coefficient. Looked at in another way, alpha did not differentiate between the two sources of observed inter-item covariances. That is, with test length constant, alpha increased as both true score and error score correlations approached unity.In addition, this dissertation demonstrates that confirmatory factor analysis (CFA) can be used for empirically estimating error score covariances. It was found that the CFA estimates were fairly good approximations to the spread and magnitude of the covariances that were artificially induced among error scores. Finally, a formula for alpha is provided which can be used in practical test or scale construction for adjusting down the value of a given alpha which has been inflated with positive error covariances.


Education, Mathematics; Education, Tests and Measurements; Education, Educational Psychology

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