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On the estimators of model-based and maximal reliability

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  • Ogasawara, Haruhiko

Abstract

Four estimators of the reliability for a composite score based on the factor analysis model and five estimators of the maximal reliability for the composite are presented. When the Wishart maximum likelihood is used for the estimation of the model parameters, it is shown that the five estimators of maximal reliability are the same. Asymptotic cumulants of the estimators and their logarithmic transformations are derived under arbitrary distributions with possible model misspecification. The theoretical results considering model misspecification when a model does not hold are shown to be closer to their simulated values than those neglecting model misspecification. Simulations of the confidence intervals using the normal approximation based on the asymptotically distribution-free theory and the asymptotic expansion by Hall's method with variable transformation are performed.

Suggested Citation

  • Ogasawara, Haruhiko, 2009. "On the estimators of model-based and maximal reliability," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1232-1244, July.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:6:p:1232-1244
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    References listed on IDEAS

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    1. Karl Jöreskog & Arthur Goldberger, 1972. "Factor analysis by generalized least squares," Psychometrika, Springer;The Psychometric Society, vol. 37(3), pages 243-260, September.
    2. Ogasawara, Haruhiko, 2007. "Asymptotic expansions of the distributions of estimators in canonical correlation analysis under nonnormality," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1726-1750, October.
    3. Haruhiko Ogasawara, 2007. "Higher-order approximations to the distributions of fit indexes under fixed alternatives in structural equation models," Psychometrika, Springer;The Psychometric Society, vol. 72(2), pages 227-243, June.
    4. Bert Green, 1950. "A note on the calculation of weights for maximum battery reliability," Psychometrika, Springer;The Psychometric Society, vol. 15(1), pages 57-61, March.
    5. Ogasawara, Haruhiko, 2006. "Asymptotic expansion of the sample correlation coefficient under nonnormality," Computational Statistics & Data Analysis, Elsevier, vol. 50(4), pages 891-910, February.
    6. Charles Mosier, 1943. "On the reliability of a weighted composite," Psychometrika, Springer;The Psychometric Society, vol. 8(3), pages 161-168, September.
    7. Ogasawara, Haruhiko, 2007. "Higher-order Estimation Error in Structural Equation Modeling," 商学討究 (Shogaku Tokyu), Otaru University of Commerce, vol. 57(4), pages 131-160.
    8. C. Rao, 1955. "Estimation and tests of significance in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 20(2), pages 93-111, June.
    9. Heng Li, 1997. "A unifying expression for the maximal reliability of a linear composite," Psychometrika, Springer;The Psychometric Society, vol. 62(2), pages 245-249, June.
    10. M. Knott & D. Bartholomew, 1993. "Constructing measures with maximum reliability," Psychometrika, Springer;The Psychometric Society, vol. 58(2), pages 331-338, June.
    11. P. Bentler, 1968. "Alpha-maximized factor analysis (alphamax): Its relation to alpha and canonical factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 33(3), pages 335-345, September.
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    Cited by:

    1. Ogasawara, Haruhiko, 2009. "Supplement to Ogasawara's papers on "the ADF pivotal statistic", "mean and covariance structure analysis", and "maximal reliability"," 商学討究 (Shogaku Tokyu), Otaru University of Commerce, vol. 60(1), pages 21-44.

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