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Inferential Procedures for Correlation Coefficients Corrected for Attenuation

Published online by Cambridge University Press:  01 January 2025

A. Ralph Hakstian ,
Marsha L. Schroeder  and
W. Todd Rogers
A. Ralph Hakstian*
Affiliation:
University of British Columbia
Marsha L. Schroeder
Affiliation:
University of British Columbia
W. Todd Rogers
Affiliation:
University of British Columbia
*
Requests for reprints should be sent to A. Ralph Hakstian, Department of Psychdlogy, University of British Columbia, Vancouver, B. C., CANADA V6T lW5.
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Abstract

A model and computational procedure based on classical test score theory are presented for determination of a correlation coefficient corrected for attenuation due to unreliability. Next, variance-covariance expressions for the sample estimates defined earlier are derived, based on application of the delta method. Results of a Monte Carlo study are presented in which the adequacy of the derived expressions was assessed for a large number of data forms and potential hypotheses encountered in the behavioral sciences. It is shown that, based on the proposed procedures, confidence intervals for single coefficients are reasonably precise. Two-sample hypothesis tests, for both independent and dependent samples, are also accurate. However, for hypothesis tests involving a larger number of coefficients than two—both independent and dependent—the proposed procedures require largens for adequate precision. Results of a preliminary power analysis reveal no serious loss in efficiency resulting from correction for attenuation. Implications for practice are discussed.

Keywords

correction for attenuationunreliabilitysampling variance and covariancedelta methodhypothesis-testing
Type
Original Paper
Information
Psychometrika , Volume 53 , Issue 1 , March 1988 , pp. 27 - 43
Copyright
Copyright © 1988 The Psychometric Society

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Footnotes

Support for the research reported in this article was provided by the Natural Sciences and Engineering Research Council of Canada. The authors acknowledge with thanks the constructive comments of the editor and three anonymous reviewers.

References

Block, J. (1963). The equivalence of measures and the correction for attenuation. Psychological Bulletin, 60, 152–156.CrossRefGoogle ScholarPubMed
Bobko, P., Rieck, A. (1980). Large sample estimators for standard errors of functions of correlation coefficients. Applied Psychological Measurement, 4, 385–398.CrossRefGoogle Scholar
Cochran, W. G. (1970). Some effects of errors of measurement on multiple correlations. Journal of the American Statistical Association, 65, 22–35.CrossRefGoogle Scholar
Cronbach, L. J. (1971). Test validation. In Thorndike, R. L. (Eds.), Educational measurement 2nd ed., (pp. 443–507). Washington, DC: American Council on Education.Google Scholar
Cureton, E. E. (1965). Reliability and validity: Basic assumptions and experimental designs. Educational and Psychological Measurement, 25, 327–346.CrossRefGoogle Scholar
Efron, B., Gong, G. (1983). A leisurely look at the bootstrap, the jackknife, and cross-validation. The American Statistician, 37, 36–48.CrossRefGoogle Scholar
Forsyth, R. A. (1967). An investigation of empirical sampling distributions of correlation coefficients corrected for attenuation. Unpublished doctoral dissertation, University of Iowa, Iowa City.Google Scholar
Forsyth, R. A., Feldt, L. S. (1969). An investigation of empirical sampling distributions of correlations corrected for attenuation. Educational and Psychological Measurement, 29, 61–72.CrossRefGoogle Scholar
Gulliksen, H. (1950). Theory of mental tests, New York: Wiley.CrossRefGoogle Scholar
Hakstian, A. R., Bennet, R. W. (1978). Validity studies using the Comprehensive Ability Battery (CAB): Relationships with the DAT and GATB. Educational and Psychological Measurement, 38, 1103–1115.CrossRefGoogle Scholar
Knuth, D. E. (1968). The art of computer programming: Vol. W. Seminumerical algorithms, Reading, MA: Addison-Wesley.Google Scholar
Kristof, W. (1982). Contributions to the analysis of correlation coefficients. Unpublished doctoral dissertation, University of South Africa, Pretoria.Google Scholar
Lord, F. M. (1957). A significance test for the hypothesis that two variables measure the same trait except for errors of measurement. Psychometrika, 22, 207–220.CrossRefGoogle Scholar
Lord, F. M. (1970). Problems arising from the unreliability of the measuring instrument. In DuBois, P. H., Mayo, G. D. (Eds.), Research strategies for evaluating training, Chicago: Rand McNally.Google Scholar
Lord, F. M., Novick, M. R. (1968). Statistical theories of mental test scores (pp. 79–93). Reading, MA: Addison-Wesley.Google Scholar
Magnusson, D., Backteman, G. (1978). Longitudinal stability of person characteristics: Intelligence and creativity. Applied Psychological Measurement, 2, 481–490.CrossRefGoogle Scholar
Marascuilo, L. A. (1966). Large-sample multiple comparisons. Psychological Bulletin, 65, 280–290.CrossRefGoogle ScholarPubMed
Morrison, D. F. (1976). Multivariate statistical methods 2nd ed.,, New York: McGraw-Hill.Google Scholar
Pearson, K., Filon, L. N. G. (1898). Mathematical contributions to the theory of evolution: IV. On the probable errors of frequency constants and on the influence of random selection on variation and correlation. Philosophical Transactions of the Royal Society of London, Series A, 191, 229–311.Google Scholar
Rao, C. R. (1973). Linear statistical inference and its applications 2nd ed.,, New York: Wiley.CrossRefGoogle Scholar
Rogers, W. T. (1976). Jackknifing disattenuated correlations. Psychometrika, 41, 121–133.CrossRefGoogle Scholar
Schmidt, F. L., Hunter, J. E., Caplan, J. R. (1981). Validity generalization results for two job groups in the petroleum industry. Journal of Applied Psychology, 66, 261–273.CrossRefGoogle Scholar
Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, 15, 72–101.CrossRefGoogle Scholar
Spearman, C. (1907). Demonstration of formulae for true measurement of correlation. American Journal of Psychology, 18, 161–169.CrossRefGoogle Scholar
Spearman, C. (1910). Correlation calculated from faulty data. British Journal of Psychology, 3, 271–295.Google Scholar
Steiger, J. H., Hakstian, A. R. (1982). The asymptotic distribution of elements of a correlation matrix: theory and application. British Journal of Mathematical and Statistical Psychology, 35, 208–215.CrossRefGoogle Scholar