Hostname: page-component-89b8bd64d-z2ts4 Total loading time: 0 Render date: 2026-05-08T23:23:09.630Z Has data issue: false hasContentIssue false
  • English
  • Français

Fisher Transformations for Correlations Corrected for Selection and Missing Data

Published online by Cambridge University Press:  01 January 2025

Jorge L. Mendoza
Jorge L. Mendoza*
Affiliation:
University of Oklahoma
*
Requests for reprints should be sent to Jorge L. Mendoza, Department of Psychology, Dale Hall, University of Oklahoma, Norman OK 73019.
Get access Rights & Permissions [Opens in a new window]

Abstract

The validity of a test is often estimated in a nonrandom sample of selected individuals. To accurately estimate the relation between the predictor and the criterion we correct this correlation for range restriction. Unfortunately, this corrected correlation cannot be transformed using Fisher's Z transformation, and asymptotic tests of hypotheses based on small or moderate samples are not accurate. We developed a Fisher r to Z transformation for the corrected correlation for each of two conditions: (a) the criterion data were missing due to selection on the predictor (the missing data were MAR); and (b) the criterion was missing at random, not due to selection (the missing data were MCAR). The two Z transformations were evaluated in a computer simulation. The transformations were accurate, and tests of hypotheses and confidence intervals based on the transformations were superior to those that were not based on the transformations.

Keywords

selectionmissing datarestriction in in rangevalidationcorrelationZ transformation

Information

Type
Original Paper
Information
Psychometrika , Volume 58 , Issue 4 , December 1993 , pp. 601 - 615
Copyright
Copyright © 1993 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

I would like to thank Alan Nicewander for his invaluable help in this project. Also, I would like to thank the referees and the editors for their thoughtful comments.

References

Anderson, T. W. (1957). Maximum likelihood estimates for the multivariate normal distribution when some of the observations are missing. Journal of the American Statistical Association, 52, 200–203.CrossRefGoogle Scholar
Anderson, T. W. (1984). An introduction to multivariate statistical analysis 2nd ed.,, New York, NY: Wiley & Sons.Google Scholar
Bobko, P., & Rieck, A. (1980). Large sample estimates for standard errors of functions of correlation coefficients. Applied Psychological Measurement, 4, 385–398.CrossRefGoogle Scholar
Cohen, A. C. (1955). Restriction and selection in samples from bivariate normal distributions. Journal of the American Statistical Association, 50, 884–893.Google Scholar
Cohen, A. C. (1991). Truncated and censored samples: Theory and applications, New York, NY: Marcel Dekker.Google Scholar
Ghiselli, E. E. (1966). The validity of occupational aptitude tests, New York, NY: Wiley.Google Scholar
Gross, A. L. (1990). A maximum likelihood approach to test validation with missing and censored dependent variables. Psychometrika, 3, 533–549.CrossRefGoogle Scholar
Heckman, J. J. (1979). Sample selection bias as a specification error. Econometrica, 47, 153–161.CrossRefGoogle Scholar
IMSL Library (1987). User's manual stat/library, Houston, TX: Author.Google Scholar
Little, R. J. A., & Rubin, D. B. (1987). Statistical analysis with missing data, New York, NY: John Wiley & Sons.Google Scholar
Lord, F. M., & Novick, M. R. (1968). Statistical theories of mental test scores, Reading, MA: Addison-Wesley.Google Scholar
Pearson, K. (1903). Mathematical contributions to the theory of evolution XI. On the influence of natural selection on the variability and correlation of organs. Philosophical Transactions of the Royal Society, London, Series A, 200, 1–66.CrossRefGoogle Scholar
Shohoji, T., Yamashita, Y., & Tarumi, T. (1981). Generalization of Fisher's z-transformation. Journal of Statistical Planning and Inference, 5, 347–354.CrossRefGoogle Scholar
Wolfram, Stephen (1991). Mathematica: A system for doing mathematics by computers, Reading, MA: Addison-Wesley.Google Scholar