Abstract
Bootstrap and jackknife techniques are used to estimate ellipsoidal confidence regions of group stimulus points derived from INDSCAL. The validity of these estimates is assessed through Monte Carlo analysis. Asymptotic estimates of confidence regions based on a MULTISCALE solution are also evaluated. Our findings suggest that the bootstrap and jackknife techniques may be used to provide statements regarding the accuracy of the relative locations of points in space. Our findings also suggest that MULTISCALE asymptotic estimates of confidence regions based on small samples provide an optimistic view of the actual statistical reliability of the solution.
Similar content being viewed by others
References
Carroll, J. D. & Arabie, P. (1980). Multidimensional scaling. In M. R. Rosenzweig & L. W. Porter (Eds.),Annual Review of Psychology. Palo Alto, CA: Annual Reviews.
Carroll, J. D. & Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via anN-way generalization of Eckart-Young decomposition.Psychometrika, 35, 283–319.
Carroll, J. D. & Wish, M. (1974). Models and methods for three-way multidimensional scaling. In D. H. Krantz, R. C. Atkinson, R. D. Luce, & P. Suppes (Eds.),Contemporary developments in mathematical psychology, Vol. II. San Francisco: Freeman.
Clarkson, D. B. (1979). Estimating the standard errors of rotated factor loadings jackknifing.Psychometrika, 44, 297–314.
Efron, B. (1979). Bootstrap methods: Another look at the jackknife.Annals of Statistics, 7, 1–26.
Efron, B. (1982). The jackknife, the bootstrap and other resampling plans.SIAM, monograph #38, CBMS-NSF.
Efron, B. & Gong, G. (1983). A leisurely look at the bootstrap, the jackknife, and cross-validation.The American Statistician, 37, 36–48.
Hartigan, J. A. (1969). Using subsample values as typical values.Journal of the American Statistical Association, 64, 1303–1317.
Heiser, W. (1981). Unfolding analysis of proximity data. Ph.D. dissertation, Department of Datatheory, University of Leiden.
Horan, C. B. (1969). Multidimensional scaling: Combining observations when individuals have different perceptual structures.Psychometrika, 34, 139–165.
Miller, R. G. (1974). The jackknife—a review.Biometrika, 61, 1–15.
Mosteller, F. & Tukey, J. W. (1968). Data analysis, including statistics. In G. Lindzey and E. Aronson (Eds.),Handbook of Social Psychology. Reading, Mass: Addison-Wesley.
Pennell, R. (1972). Routinely computable confidence intervals for factor loadings using the “jackknife.”British Journal of Mathematical and Statistical Psychology, 25, 107–114.
Quenouille, M. H. (1949). Approximate tests of correlation in time series.Journal of the Royal Statistical Society, Ser. B,11, 18–84.
Quenouille, M. H. (1956). Notes on bias in estimation.Biometrika, 43, 353–360.
Ramsay, J. O. (1968). Economical method of analyzing perceived color differences.Journal of the Optical Society of America, 58, 19–22.
Ramsay, J. O. (1977). Maximum likelihood estimation in multidimensional scaling.Psychometrika, 42, 241–266.
Ramsay, J. O. (1978a). Confidence regions for multidimensional scaling analysis.Psychometrika, 43, 145–160.
Ramsay, J. O. (1978b).MULTISCALE: Four programs for multidimensional scaling by the method of maximum likelihood. Chicago: National Educational Resources, Inc.
Ramsay, J. O. (1980). Some small sample results for maximum likelihood estimation in multidimensional scaling.Psychometrika, 45, 139–144.
Ramsay, J. O. (1982a).MULTISCALE II Manual.
Ramsay, J. O. (1982b). Some statistical approaches to multidimensional scaling data.Journal of the Royal Statistical Society, Ser. A,145, 285–312.
Tatsuoka, M. M. (1971).Multivariate analysis: Techniques for educational and psychological research. New York: Wiley.
Torgerson, W. S. (1958).Theory and methods of scaling. New York: Wiley.
Tukey, J. W. (1958). Bias and confidence in not-quite so large samples (Abstract).Annals of Mathematical Statistics, 29, 614.
Weinberg, S. L., Carroll, J. D., & Cohen, H. S. (1981).Estimating confidence regions for INDSCAL-derived group stimulus points using jackknifing. Unpublished manuscript, Bell Laboratories.
Weinberg, S. L., Carroll, J. D., & Cohen, H. S. (1982).Confidence regions for INDSCAL using the jackknife and bootstrap techniques. Unpublished manuscript, Bell Laboratories.
Wish, M. & Carroll, J. D. (1982). Multidimensional scaling and its applications. In P. R. Krishnaiah and L. N. Kanal (Eds.),Handbook of Statistics, Vol. 2, Amsterdam: North Holland Publishing Company.
Author information
Authors and Affiliations
Additional information
The authors wish to thank Geert DeSoete, Richard A. Harshman, William Heiser, Jon Kettenring, Joseph B. Kruskal, Jacqueline Meulman, James O. Ramsay, John W. Tukey, Paul A. Tukey, and Mike Wish.
Sharon L. Weinberg is a consultant at AT&T Bell Laboratories, Murray Hill, New Jersey 07974.
Rights and permissions
About this article
Cite this article
Weinberg, S.L., Carroll, J.D. & Cohen, H.S. Confidence regions for INDSCAL using the jackknife and bootstrap techniques. Psychometrika 49, 475–491 (1984). https://doi.org/10.1007/BF02302586
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02302586