Behavior Research Methods

, Volume 39, Issue 2, pp 318–325 | Cite as

Applying the bootstrap to the multivariate case: Bootstrap component/factor analysis

  • Linda Reichwein Zientek
  • Bruce Thompson


The bootstrap method, which empirically estimates the sampling distribution for either inferential or descriptive sstatistical purposes, can be applied to the multivariate case. When conducting bootstrap component, or factor, analysis, resampling results must be located in a common factor space before summary statistics for each estimated parameter can be computed. The present article describes a strategy for applying the bootstrap method to conduct either a bootstrap component or a factor analysis with a program syntax for SPSS. The Holzinger–Swineford data set is employed to make the discussion more concrete.


Bootstrap Method Multivariate Case Bootstrap Estimate Target Matrix Multivariate Behavioral Research 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Anderson, D. R., Burnham, K. P., &Thompson, W. (2000). Null hypothesis testing: Problems, prevalence, and an alternative.Journal of Wildlife Management,64, 912–923.CrossRefGoogle Scholar
  2. Capraro, M. M. (2005). An introduction to confidence intervals for both statistical estimates and effect sizes.Research in the Schools,12, 22–32.Google Scholar
  3. Carver, R. P. (1978). The case against statistical significance testing.Harvard Educational Review,48, 378–399.Google Scholar
  4. Chatterjecc, S. (1984). Variance estimation in factor analysis: An application of the bootstrap.British Journal of Mathematical & Statistical Psychology,37, 252–262.Google Scholar
  5. Cohen, J. (1994). The earth is round (p<.05).American Psychologist,49, 997–1003.CrossRefGoogle Scholar
  6. Diaconis, P., & Efron, B. (1983). Computer-intensive methods in statistics.Scientific American,248, 116–130.CrossRefGoogle Scholar
  7. Gorsuch, R. L. (1983).Factor analysis (2nd ed.). Hillsdale, NJ: Erlbaum.Google Scholar
  8. Guthrie, A. C. (2001, February).Using bootstrap methods with popular statistical programs. Paper presented at the annual meeting of the Southwest Educational Research Association, New Orleans.Google Scholar
  9. Guttman, L. (1954). Some necessary conditions for common-factor analysis.Psychometrika,19, 149–161.CrossRefGoogle Scholar
  10. Harlow, L. L., Mulaik, S. A., &Steiger, J. H. (Eds.), (1997).What if there were no significance tests? Mahwah, NJ: Erlbaum.Google Scholar
  11. Henson, R. K., &Roberts, J. K. (2007). Use of exploratory factor analysis in published research: Common errors and some comment on improved practice.Educational & Psychological Measurement,66, 393–416.CrossRefGoogle Scholar
  12. Holzinger, K. J., &Swineford, F. (1939).A study in factor analysis: The stability of a bi-factor solution. Chicago: University of Chicago Press.Google Scholar
  13. Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis.Psychometrika,30, 179–185.CrossRefPubMedGoogle Scholar
  14. Ichikawa, M., &Konishi, S. (1995). Application of the bootstrap method in factor analysis.Psychometrika,60, 77–93.CrossRefGoogle Scholar
  15. Kline, R. B. (2004).Beyond significance testing: Reforming data analysis methods in behavioral research. Washington, DC: American Psychological Association.CrossRefGoogle Scholar
  16. Kline, R. B. (2005).Principles and practice of structural equation modeling (2nd ed.). New York: Guilford.Google Scholar
  17. Lambert, Z. V., Wildt, A. R., &Durand, R. M. (1991). Approximating confidence intervals for factor loadings.Multivariate Behavioral Research,26, 421–434.CrossRefGoogle Scholar
  18. Lorenzo-Seva, U., &Ferrando, P. J. (2003). IMINCE: An unrestricted factor-analysis-based program for assessing measurement invariance.Behavior Research Methods, Instruments, & Computers,35, 318–321.CrossRefGoogle Scholar
  19. Mulaik, S. A. (edEd.) (1992). Theme issue on principal components analysis.Multivariate Behavioral Research,27 (3).Google Scholar
  20. Pedhazur, E. J. (1997).Multiple regression in behavioral research: Explanation and prediction (3rd ed.). Stamford, CT: Thomson Learning.Google Scholar
  21. Raykov, T., &Little, T. D. (1999). A note on Procrustes rotation in exploratory factor analysis: A computer intensive approach to goodness-of-fit evaluation.Educational & Psychological Measurement,59, 47–57.Google Scholar
  22. Scott, R. L., Sexton, D., &Thompson, B. (1989). Structure of a short form of the questionnaire on resources and stress: A bootstrap factor analysis.Educational & Psychological Measurement,49, 409–419.CrossRefGoogle Scholar
  23. Smith, A. D., & Henson, R. K. (2000, January).State of the art in statistical significance testing: A review of the APA Task Force on Statistical Inference. Paper presented at the annual meeting of the Southwest Educational Research Association, Dallas, TX.Google Scholar
  24. Steiger, J. H. (2004). Beyond theF test: Effect size, confidence intervals, and tests of close fit in the analysis of variance and contrast analysis.Psychological Methods,9, 164–182.CrossRefPubMedGoogle Scholar
  25. Thompson, B. (1988). Program FACSTRAP: A program that computes bootstrap estimates of factor structure.Educational & Psychological Measurement,48, 681–686.CrossRefGoogle Scholar
  26. Thompson, B. (1993). The use of statistical significance tests in research: Bootstrap and other alternatives.Journal of Experimental Education,61, 361–377.Google Scholar
  27. Thompson, B. (1994). The pivotal role of replication in psychological research: Empirically evaluating the replicability of sample results.Journal of Personality,62, 157–176.CrossRefGoogle Scholar
  28. Thompson, B. (1995). Exploring the replicability of a study’s results: Bootstrap statistics for the multivariate case.Educational & Psychological Measurement,55, 84–94.CrossRefGoogle Scholar
  29. Thompson, B. (1996). AERA editorial policies regarding statistical significance testing: Three suggested reforms.Educational Researcher,25(2), 26–30.Google Scholar
  30. Thompson, B. (2002). “Statistical,” “practical,” and “clinical”: How many kinds of significance do counselors need to consider?Journal of Counseling & Development,80, 64–71.Google Scholar
  31. Thompson, B. (2004).Exploratory and confirmatory factor analysis: Understanding concepts and applications. Washington, DC: American Psychological Association.CrossRefGoogle Scholar
  32. Thompson, B. (2006).Foundations of behavioral statistics: An insightbased approach. New York: Guilford.Google Scholar
  33. Wilkinson, L., &Task Force on Statistical Inference (1999). Statistical methods in psychology journals: Guidelines and explanations.American Psychologist,54, 594–604.CrossRefGoogle Scholar
  34. Zwick, W. R., &Velicer, W. F. (1986). Factors influencing five rules for determining the number of components to retain.Psychological Bulletin,99, 432–442.CrossRefGoogle Scholar

Copyright information

© Psychonomic Society, Inc. 2007

Authors and Affiliations

  • Linda Reichwein Zientek
    • 1
  • Bruce Thompson
    • 2
    • 3
  1. 1.Department of MathematicsSam Houston State UniversityHuntsville
  2. 2.Texas A&M UniversityCollege Station
  3. 3.Baylor College of MedicineHouston

Personalised recommendations