# MEASUREMENT INVARIANCE: CONCEPT AND IMPLEMENTATION

## Main Article Content

## Abstract

An empirical evidence for independent samples of a population regarding measurement invariance implies that factor structure of a measurement tool is equal across these samples; in other words, it measures the intended psychological trait within the same structure. In this case, the evidence of construct validity would be strengthened within the frame of the scores obtained from the tool. When measurement invariance is not supported, the researchers should consider the possibility of the different factor designs for each group. Ignoring such a situation brings forward the probability about differentiation of the trait(s) measured by measurement tool for that/those group(s), so it causes to suspect the validity of the scores obtained from the tool. The aim of this study is to examine measurement invariance in the context of the conceptual foundations of multi-group confirmatory factor analysis, and discuss the subject through the results from two hypothetical data set that one supports measurement invariance, but the other does not. As a result of analysis performed in this direction, it is determined that the five-factor design derived from the first data set is equal across the groups in the majors of science, health, and social science. It is also concluded that the three-factor design obtained from the secondary data set is not equal for female and male groups. Besides, the exploratory factor analysis performed for female and male groups separately shows that the three-factor design of the tool is valid for females, but the number of factors was four in males. When the factor design for male group is examined, it is determined that the three items in the second factor separate significantly. That leads to the conclusion that it is crucial to test measurement invariance in studies regarding the determination of the psychometric properties of the tool.

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## References

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Dunn, G., Everitt, B. & Pickles, A. (1993). Modeling covariances and latent variables using EQS. (First Edition). London: Chapman & Hall.

Everitt, B. S. & Howell, D. C. (2005). Encyclopedia of statistics in behavioral sciences. Chichester: John Wiley & Sons, Ltd.

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Harrington, D. (2009). Confirmatory factor analysis. (First Edition). New York: Oxford University Press, Inc.

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Jöreskog, K. G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36, 409-426.

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Kline, P. (2000). The handbook of psychological testing. (Second Edition). London: Taylor & Francis Group.

Kline, R. B. (2005). Principles and practice of structural equation modeling. (Second Edition). New York: Guilford Publications, Inc.

Lee, S. Y. & Leung, T. K. (1982). Covariance structure analysis in several populations. Psychometrika, 47, 297-308.

Marcoulides, G. A. & Schumacker, R. E. (1996). Advanced structural equation modeling: ıssues and techniques. (First Edition). New Jersey: Lawrence Erlbaum Associates, Inc.

Maruyama, G. M. (1998). Basics of structural equation modeling. (First Edition). Thousand Oaks, CA: SAGE Publications, Inc.

Meade, A. W., Michels, L. C., & Lautenschlager, G. J. (2007). Are internet and paper-and-pencil personality tests truly comparable? An experimental design measurement invariance study. Organizational Research Methods, 10, 322-345.

Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58, 525-543.

Nunnaly, J. C. & Bernstein, I. H. (1994). Psychometric theory. (Third Edition). New York: McGraw-Hill, Inc.

Raykov, T. & Marcoulides, G. A. (2008). An introduction to applied multivariate analysis (First Edition). New York: Taylor & Francis Group.

Rosenthal, R & Rosnow, R. L. (2008). Essential of behavioral research. (Third Edition). New York: McGraw-Hill, Inc.

Sass, D. A., Schmitt, T. A. & Marsh H. W. (2014). Evaluating model fit with ordered categorical data within a measurement invariance framework: A comparison of estimators. Structural Equation Modeling: A Multidisciplinary Journal, 21, 167-180.

Satorra, A. & Bentler, P. M. (2011). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika: A Journal of Quantitative Psychology, 66(4), 507-514.

Schmitt, N. & Kuljanin, G. (2008). Measurement invariance: Review of practice and implications. Human Resource Management Review, 18, 210-222.

Schumacker, R. E. & Lomax, R. G. (1996). A beginner’s guide to structural equation modeling. (First Edition). New Jersey: Lawrence Erlbaum Associates, Inc.

Spini, D. (2003). Measurement equivalence of 10 value types from the Schwartz value survey across 21 countries. Journal of Cross-Cultural Psychology, 34(1), 3-23.

Stapleton, C. D. (1997). Basic contepts and procedures of confirmatory factor analysis. Austin: The Annual Meeting of the Southwest Educational Research Association.

Steiger, J. H. (2007). Understanding the limitations of global fit assessment in structural equation modeling. Personality and Individual Differences, 42, 893-898.

Stevens, J. (1996). Applied multivariate statistics for social sciences (Third Edition). New Jersey: Lawrence Erlbaum Associates, Inc.

Tatlıdil, H. (1992). Uygulamalı çok değişkenli istatistiksel analiz. (Birinci Baskı) Ankara: Engin Yayınları.

Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications. (First Edition). Washington: American Psychological Association.

Toit, M. & Toit, S. (2001). Interactive LISREL: User’s guide. Lincolnwood: Scientific Software International, Inc.

Tucker, L. R. & MacCallum, R. C. (1997). Exploratory factor analysis. (Online Edition) Web: http://quantrm2.psy.ohio-state.edu/maccallum/factornew.htm adresinden 19 Şubat 2007’de alınmıştır.

Ullman, J. B. (2001). Structural equation modeling. In B. G. Tabachnick & L. S. Fidell (Eds.), Using multivariate statistics, pp. 653-771. Needham Heights, MA: Allyn & Bacon.

Urbina, S. (2004). Essentials of psychological testing. (First Edition). New Jersey: Wiley & Sons, Inc.

Vandenberg, R. J. & Lance, C. E. (2000). A Review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational Research Methods, 3(1), 4-70.

Westen, D. & Rosenthal, R. (2005). Improving construct validity: Cronbach, Meehl, and neurath’s ship. Psychological Assessment, 17(4), 409-41.

Widaman, K. F., & Reise, S. P. (1997). Exploring the measurement invariance of psychological instruments: Applications in the substance use domain. In K. J. Bryant, M. Windle, & S. G. West (Eds.), The science of prevention: Methodological advances from alcohol and substance abuse research, (pp. 281-324). Washington, DC: American Psychological Association.

Wu. A. D., Li, Z. & Zumbo, B. D. (2007). Decoding the meaning of factorial invariance and updating the practice of multigroup confirmatory factor analysis: A demonstration with TIMSS data. Practical Assessment, Research and Evaluation, 12(3), 1-26.

Brown, T. A. (2006). Confirmatory factor analysis for applied research. (First Edition). New York: Guilford Publications, Inc.

Büyüköztürk, Ş. (2002). Faktör analizi: Temel kavramlar ve ölçek geliştirmede kullanımı. Kuram ve Uygulamada Eğitim Yönetimi, 32, 470-483.

Büyüköztürk, Ş., Kılıç Çakmak, E., Akgün, Ö. E., Karadeniz, Ş. ve Demirel, F. (2012). Bilimsel araştırma yöntemleri. (11. Baskı). Ankara: Pegem Akademi.

Büyüköztürk, Ş. (2014). Sosyal bilimler için veri analizi el kitabı: İstatistik, araştırma deseni, SPSS uygulamaları ve yorum. (On Dokuzuncu Baskı). Ankara: Pegem Yayıncılık.

Byrne, B. M. (2010). Structural equation modeling with AMOS: Basic concepts, applications, and programming. (Second Edition). New York: Taylor & Francis Group.

Byrne, B. M. (2008). Testing for multigroup equivalence of a measuring instrument: A walk through the process. Psicothema, 20(4), 872-882.

Byrne, B. M. (2006). Structural equation modeling with EQS and EQS/Windows: Basic concepts, applications, and programming. (Second Edition). California: Sage Publications, Inc.

Byrne, B. M. & Stewart, S. M. (2006). Teacher's corner: The MACS approach to testing for multigroup invariance of a second-order structure: A walk through the process. Structural Equation Modeling: A Multidisciplinary Journal, 13(2), 287-321.

Chen, F. F. (2007). Sensitivity of goodness of fit indexes to lack of measurement invariance. Structural Equation Modeling, 14(3), 464-504.

Cheung, G., & Rensvold, R. (2000). Testing measurement invariance using critical values of fit indices: A Monte Carlo study. Retrieved May 20, 2004, from http://www.aom.pace.edu/rmd/cheung_files/ cheung.htm

Cheung, G. W. & Rensvold, R. B. (2002). Evaluating goodness–of–fit indexes for testing measurement invariance. Structural Equation Modeling, 9 (2), 233-255.

Crocker, L. & Algina, J. (1986). Introduction to classical and modern test theory. (First Edition). Orlando: Holt, Rinehart and Winston, Inc.

Cronbach, L. J. & Meehl, P. E. (1955). Construct validity in psychological tests. Psychological Bulletin, 52, 281-302.

Diekhoff, G. (1992). Statistics for the social and behavioral sciences: Univariate, bivariate, and multivariate. (First Edition). Dubuque, IA: William C. Brown Publishers.

Dimitrov, D. M. (2010). Testing for factorial invariance in the context of construct validation. Measurement and Evaluation in Counseling and Development, 43, 121-149.

Dunn, G., Everitt, B. & Pickles, A. (1993). Modeling covariances and latent variables using EQS. (First Edition). London: Chapman & Hall.

Everitt, B. S. & Howell, D. C. (2005). Encyclopedia of statistics in behavioral sciences. Chichester: John Wiley & Sons, Ltd.

Floyd, F. J. & Widaman, K. F. (1995). Factor analysis in the development and refinement of clinical assessment instruments. Psychological Assessment, 7(3), 286-299.

Gable, R. K. & Wolf, M. B. (2001). Instrument development in the affective domain: Measuring attitudes, and values in corporate and scholl settings. (Second Edition). London: Kluwer Academic Publishers.

Green, S. B., Salkind, N. J. & Akey, T. M. (1997). Using SPSS for windows: Analyzing and understanding data. New Jersey: Prentice Hall, Inc.

Gregorich, S. (2006). Do self report instruments allow meaningful comparisons across diverse population groups? Testing measurement invariance using the confirmatory factor analysis framework. Medical Care, 44(11-3), 78-94.

Gorsuch, R. L. (1974). Factor analysis. (First Edition). Philadelphia: W. B. Saunders Company.

Harrington, D. (2009). Confirmatory factor analysis. (First Edition). New York: Oxford University Press, Inc.

Jonson, J. L. & Plake, B. S. (1998). A historical comparison of validity standards and validity practices. Educational and Psychological Measurement, 58(5), 736-754.

Jöreskog, K. G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36, 409-426.

Jöreskog, K. G. & Sörbom, D. (1993). LISREL 8: Structural equation modeling with the SIMPLIS command language. Lincolnwood: Scientific Software International, Inc.

Jöreskog, K. G. & Sörbom, D. (2001). LISREL 8: User’s reference guide. Lincolnwood: Scientific Software International, Inc.

Jöreskog, K. G., Sörbom, D., Toit, M. & Toit, S. (2000). LISREL 8: New statistical features. Lincolnwood: Scientific Software International, Inc.

Kline, P. (2000). The handbook of psychological testing. (Second Edition). London: Taylor & Francis Group.

Kline, R. B. (2005). Principles and practice of structural equation modeling. (Second Edition). New York: Guilford Publications, Inc.

Lee, S. Y. & Leung, T. K. (1982). Covariance structure analysis in several populations. Psychometrika, 47, 297-308.

Marcoulides, G. A. & Schumacker, R. E. (1996). Advanced structural equation modeling: ıssues and techniques. (First Edition). New Jersey: Lawrence Erlbaum Associates, Inc.

Maruyama, G. M. (1998). Basics of structural equation modeling. (First Edition). Thousand Oaks, CA: SAGE Publications, Inc.

Meade, A. W., Michels, L. C., & Lautenschlager, G. J. (2007). Are internet and paper-and-pencil personality tests truly comparable? An experimental design measurement invariance study. Organizational Research Methods, 10, 322-345.

Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58, 525-543.

Nunnaly, J. C. & Bernstein, I. H. (1994). Psychometric theory. (Third Edition). New York: McGraw-Hill, Inc.

Raykov, T. & Marcoulides, G. A. (2008). An introduction to applied multivariate analysis (First Edition). New York: Taylor & Francis Group.

Rosenthal, R & Rosnow, R. L. (2008). Essential of behavioral research. (Third Edition). New York: McGraw-Hill, Inc.

Sass, D. A., Schmitt, T. A. & Marsh H. W. (2014). Evaluating model fit with ordered categorical data within a measurement invariance framework: A comparison of estimators. Structural Equation Modeling: A Multidisciplinary Journal, 21, 167-180.

Satorra, A. & Bentler, P. M. (2011). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika: A Journal of Quantitative Psychology, 66(4), 507-514.

Schmitt, N. & Kuljanin, G. (2008). Measurement invariance: Review of practice and implications. Human Resource Management Review, 18, 210-222.

Schumacker, R. E. & Lomax, R. G. (1996). A beginner’s guide to structural equation modeling. (First Edition). New Jersey: Lawrence Erlbaum Associates, Inc.

Spini, D. (2003). Measurement equivalence of 10 value types from the Schwartz value survey across 21 countries. Journal of Cross-Cultural Psychology, 34(1), 3-23.

Stapleton, C. D. (1997). Basic contepts and procedures of confirmatory factor analysis. Austin: The Annual Meeting of the Southwest Educational Research Association.

Steiger, J. H. (2007). Understanding the limitations of global fit assessment in structural equation modeling. Personality and Individual Differences, 42, 893-898.

Stevens, J. (1996). Applied multivariate statistics for social sciences (Third Edition). New Jersey: Lawrence Erlbaum Associates, Inc.

Tatlıdil, H. (1992). Uygulamalı çok değişkenli istatistiksel analiz. (Birinci Baskı) Ankara: Engin Yayınları.

Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications. (First Edition). Washington: American Psychological Association.

Toit, M. & Toit, S. (2001). Interactive LISREL: User’s guide. Lincolnwood: Scientific Software International, Inc.

Tucker, L. R. & MacCallum, R. C. (1997). Exploratory factor analysis. (Online Edition) Web: http://quantrm2.psy.ohio-state.edu/maccallum/factornew.htm adresinden 19 Şubat 2007’de alınmıştır.

Ullman, J. B. (2001). Structural equation modeling. In B. G. Tabachnick & L. S. Fidell (Eds.), Using multivariate statistics, pp. 653-771. Needham Heights, MA: Allyn & Bacon.

Urbina, S. (2004). Essentials of psychological testing. (First Edition). New Jersey: Wiley & Sons, Inc.

Vandenberg, R. J. & Lance, C. E. (2000). A Review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational Research Methods, 3(1), 4-70.

Westen, D. & Rosenthal, R. (2005). Improving construct validity: Cronbach, Meehl, and neurath’s ship. Psychological Assessment, 17(4), 409-41.

Widaman, K. F., & Reise, S. P. (1997). Exploring the measurement invariance of psychological instruments: Applications in the substance use domain. In K. J. Bryant, M. Windle, & S. G. West (Eds.), The science of prevention: Methodological advances from alcohol and substance abuse research, (pp. 281-324). Washington, DC: American Psychological Association.

Wu. A. D., Li, Z. & Zumbo, B. D. (2007). Decoding the meaning of factorial invariance and updating the practice of multigroup confirmatory factor analysis: A demonstration with TIMSS data. Practical Assessment, Research and Evaluation, 12(3), 1-26.