Research Article
BibTex RIS Cite

Kategorik Veride Faktör Analizi İçin Kullanılabilecek Alternatif Bir Korelasyon Matrisi: Goodman-Kruskal Gamma

Year 2021, Volume: 54 Issue: 54, 151 - 168, 30.06.2021
https://doi.org/10.15285/maruaebd.853905

Abstract

Açımlayıcı faktör analizi (AFA) sosyal bilimler alanında ölçeklerden elde edilen verilerin yapı geçerliğine yönelik kanıt toplama sürecinde sıklıkla kullanılmaktadır. Veriler kategorik olduğunda polikorik/tetrakorik korelasyon matrisiyle analizler gerçekleştirilirken veriler sürekli olduğunda Pearson korelasyon matrisiyle analizler gerçekleştirilmektedir. Ancak bazı durumlarda polikorik korelasyon matrisi kullanıldığında modelde yakınsama sağlanamamakta Pearson korelasyon matrisi kullanıldığında ise faktör yükleri olması gerekenden daha düşük kestirilmektedir. Bu nedenle polikorik ve Pearson korelasyon matrisine alternatif olarak Goodman-Kruskal Gamma ve Lambda katsayılarıyla gerçekleştirilen AFA sonuçlarının karşılaştırılması çalışmanın amacını oluşturmaktadır. Bu amaçla gerçekleştirilen Monte Carlo simülasyon çalışmasında; kategori sayısı, ortalama faktör yükü, örneklem büyüklüğü ve verilerin dağılımı değişkenleri simülasyon koşulu olarak belirlenmiştir. Araştırma sonucunda bazı koşullarda polikorik korelasyon matrisiyle gerçekleştirilemeyen AFA kestirimlerinin Goodman-Kruskal Gamma katsayısıyla oluşturulan matrisle yapılabildiği gözlenmiştir. Lambda katsayısı kullanıldığında ise bazı koşullarda kestirim yapılamamıştır. Kestirim yapılan koşullarda ise genellikle faktör yükleri olduğundan düşük kestirilmiştir. Kategori sayısının artmasıyla Goodman-Kruskal Gamma katsayısından elde edilen sonuçların daha az yanlı olduğu gözlenmiştir. Araştırmacılara kategorik veriyle AFA gerçekleştirilirken Goodman-Kruskal Gamma katsayısından elde edilecek sonuçların da incelemesi önerilebilir.

References

  • Bandalos, D. L. ve Leite, W. (2013). Use of Monte Carlo studies in structural equation modeling research. G. R. Hancock ve R. O. Mueller (Ed.), Structural equation modeling: A second course içinde (2nd ed.). Charlotte, NC: Information Age.
  • Baykul, Y. (2010). Eğitimde ve psikolojide ölçme: Klasik test teorisi ve uygulaması (2. Baskı.). Ankara: Pegem Akademi.
  • Beauducel, A. ve Herzberg, P. Y. (2006). On the performance of maximum likelihood versus means and variance adjusted weighted least squares estimation in CFA. Structural Equation Modeling: A Multidisciplinary Journal, 13(2), 186–203. doi:10.1207/s15328007sem1302_2
  • Brown, T. A. (2015). Confirmatory factor analysis for applied research (2nd ed.). New York: The Guilford.
  • Büyüköztürk, Ş. (2013). Sosyal bilimler için veri analizi el kitabı: İstatistik, araştırma deseni, SPSS uygulamaları ve yorum (18. Baskı.). Ankara: Pegem Akademi.
  • Çokluk, Ö., Şekercioğlu, G. ve Büyüköztürk, Ş. (2012). Sosyal bilimler için çok değişkenli istatistik SPSS ve LISREL uygulamaları (2. Baskı.). Ankara: Pegem Akademi.
  • Cooper, C. (2019). Psychological testing: Theory and practice. Abingdon, Oxon: Routledge.
  • Costello, A. B. ve Osborne, J. W. (2005). Best practices in exploratory factor analysis: Four recommendations for getting the most from your analysis. Practical Assessment, Research & Evaluation, 10(7), 27–29. doi:10.1.1.110.9154
  • Crocker, L. ve Algina, J. (2008). Introduction of classical and modern test theory. Ohio: Cengage Learning.
  • Depaoli, S. ve Scott, S. (2015). Frequentist and bayesian estimation of CFA measurement models with mixed item response types: A monte carlo investigation. Structural Equation Modeling: A Multidisciplinary Journal, (September), 1–16. doi:10.1080/10705511.2015.1044653
  • Erkuş, A. (2014). Psikolojide ölçme ve ölçek geliştirme-I: Temel kavramlar ve işlemler (2nd ed.). Ankara: Pegem Akademi.
  • Finney, S. J. ve DiStefano, C. (2013). Nonnormal and categorical data in structural equation modeling. G. R. Hancock ve R. O. Mueller (Ed.), Structural equation modeling: A second course içinde (2nd ed., ss. 439–492). Charlotte, NC: IAP.
  • Flora, D. B. ve Curran, P. J. (2004). An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data. Psychological Methods, 9(4), 466–491. doi:10.1037/1082-989X.9.4.466
  • Flora, D. B., Finkel, E. J. ve Foshee, V. A. (2003). Higher order factor structure of a self-control test: Evidence from confirmatory factor analysis with polychoric correlations. Educational and Psychological Measurement, 63(1), 112–127. doi:10.1177/0013164402239320
  • Forero, C. G., Maydeu-Olivares, A. ve Gallardo-Pujol, D. (2009). Factor analysis with ordinal indicators: A monte carlo study comparing DWLS and ULS estimation. Structural Equation Modeling: A Multidisciplinary Journal, 16(4), 625–641. doi:10.1080/10705510903203573
  • Goodman, L. A. ve Kruskal, W. H. (1954). Measures of association for cross classifications. Journal of the American Statistical Association, 49(268), 732–764. doi:10.1080/01621459.1954.10501231
  • Goodman, L. A. ve Kruskal, W. H. (1979). Measures of Association for Cross Classifications. Springer Series in Statistics. New York, NY: Springer. doi:10.1007/978-1-4612-9995-0
  • Gorsuch, R. L. (1974). Factor analysis. Toronto: W. B. Saunders.
  • Hair, J. F., Black, W. C., Babin, B. J. ve Anderson, R. E. (2009). Multivariate data analysis (7th ed.). Boston: Pearson.
  • Harrington, D. (2009). Confirmatory factor analysis. New York: Oxford University.
  • Harwell, M., Stone, C. A., Hsu, T.-C. ve Kirisci, L. (1996). Monte carlo studies in item response theory. Applied Psychological Measurement, 20(2), 101–125. doi:10.1177/014662169602000201
  • Hauke, J. ve Kossowski, T. (2011). Comparison of values of Pearson’s and Spearman’s correlation coefficients on the same sets of data. Quaestiones Geographicae, 30(2), 87–93. doi:10.2478/v10117-011-0021-1
  • Holgado–Tello, F. P., Chacón–Moscoso, S., Barbero–García, I. ve Vila–Abad, E. (2010). Polychoric versus Pearson correlations in exploratory and confirmatory factor analysis of ordinal variables. Quality & Quantity, 44(1), 153–166. doi:10.1007/s11135-008-9190-y
  • Jöreskog, K. G. (1994). On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika, 59(3), 381–389. doi:10.1007/BF02296131
  • Jöreskog, K. G. ve Sörbom, D. (1993). Lisrel 8: Structural equation modeling with the SIMPLIS command language. Lincolnwood: Scientific Software International Inc.
  • Kilic, A. F., Uysal, I. ve Atar, B. (2020). Comparison of confirmatory factor analysis estimation methods on binary data. International Journal of Assessment Tools in Education, 7(3), 451–487. doi:10.21449/ijate.660353
  • Kılıç, A. F. ve Uysal, İ. (2019). Comparison of factor retention methods on binary data: A simulation study. Turkish Journal of Education, 8(3), 160–179. doi:10.19128/turje.518636
  • Kılıç, A. F., Uysal, İ. ve Atar, B. (2017). Doğrulayıcı faktör analizinde kullanılan kestirim yöntemlerinin karşılaştırmalı olarak incelenmesi. IV th International Eurasian Educational Research Congress içinde (ss. 1289–1290). Denizli. http://ejercongress.org/pdf/bildiriozetleri2017ejer.pdf adresinden erişildi.
  • Kılıç, A. F., Uysal, İ. ve Doğan, N. (2018). Simülasyon çalışmalarında replikasyon sayısının üretilen veri setlerine etkisi. 27. Uluslararası Eğitim Bilimleri Kongresi içinde . Antalya.
  • Kvålseth, T. O. (2017). An alternative measure of ordinal association as a value-validity correction of the Goodman–Kruskal gamma. Communications in Statistics - Theory and Methods, 46(21), 10582–10593. doi:10.1080/03610926.2016.1239114
  • Kvålseth, T. O. (2018). Measuring association between nominal categorical variables: an alternative to the Goodman–Kruskal lambda. Journal of Applied Statistics, 45(6), 1118–1132. doi:10.1080/02664763.2017.1346066
  • Li, C.-H. (2016a). Confirmatory factor analysis with ordinal data: Comparing robust maximum likelihood and diagonally weighted least squares. Behavior Research Methods, 48(3), 936–949. doi:10.3758/s13428-015-0619-7
  • Li, C.-H. (2016b). The performance of ML, DWLS, and ULS estimation with robust corrections in structural equation models with ordinal variables. Psychological Methods, 21(3), 369–387. doi:10.1037/met0000093
  • Lozano, L. M., García-Cueto, E. ve Muñiz, J. (2008). Effect of the number of response categories on the reliability and validity of rating scales. Methodology, 4(2), 73–79. doi:10.1027/1614-2241.4.2.73
  • Maturi, T. A. ve Elsayigh, A. (2010). A comparison of correlation coefficients via a three-step bootstrap approach. Journal of Mathematics Research, 2(2), 3–10.
  • Morata-Ramirez, M. de los A. ve Holgado-Tello, F. P. (2013). Construct validity of likert scales through confirmatory factor analysis: A simulation study comparing different methods of estimation based on Pearson and polychoric correlations. International Journal of Social Science Studies, 1(1), 54--61. doi:10.11114/ijsss.v1i1.27
  • Nunnally, J. C. ve Bernstein, I. H. (1994). Psychometric theory (3rd. ed.). New York, NY: McGraw-Hill.
  • Oranje, A. (2003). Comparison of estimation methods in factor analysis with categorized variables: Applications to NEAP data. Paper presented at the Annual Meeting of the National Council on Measurement in Education (Chicago, IL, April 21-25, 2003).
  • R Core Team. (2018). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.r-project.org/. adresinden erişildi.
  • Revelle, W. (2020). psych: Procedures for psychological, psychometric, and personality research. Evanston, Illinois. https://cran.r-project.org/package=psych adresinden erişildi.
  • Rhemtulla, M., Brosseau-Liard, P. É. ve Savalei, V. (2012). When can categorical variables be treated as continuous? A comparison of robust continuous and categorical SEM estimation methods under suboptimal conditions. Psychological Methods, 17(3), 354–373. doi:10.1037/a0029315
  • Rosseel, Y. (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1–36.
  • Signorell, A., Aho, K., Alfons, A., Anderegg, N., Aragon, T., Arachchige, C., … Zeileis, A. (2020). DescTools: Tools for descriptive statistics. https://cran.r-project.org/package=DescTools adresinden erişildi.
  • Tabachnik, B. G. ve Fidell, L. S. (2012). Using multivariate statistics (6th ed.). Boston: Pearson.
  • Timmerman, M. E. ve Lorenzo-Seva, U. (2011). Dimensionality assessment of ordered polytomous items with parallel analysis. Psychological Methods, 16(2), 209–220. doi:10.1037/a0023353
  • Trierweiler, T. (2009). An evaluation of estimation methods in confirmatory factor analytic models with ordered categorical data in LISREL. (Doctoral dissertation). Fordhame University, New York.
  • Tuğran, E., Kocak, M., Mirtagioğlu, H., Yiğit, S. ve Mendes, M. (2015). A simulation based comparison of correlation coefficients with regard to type I error rate and power. Journal of Data Analysis and Information Processing, 03(03), 87–101. doi:10.4236/jdaip.2015.33010
  • West, S. G., Finch, J. F. ve Curran, P. J. (1995). Structural equation models with non-normal variables: Problems and remedies. R. H. Hoyle (Ed.), Structural equation modeling: Concepts, issues, and applications içinde . Thousand Oaks, CA: Sage.

An Alternative Correlation Matrix Based Factor Analysis for Categorical Data: Goodman and Kruskal’s Gamma

Year 2021, Volume: 54 Issue: 54, 151 - 168, 30.06.2021
https://doi.org/10.15285/maruaebd.853905

Abstract

Exploratory factor analysis (EFA) is a frequently used method in social sciences while gathering evidence for the construct validity of data obtained from scales. When the data are categorical, EFA is performed using the polychoric/tetrachoric correlation matrix, while analyses are performed using the Pearson correlation matrix when the data are continuous. However, in some cases when the polychoric correlation matrix is used, non-convergence issues can emerge, and factor loadings can be underestimated when the Pearson correlation matrix is used. Therefore, this study aims to compare the EFA results obtained from Goodman and Kruskal’s lambda and gamma coefficients as an alternative correlation matrix with the results obtained from the polychoric and Pearson correlation matrix. In the Monte Carlo simulation study carried out for this purpose, variables such as number of categories, average factor loading, sample size, and distribution of variables were determined as the simulated conditions. As a result of the study, in some conditions unachievable using the polychoric correlation matrix, estimations are observed able to be made with the matrix formed by Goodman and Kruskal’s gamma coefficient. In certain conditions using the lambda coefficient, the model did not converge. Lambda had underestimated factor loadings in the converged data sets. The results obtained from Goodman and Kruskal’s gamma are also observed to be less biased as the number of categories increases. Researchers can be recommended to examine the results obtained from Goodman and Kruskal’s gamma while performing EFA with categorical data.

References

  • Bandalos, D. L. ve Leite, W. (2013). Use of Monte Carlo studies in structural equation modeling research. G. R. Hancock ve R. O. Mueller (Ed.), Structural equation modeling: A second course içinde (2nd ed.). Charlotte, NC: Information Age.
  • Baykul, Y. (2010). Eğitimde ve psikolojide ölçme: Klasik test teorisi ve uygulaması (2. Baskı.). Ankara: Pegem Akademi.
  • Beauducel, A. ve Herzberg, P. Y. (2006). On the performance of maximum likelihood versus means and variance adjusted weighted least squares estimation in CFA. Structural Equation Modeling: A Multidisciplinary Journal, 13(2), 186–203. doi:10.1207/s15328007sem1302_2
  • Brown, T. A. (2015). Confirmatory factor analysis for applied research (2nd ed.). New York: The Guilford.
  • Büyüköztürk, Ş. (2013). Sosyal bilimler için veri analizi el kitabı: İstatistik, araştırma deseni, SPSS uygulamaları ve yorum (18. Baskı.). Ankara: Pegem Akademi.
  • Çokluk, Ö., Şekercioğlu, G. ve Büyüköztürk, Ş. (2012). Sosyal bilimler için çok değişkenli istatistik SPSS ve LISREL uygulamaları (2. Baskı.). Ankara: Pegem Akademi.
  • Cooper, C. (2019). Psychological testing: Theory and practice. Abingdon, Oxon: Routledge.
  • Costello, A. B. ve Osborne, J. W. (2005). Best practices in exploratory factor analysis: Four recommendations for getting the most from your analysis. Practical Assessment, Research & Evaluation, 10(7), 27–29. doi:10.1.1.110.9154
  • Crocker, L. ve Algina, J. (2008). Introduction of classical and modern test theory. Ohio: Cengage Learning.
  • Depaoli, S. ve Scott, S. (2015). Frequentist and bayesian estimation of CFA measurement models with mixed item response types: A monte carlo investigation. Structural Equation Modeling: A Multidisciplinary Journal, (September), 1–16. doi:10.1080/10705511.2015.1044653
  • Erkuş, A. (2014). Psikolojide ölçme ve ölçek geliştirme-I: Temel kavramlar ve işlemler (2nd ed.). Ankara: Pegem Akademi.
  • Finney, S. J. ve DiStefano, C. (2013). Nonnormal and categorical data in structural equation modeling. G. R. Hancock ve R. O. Mueller (Ed.), Structural equation modeling: A second course içinde (2nd ed., ss. 439–492). Charlotte, NC: IAP.
  • Flora, D. B. ve Curran, P. J. (2004). An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data. Psychological Methods, 9(4), 466–491. doi:10.1037/1082-989X.9.4.466
  • Flora, D. B., Finkel, E. J. ve Foshee, V. A. (2003). Higher order factor structure of a self-control test: Evidence from confirmatory factor analysis with polychoric correlations. Educational and Psychological Measurement, 63(1), 112–127. doi:10.1177/0013164402239320
  • Forero, C. G., Maydeu-Olivares, A. ve Gallardo-Pujol, D. (2009). Factor analysis with ordinal indicators: A monte carlo study comparing DWLS and ULS estimation. Structural Equation Modeling: A Multidisciplinary Journal, 16(4), 625–641. doi:10.1080/10705510903203573
  • Goodman, L. A. ve Kruskal, W. H. (1954). Measures of association for cross classifications. Journal of the American Statistical Association, 49(268), 732–764. doi:10.1080/01621459.1954.10501231
  • Goodman, L. A. ve Kruskal, W. H. (1979). Measures of Association for Cross Classifications. Springer Series in Statistics. New York, NY: Springer. doi:10.1007/978-1-4612-9995-0
  • Gorsuch, R. L. (1974). Factor analysis. Toronto: W. B. Saunders.
  • Hair, J. F., Black, W. C., Babin, B. J. ve Anderson, R. E. (2009). Multivariate data analysis (7th ed.). Boston: Pearson.
  • Harrington, D. (2009). Confirmatory factor analysis. New York: Oxford University.
  • Harwell, M., Stone, C. A., Hsu, T.-C. ve Kirisci, L. (1996). Monte carlo studies in item response theory. Applied Psychological Measurement, 20(2), 101–125. doi:10.1177/014662169602000201
  • Hauke, J. ve Kossowski, T. (2011). Comparison of values of Pearson’s and Spearman’s correlation coefficients on the same sets of data. Quaestiones Geographicae, 30(2), 87–93. doi:10.2478/v10117-011-0021-1
  • Holgado–Tello, F. P., Chacón–Moscoso, S., Barbero–García, I. ve Vila–Abad, E. (2010). Polychoric versus Pearson correlations in exploratory and confirmatory factor analysis of ordinal variables. Quality & Quantity, 44(1), 153–166. doi:10.1007/s11135-008-9190-y
  • Jöreskog, K. G. (1994). On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika, 59(3), 381–389. doi:10.1007/BF02296131
  • Jöreskog, K. G. ve Sörbom, D. (1993). Lisrel 8: Structural equation modeling with the SIMPLIS command language. Lincolnwood: Scientific Software International Inc.
  • Kilic, A. F., Uysal, I. ve Atar, B. (2020). Comparison of confirmatory factor analysis estimation methods on binary data. International Journal of Assessment Tools in Education, 7(3), 451–487. doi:10.21449/ijate.660353
  • Kılıç, A. F. ve Uysal, İ. (2019). Comparison of factor retention methods on binary data: A simulation study. Turkish Journal of Education, 8(3), 160–179. doi:10.19128/turje.518636
  • Kılıç, A. F., Uysal, İ. ve Atar, B. (2017). Doğrulayıcı faktör analizinde kullanılan kestirim yöntemlerinin karşılaştırmalı olarak incelenmesi. IV th International Eurasian Educational Research Congress içinde (ss. 1289–1290). Denizli. http://ejercongress.org/pdf/bildiriozetleri2017ejer.pdf adresinden erişildi.
  • Kılıç, A. F., Uysal, İ. ve Doğan, N. (2018). Simülasyon çalışmalarında replikasyon sayısının üretilen veri setlerine etkisi. 27. Uluslararası Eğitim Bilimleri Kongresi içinde . Antalya.
  • Kvålseth, T. O. (2017). An alternative measure of ordinal association as a value-validity correction of the Goodman–Kruskal gamma. Communications in Statistics - Theory and Methods, 46(21), 10582–10593. doi:10.1080/03610926.2016.1239114
  • Kvålseth, T. O. (2018). Measuring association between nominal categorical variables: an alternative to the Goodman–Kruskal lambda. Journal of Applied Statistics, 45(6), 1118–1132. doi:10.1080/02664763.2017.1346066
  • Li, C.-H. (2016a). Confirmatory factor analysis with ordinal data: Comparing robust maximum likelihood and diagonally weighted least squares. Behavior Research Methods, 48(3), 936–949. doi:10.3758/s13428-015-0619-7
  • Li, C.-H. (2016b). The performance of ML, DWLS, and ULS estimation with robust corrections in structural equation models with ordinal variables. Psychological Methods, 21(3), 369–387. doi:10.1037/met0000093
  • Lozano, L. M., García-Cueto, E. ve Muñiz, J. (2008). Effect of the number of response categories on the reliability and validity of rating scales. Methodology, 4(2), 73–79. doi:10.1027/1614-2241.4.2.73
  • Maturi, T. A. ve Elsayigh, A. (2010). A comparison of correlation coefficients via a three-step bootstrap approach. Journal of Mathematics Research, 2(2), 3–10.
  • Morata-Ramirez, M. de los A. ve Holgado-Tello, F. P. (2013). Construct validity of likert scales through confirmatory factor analysis: A simulation study comparing different methods of estimation based on Pearson and polychoric correlations. International Journal of Social Science Studies, 1(1), 54--61. doi:10.11114/ijsss.v1i1.27
  • Nunnally, J. C. ve Bernstein, I. H. (1994). Psychometric theory (3rd. ed.). New York, NY: McGraw-Hill.
  • Oranje, A. (2003). Comparison of estimation methods in factor analysis with categorized variables: Applications to NEAP data. Paper presented at the Annual Meeting of the National Council on Measurement in Education (Chicago, IL, April 21-25, 2003).
  • R Core Team. (2018). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.r-project.org/. adresinden erişildi.
  • Revelle, W. (2020). psych: Procedures for psychological, psychometric, and personality research. Evanston, Illinois. https://cran.r-project.org/package=psych adresinden erişildi.
  • Rhemtulla, M., Brosseau-Liard, P. É. ve Savalei, V. (2012). When can categorical variables be treated as continuous? A comparison of robust continuous and categorical SEM estimation methods under suboptimal conditions. Psychological Methods, 17(3), 354–373. doi:10.1037/a0029315
  • Rosseel, Y. (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1–36.
  • Signorell, A., Aho, K., Alfons, A., Anderegg, N., Aragon, T., Arachchige, C., … Zeileis, A. (2020). DescTools: Tools for descriptive statistics. https://cran.r-project.org/package=DescTools adresinden erişildi.
  • Tabachnik, B. G. ve Fidell, L. S. (2012). Using multivariate statistics (6th ed.). Boston: Pearson.
  • Timmerman, M. E. ve Lorenzo-Seva, U. (2011). Dimensionality assessment of ordered polytomous items with parallel analysis. Psychological Methods, 16(2), 209–220. doi:10.1037/a0023353
  • Trierweiler, T. (2009). An evaluation of estimation methods in confirmatory factor analytic models with ordered categorical data in LISREL. (Doctoral dissertation). Fordhame University, New York.
  • Tuğran, E., Kocak, M., Mirtagioğlu, H., Yiğit, S. ve Mendes, M. (2015). A simulation based comparison of correlation coefficients with regard to type I error rate and power. Journal of Data Analysis and Information Processing, 03(03), 87–101. doi:10.4236/jdaip.2015.33010
  • West, S. G., Finch, J. F. ve Curran, P. J. (1995). Structural equation models with non-normal variables: Problems and remedies. R. H. Hoyle (Ed.), Structural equation modeling: Concepts, issues, and applications içinde . Thousand Oaks, CA: Sage.
There are 48 citations in total.

Details

Primary Language Turkish
Subjects Studies on Education
Journal Section Articles
Authors

Abdullah Faruk Kılıç 0000-0003-3129-1763

Early Pub Date July 15, 2021
Publication Date June 30, 2021
Acceptance Date May 21, 2021
Published in Issue Year 2021 Volume: 54 Issue: 54

Cite

APA Kılıç, A. F. (2021). Kategorik Veride Faktör Analizi İçin Kullanılabilecek Alternatif Bir Korelasyon Matrisi: Goodman-Kruskal Gamma. Marmara Üniversitesi Atatürk Eğitim Fakültesi Eğitim Bilimleri Dergisi, 54(54), 151-168. https://doi.org/10.15285/maruaebd.853905