Abstract
When sample sizes are too small to support multiple-group models, an alternative method to evaluate measurement invariance is restricted factor analysis (RFA), which is statistically equivalent to the more common multiple-indicator multiple-cause (MIMIC) model. Although these methods traditionally were capable of detecting only uniform measurement bias, RFA can be extended with latent moderated structural equations (LMS) to assess nonuniform measurement bias. As LMS is implemented in limited structural equation modeling (SEM) computer programs (e.g., Mplus), we propose the use of the product indicator (PI) method in RFA models, which is available in any SEM software. Using simulated data, we illustrate how to apply this method to test for measurement bias, and we compare the conclusions with those reached using LMS in Mplus. Both methods obtain comparable results, indicating that the PI method is a viable alternative to LMS for researchers without access to SEM software featuring LMS.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In traditional RFA models, common-factor variances are also assumed to be equal across groups. However, when extending RFA to include a latent interaction factor with product indicators (described immediately following), differences in common-factor variances can be captured by the covariance between the common factor and the latent interaction factor.
- 2.
LMS is also available in the open-source R package nlsem (Umbach et al. 2017), but the implementation is very limited. It is not possible to test measurement bias using RFA models in the nlsem package, so we do not consider it further.
- 3.
In the case of a dummy-coded indicator, the mean is the proportion of the sample in Group 1. Mean-centering does not affect the variance, so a 1-unit increase in a mean-centered dummy code still represents a comparison of Group 1 to Group 0, just as the original dummy code does.
References
Akaike, H. (1973). Information theory as an extension of the maximum likelihood principle. In B. N. Petrov & F. Csaki (Eds.), Second international symposium on information theory (pp. 267–281). Budapest, Hungary: Akademia Kiado. https://doi.org/10.1007/978-1-4612-1694-0_15.
Barendse, M. T., Oort, F. J., & Garst, G. J. A. (2010). Using restricted factor analysis with latent moderated structures to detect uniform and nonuniform measurement bias: A simulation study. Advances in Statistical Analysis, 94, 117–127. https://doi.org/10.1007/s10182-010-0126-1.
Barendse, M. T., Oort, F. J., Werner, C. S., Ligtvoet, R., & Schermelleh-Engel, K. (2012). Measurement bias detection through factor analysis. Structural Equation Modeling, 19, 561–579. https://doi.org/10.1080/10705511.2012.713261.
Henseler, J., & Chin, W. W. (2010). A comparison of approaches for the analysis of interaction effects between latent variables using partial least squares path modeling. Structural Equation Modeling, 17, 82–109. https://doi.org/10.1080/10705510903439003.
Kenny, D. A., & Judd, C. M. (1984). Estimating the nonlinear and interactive effects of latent variables. Psychological Bulletin, 96, 201–210. https://doi.org/10.1037/0033-2909.96.1.201.
Lin, G.-C., Wen, Z., Marsh, H. W., & Lin, H.-S. (2010). Structural equation models of latent interactions: Clarification of orthogonalizing and double-mean-centering strategies. Structural Equation Modeling, 17, 374–391. https://doi.org/10.1080/10705511.2010.488999.
Little, T. D., Bovaird, J. A., & Widaman, K. F. (2006). On the merits of orthogonalizing powered and product terms: Implications for modeling interactions among latent variables. Structural Equation Modeling, 13, 497–519. https://doi.org/10.1207/s15328007sem1304_1.
Marsh, H. W., Wen, Z., & Hau, K.-T. (2004). Structural equation models of latent interactions: Evaluation of alternative estimation strategies and indicator construction. Psychological Methods, 9, 275–300. https://doi.org/10.1037/1082-989X.9.3.275.
Mellenbergh, G. J. (1989). Item bias and item response theory. International Journal of Educational Research, 13, 127–143. https://doi.org/10.1016/0883-0355(89)90002-5.
Muthén, L. K., & Muthén, B. O. (2012). Mplus user’s guide (7th ed.). Los Angeles, CA: Muthén & Muthén.
Oort, F. J. (1992). Using restricted factor analysis to detect item bias. Methodika, 6, 150–160. https://doi.org/10.1007/s10182-010-0126-1.
Oort, F. J. (1998). Simulation study of item bias detection with restricted factor analysis. Structural Equation Modeling, 5, 107–124. https://doi.org/10.1080/10705519809540095.
R Core Team. (2016). R: A language and environment for statistical computing [Computer software]. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from the comprehensive R archive network (CRAN). https://www.R-project.org/.
Rosseel, Y. (2012). lavaan : An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1–36. https://doi.org/10.18637/jss.v048.i02.
Satorra, A., & Bentler, P. M. (2001). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66, 507–514. https://doi.org/10.1007/BF02296192.
Schwartz, G. (1978). Estimating the dimensions of a model. Annals of Statistics, 6, 461–464. https://doi.org/10.1214/aos/1176344136.
semTools Contributers. (2016). semTools : Useful tools for structural equation modeling [Compute software]. Retrieved from https://CRAN.R-project.org/package=semTools.
Umbach, N., Naumann, K., Brandt, H., & Kelava, A. (2017). Fitting nonlinear structural equation models in R with package nlsem . Journal of Statistical Software, 77(7), 1–20. https://doi.org/10.18637/jss.v077.i07.
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, 4–70. https://doi.org/10.1177/109442810031002.
Woods, C. M., & Grimm, K. J. (2011). Testing for nonuniform differential item functioning with multiple indicator multiple cause models. Applied Psychological Measurement, 35, 339–361. https://doi.org/10.1177/0146621611405984.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Kolbe, L., Jorgensen, T.D. (2018). Using Product Indicators in Restricted Factor Analysis Models to Detect Nonuniform Measurement Bias. In: Wiberg, M., Culpepper, S., Janssen, R., González, J., Molenaar, D. (eds) Quantitative Psychology. IMPS 2017. Springer Proceedings in Mathematics & Statistics, vol 233. Springer, Cham. https://doi.org/10.1007/978-3-319-77249-3_20
Download citation
DOI: https://doi.org/10.1007/978-3-319-77249-3_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-77248-6
Online ISBN: 978-3-319-77249-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)