Hostname: page-component-5b777bbd6c-v47t2 Total loading time: 0 Render date: 2025-06-19T22:32:10.451Z Has data issue: false hasContentIssue false
  • English
  • Français

Profile Likelihood-Based Confidence Intervals and Regions for Structural Equation Models

Published online by Cambridge University Press:  01 January 2025

Jolynn Pek  and
Hao Wu
Jolynn Pek*
Affiliation:
York University
Hao Wu
Affiliation:
Boston College
*
Correspondence should be made to Jolynn Pek, Department of Psychology, York University, 322 Behavioural Science Building, 4700 Keele Street, Toronto, ON M3J 1P3 Canada. Email: pek@yorku.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Structural equation models (SEM) are widely used for modeling complex multivariate relationships among measured and latent variables. Although several analytical approaches to interval estimation in SEM have been developed, there lacks a comprehensive review of these methods. We review the popular Wald-type and lesser known likelihood-based methods in linear SEM, emphasizing profile likelihood-based confidence intervals (CIs). Existing algorithms for computing profile likelihood-based CIs are described, including two newer algorithms which are extended to construct profile likelihood-based confidence regions (CRs). Finally, we illustrate the use of these CIs and CRs with two empirical examples, and provide practical recommendations on when to use Wald-type CIs and CRs versus profile likelihood-based CIs and CRs. OpenMx example code is provided in an Online Appendix for constructing profile likelihood-based CIs and CRs for SEM.

Keywords

confidence intervalsconfidence regionsprofile likelihoodWaldmulti-parameter
Type
Original Paper
Information
Psychometrika , Volume 80 , Issue 4 , December 2015 , pp. 1123 - 1145
Copyright
Copyright © 2015 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Electronic supplementary material The online version of this article (doi:10.1007/s11336-015-9461-1) contains supplementary material, which is available to authorized users.

References

American Educational Research Association. (2006). Standards for reporting on empirical social science research in AERA publications American Educational Research Association. Educational Researcher, 35, 33–40. doi:10.3102/0013189X035006033.CrossRefGoogle Scholar
American Psychological Association. (2001). Publication manual of the American Psychological Association (6th ed.). Washington, DC: American Psychological Association.Google Scholar
Aptech Systems, Inc. (2002). GAUSS user guide [Computer software manual]. Maple Valley, WA: Aptech Systems Inc.Google Scholar
Arbuckle, J. L. (2006). Amos 7.0 user’s guide [Computer software manual]. Chicago, IL: SPSS.Google Scholar
Baillargeon, S., & Rivest, L.-P. (2007). Rcapture: Loglinear models for capture-recapture in R. Journal of Statistical Software, 19, 1–31. Retrieved from http://www.jstatsoft.org/v19/i05.CrossRefGoogle Scholar
Baron, R.M., & Kenny, D.A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173–1182. doi:10.1037/0022-3514.51.6.1173.CrossRefGoogle ScholarPubMed
Basu, D. (1977). On the elimination of nuisance parameters. Journal of the American Statistical Association, 72, 355–366. doi:10.2307/2286800.CrossRefGoogle Scholar
Bauer, D.J. (2003). Estimating multilevel linear models as structural equation models. Journal of Educational and Behavioral Statistics, 28, 135–167. doi:10.3102/10769986028002135.CrossRefGoogle Scholar
Benjamini, Y. (2010). Simultaneous and selective inference: Current successes and future challenges. Biometrical Journal, 52, 708–721. doi:10.1002/bimj.200900299.CrossRefGoogle ScholarPubMed
Bentler, P. M. (2006). EQS 6 structural equations program manual [Computer software manual]. Encino, CA: Multivariate Software Inc.Google Scholar
Biesanz, J.C., Falk, C.F., & Savalei, V. (2010). Assessing mediational models: Testing and interval estimation for indirect effects. Multivariate Behavioral Research, 45, 661–701. doi:10.1080/00273171.2010.498292.CrossRefGoogle ScholarPubMed
Bock, R. D., & Lieberman, M. (1970). Fitting a response model from dichotomously scored items. Psychometrika, 35, 179–197. doi:10.1007/BF02291262.CrossRefGoogle Scholar
Boker, S., Neale, M., Maes, H., Wilde, M., Spiegel, M., Brick, T., et al. (2011). OpenMx: An open source extended structural equation modeling framework. Psychometrika, 76, 306–317. doi:10.1007/s11336-010-9200-6.CrossRefGoogle Scholar
Bollen, K.A. (1989). Structural equation models with latent variables. New York: Wiley.CrossRefGoogle Scholar
Bollen, K.A., & Stine, R. (1990). Direct and indirect effects: Classical and bootstrap estimates of variability. Sociological Methodology, 20, 115–140. doi:10.2307/271084.CrossRefGoogle Scholar
Branscombe, N.R., Schmitt, M.T., & Harvey, R.D. (1999). Perceiving pervasive discrimination among African Americans: Implications for group identification and wellbeing. Journal of Personality and Social Psychology, 77, 135–149. doi:10.1037/0022-3514.77.1.135.CrossRefGoogle Scholar
Branscombe, N. R., Wann, D. L., Noel, J. G., & Coleman, J. (1993). In-group or out-group extemity: Importance of the threatened social identity. Personality and Social Psychology Bulletin, 19, 381–388. doi:10.1177/0146167293194003.CrossRefGoogle Scholar
Brent, R.P. (1973). Algorithms for minimization without derivatives. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83. doi:10.1111/j.2044-8317.1984.tb00789.x.CrossRefGoogle Scholar
Browne, M. W., & Arminger, G. (1995). Specification and estimation of mean- and covariance-structure models. In G. Arminger, C. C. Clogg, & M. E. Sobel (Eds.), Handbook of statistical modeling for the social and behavioral sciences (pp. 185–249). New York, NY: Springer.Google Scholar
Cheung, M.W.L. (2007). Comparison of approaches to constructing confidence intervals for mediating effects using structural equation models. Structural Equation Modeling, 14, 227–246. doi:10.1080/10705510709336745.CrossRefGoogle Scholar
Cheung, M.W.L. (2009). Constructing approximate confidence intervals for parameters with structural equation models. Structural Equation Modeling, 16, 267–294. doi:10.1080/10705510902751291.CrossRefGoogle Scholar
Chow, S-M, Ho, M-HR, Hamaker, E.L., & Dolan, C.V. (2010). Equivalence and differences between structural equation modeling and state-space modeling techniques. Structural Equation Modeling, 17, 303–332. doi:10.1080/10705511003661553.CrossRefGoogle Scholar
Cohen, J. The earth is round (p<.05)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p < .05)$$\end{document} (1994). American Psychologist. 49, 997–1003. doi:10.1037/0003-066X.49.12.997.CrossRefGoogle Scholar
Cook, R.D., & Weisberg, S. (1990). Confidence curves in nonlinear regression. Journal of the American Statistical Association, 85, 544–551. doi:10.1080/01621459.1990.10476233.CrossRefGoogle Scholar
Cox, D. R., & Hinkley, D. V. (1974). Theoretical statistics. London: Chapman & Hall.CrossRefGoogle Scholar
Curran, P. J. (2003). Have multilevel models been structural equation models all along? Multivariate Behavioral Research, 38, 529–569. doi:10.1207/s15327906mbr3804_5.CrossRefGoogle Scholar
Dennis, J.E., Gay, D.M., & Walsh, R.E. (1981). An adaptive nonlinear least-squares algorithm. ACM Transactions on Mathematical Software, 7, 348–368. doi:10.1145/355958.355965.CrossRefGoogle Scholar
DiCiccio, T. J., & Tibshirani, R. (1991). On the implementation of profile likelihood methods (Tech. Rep. No. 9107). Toronto, ON: University of Toronto, Department of Statistics.Google Scholar
du Toit, S.H., & Cudeck, R. (2009). Estimation of the nonlinear random coefficient model when some random effects are separable. Psychometrika, 74, 65–82. doi:10.1007/s11336-008-9107-7.CrossRefGoogle Scholar
Falk, C.F., & Biesanz, J.C. (2015). Inference and interval estimation for indirect effects with latent variable models. Structural Equation Modeling, 22, 24–38. doi:10.1080/10705511.2014.935266.CrossRefGoogle Scholar
Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London, Series A, 22, 309–368. doi:10.1098/rsta.1922.0009.CrossRefGoogle Scholar
Gonzalez, R., & Griffn, D. (2001). Testing parameters in structural equation modeling: Every “one” matters. Psychological Methods, 6, 258–269. doi:10.1037/1082-989X.6.3.258.CrossRefGoogle Scholar
Hancock, G. R., & Choi, J. (2006). A vernacular for linear latent growth models. Structural Equation Modeling, 13, 352–377. doi:10.1207/s15328007sem1303_2.CrossRefGoogle Scholar
Insightful Corporation. (2007). S-PLUS 8 for Windows user’s guide [Computer software manual]. Seattle, WA: Insightful Corporation.Google Scholar
Jöreskog, K. G., & Sörbom, D. (2006). LISREL 8.8 for windows [Computer software manual]. Skokie, IL: Scientific Software International Inc.Google Scholar
Kalbfleisch, J. D., & Sprott, D. A. (1970). Application of likelihood methods to models involving large numbers of parameters. Journal of the Royal Statistical Society. Series B, 32, 175–208. Retrieved from http://www.jstor.org/stable/2984524.Google Scholar
Lehmann, E.L., & Romano, J.P. (2006). Testing statistical hypotheses (3rd ed.). New York: Springer.Google Scholar
MacCallum, R. C., Browne, M. W., & Lee, T. (2009). Fungible parameter estimates in structural equation modeling. Paper presented at the Society of Multivariate Experimental Research. Salishan Resort, OR.Google Scholar
MacCallum, R. C., Lee, T., & Browne, M. W. (2012). Fungible parameter estimates in latent curve models. In M. C. Edwards & R. C. MacCallum (Eds.), Current topics in the theory and application of latent variable models (pp. 183–197). New York, NY: Routledge.Google Scholar
MacKinnon, D.P., Lockwood, C.M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39, 99–128. doi:10.1207/s15327906mbr3901_4.CrossRefGoogle ScholarPubMed
McGraw, K.O., & Wong, S. (1996). Forming inferences about some intraclass correlation coefficients. Psychological Methods, 1, 30–46. doi:10.1037/1082-989X.1.1.30.CrossRefGoogle Scholar
Meehl, P.E. (1997). The problem is epistemology, not statistics: Replace significance tests by confidence intervals and quantify accuracy of risky numerical predictions. In Harlow, L.L., Mulaik, S.A., & Steiger, J.H. (Eds.), What if there were no significance tests (pp. 393–425). Mahwah, NJ: Erlbaum.Google Scholar
Meeker, W.Q., & Escobar, L.A. (1995). Teaching about approximate confidence regions based on maximum likelihood estimation. The American Statistician, 49, 48–53. doi:10.1080/00031305.1995.10476112.CrossRefGoogle Scholar
Muthén, L. K., & Muthén, B. O. (2012). Mplus user’s guide (7th ed.) [Computer software manual]. Los Angeles, CA: Muthéen & Muthéen.Google Scholar
Neale, M. C., Boker, S. M., Xie, G., & Maes, H. H. (2003). Mx: Statistical modeling (6th ed.) [Computer software manual]. Richmond, VA: Department of Psychiatry.Google Scholar
Neale, M. C., Hunter, M. D., Pritkin, J., Zahery, M., Brick, T. R., Kirkpatrick, R. M., et al. (2015). OpenMx 2.0: Extended structural equation and statistical modeling. Psychometrika. doi:10.1007/s11336-014-9435-8.CrossRefGoogle Scholar
Neale, M.C., & Miller, M.B. (1997). The use of likelihood-based confidence intervals in genetic models. Behavior Genetics, 27, 113–120. doi:10.1023/A:1025681223921.CrossRefGoogle ScholarPubMed
Neyman, J., & Pearson, E.S. (1928). On the use and interpretation of certain test criteria for purposes of statistical inference: Part I. Biometrika, 20A, 175–240. doi:10.1093/biomet/20A.1-2.175.Google Scholar
Nickerson, R.S. (2000). Null hypothesis significance testing: A review of an old and continuing controversy. Psychological Methods, 5, 241–301. doi:10.1037/1082-989X.5.2.241.CrossRefGoogle ScholarPubMed
Pawitan, Y. (2001). In all likelihood: Statistical modelling and inference using likelihood. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Preacher, K. J., & Hancock, G. R. (2012). On interpretable reparameterizations of linear and nonlinear latent growth curve models. In J. R. Harring & G. R. Hancock (Eds.), Advances in longitudinal methods in the social and behavioral sciences (pp. 25–58). Charlotte, NC: Information Age Publishing.Google Scholar
Preacher, K. J., & Hancock, G. R. (2015). Meaningful aspects of change as novel random coefficients: A general method for reparameterizing longitudinal models. Psychological Methods, 20, 84–101. doi:10.1037/met0000028.CrossRefGoogle Scholar
Press, W.H., Teukolsky, S.A., Vetterling, W.T., & Flannery, B.P. (1992). Numerical recipes in Fortran 77: The art of scientific computing. Cambridge, UK: Cambridge University Press.Google Scholar
R Development Core Team. (2013). R: A language and environment for statistical computing [Computer software manual]. Vienna, Austria. Retrieved from http://www.R-project.org.Google Scholar
Raykov, T., & Marcoulides, G. A. (2004). Using the delta method for approximate interval estimation of parameter functions in SEM. Structural Equation Modeling, 11, 621–637. doi:10.1207/s15328007sem1104_7.CrossRefGoogle Scholar
Ritter, C., & Bates, D. M. (1993). Profile methods (Tech. Rep. No. 93–31). Louvain-la-Neuve, Beligium: Institut de Statistique, Universie Catholique de Louvain.Google Scholar
Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48, 1–36. Retrieved from http://www.jstatsoft.org/v48/i02/paper.Google Scholar
Satorra, A., & Bentler, P. (2001). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66(4), 507–514. doi:10.1007/BF02296192.CrossRefGoogle Scholar
Satterthwaite, F. (1941). Synthesis of variance. Psychometrika, 6(5), 309–316. doi:10.1007/BF02288586.CrossRefGoogle Scholar
Savalei, V. (2014). Understanding robust corrections in structural equation modeling. Structural Equation Modeling, 21, 149–160. doi:10.1080/10705511.2013.824793.CrossRefGoogle Scholar
Scheffé, H. (1953). A method for judging all constrasts in the analysis of variance. Biometrika, 40, 87–110. doi:10.1093/biomet/40.1-2.87.CrossRefGoogle Scholar
Schmitt, M.T., Branscombe, N.R., Kobrynowicz, D., & Owen, S. (2002). Perceiving discrimination agaist one’s gender group has different implications for well-being in women and men. Personality and Social Psychology Bulletin, 28, 197–210. doi:10.1177/0146167202282006.CrossRefGoogle Scholar
Schoenberg, R. (1995). CML users guide [Computer software manual]. Maple Valley, WA: Aptech Systems Inc.Google Scholar
Schoenberg, R. (1997). Constrained maximum likelihood. Computational Economics, 10, 251–266. doi:10.1023/A:1008669208700.CrossRefGoogle Scholar
Sobel, M.E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology, 13, 290–312. doi:10.2307/270723.CrossRefGoogle Scholar
Sprott, D. (1980). Maximum likelihood in small samples: Estimation in the presence of nuisance parameters. Biometrika, 67, 515–523. doi:10.1093/biomet/67.3.515.CrossRefGoogle Scholar
Steiger, J. H. (1980). Testing pattern hypotheses on correlation matrices: Alternative statistics and some empirical results. Multivariate Behavioral Research, 15, 335–352. doi:10.1207/s15327906mbr1503_7.CrossRefGoogle Scholar
Stoolmiller, M. (1994). Antisocial behavior, delinquent peer association, and unsupervised wandering for boys: Growth and change from childhood to early adolescence. Multivariate Behavioral Research, 29, 263–288. doi:10.1207/s15327906mbr2903_4.CrossRefGoogle Scholar
Stryhn, H., & Christensen, J. (2003). Confidence intervals by the profile likelihood method, with applications in veterinary epidemiology. In Proceedings of the 10th international symposium on veterinary epidemiology and economics. Vina del Mar, Chile.Google Scholar
Venzon, D., & Moolgavkar, S. (1988). A method for computing profile-likelihood-based confidence intervals. Applied Statistics, 37, 87–94. doi:10.2307/2347496.CrossRefGoogle Scholar
Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society, 54, 426–482. doi:10.2307/1990256.CrossRefGoogle Scholar
Waller, N.G. (2008). Fungible weights in multiple regression. Psychometrika, 73, 691–703. doi:10.1007/s11336-008-9066-z.CrossRefGoogle Scholar
Wilkinson, L., & Task Force on Statistical Inference. (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594–604. doi:10.1037/0003-066X.54.8.594.CrossRefGoogle Scholar
Wu, H., & Neale, M.C. (2012). Adjusted confidence intervals for a bounded parameter. Behavior Genetics, 42, 886–898. doi:10.1007/s10519-012-9560-z.CrossRefGoogle ScholarPubMed
Supplementary material: File

Pek and Wu supplementary material

Pek and Wu supplementary material
Download Pek and Wu supplementary material(File)
File 126 KB