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A Propensity Score Adjustment for Multiple Group Structural Equation Modeling

Published online by Cambridge University Press:  01 January 2025

Takahiro Hoshino ,
Hiroshi Kurata  and
Kazuo Shigemasu
Takahiro Hoshino*
Affiliation:
The University of Tokyo
Hiroshi Kurata
Affiliation:
The University of Tokyo
Kazuo Shigemasu
Affiliation:
The University of Tokyo
*
Requests for reprints should be sent to Takahiro Hoshino, Department of Cognitive and Behavioral Science, The University of Tokyo, 4-3-5 yanaka, Taito-ku, Tokyo, 110-0001, Japan. E-mail: bayesian@jasmine.ocn.ne.jp
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Abstract

In the behavioral and social sciences, quasi-experimental and observational studies are used due to the difficulty achieving a random assignment. However, the estimation of differences between groups in observational studies frequently suffers from bias due to differences in the distributions of covariates. To estimate average treatment effects when the treatment variable is binary, Rosenbaum and Rubin (1983a) proposed adjustment methods for pretreatment variables using the propensity score.

However, these studies were interested only in estimating the average causal effect and/or marginal means. In the behavioral and social sciences, a general estimation method is required to estimate parameters in multiple group structural equation modeling where the differences of covariates are adjusted.

We show that a Horvitz-Thompson-type estimator, propensity score weighted M estimator (PWME) is consistent, even when we use estimated propensity scores, and the asymptotic variance of the PWME is shown to be less than that with true propensity scores.

Furthermore, we show that the asymptotic distribution of the propensity score weighted statistic under a null hypothesis is a weighted sum of independent χ12 variables.

We show the method can compare latent variable means with covariates adjusted using propensity scores, which was not feasible by previous methods. We also apply the proposed method for correlated longitudinal binary responses with informative dropout using data from the Longitudinal Study of Aging (LSOA). The results of a simulation study indicate that the proposed estimation method is more robust than the maximum likelihood (ML) estimation method, in that PWME does not require the knowledge of the relationships among dependent variables and covariates.

Keywords

causal inferencestructural equation modelingobservational studytwo step estimationM estimationunconfoundedness
Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 4 , December 2006 , pp. 691 - 712
Copyright
Copyright © 2007 The Psychometric Society

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References

Amemiya, T. (1985). Advanced econometrics, Cambridge, MA: Harvard University Press.Google Scholar
Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Petrox, B.N., Caski, F. (Eds.), Proceedings of the Second International Symposium on Information Theory (pp. 267–281). Budapest: Akademiai Kiado.Google Scholar
Bentler, P.M., Bonett, D.G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88, 588–606.CrossRefGoogle Scholar
Bentler, P.M. (1983). Some contributions to efficient statistics in structural equation models: Specification and estimation of moment structure. Psychometrika, 48, 493–517.CrossRefGoogle Scholar
Cheng, P.E. (1994). Nonparametric estimation of mean functionals with data missing at random. Journal of the American Statistical Association, 89, 81–87.CrossRefGoogle Scholar
Clarke, B.R. (1983). Uniqueness and Fŕechet differentiability of functional solutions to maximum likelihood-type equations. Annals of Statistics, 11, 1196–1205.CrossRefGoogle Scholar
Duncan, G.M. (1987). A simplified approach to M-estimation with application to two-stage estimators. Journal of Econometrics, 34, 373–389.CrossRefGoogle Scholar
Foutz, R.V., Srivastava, R.C. (1977). The performance of the likelihood ratio test when the model is incorrect. Annals of Statistics, 5, 1183–1194.CrossRefGoogle Scholar
Gong, G., Samaniego, F.J. (1981). Pseudo maximum likelihood estimation: Theory and applications. Annals of Statistics, 9, 861–869.CrossRefGoogle Scholar
Heckman, J.J., Ichimura, H., Todd, P. (1997). Matching as an econometric evaluation estimator: Evidence from evaluating a job training program. Review of Economic Studies, 64, 605–654.CrossRefGoogle Scholar
Hill, A.B. (1965). The environment and disease: Association or causation?. Proceedings of the Royal Society of Medicine, 58, 295–300.CrossRefGoogle ScholarPubMed
Hirano, K., Imbens, G., Ridder, G. (2003). Efficient estimation of average treatment effects using the estimated propensity score. Econometrica, 71, 1161–1189.CrossRefGoogle Scholar
Hoffer, T.B. (1992). Middle school ability grouping and student achievement in science and mathematics. Educational Evaluation and Policy Analysis, 14, 205–227.CrossRefGoogle Scholar
Horvitz, D.G., Thompson, D.J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.CrossRefGoogle Scholar
Huber, P.J. (1967). The behavior of the maximum likelihood estimates under nonstandard conditions. Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, 1, 221–233.Google Scholar
Huber, P.J. (1981). Robust statistics, New York: Wiley.CrossRefGoogle Scholar
Imbens, G.W. (2000). The role of the propensity score in estimating dose-response functions. Biometrika, 87, 706–710.CrossRefGoogle Scholar
Jöreskog, K.G. (1970). A general method for the analysis of covariance structure. Biometrika, 57, 239–251.CrossRefGoogle Scholar
Jöreskog, K.G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36, 409–426.CrossRefGoogle Scholar
Johnson, N.L., Kotz, S. (1970). Distributions in statistics: Continuous univariate distributions (Vol. 2), New York: Wiley.Google Scholar
Kano, Y., Berkane, M., Bentler, P.M. (1993). Statistical inference based on pseudo-maximum likelihood estimators in elliptical populations. Journal of the American Statistical Association, 88, 135–143.CrossRefGoogle Scholar
Kaplan, D. (1999). An extension of the propensity score adjustment method for the analysis of group differences in MIMIC models. Multivariate Behavioral Research, 34, 467–492.CrossRefGoogle ScholarPubMed
Lawless, J.F., Kalbfleisch, J.D., Wild, C.J. (1999). Semiparametric methods for response-selective and missing data problems in regression. Journal of the Royal Statistical Society, Series B, 61, 413–438.CrossRefGoogle Scholar
Lehmann, E.L., Casella, G. (1998). Theory of point estimation 2nd. ed.,, New York: Springer-Verlag.Google Scholar
Liang, K.-Y., Self, S.G. (1996). On the asymptotic behavior of the pseudolikelihood ratio test statistic. Journal of the Royal Statistical Society, Series B, 58, 785–796.CrossRefGoogle Scholar
Lipsitz, S.R., Ibrahim, J.G., Zhao, L.P. (1999). A weighted estimating equation for missing covariate data with properties similar to maximum likelihood. Journal of the American Statistical Association, 94, 1147–1160.CrossRefGoogle Scholar
McArdle, J.J. (1986). Latent variable growth within behavior genetic models. Behavior Genetics, 16, 163–200.CrossRefGoogle ScholarPubMed
Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58, 525–543.CrossRefGoogle Scholar
Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115–132.CrossRefGoogle Scholar
Meredith, W., Tisak, J. (1990). Latent curve analysis. Psychometrika, 55, 107–122.CrossRefGoogle Scholar
National Center for Education Statistics (1988). National educational longitudinal study of 1988. Washington, DC.Google Scholar
Parke, W.R. (1986). Pseudo maximum likelihood estimation: The asymptotic distribution. Annals of Statistics, 14, 355–357.CrossRefGoogle Scholar
Pierce, D.A. (1982). The asymptotic effect of substituting estimators for parameters in certain types of statistics. Annals of Statistics, 10, 475–478.CrossRefGoogle Scholar
Robins, J.M., Rotnitzky, A., Zhao, L.-P. (1995). Analysis of semiparametric regression models for repeated outcomes in the presence of missing data. Journal of the American Statistical Association, 90, 106–121.CrossRefGoogle Scholar
Rosenbaum, P.R. (1984). From association to causation in observational studies: The role of tests of strongly ignorable treatment assignment. Journal of the American Statistical Association, 79, 41–48.CrossRefGoogle Scholar
Rosenbaum, P.R. (1987). Model-based direct adjustment. Journal of the American Statistical Association, 82, 387–394.CrossRefGoogle Scholar
Rosenbaum, P.R., Rubin, D.B. (1983a). The central role of the propensity score in observational studies for causal effects. Biometrika, 70, 41–55.CrossRefGoogle Scholar
Rosenbaum, P.R., Rubin, D.B. (1983b). Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. Journal of the Royal Statistical Society, Series B, 45, 212–218.CrossRefGoogle Scholar
Rotnitzky, A., Robins, J.M. (1995). Semiparametric regression estimation in the presence of dependent censoring. Biometrika, 82, 805–820.CrossRefGoogle Scholar
Rubin, D. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66, 688–701.CrossRefGoogle Scholar
Rubin, D.B. (1976). Inference and missing data. Biometrika, 63, 581–592.CrossRefGoogle Scholar
Rubin, D.B. (1977). Assignment to treatment group on the basis of a covariate. Journal of Educational Psychology, 66, 688–701.CrossRefGoogle Scholar
Rubin, D.B. (1985). The use of propensity scores in applied Bayesian inference. In Bernardo, J.M., Gedroot, M.H., Lindley, D.V., Smith, A.F.M. (Eds.), Bayesian statistics Vol. 2 (pp. 463–472). Amsterdam: North-Holland, Elsevier Science.Google Scholar
Sammel, M.D., Ryan, L.M., Legler, J.M. (1997). Latent variable models for mixed discrete and continuous outcomes. Journal of the Royal Statistical Society, Series B, 59, 667–678.CrossRefGoogle Scholar
Sörbom, D. (1974). A general method for studying differences in factor means and factor structures between groups. British Journal of Mathematical and Statistical Psychology, 27, 229–239.CrossRefGoogle Scholar
Tanaka, J.S., Huba, G.J. (1985). A general coefficient of determination for covariance structure models under arbitrary GLS estimation. British Journal of Mathematical and Statistical Psychology, 42, 233–239.CrossRefGoogle Scholar
Tanaka, J.S. (1993). Multifaceted conceptions of fit in structural equation models. In Bollen, K.A., Long, J.S. (Eds.), Testing structural equation models (pp. 10–39). Newbury Park, CA: Sage.Google Scholar
Ten Have, T.R., Reboussin, B.A., Miller, M.E., Kunselman, A. (2002). Mixed effects logistic regression models for multiple longitudinal binary function limitation responses with informative drop-out and confounding by baseline outcomes. Biometrics, 58, 127–144.CrossRefGoogle Scholar
van der Laan, M.J., Robins, J.M. (2002). Unified methods for censored longitudinal data and causality, New York: Springer-Verlag.Google Scholar
Warren, J.R., Lee, J.C. (2003). The impact of adolescent employment on high school dropout: Differences by individual and labor-market characteristics. Social Science Research, 32, 98–128.CrossRefGoogle Scholar
Yuan, K.-H., Bentler, P.M. (1997). Mean and covariance structure analysis: Theoretical and practical improvements. Journal of the American Statistical Association, 92, 767–774.CrossRefGoogle Scholar