# Some contributions to efficient statistics in structural models: Specification and estimation of moment structures

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## Abstract

Current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments (e.g., means and covariances), and to distribution theory based on multivariate normality. In psychometrics the multinormality assumption is often incorrect, so that statistical tests on parameters, or model goodness of fit, will frequently be incorrect as well. It is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary. Structural representations are developed for generalizations of the Bentler-Weeks, Jöreskog-Keesling-Wiley, and factor analytic models. Some asymptotically distribution-free efficient estimators for such arbitrary structural models are developed. Limited information estimators are obtained as well. The special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail. The argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory, which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds. Many open research areas are described.

## Key words

structural equations covariance structures moment structures errors in variables factor analysis minimum chi square generalized least squares elliptical distributions asymptotically distribution free## Preview

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## References

- Aigner, D. J., & Goldberger, A. S.
*Latent variables in socioeconomic models*. Amsterdam: North-Holland, 1977.Google Scholar - Aigner, D. J., Hsiao, C., Kapteyn, A., & Wansbeek, T. Latent variable models in econometrics. In Z. Griliches & M. D. Intriligator (Eds.),
*Handbook of econometrics*. Amsterdam: North-Holland, 1983.Google Scholar - Akahira, M., & Takeuchi, K.
*Asymptotic efficiency of statistical estimators: Concepts and higher order asymptotic efficiency*. New York: Springer, 1981.Google Scholar - Bagozzi, R. P.
*Causal models in marketing*. New York: Wiley, 1980.Google Scholar - Barankin, E. W., & Gurland, J. On asymptotically normal, efficient estimators.
*University of California Publications in Statistics*, 1951,*1*, 89–130.Google Scholar - Bartholomew, D. J. Latent variable models for ordered categorical data.
*Journal of Econometrics*, 1983,*22*, 229–243.Google Scholar - Bentler, P. M. Multistructure statistical model applied to factor analysis.
*Multivariate Behavioral Research*, 1976,*11*, 3–25.Google Scholar - Bentler, P. M. Multivariate analysis with latent variables: Causal modeling.
*Annual Review of Psychology*, 1980,*31*, 419–456.Google Scholar - Bentler, P. M. Linear systems with multiple levels and types of latent variables. In K. G. Jöreskog & H. Wold (Eds.),
*Systems under indirect observation: Causality, structure, prediction. Part I*. Amsterdam: North-Holland, 1982. Pp. 101–130. (a)Google Scholar - Bentler, P. M. Confirmatory factor analysis via noniterative estimation: A fast, inexpensive method.
*Journal of Marketing Research*, 1982,*19*, 417–424. (b)Google Scholar - Bentler, P. M. Simultaneous equations as moment structure models: With an introduction to latent variable models.
*Journal of Econometrics*, 1983,*22*, 13–42. (a)Google Scholar - Bentler, P. M. The EQS approach to structural equation models for normal and nonnormal continuous variables. In C. Möbus & W. Schneider (Eds.),
*Kausalmodelle in den Sozialwissenschaften—zur Analyse von Längschnitt und experimentellen Daten mit LISREL und verwandten Methoden*. Bern, Switzerland: Huber, in press, 1983. (b)Google Scholar - Bentler, P. M.
*Theory and implementation of EQS, a structural equations program*. Technical Report, BMDP Statistical Software, 1983. (c)Google Scholar - Bentler, P. M., & Bonett, D. G. Significance tests and goodness of fit in the analysis of covariance structures.
*Psychological Bulletin*, 1980,*88*, 588–606.Google Scholar - Bentler, P. M., & Dijkstra, T. Efficient estimation via linearization in structural models. In P. R. Krishnaiah (Ed.),
*Multivariate analysis VI*. Amsterdam: North-Holland, in press, 1983.Google Scholar - Bentler, P. M., & Lee, S. Y. Matrix derivatives with chain rule and rules for simple, Hadamard, and Kronecker products.
*Journal of Mathematical Psychology*, 1978,*17*, 255–262.Google Scholar - Bentler, P. M., & Lee, S. Y. A statistical development of three-mode factor analysis.
*British Journal of Mathematical and Statistical Psychology*, 1979,*32*, 87–104.Google Scholar - Bentler, P. M., & Lee, S. Y. Covariance structures under polynomial constraints: Applications to correlation and alpha-type structural models.
*Journal of Educational Statistics*, 1983,*8*, 207–222.Google Scholar - Bentler, P. M., & Peeler, W. H. Models of female orgasm.
*Archives of Sexual Behavior*, 1979,*8*, 405–423.Google Scholar - Bentler, P. M., & Speckart, G. Models of attitude-behavior relations.
*Psychological Review*, 1979,*86*, 452–464.Google Scholar - Bentler, P. M., & Speckart, G. Attitudes “cause” behaviors: A structural equation analysis.
*Journal of Personality and Social Psychology*, 1981,*40*, 226–238.Google Scholar - Bentler, P. M., & Weeks, D. G. Interrelations among models for the analysis of moment structures.
*Multivariate Behavioral Research*, 1979,*14*, 169–185.Google Scholar - Bentler, P. M., & Weeks, D. G. Linear structural equations with latnet variables.
*Psychometrika*, 1980,*45*, 289–308.Google Scholar - Bentler, P. M., & Weeks, D. G. Multivariate analysis with latent variables In P. R. Krishnaiah & L. Kanal (Eds.),
*Handbook of statistics, Vol. 2*. Amerdam: North-Holland, 1982. Pp. 747–771.Google Scholar - Beran, R. Testing for ellipsoidal symmetry of a multivariate density.
*The Annals of Statistics*, 1979,*7*, 150–162.Google Scholar - Berkson, J. Minimum chi-square, not maximum likelihood!
*The Annals of Statistics*, 1980,*8*, 457–487.Google Scholar - Boomsma, A.
*On the robustness of LISREL (maximum likelihood estimation) against small sample size and nonnormality*. Ph.D. dissertation, University of Groningen, Groningen, 1983.Google Scholar - Browne, M. W. Generalized least squares estimators in the analysis of covariance structures.
*South African Statistical Journal*, 1974,*8*, 1–24.Google Scholar - Browne, M. W. Covariance structures. In D. M. Hawkins (Ed.),
*Topics in applied multivariate analysis*. London: Cambridge University Press, 1982. Pp. 72–141.Google Scholar - Browne, M. W.
*The decomposition of multitrait-multimethod matrices*. Research Report 83/2. Department of Statistics and Operations Research, University of South Africa, 1983.Google Scholar - Cambanis, S., Huang, S., & Simons, G. On the theory of elliptically contoured distributions.
*Journal of Multivariate Analysis*, 1981,*11*, 368–385.Google Scholar - Chmielewski, M. A. Elliptically symmetric distributions: A review and bibliography.
*International Statistical Review*, 1981,*49*, 67–74.Google Scholar - Cook, M. B. Bi-variate
*k*-statistics and cumulants of their joint sampling distribution.*Biometrika*, 1951,*38*, 179–195.Google Scholar - de Leeuw, J. Models and methods for the analysis of correlation coefficients.
*Journal of Econometrics*, 1983,*22*, 113–137.Google Scholar - Dempster, A. P., Laird, N. M., & Rubin, D. B. Iteratively reweighted least squares for linear regression when errors are normal/independent distributed. In P. R. Krishnaiah (Ed.),
*Multivariate Analysis-V*. Amsterdam: North-Holland, 1980. Pp. 35–57.Google Scholar - Dijkstra, T.
*Latent variables in linear stochastic models*. Ph.D. thesis, University of Groningen, 1981.Google Scholar - Dijkstra, T. Some comments on maximum likelihood and partial least squares methods.
*Journal of Econometrics*, 1983,*22*, 67–90.Google Scholar - Duncan, O. D.
*Introduction to structural equation models*. New York: Academic, 1975.Google Scholar - DuToit, S. H. C.
*The Analysis of growth curves*. Ph.D. Thesis, University of South Africa, 1979.Google Scholar - Efron, B.
*The jackknife, the bootstrap, and other resampling plans*. Philadelphia: SIAM, 1982.Google Scholar - Fang, C., & Krishnaiah, P. R. Asymptotic distributions of functions of the eigenvalues of some random matrices for nonnormal populations.
*Journal of Multivariate Analysis*, 1982,*12*, 39–63.Google Scholar - Ferguson, T. S. A method of generating best asymptotically normal estimates with application to the estimation of bacterial densities.
*Annals of Mathematical Statistics*, 1958,*29*, 1046–1062.Google Scholar - Hägglund, G. Factor analysis by instrumental variables methods.
*Psychometrika*, 1982,*47*, 209–222.Google Scholar - Hartley, H. O., & Booker, A. Nonlinear least squares estimation.
*Annals of Mathematical Statistics*, 1965,*36*, 638–650.Google Scholar - Huba, G. J., & Bentler, P. M. A developmental theory of drug use: Derivation and assessment of a causal modeling approach. In P. B. Baltes & O. G. Brim, Jr. (Eds.),
*Lifespan development and behavior, Vol. 4*. New York: Academic, 1982. Pp. 147–203. (a)Google Scholar - Huba, G. J., & Bentler, P. M. On the usefulness of latent variable causal modeling in testing theories of naturally occurring events (including adolescent drug use): A rejoinder to Martin.
*Journal of Personality and Social Psychology*, 1982,*43*, 604–611. (b)Google Scholar - Huba, G. J., & Bentler, P. M. Causal models of the development of law abidance and its relationship to psychosocial factors and drug use. In W. S. Laufer & J. M. Day (Eds.),
*Personality theory, moral development, and criminal behavior*. Lexington, Mass.: Heath, 1983. Pp. 165–215. (a)Google Scholar - Huba, G. J., & Bentler, P. M. Test of a drug use causal model using asymptotically distribution-free methods.
*Journal of Drug Education*, 1983,*13*, 3–14. (b)Google Scholar - Huba, G. J., & Bentler, P. M.
*Antecedents and consequences of adolescent drug use: A psychosocial study of development using a causal modeling approach*. New York: Plenum, in preparation, 1984.Google Scholar - Hsu, P. L. The limiting distribution of functions of sample means and applications to testing hypotheses.
*Proceedings of the First Berkeley Symposium in Mathematical Statistics and Probability*, 1949. Pp. 359–402.Google Scholar - Ibragimov, I. A., & Has'minskii, R. Z.
*Statistical estimation: Asymptotic theory*. New York: Springer, 1981.Google Scholar - James, L. R., Mulaik, S. A., & Brett, J. M.
*Causal analysis: Assumptions, models, and data*. Beverly Hills: Sage, 1982.Google Scholar - Jennrich, R. I., & Moore, R. H. Maximum likelihood estimation by means of nonlinear least squares.
*Proceedings of ASA: Statistical Computing*, 1975, 57–65.Google Scholar - Jöreskog, K. G. Structural analysis of covariance and correlation matrices.
*Psychometrika*, 1978,*43*, 443–477.Google Scholar - Jöreskog, K. G., & Goldberger, A. S. Factor analysis by generalized least squares.
*Psychometrika*, 1972,*37*, 243–260.Google Scholar - Jöreskog, K. G., & Sörbom, D.
*LISREL V user's guide*. Chicago: International Educational Services, 1981.Google Scholar - Jorgensen, B. Maximum likelihood estimation and large sample inference for generalized linear and nonlinear regression models.
*Biometrika*, 1983,*70*, 19–28.Google Scholar - Kagan, A. M., Linnik, Y. V., & Rao, C. R.
*Characterization problems in mathematical statistics*. New York: Wiley, 1973.Google Scholar - Kaplan, E. L. Tensor notation and the sampling cumulants of
*k*-statistics.*Biometrika*, 1952,*39*, 319–323.Google Scholar - Lee, S. Y., & Bentler, P. M. Some asymptotic properties of constrained generalized least squares estimation in covariance structure models.
*South African Statistical Journal*, 1980,*14*, 121–136.Google Scholar - Lee, S. Y., & Jennrich, R. I. A study of algorithms for covariance structure analysis with specific comparisons using factor analysis.
*Psychometrika*, 1979,*44*, 99–113.Google Scholar - Lehmann, E. L.
*Theory of point estimation*. New York: Wiley, 1983.Google Scholar - Maddala, G. S.
*Limited-dependent and qualitative variables in econometrics*. Cambridge: Cambridge University Press, 1983.Google Scholar - Malinvaud, E.
*Statistical methods of econometrics*. Amsterdam: North-Holland, 1980.Google Scholar - Manski, C. F., & McFadden, D.
*Structural analysis of discrete data with econometric applications*. Cambridge, MA: The MIT Press, 1981.Google Scholar - Mardia, K. V. Measures of multivariate skewness and kurtosis with applications.
*Biometrika*, 1970,*57*, 519–530.Google Scholar - Mardia, K. V. Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies.
*Sankhya*, 1974,*B36*, 115–128.Google Scholar - McDonald, R. P. A simple comprehensive model for the analysis of covariance structures.
*British Journal of Mathematical and Statistical Psychology*, 1978,*31*, 59–72.Google Scholar - Mooijaart, A.
*Factor analysis for non-normal variables*. Technical report, Department of Psychology, UCLA, December 1982.Google Scholar - Moore, D. S., & Stubblebine, J. B. Chi-square tests for multivariate normality with application to common stock prices.
*Communications in Statistics-Theory & Methods*, 1981,*A10*, 713–738.Google Scholar - Muirhead, R. J.
*Aspects of multivariate statistical theory*. New York: Wiley, 1982.Google Scholar - Muirhead, R. J., & Waternaux, C. M. Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations.
*Biometrika*, 1980,*67*, 31–43.Google Scholar - Muthén, B. Latent variable structural equation modeling with categorical data.
*Journal of Econometrics*, 1983,*22*, 43–65.Google Scholar - Nel, D. G. On matrix differentition in statistics.
*South African Statistical Journal*, 1980,*14*, 137–193.Google Scholar - Newcomb, M. D., & Bentler, P. M. Dimensions of subjective female orgasmic responsiveness.
*Journal of Personality and Social Psychology*, 1983,*44*, 862–873.Google Scholar - Neyman, J. Contributions to the theory of the
*χ*^{2}test.*Berkeley Symposium on Mathematical Statistics and Probability*. Berkeley: University of California, 1949, Pp. 239–273.Google Scholar - Pal, M. Consistent moment estimators of regression coefficients in the presence of errors in variables.
*Journal of Econometrics*, 1980,*14*, 349–364.Google Scholar - Parr, W. C. Minimum distance estimation: A bibliography.
*Communications in Statistics: Theory and Methods*, 1981,*A10*, 1205–1224.Google Scholar - Pfanzagl, J.
*Contributions to a general asymptotic statistical theory*. New York: Springer-Verlag, 1982.Google Scholar - Phillips, P. C. B. The iterated minimum distance estimator and the quasi-maximum likelihood estimator.
*Econometrica*, 1976,*44*, 449–460.Google Scholar - Rao, C. R. Characterization of the distribution of random variables in linear structural relations.
*Sankhya*, 1966,*28A*, 251–260.Google Scholar - Rao, C. R.
*Linear statistical inference and its applications*. New York: Wiley, 1973.Google Scholar - Reiersøl, O. Identifiability of a linear relation between variables which are subject to error.
*Econometrica*, 1950,*18*, 375–389.Google Scholar - Rothenberg, T. J. Approximating the distributions of econometric estimators and test statistics. In Z. Griliches & M. D. Intriligator (Eds.),
*Handbook of econometrics*. Amsterdam: North-Holland, 1983.Google Scholar - Rothenberg, T. J., & Leenders, C. T. Efficient estimation of simultaneous equation systems.
*Econometrica*, 1964,*32*, 57–76.Google Scholar - Schwager, S. J., & Margolin, B. H. Detection of multivariate normal outliers.
*The Annals of Statistics*, 1982,*10*, 943–954.Google Scholar - Shapiro, A. Asymptotic distribution theory in the analysis of covariance structures.
*South African Statistical Journal*, 1983,*17*, 33–81.Google Scholar - Sieber, M., & Bentler, P. M. Kausalmodelle zur Persönlichkeit und dem späteren Konsum legaler und illegaler Drogen.
*Schweizerische Zeitschrift für Psychologie und Ihre Anwendungen*, 1982,*41*, 1–15.Google Scholar - Sörbom, D. Structural equation models with structured means. In K. G. Jöreskog & H. Wold (Eds.),
*Systems under indirect observation: Causality, structure, prediction. I*. Amsterdam: North-Holland, 1982. Pp. 183–195.Google Scholar - Spearman, C. The proof and measurement of association between two things.
*American Journal of Psychology*, 1904,*15*, 72–101.Google Scholar - Speckart, G., & Bentler, P. M. Application of attitude-behavior models to varied content domains.
*Academic Psychology Bulletin*, 1982,*4*, 453–466.Google Scholar - Steiger, J. H., & Hakstian, A. R. The asymptotic distribution of elements of a correlation matrix: Theory and application.
*British Journal of Mathematical and Statistical Psychology*, 1982,*35*, 208–215.Google Scholar - Tanaka, J. S.
*Some results on the estimation of covariance structure models*. Ph.D. thesis, University of California, Los Angeles, 1983.Google Scholar - Van Praag, B. M. S., Dijkstra, T. K., & Van Velzen, J.
*Least squares theory based on general distributional assumptions with an application to the incomplete observations problem*. Technical Report 83.09, Leyden University, 1983.Google Scholar - Wald, A. Tests of statistical hypotheses concerning several parameters when the number of observations is large.
*Transactions of the American Mathematical Society*, 1943,*54*, 426–482.Google Scholar - Wedderburn, R. W. M. Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method.
*Biometrika*, 1974,*61*, 439–447.Google Scholar - Wiley, D. E. The identification problem for structural equation models with unmeasured variables. In A. S. Goldberger & O. D. Duncan (Eds.),
*Structural equation models in the social sciences*. New York: Academic, 1973. Pp. 69–83.Google Scholar