Practical problems that are frequently encountered in applications of covariance structure analysis are discussed and solutions are suggested. Conceptual, statistical, and practical requirements for structural modeling are reviewed to indicate how basic assumptions might be violated. Problems associated with estimation, results, and model fit are also mentioned. Various issues in each area are raised, and possible solutions are provided to encourage more appropriate and successful applications of structural modeling.
Get full access to this article
View all access options for this article.
References
1.
AITCHISON, J.
and D. C. SILVEY (1958) “Maximum likelihood estimation of parameters subject to restraints.”Annals of Mathematical Statistics29: 813-828.
2.
AITKIN, M.
and D. R. RUBIN (1985) “Estimation and hypothesis testing in finite mixture models.”J. of the Royal Stat. Society47: 67-75.
3.
AMEMIYA, Y. (1985) On the Goodness-Of-Fit Tests for Linear Statistical Relationships. Technical Report No. 10, Econometric Workshop. Stanford: Stanford University.
4.
ANDERSON, J. C.
and D. W. GERBING (1984) “The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analyses.”Psychometrika49: 155-173.
5.
ARMINGER, G.
(1986) “Linear stochastic differential equation models for panel data with unobserved variables.”Sociological Methodology, 16: 187-212.
6.
ARMINGER, G.
and U. KUSTERS (1986) “Latent trait and correlation models with indicators of mixed measurement level.” Manuscript under editorial review.
7.
BAUMRIND, D.
(1983) “Specious causal attributions in the social sciences: the reformulated stepping-stone theory of heroin use as exemplar.”J. of Personality and Social Psychology45: 1289-1298.
8.
BEARDEN, W. O.
, S. SHARMA, and J. E. TEEL (1982) “Sample size effects on chi-square and other statistics used in evaluating causal models.”J. of Marketing Research19: 425-430.
9.
BENTLER, P. M.
(1983) “Some contributions to efficient statistics in structural models: specification and estimation of moment structures.”Psychometrika48: 493-517.
10.
BENTLER, P. M.
(1985) Theory and Implementation of EQS, a Structural Equations Program. Los Angeles: BMDP Statistical Software.
11.
BENTLER, P. M.
(1986a) “Structural modeling and Psychometrika: an historical perspective on growth and achievements.”Psychometrika51: 35-51.
12.
BENTLER, P. M.
(1986b) Lagrange Multiplier and Wald Tests for EQS and EQS/PC. Los Angeles: BMDP Statistical Software.
13.
BENTLER, P. M.
(1987a) “Drug use and personality in adolescence and young adulthood: structural models with nonnormal variables.”Child Development.
14.
BENTLER, P. M.
(1987b) “Structural modeling and the scientific method: comments on Freedman's critique.”J. of Educ. Statistics12: 151-157.
15.
BENTLER, P. M.
and M. BERKANE (1985) “Developments in the elliptical theory generalization of normal multivariate analysis,” pp. 291-295 in Proceedings of the Social Statistics Section, American Statistical Association.
16.
BENTLER, P. M.
and D. G. BONETT (1980) “Significance tests and goodness of fit in the analysis of covariance structures.”Psych. Bull.88: 588-606.
17.
BENTLER, P. M.
and C. P. CHOU (1986) “Statistics for parameter expansion and contraction in structural models.” Presented at the meeting of the American Educational Research Association, San Francisco.
18.
BENTLER, P. M.
, C. P. CHOU, and S. Y. LEE (1986) “Distributional consequences of linear latent variable structures.” Under editorial review.
19.
BENTLER, P. M.
and T. DIJKSTRA (1985) “Efficient estimation via linearization in structural models, pp. 9-42 in P. R. Krishnaiah (ed.) Multivariate Analysis VI. Amsterdam: North-Holland.
20.
BENTLER, P. M.
and E. H. FREEMAN (1983) “Tests for stability in linear structural equation systems.”Psychometrika48: 143-145.
21.
BENTLER, P. M.
and S. Y. LEE (1983) “Covariance structures under polynomial constraints: applications to correlation and alpha-type structural models.”J. of Educ. Statistics8: 207-222, 315-317.
22.
BENTLER, P. M.
, S. Y. LEE, and J. WENG (1987) “Multiple population covariance structure analysis under arbitrary distribution theory.”Communications in Statistics.
23.
BENTLER, P. M.
and W. PEELER (1979) “Models of female orgasm.”Archives of Sexual Behavior8: 405-423.
24.
BENTLER, P. M.
and J. S. TANAKA (1983) “Problems with Em algorithms for ML factor analysis.”Psychometrika48: 247-251.
25.
BENTLER, P. M.
and D. G. WEEKS (1980) “Linear structural equations with latent variables.”Psychometrika45: 289-308.
26.
BERKANE, M.
and P. M. BENTLER (1987) “Distribution of kurtoses, with estimators and tests of homogeneity of kurtosis.”Statistics and Probability Letters5: 201-207.
27.
BERKANE, M.
and P. M. BENTLER (1987b) “Characterizing parameters of multivariate elliptical distributions.”Communication in Statistics-Simulation16: 193-198.
28.
BOCK, R. D.
and M. AITKIN (1981) “Marginal maximum likelihood estimation of item parameters: application of an EM algorithm.”Psychometrika46: 443-459.
29.
BOLLEN, K. A.
(1987) “Outliers and improper solutions: a confirmatory factor analysis example.”Soc. Methods & Research15:375-384.
30.
BOLLEN, K. A.
and K. G. JÖRESKOG (1985) “Uniqueness does not imply identification: a note on confirmatory factor analysis.”Soc. Methods & Research14: 155-163.
31.
BONETT, D. G.
, J. A. WOODWARD, and P. M. BENTLER (1985) “Some extensions of a linear model for categorical variables.”Biometrics41: 745-750.
32.
BOOMSMA, A.
(1983) On the Robustness of LISREL (Maximum Likelihood Estimation) Against Small Sample Size and Nonnormality. Amsterdam: Sociometric Research Foundation.
33.
BOOMSMA, A.
(1985) “Nonconvergence, improper solutions, and starting values in LISREL maximum likelihood estimation.”Psychometrika50: 229-242.
34.
BOWDEN, R. J.
(1986) “Self-selection biases in correlational studies based on questionnaires.”Psychometrika51: 313-325.
35.
BROWNE, M. W.
(1982) “Covariance structures,” pp. 72-141 in D. M. Hawkins (ed.) Topics in Applied Multivariate Analysis. London: Cambridge Univ. Press.
36.
BROWNE, M. W.
(1984) “Asymptotically distribution free methods for the analysis of covariance structures.”British J. of Mathematical and Stat. Psychology37: 62-83.
37.
BROWNE, M. W.
(1985) “Robustness of the likelihood ratio rest of fit of a factor analysis model against nonnormality distributed common factors.” Presented at 4th European meeting of the Psychometric Society and Classification Societies, Cambridge, July.
38.
BUSE, A.
(1982) “The likelihood ratio, Wald, and Lagrange Multiplier tests: an expository note. Amer. Statistician36: 153-157.
39.
BYE, B. V.
, S. J. GALLICCHIO, and J. M. DYKACZ (1985) “Multiple-indicator, multiple-cause models for a single latent variable with ordinal indicators.”Soc. Methods & Research13: 487-509.
40.
CHAMBERLAIN, G.
(1982) “Multivariate regression models for panel data.”Journal of Econometrics18:5-46.
41.
CHATTERJEE, S.
and A. S. HADI (1986) “Influential observations, high leverage points, and outliers in linear regression.”Statistical Science1: 379-416.
42.
CLIFF, N.
(1983) “Some cautions concerning the application of causal modeling methods.”Multivariate Behavioral Research18: 115-126.
43.
COLLINS, L. M.
, N. CLIFF, D. J. McCORMICK, and J. L. ZATKIN (1986) “Factor recovery in binary data sets: a simulation.”Multivariate Behavioral Research21: 377-391.
44.
COMREY, A.
(1985) “A method for removing outliers to improve factor analytic results.”Multivariate Behavioral Research20: 273-282.
45.
CUDECK, R.
and M. W. BROWNE (1983) “Cross-validation of covariance structures.”Multivariate Behavioral Research18: 147-167.
46.
ETEZADI-AMOLI, J.
and R. P. McDONALD (1983) “A second generation non-linear factor analysis.”Psychometrika48: 315-342.
47.
FREEDMAN, D. A.
(1985) “Statistics and the scientific method,” pp. 345-366 in W. Mason and S. Fienberg (eds.) Cohort Analysis in Social Research. New York: Springer.
48.
FREEDMAN, D. A.
(1987) “As others see us: a case study in path analysis.”J. of Educ. Statistics12:101-128.
49.
GERBING, D. W.
and J. C. ANDERSON (1985) “The effects of sampling error and model characteristics on parameter estimation for maximum likelihood confirmatory factor analysis.”Multivariate Behavioral Research20: 255-271.
50.
GERBING, D. W.
and J. C. ANDERSON (1987) “Improper solutions in the analysis of covariance structures: their interpretability and a comparison of alternative respecifications.”Psychometrika52:99-111.
51.
GEWELE. J. F.
and K. J. SINGLETON (1980) “Interpreting the likelihood ratio statistic in factor models when the sample size is small.”J. of the Amer. Stat. Assn.75: 133-137.
52.
GOLDSCHMID, M.
and P. M. BENTLER (1968) Concept Assessment Kit: Conservation. San Diego, CA: Educational and Industrial Testing Service.
53.
GOLLOB, H. F.
and C. S. REICHARDT (1987) “Taking account of time lags in causal models.”Child Development58: 80-92.
54.
HARLOW, L. L. (1985) “Behavior of some elliptical theory estimators with nonnormal data in a covariance structures framework: A Monte Carlo study.” Ph.D. dissertation, University of California, Los Angeles.
55.
HEISE, D. R.
(1986) “Estimating nonlinear models: correcting for measurement error.”Soc. Methods & Research14: 447-472.
56.
JÖRESKOG, K. G.
(1977) “Structural equation models in the social sciences: specification, estimation and testing,” pp. 265-287 in P. R. Krishnaiah (ed.) Applications of Statistics. Amsterdam: North-Holland.
57.
JÖRESKOG, K. G.
and D. SÖRBOM (1984) LISREL VI User's Guide. Mooresville, IN: Scientific Software.
58.
KENNY, D. A.
(1979) Correlation and Causality. New York: John Wiley.
59.
KENNY, D. A.
and C. M. JUDD (1984) “Estimating the nonlinear and interactive effects of latent variables.”Psych. Bull.96: 201-210.
60.
LEAMER, E. E.
(1978) Specification Searches: Ad Hoc Inference with Nonexperimental Data. New York: John Wiley.
61.
LEE, S. Y.
(1985a) “Analysis of covariance and correlation structures.”Computational Statistics & Data Analysis2: 279-295.
62.
LEE, S. Y.
(1985b) “On testing functional constraints in structural equation models.”Biometrika72: 125-131.
63.
LEE, S. Y.
(1985c) “Maximum likelihood estimation of polychoric correlation in rXsXt contingency tables.”J. of Stat. Computation and Simulation23: 53-67.
64.
LEE, S. Y.
(1986) “Estimation for structural equation models with missing data.”Psychometrika51: 93-99.
65.
LEE, S. Y.
and P. M. BENTLER (1980) “Some asymptotic properties of constrained generalized least squares estimation in covariance structure models.”South African Stat. J.14: 121-136.
66.
LEE, S. Y.
and W. Y. POON (1986) “Maximum likelihood estimation of polyserial correlations.”Psychometrika51: 113-121.
67.
LEE, S. Y.
, W. Y. POON and P. M. BENTLER (1987) “Analysis of structural equation models with polytomous variables.”
68.
LEE, S. Y.
and K. L. TSUI (1982) “Covariance structure analysis in several populations.”Psychometrika47: 297-308.
69.
LITTLE, R.J.A.
(1985) “A note about models for selectivity bias.”Econometrica53: 1469-1474.
70.
LITTLE, R.J.A.
(1986) “Robust estimation of the mean and covariance matrix from data with missing values.” Under editorial review.
71.
LONG, J. S.
(1983) Covariance Structure Models: An Introduction to LISREL. Newbury Park, CA: Sage.
72.
MacCALLUM, R.
(1986) “Specification searches in covariance structure analyses.”Psych. Bull.100: 107-120.
73.
MARDIA, K. V.
(1970) “Measures of multivariate skewness and kurtosis with applications.”Biometrika57: 519-530.
74.
McDONALD, R. P.
and W. R. KRANE (1979) “A Monte Carlo study of local identifiability and degrees of freedom in the asymptotic likelihood ratio test.”British J. of Mathematical Stat. Psychology32: 121-132.
75.
MOOIJAART, A.
(1985) “Factor analysis for nonnormal variables.”Psychometrika50: 323-342.
76.
MOOIJAART, A.
and P. M. BENTLER (1986a) Random polynomial factor analysis. In E. Didayet al. (eds.) Data Analysis and Informatics. Amsterdam: North-Holland.
77.
MOOIJAART, A.
and P. M. BENTLER (1986b) “Robustness of M.L. estimates in exploratory factor analysis.” Under editorial review.
78.
MUTHEN, B.
(1984) “A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators.”Psychometrika49: 115-132.
79.
MUTHEN, B.
and D. KAPLAN (1985) “A comparison of some methodologies for the factor analysis of non-normal Likert variables.”British J. of Mathematical and Stat. Psychology38: 171-189.
80.
NEWCOMB, M. D.
and P. M. BENTLER (1983) “Dimensions of subjective female orgasmic repsonsiveness.”J. of Personality and Social Psychology44: 862-873.
81.
OLSSON, U.
(1979) “On the robustness of factor analysis against crude classification of the observations.”Multivariate Behavioral Research14: 485-500.
82.
REICHARDT, C. S.
and H. F. GOLLOB (1986) “Satisfying the constraints of causal modeling,” pp. 91-107 in W.M.K. Trochim (ed.) Advances in Quasi-Experimental Design and Analysis. San Francisco: Jossey-Bass.
83.
RINDSKOPF, D.
(1983) “Parameterizing inequality constraints on unique variances in linear structural models.”Psychometrika48: 73-83.
84.
RINDSKOPF, D.
(1984) “Structural equation models: empirical identification, Heywood cases, and related problems.”Soc. Methods & Research13: 109-119.
85.
RUBIN, D. B.
and D. T. THAYER (1982) “EM algorithms for ML factor analysis.”Psychometrika47: 69-76.
86.
SATORRA, A.
(1986) “Alternative test criteria in covariance structure analysis: a unified approach.” Under editorial review.
87.
SATORRA, A.
and P. M. BENTLER (1986) “Some robustness properties of goodness of fit statistics in covariance structure analysis.”Proceedings of the Business & Economic Statistics Section, American Statistical Association, 549-554.
88.
SHAPIRO, A.
(1985) “Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints.”Biometrika72: 133-144.
89.
SHAPIRO, A.
(1986) “Asymptotic theory of overparameterized structural models.”J. of Amer. Stat. Assn.81: 142-149.
90.
SILVEY, S. D.
(1959) “The Lagrangian multiplier test.”Annals of Mathematical Statistics30: 389-407.
91.
SÖRBOM, D.
(1982) “Structural equation models with structured means,” pp. 183-195 in K. G. Joreskog and H. Wold (eds.) Systems Under Indirect Observation. Amsterdam: North-Holland.
92.
SPEARMAN, C.
(1904) “`General intelligence” objectively determined and measured.”Amer. J. of Psychology15: 201-292.
93.
STEIGER, J. H.
, A. SHAPIRO, and M. W. BROWNE (1985) “On the multivariate asymptotic distribution of sequential chi-square statistics.”Psychometrika50: 253-264.
94.
STELZL, I.
(1986) “Changing a causal hypothesis without changing the fit: some rules for generating equivalent path models.”Multivariate Behavioral Research21: 309-331.
95.
STROTZ, R. H.
and H.O.A. WOLD (1960) “Recursive vs. nonrecursive systems: an attempt at synthesis.”Econometrica33: 1-41.
96.
TANAKA, J. S. (1984) “Some results on the estimation of covariance structure models.” Ph.D. thesis, University of California, Los Angeles.
97.
TANAKA, J. S.
and P. M. BENTLER (1985) “Quasi-likelihood estimation in asymptotically efficient covariance structure models.”1984 Proceedings of the American Statistical Association, Social Statistics Section, 658-662.
98.
THEIL, H.
and K. LAITINEN (1980) “Singular moment matrices in applied econometrics,” pp. 629-649 in P. R. Krishnaiah (ed.) Multivariate Analysis V. Amsterdam: North-Holland.
99.
THORNDIKE, R. L.
(1985) “The central role of general ability in prediction.”Multivariate Behavioral Research20: 241-254.
100.
TYLER, D. E.
(1983) “Robustness and efficiency properties of scatter matrices.”Biometrika70: 411-420.
101.
VAN PRAAG, B.M.S.
, T. K. DIJKSTRA, and J. VAN VELZEN (1985) “Least squares theory based on general distributional assumptions with an application to the incomplete observations problem.”Psychometrika50: 25-36.
102.
VELICER, W. F.
and J. L. FAVA (1987) “An evaluation of the effects of variable sampling on component, image, and factor analysis.”Multivariate Behavioral Research22: 193-209.
103.
WALD, A.
(1943) “Tests of statistical hypotheses concerning several parameters where the number of observations is large.”Transactions of the Amer. Mathematical Society54: 426-482.
104.
WIGGINS, J. S.
(1972) Personality and Prediction: Principles of Personality Assessment. Reading, MA: Addison-Wesley.
105.
WRIGHT, S.
(1934) “The method of path coefficients.”Annals of Mathematical Statistics5: 161-215.