Psychology 702           Moment Structure Modeling           Spring, 2000

Course Description:

This course provides a general introduction and overview of the most common methods of moment structure modeling. Moment structure analysis is a very general term that covers a variety of more or less well-known statistical models. Largely, the focus of the course will be on the assessment of linear structural relationship models, such as path analysis, exploratory and confirmatory factor analysis, cross-lagged panel analysis, simplex and circumplex models, Markov models, and various alternatives to the analysis of variance and covariance. Both least squares and maximum likelihood methods of model specification and testing will be examined, and the parallels between the two, and the advantages and disadvantages of each, will be explored.

Course Objectives:

The approach taken in this course will be to focus on a heuristic understanding of the underlying statistical principles, rather than on the complexity of the estimation process. Some coverage will be given to both the least squares and maximum likelihood methods of estimation and significance testing. However, the major emphasis in the course will be on methods of model specification, and the development and testing of appropriate hypothetical model(s) within the context of the various statistical models.

Texts:

Hayduk, L. A. (1987). Structural equation modeling with LISREL: Essentials and advances. Baltimore: Johns Hopkins University Press.

Kenny, D.A. (1979). Correlation and causality. New York: Wiley-Interscience.

Dwyer, J.H. (1987). Statistical models for the social and behavioral sciences. New York: Oxford.

Assignments:

For those of you who are taking this course for credit, there will be periodic assignments. These assignments will focus primarily on what we are discussing each week, although considerable residual knowledge from prior weeks may be necessary. The assignments will largely involve the use of the computer to solve various problems and your interpretation of the results, written on the printout. You must turn in your own work product.

Computer Usage:

I do not expect you to know how computers work, or how to program. However, I do expect that you have some experience interacting with computers, and have had some knowledge of statistical packages, such as SPSS-X, BMDP, SAS, and so on. We will be spending only a minimal amount of time on such issues, so if you are not knowledgeable about computers, be prepared for a time-consuming introduction.

Grading:

Your grade will be based on your assignments and a final exam. In order to get an A in the course, you must hand in your weekly assignments on time and they must have been done correctly. If they are incorrect, you can redo them after I have returned them to you, and turn them in again (and again,...). If you do so, you will still be able to earn an A. In addition, you must get at least a B on the final exam. A grade of C on the final exam would result in a course grade of B, assuming all homeworks have been turned in on time, and (eventually) correctly. A grade of D on the final exam would result in a course grade of C, again, assuming all homeworks have been turned in on time, and (eventually) correctly. Failure to turn in a homework on time will result in the loss of this one-letter grade bonus.

Moreover, failure to complete all homeworks by the due date for the final exam will result in the loss of a letter grade.

Schedule of Assignments:

I'm not going to set a specific schedule for completing topics. This will largely depend on the speed with which we master each topic, and the interests of the class. We will begin with a discussion of ordinary least-squares regression, and then an introduction to maximum likelihood methods of estimation and significance testing for the regression model. This will be followed by an introduction to least-squares path analysis with measured variables. We will then turn to a brief introduction to exploratory factor analysis, and then confirmatory maximum likelihood models. This will lead us to the most general model we will probably discuss, linear structural relations with unmeasured variables. We will discuss non-recursive as well as recursive models here (i.e., "feedback" models). We will also consider the multitrait-multimethod models. From here we will turn to panal analysis, again with both measured and then unmeasured variables. We will then wrap up with a more general discussion of Markov models, evaluation of simplex and circumplex models, linear structural relationship models in planned experiments, and alternatives to the analysis of variance and covariance, both univariate and multivariate.

Multivariate Statistics:

Harris, R.J. (1985). A primer of multivariate statistics (Second edition). New York: Academic Press.

Morrison, D.F. (1976). Multivariate statistical methods (Second edition). New York: McGraw-Hill.

Timm, N.H. (1975). Multivariate statistics: With applications in education and psychology. Monterey, CA: Brooks/Cole.

Regression:

Cohen, J., & Cohen, P. (1983). Applied multiple regression/correlation analysis for the behavioral sciences (Second edition). New York: Erlbaum.

Namboodiri, N.K., Carter,L.F., & Blalock, H.M., Jr. (1975). Applied multivariate analysis and experimental designs. New York: McGraw-Hill.

Pedhazur, E.J. (1982). Multiple regression in behavioral research. New York: Holt, Rinehart and Winston.

Factor analysis:

Gorsuch, R.L. (1983). Factor analysis. (Second edition). Hillsdale, N.J.: Erlbaum.

Harman, H.H. (1967). Modern factor analysis. Chicago Press.

Jöreskog, K.G. (1963). Statistical estimation in factor analysis. Stockholm: Almqvist & Wiskell.

Lawley, D.N., & Maxwell, A.E. (1971). Factor analysis as a statistical method. (Second edition). New York: American Elsevier.

Mulaik, S.A. (1972). The foundation of factor analysis. New York: McGraw-Hill.

Measurement:

Blalock, H.M., Jr. (1974). Measurement in the social sciences. Chicago: Aldine.

Blalock, H.M., Jr. (1982). Conceptualization and measurement in the social sciences. Beverly Hills, CA: Sage.

Lord, F.M., & Novick, M.E. (1968). Statistical theories of mental test scores. Reading, MA: Addison-Wesley.

Nunnally, J. (1978). Psychometric theory. New York: McGraw-Hill.

Panel models:

Kessler, R.C., & Greenberg, D.F. (1981). Linear panel analysis: Models of quantitative change. New York: Academic Press.

Causal modelling:

Aigner, D.J., & Goldberger, A.S. (1977). Latent variables in socio-economic models. Amsterdam: North Holland.

Blalock, H.M., Jr. (1971). Causal models in the social sciences. Chicago: Aldine-Atherton.

Duncan, O.D. (1975). Introduction to structural equation models. New York: Academic Press.

Dwyer, J.H. (1983). Statistical models for the social and behavioral sciences. New York: Oxford.

Goldberger, A.S., & Duncan, O.D. (1973). Structural equation models in the social sciences. New York: Seminar Press.

Heise, D.R. (1975). Causal analysis. New York: Wiley-Interscience.

Kenny, D.A. (1979). Correlation and causality. New York: Wiley-Interscience.

James, L.R., Mulaik, S.A., & Brett, J.M. (1982). Causal analysis: Assumptions, models, and data. Beverly Hills, CA: Sage.

Loehlin, J.C. (1987). Latent variable models: Introduction to factor, path, and structural analysis. Hillsdale, NJ: Lawrence Erlbaum.

McDonald, R.P. (1985). Factor analysis and related methods. Hillsdale, NJ: Lawrence Erlbaum.

Factor Analysis:

Lawley, D.N., & Maxwell, A.E. (1964). Factor transformation methods. British Journal of Mathematical and Statistical Psychology, 17, 97-103.

Meredith, W. (1964). Notes on factorial invariance. Psychometrika, 29, 177-185.

Meredith, W. (1964). Rotation to achieve factorial invariance. Psychometrika, 36, 187-206.

Meredith, W. (1993). Measurement invariance, factor analysis, and factorial invariance. Psychometrika, 58, 525-543.

Jöreskog, K.G. (1967). Some contributions to maximum likelihood factor analysis. Psychometrika, 32, 443-482.

Jöreskog, K.G., & Lawley, D.N. (1968). New methods in maximum

likelihood factor analysis. British Journal of Mathematical and Statistical Psychology, 21, 85-96.

Jöreskog, K.G. (1969). A general approach to confirmatory

maximum likelihood factor analysis. Psychometrika, 34, 183-202.

Jöreskog, K.G., & Goldberger, A.S. (1972). Factor analysis by generalized least squares. Psychometrika, 37, 243-2650.

Archer, C.O., & Jennrich, R.I. (1973). Standard errors for rotated factor loadings. Psychometrika, 581-592.

Jennrich, R.I., (1973). Standard errors for obliquely rotated factor loadings. Psychometrika, 38, 593-604.

Jöreskog, K.G. (1976). Factor analysis by least squares and maximum likelihood methods. In K. Enslein, A.Ralston, & M.S. Wilf (Eds.), Statistical methods for digital computers. New York: Wiley, 125-165.

Bentler, P.M. (1976). Multistructure statistical model applied to factor analysis. Multivariate Behavioral Research, 11, 3-25.

McDonald, R.P., & Mulaik, S.A. (1979). Determinacy of common factors: A nontechnical review. Psychological Bulletin, 86, 297-306.

Mulaik, S.A. (1987). A brief history of the philosophical foundations of exploratory factor analysis. Multivariate Behavioral Research, 22, 267-305.

Browne, M.W. (1988). Properties of the maximum likelihood solution in factor analysis regression. Psychometrika, 53, 585-590.

MacCallum, R.C., & Tucker, L.R. (1991). Representing sources of error in the common-factor model: Implications for theory and practice. Psychological Bulletin, 109, 502-511.

Kano, Y., Ihara, M. (1994). Identification of inconsistent variates in factor analysis. Psychometrika, 59, 5-20.

Cudeck, R., & O'Dell, L.L. (1994). Applications of standard error estimates in unrestricted factor analysis: Significance tests for factor loadings and correlations. Psychological Bulletin, 115, 475-487.

Structural Equation Models:

Bock, R.D., & Bargmann, R.E. (1966). Analysis of covariance structures. Psychometrika, 31, 507-534.

Duncan, O.D. (1966). Path analysis: Sociological examples. American Journal of Sociology, 72, 1-16.

Anderson, T.W. (1969). Statistical inference for covariance matrices with linear structure. In P.R. Krishnaiah (Ed.), Multivariate analysis - II. New York: Academic Press, 55-66.

Mukherjee, B.N. (1970). Likelihood ratio tests of statistical hypotheses associated with patterned covariance matrices in psychology. British Journal of Mathematical and Statistical Psychology, 23, 120.

Jöreskog, K.G. (1970). A general method for analysis of covariance structures. Biometrika, 57, 239-251.

Werts, C.E., & Linn, R.L. (1970). Path analysis: Psychological examples. Psychological Bulletin, 74, 193-212.

Hauser, R.M. & Goldberger, A.S. (1971). The treatment of unobservable variables in path analysis. In H.L. Costner (Ed.), Sociological methodology, San Francisco: Jossey-Bass, 81-117.

Goldberger, A.S. (1972). Structural equation methods in the social sciences. Econometrica, 40, 979-1001.

Anderson, T.W. (1973). Asymptotically efficient estimation of covariance matrices with linear structure. Annals of Statistics, 135-141.

Jöreskog, K.G. (1973). A general method for estimating a linear structural equation system. In A.S. Goldberger & O.D. Duncan (Eds.), Structural equation models in the social sciences. New York: Seminar Press, 85-112.

Jöreskog, K.G. (1973). Analysis of covariance structures. In P. Krishnaiah (Ed.), Multivariate analysis - III. New York: Academic Press.

Jöreskog, K.G. (1974). Analyzing psychological data by structural analysis of covariance matrices. In R.C. Atkinson, D.H. Krantz, R.D. Luce, & P. Suppes (Eds.), Contemporary developments in mathematical psychology - Volume II. San Francisco: W.H. Freeman & Company, 1-56.

Griliches, Z. (1974). Errors in variables and other unobservables. Econometrics, 42, 971-998.

Wiley, D.E. (1974). 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: Seminar Press, 69-83.

Jöreskog, K.G., & Goldberger, A.S. (1975). Estimation of a model with multiple indicators and multiple causes of a single latent variable. Journal of the American Statistical Association, 10, 631-639.

Geraci, V.J. (1976). Identification of simultaneous equation models with measurement error. Journal of Econometrics, 4, 263-283.

Jöreskog, K.G. (1977). Structural equation models in the social sciences: Specification, estimation and testing. In P.R. Krishnaiah (Ed.), Applications of statistics. Amsterdam: North Holland Publishing Company, 263-286.

Rock, D.A., Werts, C.E., Linn, R.L., & Jöreskog, K.G. (1977). A maximum likelihood solution to the errors in variables and errors in equations model. Multivariate Behavioral Research, 12, 187-197.

Jöreskog, K.G. (1978). Structural analysis of covariance and correlation matrices. Psychometrika, 443, 443-472.

McDonald, R.P. (1978). A simple comprehensive model for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 31, 59-72.

Sörbom, D. & Jöreskog, K.G. (1979). The use of LISREL in sociological model building. In E. Borgatta and D.J. Jackson (Eds.), Factor analysis and measurement in sociological research: A multidimensional perspective. Beverly Hills, CA: Sage.

Bentler, P.M., & Weeks, D.G. (1980). Linear structural equations with latent variables. Psychometrika, 45, 289-308.

Bentler, P.M. (1980). Multivariate analysis with latent

variables: Causal modeling. Annual Review of Psychology, 31, 419-456.

Bentler, P.M. (1983). Some contributions to efficient statistics in structural models: Specification and estimation of moment structure. Psychometrika, 48, 493-518.

Longitudinal Models:

Jöreskog, K.G. (1970). Estimation and testing of simplex models. British Journal of Mathematical and Statistical Psychology, 23, 121-145.

Sörbom, D. (1975). Detection of correlated errors in longitudinal data. British Journal of Mathematical and Statistical Psychology, 28, 138-151.

Jöreskog, K.G. & Sörbom, D. (1977). Statistical models and methods for analysis of longitudinal data. In D.J. Aigner & A.S. Goldberger (Eds.), Latent variables in socio-economic models. Amsterdam: North Holland Publishing Company, 235-285.

Jöreskog, K.G. (1978). An econometric model for multivariate panel data. Annales de ITNSEE, 30-31, 355-366.

Olsson, U., & Bergman, L.R. (1977). A longitudinal factor model for studying change in ability structure. Multivariate Behavioral Research, 12, 221-242.

Werts, C.E., Linn, R.L., & Jöreskog, K.G. (1977). A simplex model for analyzing academic growth. Educational and Psychological Measurement, 37, 745-756.

Werts, C.E., Linn, R.L., & Jöreskog, K.G. (1978). Reliability of college grades from longitudinal data. Educational and Psychological Measurement, 38, 89-95.

Jöreskog, K.G., & Sörbom, D. (1979). Simultaneous analysis of longitudinal data from several cohorts. Paper presented at the SSRC Conference on Analyzing Longitudinal Data for Age, Period and Cohort Effects in Snowmass, Colorado, June 18-20.

Jöreskog, K.G. (1979). Statistical estimation of structural models in longitudinal-developmental investigations. In J.R. Nesselroade & P.B. Baltes (Eds.), Longitudinal research in the study of behavior and development. New York: Academic Press, 303-351.

Labouvie, E.W. (1981). The study of multivariate change structures: A conceptual perspective. Multivariate Behavioral Research, 16, 23-25.

Bentler, P.M., & Freeman, E.H. (1983). Tests for stability in linear structural systems. Psychometrika, 48, 143-146.

Cudeck, R. (1986). A note on structural models for the circumplex. Psychometrika, 51, 143-147.

Goldstein, H., & McDonald, R.P. (1988). A general model for the analysis of multilevel data. Psychometrika, 53, 455-467.

Raykov, T. (1993). A sturctural equation model for measuring residualized change and discerning patterns of growth or decline. Applied Psychological Measurement, 17, 53-71.

Raykov, T. (1994). Studying correlates and predictors of longitudinal change using structural equation modeling. Applied Psychological Measurement, 18, 63-77.

Willett, J.B., & Sayer, A.G. (1994). Using covariance structure analysis to detect correlates and predictors of individual change over time. Psychological Bulletin, 116, 363-381.

Multiple-population Models:

Jöreskog, K.G. (1971). Simultaneous factor analysis in several populations. Psychometrika, 36, 409-426.

Sordom, D.A. (1974). A general method for studying differences in factor means and factor structure between groups. British Journal of Mathematical and Statistical Psychology, 27, 229-239.

Sörbom D. (1979). Structural equation models with structured means. Paper presented at the conference Systems under Indirect Observation: Causality, Structure and Prediction at Cartigny, Switzerland, October 18-20.

Muthén, B.O. (1989). Latent variable modeling in heterogeneous populations. Psychometrika, 54, 557-585.

Byrne, B.M., Shavelson, R.J., & Muthen, B. (1989). Testing for the equivalence of factor covariance and mean structures: The issue of partial measurement invariance. Psychological Bulletin, 105, 467-477.

Cole, D.A., Maxwell, S.E., Arvey, R., & Salas, E. (1993). Mutivariate group comparisons of variable systems: MANOVA and structural equation modeling. Psychological Bulletin, 114, 174-184.

Dolan, C.V., & Molenaar, P.C.M. (1994). Testing specific hypotheses concerning latent group difference in multi-group covariance structure analysis with structured means. Multivariariate Behavioral Research, 29, 203-222.

Measurement models:

Goldberger, A.S. (1970). Econometrics and psychometrics: A survey of communalities. Psychometrika, 35, 83-107.

Jöreskog, K.G. (1971). Statistical analysis of sets of congeneric tests. Psychometrika, 36, 109-133.

Jöreskog, K.G., & Sörbom, D. (1976). Statistical models and methods for test-retest situations. In D.N.M. de Gruijter & L. J. Th. van der Kamp (Eds.), Advances in psychological and educational measurement. New York: Wiley, 135-157.

Werts, C.E., Rock, D.A., Linn, R.L., & Jöreskog, K.G. (1977). Validating psychometric assumptions within and between populations. Educational and Psychological Measurement, 37, 863-871.

Wheaton, B., Muthen, B., Alwin, D., & Summers, G. (1977). Assessing reliability and stability in panel models. In D.R. Heise (Ed.), Sociological Methodology, 1977. San Francisco: Jossey-Bass, 84-136.

Anderson, E.B. (1985). Estimating latent correlations between repeated testings. Psychometrika, 50, 3-16.

Schriesheim, C.A., Solomon, E., & Kopelman, R.E. (1989). Grouped versus randomized format: An investigation of scale convergent and discriminant validity using LISREL confirmatory factor analysis. Applied Psychological Measurement, 13, 19-32.

Millsap, R.E., & Everson, H. (1991). Confirmatory measurement model comparisons using latent means. Multivariate Behavioral Research, 26, 479-497.

O'Grady, K.E., & Medoff, D.R. (1991). Rater reliability: A maximum likelihood confirmatory factor-analytic approach. Multivariate Behavioral Research, 26, 363-387.

Labouvie, E., & Ruetsch, C. (1995). Testing for equivalence of measurement scales: Simple structure and metric invariance reconsidered. Multivariate Behavioral Research, 30, 63-76. [cf. all following commentaries.]

Goodness-of-fit:

Cudeck, R., & Browne, N.W. (1983). Cross-validation of covariance structures. Multivariate Behavioral Research, 8, 147-167.

Anderson, J.C., & Gerbing, D.W. (1984). The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analysis. Psychometrika, 49, 155-174.

Bollen, K.A. (1986). Sample size and Bentler and Bonett's nonnormed fit index. Psychometrika, 51, 375-378.

Balderjahn, I. (1988). A note to Bollen's alternative fit measure. Psychometrika, 53, 283-285.

Browne, M.W., & Cudeck, R. (1989). Single-sample cross-validation

in a structural equation model. Multivariate Behavioral Research, 24, 445-455.

Maiti, S.S., & Mukherjee, B.N. (1990). A note on distributional properties of the Jöreskog-Sörbom fit indices. Psychometrika, 55, 721-726.

Bentler, P.M. (1990). Comparative fit indices in structural models. Psychological Bulletin, 107, 238-246.

Bollen, K.A. (1990). Overall fit in covariance structure models: Two types of sample size effects. Psychological Bulletin, 107, 256-259.

McDonald, R.P., & Marsh, H.W. (1990). Choosing a multivariate model: Noncentrality and goodness of fit. Psychological Bulletin, 107, 247-255.

Goffin, R.D. (1993). A comparison of two new indices for the assessment of fit of structural equation models. Multivariate Behavioral Research, 28, 205-214.

Bandalos, D.L. (1993). Factors influencing cross-validation of confirmatory factor analysis models. Multivariate Behavioral Research, 28, 351-374.

MacCallum, R.C., Roznowski, M., Mar, C.M., & Reith, J.V. (1994). Alternative strategies for cross-validation of covariance structure models. Multivariate Behavioral Research, 29, 1-32.

Sugawara, H.W., & MacCallum, R.C. (1993) Effect of estimation method on incremental fit indexes for covariance structure models. Applied Psychological Measurement, 17, 365-377.

Model modification:

MacCallum, R. (1986). Specification searches in covariance structure modeling. Psychological Bulletin, 100, 107-129,

Dillon, W.R., Kumar, A., & Mulani, N. (1987). Offending estimates in covariance structure analysis: Comments on the causes and solutions to Heywood cases. Psychological Bulletin, 101, 126-135.

Kaplan, D. (1988). The impact of specification error on the estimation, testing, and improvement of structural equation models. Multivariate Behavioral Research, 23, 69-86.

Kaplan, D. (1989). A study of the sampling variability and z-values of parameter estimates from misspecified structural equation models. Multivariate Behavioral Research, 24, 41-58.

Kaplan, D. (1989). Model modification in covariance structure analysis: Application of the expected parameter change statistic. Multivariate Behavioral Research, 24, 285-305.

Lance, C.E. (1989). Disturbance term regression tests: A note on the computation of standard errors. Multivariate Behavioral Research, 24, 135-141.

Sörbom, D. (1989). Model modification. Psychometrika, 54, 371-384.

Bentler, P.M., & Mooijaart, A. (1989). Choice of a structural model via parsimony: A rationale based on precision. Psychological Bulletin, 106, 315-317.

Chou, C-P., & Bentler, P.M. (1990). Model modification in covariance structure modeling: A comparison among likelihood ratio, Lagrange multiplier, and Wald tests. Multivariate Behavioral Research, 25, 115-136.

Kaplan, D. (1990). Evaluating and modifying covariance structure models: A review and recommendation. Multivariate Behavioral Research, 25, 137-156.

MacCallum, R.C. (1990). The need for alternative measures of fit in covariance structure modeling. Multivariate Behavioral Research, 25, 157-162.

Bentler, P.M. (1990). Fit indexes, Lagrange multipliers, constraint changes and incomplete data in structural models. Multivariate Behavioral Research, 25, 163-172.

Steiger, J.H. (1990). Structural model evaluation and modification: An interval estimation approach. Multivariate Behavioral Research, 25, 173-180.

Bollen, K.A. (1990). A comment on model evaluation and modification. Multivariate Behavioral Research, 25, 181-186.

Tanaka, J.S. (1990). Towards the second generation of structural modeling. Multivariate Behavioral Research, 25, 187-192.

Hayduk, L.A. (1990). Should model modifications be oriented towards improving data fit or encouraging creative and analytical thinking? Multivariate Behavioral Research, 25, 193-196.

Kaplan, D. (1990). A rejoinder on evaluating and modifying covariance structure models. Multivariate Behavioral Research, 25, 197-204.

Cudeck, R., & Henly, S.J. (1991). Model selection in covariance structures analysis and the "problem" of sample size: A clarification. Psychological Bulletin, 109, 512-519.

Bollen, K., & Lennox, R. (1991). Conventional wisdom and measurement: A structural equation perspective. Psychological Bulletin, 110, 305-314.

MacCallum, R.C., Roznowski, M., & Necowitz, L.B. (1992). Model modifications in covariance structure analysis: The problem of capitalization on chance. Psychological Bulletin, 111, 490-504.

Chou, C-P., & Bentler, P.M. (1993). Invariant standardized estimated parameter change for model modification in covariance structure analysis. Multivariate Behavioral Research, 28, 97-110.

Multitrait-multimethod Models:

Marsh, H.W. (1989). Confirmatory factor analyses of multitrait-multimethod data: Many problems and a few solutions. Applied Psychological Measurement, 13, 335-361.

Wothke, W., & Browne, M.W. (1990). The direct product model for the MTMM matrix parameterized as a second order factor analysis model. Psychometrika, 55, 255-262.

Graham, J.W., & Collins, N.L. (1991). Controlling correlational bias via confirmatory factor analysis of MTMM data. Multivariate Behavioral Research, 26, 607-629.

Kenny, D.A., & Kashy, D.A. (1992). Analysis of the multitrait-multimethod matrix by confirmatory factor analysis. Psychological Bulletin, 112, 165-172.

Goffin, R.D., & Jackson, D.N. (1992). Analysis of multitrait-multirater performance appraisal data: Composite direct product model versus confirmatory factor analysis. Multivariate Behavioral Research, 27, 363-385.

Marsh, H.W., Byrne, B.M., & Craven, R. (1992). Overcoming problems in confirmatory factor analysis of MTMT data: The correlated uniqueness model and factorial invariance. Multivariate Behavioral Research, 27, 489-508.

Byrne, B.M., & Goffin, R.D. (1993). Modeling MTMM data from additive and multiplicative covariance structures: An audit of construct validity concordance. Multivariate Behavioral Research, 28, 67-96.

Marsh, H.W., & Byrne, B.M. (1993). Confirmatory factor analysis of multitrait-multimethod self-concept data: Between-group and within-group invariance constraints. Multivariate Behavioral Research, 28, 313-340.

Grayson, D., & Marsh, H.W. (1994). Identification with deficient rank loading matrices in confirmatory factor analysis. Psychometrika, 59, 121-134.

Reichart, C.S., & Coleman, S.C. (1995). The criteria for convergent and discriminant validity in a multitrait-multimethod matrix. Multivariate Behavioral Research, 30, 513-538.

Analysis of Covariance Models:

Magidson, J. (1977). Toward a causal model approach for adjusting for preexisting differences in the nonequivalent control group situation: A general alternative to ANCOVA. Evaluation Quarterly, 1, 399-420.

Sörbom, D. (1978). An alternative to the methodology for analysis of covariance. Psychometrika, 43, 381-396.

Raaijmakers, J.G.W., & Pieters, J.P.M. (1987). Measurement error and ANCOVA: Functional and structural relationship approaches. Psychometrika, 52, 521-538.

Test Criteria:

Satorra, A., & Saris, W.E. (1982). The accuracy of a procedure for calculating the power of the likelihood ratio test as used within the LISREL framework. In Middendorp, C.P., Niemoller, B., & Saris, W.E. (Eds.), Sociometric Research 1982. Amsterdam: Sociometric Research Foundation.

Sartorra, A., & Saris, W.E. (1985). Power of the likelihood ratio test in covariance structure analysis. Psychometrika. 50, 83-90.

Steiger, J.H., Shapiro, A., & Browne, M.W. (1985). On the multivariate asymptotic distribution of sequential chi-square statistics. Psychometrika, 50, 253-264.

Saris, W.E., & Satorra, A. (1987). Characteristics of structural equation models which affect the power of the likelihood ratio test. In Saris, W.E. & Gallhofer, I.N. (Eds.), Sociometric Research, Volume II. London: MacMillan.

Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach. Psychometrika, 54, 131-151.

Hu, L-T., Bentler, P.M., & Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted? Psychological Bulletin, 112, 351-362.

Chan, W., Yung, Y-F., & Bentler, P.M. (1995). A note on using an unbiased weight matrix in the ADF test statistic. Multivariate Behavioral Research, 30, 453-459.

Extensions and Generalizations:

Browne, M.W. (1977). The analysis of patterned correlation matrices by generalized least squares. British Journal of Mathematical and Statistical Psychology, 30, 113-124.

Hagglund, G. (1982). Factor analysis by instrumental variables methods. Psychometrika, 47, 209-222.

Muthen, B. (1983). Latent variable structural equation modeling with categorical data. Journal of Econometrics, 22, 43-65.

Hodapp, V., & Werdmuth, N. (1983). Decomposable models: A new look at interdependence and dependence structures in psychological research. Multivariate Behavioral Research, 18, 361-390.

Muthen, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115-132.

Lee, S-Y., & Jennrich, R.I. (1984). The analysis of structural equation models by means of derivative free nonlinear least squares. Psychometrika, 49, 521-528.

Lee, S-Y., & Poon, W. (1986). Maximum likelihood estimation of polyserial correlations. Psychometrika, 51, 113-122.

Poon, W., & Lee, S-Y. (1987). Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients. Psychometrika, 52, 409-430.

Lance, C.E., Cornwall, J.M., & Mulaik, S.A. (1988). Limited information parameter estimates for latent or mixed manifest and latent variable models. Multivariate Behavioral Research, 23 171-188.

Muthén, B., Hofacker, C. (1988). Testing the assumptions underlying tetrachoric correlations. Psychometrika, 53, 563-578.

Lee, S-Y., Poon, W-Y., & Bentler, P.M. (1989). Simultaneous analysis of multivariate polytomous variates in several groups. Psychometrika, 54, 63-73.

Arminger, G., & Schoenberg, R.J. (1989). Pseudo maximum likelihood estimation and a test for misspecification in mean and covariance structure models. Psychometrika, 54, 409-426.

Brown, R.L. (1989). Congeneric modeling of reliability using censored variables. Applied Psychological Measurement, 13, 151-159.

Lehmann, D.R., & Gupta, S. (1989). PACM: A two-stage procedure for analyzing structural models. Applied Psychological Measurement, 13, 301-321.

Fornell, C., & Rust, R.T. (1989). Incorporating prior theory in covariance structure analysis: A Bayesian approach. Psychometrika, 54, 249-259.

Browne, M.W. (1992). Automated fitting of nonstandard models. Multivariate Behavioral Research, 27, 269-300.

Lee, S-Y., Poon, W-Y., & Bentler, P.M. (1992). Structural equation models with continuous and polytomous variables. Psychometrika, 57, 89-106.

Cudeck, R., Klebe, K.J., & Henley, S.J. (1993). A simple Gauss-Newton procedure for covariance structure analysis with high-level computer languages. Psychometrika, 58, 211-232.

McDonald, R.P., & Parker, P.M. (1993). A scale-invariant treatment for recursive path models. Psychometrika, 58, 431-443.

Missing Data:

Lee, S-Y. (1986). Estimation for structural equation models with missing data. Psychometrika, 51, 93-100.

Muthen, B., Kaplan, D., & Hollis, M. (1987). On the structural equation modeling with data that are not missing completely at random. Psychometrika, 52, 431-462.

Kiiveri, H.T. (1987). An incomplete data approach to the analysis of covariance structures. Psychometrika, 52, 539-554.

McArdle, J.J. (1994). Structural factor analysis experiments with incomplete data. Multivariate Behavioral Research, 29, 409-454.

Simulations:

Olsson, U. (1979). On the robustness of factor analysis against crude classification of the observations. Multivariate Behavioral Research, 14, 485-500.

Boomsma, A. (1985). Nonconvergence, improper solutions, and starting values in LISREL maximum likelihood estimation. Psychometrika, 50, 229-242.

Gerbing, D.W., & Anderson, J.C. (1985). The effect of sampling

error and model characteristics on parameter estimation for maximum likelihood confirmatory factor analysis. Multivariate Behavioral Research, 20, 255-272.

ation Farley, J.U., & Reddy, S.K. (1987). A factorial evaluation of effects of model specification and error on parameter estimation in a structural equation model. Multivariate Behavioral Research, 22, 71-90.

Gerbing, D.W., & Anderson, J.C. (1987). Improper solutions in the analysis of covariance structures: Their interpretability and a comparison of alternative respecifications. Psychometrika, 52, 99-112.

Silvia, E.S.M., & MacCallum, R.C. (1988). Some factors affecting the success of specification searches in covariance structure modeling. Multivariate Behavioral Research, 23, 297-326.

Marsh, H.W., Balla, J.R., & McDonald, R.P. (1988). Goodness-of-fit indexes in confirmatory factor analysis: The effect of sample size. Psychological Bulletin, 103, 391-410.

Mulaik, S.A., James, L.R., Alstine, J.V., Bennett, N., Lind, S., & Stillwell, C.D. (1989). Evaluation of goodness-of-fit indices for structural equation models. Psychological Bulletin, 105, 430-445.

Stone, C.A., & Sobel, M.E. (1990). The robustness of estimates of total indirect effects in covariance structure models estimated by maximum likelihood. Psychometrika, 55, 337-352.

Interactions:

Busemeyer, J.R., & Jones, L.E. (1983). Analysis of multiplicative combination rules when the causal variables are measured with error. Psychological Bulletin, 93, 549-562.

Kenny, D.A., & Judd, C.M. (1984). Estimating the nonlinear and interactive effects of latent variables. Psychological Bulletin, 96, 201-210.

Jaccard, J., & Wan, C.K. (1995). Measurement error in the analysis of interaction effects between continuous predictors using multiple regression: Multiple indicator and structural equation approaches. Psychological Bulletin, 117, 348-357.

Lance, C.E. (1988). Residual centering, exploratory and confirmatory moderator analysis, and decomposition of effects in path models containing interactions. Applied Psychological Measurement, 12, 163-175.

Wood, P.K., & Games, P. (1990). Rationale, detection, and implications of interactions between indpendent variables and unmeasured variables in linear models. Multivariate Behavioral Research, 25, 295-311.

Ping, Jr., R.A. (1996). Latent variable interaction and quadratic effect estimation: A two-step technique using structural equation analysis. Psychological Bulletin, 119, 166-175.

Analysis of correlation matrices:

Werts, C.E., Rock, D.A., Linn, R.L., & Jöreskog, K.G. (1976). A comparison of correlations, variances, covariances and regression weights with and without measurement error. Psychological Bulletin, 83, 1007-1013.

Cudeck, R. (1989). Analysis of correlation matrices using covariance structure models. Psychological Bulletin, 105, 317-327.

Reparameterization:

Rindskopf, D. (1984). Using phantom and imaginary latent variables to parameterize constraints in linear structural models. Psychometrika, 49, 37-48.

Rindskopf, D. (1984). Linear equality restrictions in regression and loglinear models. Psychological Bulletin, 96, 597-603.

Green, D.P., & Palmquist, B.L. (1991). More "tricks of the trade": Reparameterizing LISREL models using negative variances. Psychometrika, 56, 137-145.

Commentary:

Bentler, P.M. (1978). The interdependence of theory, methodology, and empirical data: Causal modeling as an approach to construct validation. In D.B. Kandel (Ed.), Longitudinal research on drug use: Empirical findings and methodological issues. Washington, DC: Hemisphere.

Cliff, N. (1983). Some cautions concerning the application of causal modeling methods. Multivariate Behavioral Research, 8, 115-126.

Stelzl, I. (1986). Changing causal hypotheses without changing the fit: Some rules for generating equivalent path models. Multivariate Behavioral Research, 21, 309-332.

Bentler, P.M. (1986). Structural modeling and Psychometrika: An historical perspective on growth and achievements. Psychometrika, 35-52.

Otter, P.W. (1986). Dynamic structural systems under indirect observation: Identifiability and estimation aspects from a system theoretic perspective. Psychometrika, 51, 415-428.

Anderson, J.C., & Gerbing, D.W. (1988). Structural equation modeling in practice: A review and recommended two-step approach. Psychological Bulletin, 103, 411-423.

Vittadini, G. (1989). Indeterminacy problems in the LISREL model. Multivariate Behavioral Research , 24, 397-414.

Cohen, P., Cohen, J., Teresi, J., Marchi, M., & Velez, C.N. (1990). Problems in the measurement of latent variables in structural equation models. Applied Psychological Measurement, 14, 183-196.

Breckler, S.J. (1990). Applications of covariance structure modeling in Psychology: Cause for concern? Psychological Bulletin, 107, 260-273.

Stelzl, I. (1991). Rival hypotheses in linear structure modeling: Factor rotation in confirmatory factor analysis and latent path analysis. Multivariate Behavioral Research, 26, 199-225.

Luijben, T.C.W. (1991). Equivalent models in covariance structure analysis. Psychometrika, 56, 653-666.

MacCallum, R.C., Wegener, D.T., Uchino, B.N., & Fabrigar, L.R. (1993). The problem of equivalent models in applications of covariance structure analysis. Psychological Bulletin, 114, 185-199.