Mean comparison: Manifest variable versus latent variable

Psychometrika

Volumn 71, Issue 1, 2006, Pages 139-159

  Yuan, Ke Hai ^a   Bentler, Peter M ^b  

^a UNIVERSITY OF NOTRE DAME   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Asymptotic robustness; Empirical power; Likelihood ratio statistic; Noncentrality parameter; Parameter bias; Wald statistic

Indexed keywords

EID: 33744968642     PISSN: 00333123     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11336-004-1181-x     Document Type: Article

References (60)

1. 2 to variance-covariance heteroscedasticity when sample sizes are unequal and in small ratios. Psychological Bulletin, 108, 308-313.
2. Amemiya, Y., & Anderson, T.W. (1990). Asymptotic chi-square tests for a large class of factor analysis models. Annals of Statistics, 18, 1453-1463.
3. Arnold, S.F. (1981). The theory of linear models and multivariate analysis. New York: Wiley.
4. Bagozzi, R.P. (1977), Structural equation models in experimental research. Journal of Marketing Research, 14, 209-226.
5. Bagozzi, R.P., & Yi, Y. (1989). On the use of structural equation models in experimental designs. Journal of Marketing Research, 26, 271-284.
6. Bentler, P.M. (1968). Alpha-maximized factor analysis (Alphamax): Its relation to alpha and canonical factor analysis. Psychometrika, 33, 335-345.
7. Bentler, P.M. (1995). EQS structural equations program manual. Encino, CA: Multivariate Software.
8. Bentler, P.M., & Yuan, K.-H. (2000). On adding a mean structure to a covariance structure model. Educational and Psychological Measurement, 60, 326-339.
9. 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: Plenum Press.
10. Browne, M.W., & Shapiro, A. (1988). Robustness of normal theory methods in the analysis of linear latent variate models. British Journal of Mathematical and Statistical Psychology, 41, 193-208.
11. Buse, A. (1982). The likelihood ratio, Wald, and Lagrange multiplier tests: An expository note. American Statistician, 30, 153-157.
12. Byrne, B.M., Shavelson, R.J., & Muthén, B. (1989). Testing for the equivalence of factorial covariance and mean structures: The issue of partial measurement invariance. Psychological Bulletin, 105, 456-466.
13. 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.
14. Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.
15. Cole, D. A., Maxwell. S.E., Arvey, R., & Salas, E. (1993). Multivariate group comparisons of variable systems: MANOVA and structural equation modeling. Psychological Bulletin. 114, 174-184.
16. Curran, P.J. (2000). A latent curve framework for the study of developmental trajectories in adolescent substance use. In J.R. Rose, L. Chassin, C.C. Presson, & S.J. Sherman (Eds.), Multivariate applications in substance use research: New methods for new questions (pp. 1-42), Mahwah, NJ: Erlbaum.
17. Duncan, T.E., Duncan, S.C., Strycker, L.A., Li, F., & Alpert, A. (1999). An introduction to latent variable growth curve modeling: Concepts, issues, and applications. Mahwah, NJ: Erlbaum.
18. Efron, B., & Tibshirani, R.J. (1993). An introduction to the bootstrap. New York: Chapman & Hall.
19. Engle, R. (1984). Wald, likelihood ratio and Lagrange multiplier tests in econometrics. In Z. Griliches, & M.D. Intrilligator (Eds.). Handhook of econometrics (Vol. II, pp. 775-826). Amsterdam: North-Holland.
20. Ferguson, T. (1996). A course in large sample theory. London: Chapman & Hall.
21. 2 procedure and the assumption of homogeneous covariance matrices. Psychological Billetin, 86, 1255-1263.
22. Hancock, G.R. (2001). Effect size, power, and sample size determination for structured means modeling and MIMIC approaches to between-groups hypothesis testing of means on a single latent construct. Psychometrika, 66, 373-388.
23. Hancock, G.R. (2003). Fortune cookies, measurement error, and experimental design. Journal of Modern Applied Statistical Methods, 2, 293-305.
24. Hancock, G.R., Lawrence. F.R., & Nevitt, J. (2000). Type 1 error and power of latent mean methods and MANOVA in tactorially invariant and noninvariant latent variable systems. Structural Equation Modeling. 7, 534-556.
25. Hancock, G.R., & Mueller, R.O. (2001). Rethinking construct reliability. In R. Cudeck, S. du Toit. & D. Sörbom (Eds.), Structural equation modeling: Present and future (pp. 195-216). Lincolnwood. IL: Scientific Software International,
26. 2 test in multivariate analysis of variance when variance-covariance matrices are not equal, Biometrika, 51, 71-82.
27. Kano, Y. (1986). Conditions on consistency of estimators in the covariance structure model. Journal of the Japan Statistical Society, 16, 75-80.
28. 2 statistic under general distributions. American Journal of Mathematical and Management Sciences, 15, 317-341.
29. Kano, Y. (2001). Structural equation modeling for experimental data. In R. Cudeck, S.H.C. du Toit, & D. Sörbom (Eds.) Structural equation modeling: Present and future (pp. 381-402). Lincolnwood. IL: Scientific Software International.
30. Kano, Y., Bentler, P.M., & Mooijaart, A. (1993). Additional information and precision of estimators in multivariate structural models. In K. Matusita, M.L. Puri, & T. Hayakawa (Eds.), Statistical sciences and data analysis (pp. 187-196). Zeist. Netherlands: VSP International Science.
31. Kaplan, D., & George, R. (1995). A study of the power associated with testing factor mean differences under violations of factorial invariance. Structural Equation Modeling, 2, 101-118.
32. Kühnel, S.M. (1988). Testing MANOVA designs with LISREL. Sociological Methods and Research. 16, 504-523.
33. Li, H. (1997). A unifying expression for the maximal reliability of a linear composite. Psychometrika, 62, 245-249.
34. MacCallum,. R.C., Browne, M.W., & Sugawara, H.M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological Methods, 1, 130-149.
35. Mugnus, J.R., & Neudecker, H. (1999). Matrix differential calculus with applications in statistics and econometrics. New York: Wiley.
36. McArdle, J.J., & Epstein, D. (1987). Latent growth curves with developmental structure equation models. Child Development, 58, 110-133.
37. Meredith, W., & Tisak, J. (1990). Latent curve analysis. Psychometrika, 55, 107-122.
38. Muthén, B. (1989). Latent variable modeling in heterogeneous populations. Psychometrika, 54, 557-585.
39. Raykov, T. (2004). Estimation of maximal reliability: A note on a covariance structure modeling approach. British Journal of Mathematical and Statistical Psychology, 57, 21-27.
40. Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). New York: McGraw-Hill.
41. Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach. Psychometrika, 54, 131-151.
42. Satorra, A., & Bentler, P.M. (1990). Model conditions for asymptotic robustness in the analysis of linear relations. Computational Statistics & Data Analysis. 10, 235-249.
43. Satorra, A., & Saris, W. (1985). Power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50, 83-90.
44. Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance structures (a unified approach). South African Statistical Journal, 17, 33-81.
45. Shapiro, A. (1984). A note on the consistency of estimators in the analysis of moment structures. British Journal of Mathematical and Statistical Psychology, 37, 84-88.
46. 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.
47. Steiger, J.H., Shapiro, A., & Browne, M.W. (1985). On the multivariate asymptotic distribution of sequential chi-square statistics. Psychometrika, 50, 253-264.
48. Stroud, T.W.F. (1972). Fixed alternatives and Wald's formulation of the noncentral asymptotic behavior of the likelihood ratio statistic. Annals of Mathematical Statistics, 43, 447-454.
49. Wakaki, H., Yanagihara, H., & Fujikoshi, Y. (2002). Asymptotic expansions of the null distributions of test statistics for multivariate linear hypothesis under nonnormality. Hiroshima Mathematical Journal, 32, 17-50.
50. 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.
51. Yanagihara, H., & Yuan. K.-H. (2005). Three approximate solutions to the multivariate Behrens-Fisher problem. Communication in Statistics-Simulation and Computation, 34, 975-988.
52. Yuan, K.-H., & Bentler, P.M. (1998). Normal theory based test statistics in structural equation modeling. British Journal of Mathematical and Statistical Psychology, 51, 289-309.
53. Yuan, K.-H., & Bentler, P.M. (1999). On normal theory and associated test statistics in covariance structure analysis under two classes of nonnormal distributions. Statistica Sinica, 9, 831-853.
54. Yuan, K.-H., & Bentler, P.M. (2002). On robustness of the normal-theory based asymptotic distributions of three reliability coefficient estimates. Psychometrika, 67, 251-259.
55. Yuan, K.-H., & Bentler, P.M. (2004). On chi-square difference and z tests in mean and covariance structure analysis when the base model is misspecified. Educational and Psychological Measurement, 64, 737-757.
56. Yuan, K.-H., Bentler, P.M., & Kano, Y. (1997). On averaging variables in a confirmatory factor analysis model. Behaviormetrika, 24, 71-83.
57. Yuan, K.-H., & Chan, W. (2005). On nonequivalence of several procedures of structural equation modeling. Psychometrika, 70, 791-798.
58. Yuan, K.-H., Marshall, L.L., & Bentler, P.M. (2002). A unified approach to exploratory factor analysis with missing data, nonnormal data, and in the presence of outliers. Psychometrika, 67, 95-122.
59. Yuan, K.-H., Marshall, L.L., & Bentler, P.M. (2003). Assessing the effect of model misspecifications on parameter estimates in structural equation models. Sociological Methodology, 33, 241-265.
60. Yung, Y.-F., & Bentler, P.M. (1999). On added information for ML factor analysis with mean and covariance structures. Journal of Educational and Behavioral Statistics, 24, 1-20.