This article proposes a method to detect influential observations in a factor analysis model with continuous and ordinal categorical variables. The key ideas are to treat the latent factors as hypothetical missing data and then develop the diagnostic measures on the basis of the conditional expectation of the complete-data log-likelihood function in the EM algorithm. A one-step approximation is proposed to reduce the computational burden. Building blocks for achieving the diagnostic measures are computed via observations generated by the Gibbs sampler. Results from a simulation study and an illustrative real example are presented.
Get full access to this article
View all access options for this article.
References
1.
Andensen, Erling B. 1992. “Diagnostics in Categorical Data Analysis.” Journal of the Royal Statistical Society, Series B 54:781-91.
2.
Banerjee, Mousumi and Edward W. Frees. 1997. “Influence Diagnostics for Linear Longitudinal Models.” Journal of the American Statistical Association92:999-1005.
3.
Beckman, R. J. and Dennis R. Cook. 1983. “Outliers … ” Technometrics25:119-49.
4.
Belsley, David A., Edwin Kuh, and Roy E. Welsch. 1980. Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. New York: John Wiley.
5.
Bentler, Peter M. 1992. EQS: Structural Equation Program Manual. Los Angeles: BMDP Statistical Software.
6.
Bentler, Peter M. 1998. “Causal Modeling: New Interfaces and New Statistics.” In Advances in Psychological Science: Social, Personal, and Cultural Aspects, vol. 1, edited by J. G. Adair, D. Belanger, and K. L. Dion. East Sussex, UK: Psychology Press.
7.
Bentler, Peter M. and C.-P. Chou. 1993. “Some New Covariance Structural Model Improvement Statistics.” In Testing Structural Equation Models, edited by K. A. Bollen and J. S. Long, 235-55. Newbury Park, CA: Sage.
8.
Berkane, M. and Peter M. Bentler. 1988. “Estimation of Contamination Parameters and Identification of Outliers in Multivariate Data.” Sociological Methods & Research17:55-64.
9.
Bollen, Kenneth A. 1987. “Outliers and Improper Solutions: A Confirmatory Factor Analysis Example.” Sociological Methods & Research15:375-84.
10.
Bollen, Kenneth A. 1989. Structural Equations With Latent Variables. New York: Wiley.
11.
Bollen, Kenneth A. and R. W. Jackman. 1985. “Regression Diagnostics: An Expository Treatment of Outliers and Influential Cases.” Sociological Methods & Research13:510-42.
12.
Bollen, Kenneth A. and J. Scott Long. 1993. Testing Structural Equation Models. Newbury Park, CA: Sage.
13.
Chatterjee, Samprit and Ali S. Hadi. 1988. Sensitivity Analysis of Linear Regression. New York: John Wiley.
14.
Christensen, Ronald, Larry M. Pearson, and Wesley Johnson. 1992. “Case-Deletion Diagnostics for Mixed Models.” Technometrics34:38-45.
15.
Cook, Dennis R. 1977. “Detection of Influential Observations in Linear Regression.” Technometrics19:15-18.
16.
Cook, Dennis R. 1986. “Assessment of Local Influence (with discussion).” Journal of the Royal Statistical Society, Series B 48:133-69.
17.
Cook, Dennis R. and Sanford Weisberg. 1982. Residuals and Influence in Regression. New York: Chapman and Hall.
18.
Davison, A. C. and C. L. Tsai. 1992. “Regression Model Diagnostics.” International Statistics Review60:337-55.
19.
Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. “Maximum Likelihood From Incomplete Data via the EM Algorithm (with discussion).” Journal of the Royal Statistical Society, Series B 39:1-38.
20.
Gelman, Andrew. 1996. “Inference and Monitoring Convergence.” In Markov Chain Monte Carlo in Practice, edited by W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, 131-44. London: Chapman and Hall.
21.
Geman, S. and D. Geman. 1984. “Stochastic Relaxation, Gibbs Distribution, and the Bayesian Restoration of Images.” IEEE Transactions on Pattern Analysis and Machine Intelligence6:721-41.
22.
Jöreskog, K. G. and D. Sörbom. 1996. LISREL *8: Structural Equation Modeling With the SIMPLIS Command LANGUAGE. London: Scientic Software International.
23.
Lee, Sik-Yum, W. Y. Poon, and Peter M. Bentler. 1995. “A Two-Stage Estimation of Structural Equation Models With Continuous and Polytomous Variables.” British Journal of Mathematical and Statistical Psychology48:339-58.
24.
Lee, Sik-Yum and Jian-Qing Shi. 2001. “Maximum Likelihood Estimation of Two-Level Latent Variable Models With Mixed Continuous and Polytomous Data.” Biometrics57:787-94.
25.
Lee, Sik-Yum and Hong-Tu Zhu. 2000. “Statistical Analysis of Nonlinear Structural Equation Models With Continuous and Polytomous Data.” British Journal of Mathematical and Statistical Psychology53:209-32.
26.
Lesaffre, Emmanuel and Geert Verbeke. 1998. “Local Influence in Linear Mixed Models.” Biometrics54:570-82.
27.
Little, J. A. Roderrisk and Donald B. Rubin. 1997. Statistical Analysis With Missing Data. New York: Wiley.
28.
Meng, Xiao-Li and Stephen Schilling. 1996. “Fitting Full-Information Item Factor Models and an Empirical Investigation of Bridge Sampling.” Journal of the American Statistical Association91:1254-67.
29.
Meng, Xiao-Li and Wing Hung Wong. 1996. “Simulating Ratios of Normalizing Constants via a Simple Identity: A Theoretical Exploration.” Statistica Sinica6:831-60.
30.
Morisky, D. E., T. V. Tiglao, C. D. Sneed, S. B. Tempongko, J. C. Baltazar, R. Detels, and J. A. Stein. 1998. “The Effects of Establishment Practices, Knowledge and Attitudes on Condom Use Among Filipina Sex Workers.” AIDS Care10:213-20.
31.
Poon, Wai-Yin and Yat-Sun Poon. 2001. “Conditional Local Influences in Case-Weights Linear Regression.” British Journal of Mathematical and Statistical Psychology54: 177-91.
32.
Reboussin, B. A. and K. Y. Liang. 1998. “An Estimating Equation Approach for the LISCOMP Model.” Psychometrika63:165-82.
33.
Rousseeuw, Peter J. and Annick M. Leroy. 1987. Robust Regression and Outlier Detection. London: Chapman and Hall.
34.
Rubin, Donald B. and D. T. Thayer. 1982.“EMAlgorithms for Factor Analysis.” Psychometrika47:69-76.
35.
Shi, Jian-Qing and Sik-Yum Lee. 1998. “Bayesian Sampling-Based Approach for Factor Analysis Models With Continuous and Polytomous Data.” British Journal of Mathematical and Statistical Psychology51:233-52.
36.
Shi, Jian-Qing and Sik-Yum Lee. 2000. “Latent Variable Models With Mixed Continuous and Polytomous Data.” Journal of the Royal Statistical Society, Series B 62:77-87.
37.
Song, Xin-Yuan and Sik-Yum Lee. 2001. “Bayesian Estimation and Test for Factor Analysis Model With Continuous and Polytomous Data in Several Populations.” British Journal of Mathematical and Statistical Psychology54:237-63.
38.
Tanaka, Yutaka and Yoshimasa Y. Odaka. 1989. “Influential Observations in Principal Factor Analysis.” Psychometrika54:457-85.
39.
Tanaka, Yutaka, S. Watadani, and S. H. Moon. 1991. “Influence in Covariance Structure Analysis: With an Application to Confirmatory Factor Analysis.” Communications in Statistics, Series A 20:3805-21.
40.
Zhu, Hong-Tu and Sik-Yum Lee. 2001. “Local Influence for Incomplete Data Models.” Journal of the Royal Statistical Society, Series B 63:111-26.
41.
Zhu, Hong-Tu, Sik-Yum Lee, Bo-Cheng Wei, and Julie Zhou. 2001. “Case-Deletion Measures for Models With Incomplete Data.” Biometrika88:727-37.
