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An Empirical Bayesian Approach to Misspecified Covariance Structures An Empirical Bayesian Approach to Misspecified Covariance Structures Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Hao Wu, M.A., M.S. Graduate Program in Psychology The Ohio State University 2010 Dissertation Committee: Dr. Michael W. Browne, Advisor Dr. Michael C. Edwards Dr. Steven N. MacEachern Dr. Thomas E. Nygren Copyright by Hao Wu 2010 Abstract The analysis of covariance structures has been an important topic in psy- chometrics and latent variable modeling. A covariance structure is a model for the covariance matrix of observed manifest variables. It is derived from hypothesized lin- ear relationships among the manifest variables and hypothesized unobserved latent variables. The traditional approach to covariance structures has been successful when the covariance structure is correctly specified, i.e., when the population covariance matrix satisfies the given covariance structure. However, in reality, covariance struc- tures never hold exactly in the population as the hypotheses behind them are only approximations to the truth. Consequently it is necessary to model misspecification when covariance structures are analyzed. The traditional approach, nevertheless, only acknowledges and accounts for the effect of misspecification by post hoc modifications of the original approach to correctly specified covariance structures, and does not actively model the process that may have lead to misspecification. In this dissertation, we present a new approach to misspecified covariance structures in which the systematic error, identified as the process behind misspecifi- cation, is explicitly modeled along with the sampling error as a stochastic quantity ii with a distribution, and the inverse sample size for this distribution, as an unknown parameter to be estimated, gives a measure of misspecification. Analytical properties of the maximum beta likelihood (MBL) procedure implied by this approach and its limit, the maximum inverted Wishart likelihood (MIWL) procedure, are investigated and several connections with the traditional approach are found. Computer programs that give numerical implementations of these procedures are provided. Asymptotic sampling distributions of estimators given by the above two procedures are derived un- der different replication frameworks with a much weaker assumption than the usually invoked Pitman drift assumption. Sampling experiments are conducted to validate the asymptotic sampling distributions and to demonstrate the importance to account for the variations in the parameter estimates due to systematic error. iii Dedicated to my mother iv Acknowledgments I am greatly indebted to my advisor, Dr. Michael W. Browne, for his insight that initiated the current research, his intellectual support and encouragement with- out which the dissertation would not have been possible, and his careful review of the dissertation. I am also grateful to Dr. Steven N. MacEachern and Dr. Michael C. Edwards, whose insightful comments and helpful suggestions point to the practical significance of the current research, shed light on its potentials of further development, and lead to the improvement of the dissertation in terms of both the presentation of key ideas and the demonstration of key results. I wish to thank Dr. Thomas E. Nygren for serving in my dissertation com- mittee though he has a tight schedule as the vice chair of the department. This work was partially supported by NSF grant SES-0437251 from January 2008 through September 2009. v Vita 2004 ................................. B.S. Statistics, Peking University, China 2006 ................................. M.A. Psychology, The Ohio State University 2007 ................................. M.S. Statistics, The Ohio State University 2004 - present .................. Graduate Research and Teaching Associate, Department of Psychology, The Ohio State University Publications Wu, H., Myung, I. J. and Batchelder, W. H. (2010a) On the minimum description length complexity of multinomial processing tree models. Journal of Mathematical Psychology, 54, 291-303. Wu, H., Myung, I. J. and Batchelder, W. H. (2010b) Minimum description length model selection of multinomial processing tree models. Psychonomic Bulletin & Re- view, 17(3), 275-286. Fields of Study Major Field: Psychology Area of Concentration: Quantitative Psychology vi Table of Contents Page Abstract . ii Acknowledgments . v Vita . vi List of Figures . x List of Tables . xi List of Acronyms . xiii List of Notation . xiv Chapters: 1. Introduction . 1 1.1 Covariance Structures . 1 1.2 Model Misspecification . 4 1.2.1 The Issue of Misspecification . 4 1.2.2 Traditional Approach: Procedures . 4 1.2.3 Traditional Approach: Problems . 7 1.3 Overview of the Dissertation . 9 2. An Empirical Bayesian Approach: Motivation . 10 2.1 The Rationale behind the Model . 10 2.2 Statistical Formulation . 12 2.3 Previous Use of This Model . 13 2.4 A Parallel Example . 14 vii 3. Analytical Properties . 18 3.1 The Marginal Distribution . 18 3.2 Asymptotic Behavior . 19 3.2.1 Relationship to Wishart and Inverted Wishart Distributions 19 3.2.2 Relationship to the Normal Distribution . 20 3.3 The Inverted Wishart Model . 22 3.3.1 Parameter Estimation . 23 3.3.2 Bias Correction . 25 3.3.3 Usage as a Loss Function . 26 3.4 The Saturated Covariance Structure . 27 4. Computation . 29 4.1 Derivatives . 29 4.2 Approximate Hessian Matrix . 31 4.3 Computational Formulae . 33 4.4 Computation of the MIWLE . 35 5. Replication Frameworks and Consistency . 36 5.1 The Major Replication Framework . 36 5.2 An Alternative Replication Framework . 37 6. Unconditional Sampling Distribution . 39 6.1 An Additional Assumption . 39 6.2 MBLE . 40 6.3 MIWLE . 43 7. Conditional Sampling Distribution . 45 7.1 v# > 0.................................. 45 7.2 v# =0.................................. 48 7.3 A Non-Central χ2 Approximation . 49 7.4 MIWLE . 52 7.5 Confidence Intervals . 53 8. Sampling Experiments . 55 8.1 Unconditional Sampling Distribution . 55 8.1.1 n = m = 1000 . 56 viii 8.1.2 n = m = 200 . 58 8.1.3 When n and m Vary . 58 8.1.4 Interim Summary . 60 8.2 Conditional Sampling Distribution . 60 8.2.1 The Construction of a Misspecified Covariance Matrix . 61 8.2.2 n = m = 1000 . 62 8.2.3 n = m = 200 . 64 8.2.4 When n and m Vary . 65 8.2.5 Interim Summary . 67 8.3 Comparing the New Approach with the Traditional Approach . 68 9. Summary and Conclusions . 70 Bibliography . 73 Appendices: A. The Constant k ................................ 75 B. Mathematical Formulae . 77 B.1 Taylor Expansions of ln Γp(m/2) . 77 B.2 Second Derivatives of F1 and F2 .................... 81 C. Tables and Figures . 83 D. MATLAB Codes . 104 D.1 Parameter Estimation via MBL . 104 D.2 Parameter Estimation via MIWL . 112 D.3 CFA Covariance Structure and its Derivatives . 122 D.4 Construction of a Misspecified Covariance Matrix . 124 D.5 Inversion of the Non-Central χ2 Distribution . 126 ix List of Figures Figure Page C.1 Unconditional sampling distribution of ξˆ................. 95 IW C.2 Unconditional sampling distribution ofv ˆ0 andv ˆ0 . 96 C.3 Comparison of the unconditional sampling distributions of ξˆ and ξˆIW when n = m = 1000. 97 C.4 Comparison of the unconditional sampling distributions of ξˆ and ξˆIW when n = m = 200. 98 C.5 Conditional sampling distributions of ξˆ. 99 IW C.6 Conditional sampling distribution ofv ˆ0 andv ˆ0 . 100 C.7 Comparison of the conditional sampling distributions of ξˆ and ξˆIW when n = m = 1000. 101 C.8 Comparison of the conditional sampling distributions of ξˆ and ξˆIW when n = m = 200. 102 C.9 Comparison of ξˆ and ξˆW.......................... 103 x List of Tables Table Page C.1 The missing rates of 95% unconditional CIs of ξ when n = m = 1000. 83 C.2 Halflengths of 95% unconditional CIs of ξ when n = m = 1000. 84 C.3 The unconditional RMSEs of parameter estimators when n = m = 1000. 84 C.4 The missing rates of 95% unconditional CIs of ξ when n = m = 200. 85 C.5 The missing rates of 95% unconditional CBs of v. 86 C.6 The missing rates of 95% unconditional CIs of selected covariance structure parameters. 86 C.7 The average KS distances between the simulated and asymptotic un- conditional sampling distributions for both ξˆ and ξˆIW. ........ 87 C.8 The average KS distances between the simulated unconditional sam- pling distributions of ξˆ and ξˆIW...................... 87 C.9 The KS distances between the simulated and asymptotic unconditional IW sampling distributions for bothv ˆ0 andv ˆ0 . 87 C.10 The KS distances between the simulated unconditional sampling dis- IW tributions ofv ˆ0 andv ˆ0 .......................... 88 C.11 The missing rates of 95% conditional CIs of ξ when n = m = 1000. 89 C.12 Halflengths of 95% conditional CIs of ξ when n = m = 1000. 89 C.13 Comparing three estimators of Σ..................... 90 xi C.14 The conditional RMSEs of parameter estimators when n = m = 1000. 90 C.15 The missing rates of 95% conditional CIs of ξ when n = m = 200. 91 C.16 The missing rates of 95% conditional CIs of selected covariance struc- ture parameters. 91 ˆ C.17 The bias ofρ ˆ and ψ1 ........................... 92 C.18 The missing rates of 95% conditional CBs of v ............. 92 C.19 Lengths of 90% conditional CIs of v. .................. 92 C.20 The KS distances between the simulated and asymptotic conditional sampling distributions for estimators of selected parameters. 93 C.21 The KS distances between the simulated conditional sampling distri- butions
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