Direction Dependence in Statistical Modeling

WOLFGANG WIEDERMANN is Associate Professor at the University of Missouri-Columbia. He received his Ph.D. in Quantitative Psychology from the University of Klagenfurt, Austria. His primary research interests include the development of methods for causal inference, methods to determine the causal direction of dependence in observational data, and methods for person-oriented research settings. He has edited books on advances in statistical methods for causal inference (with von Eye, Wiley) and new developments in statistical methods for dependent data analysis in the social and behavioral sciences (with Stemmler and von Eye). DAEYOUNG KIM is Associate Professor of Mathematics and Statistics at the University of Massachusetts, Amherst. He received his Ph.D. from the Pennsylvania State University in Statistics. His original research interests were in likelihood inference in finite mixture modelling including empirical identifiability and multimodality, development of geometric and computational methods to delineate multidimensional inference functions, and likelihood inference in incompletely observed categorical data, followed by a focus on the analysis of asymmetric association in multivariate data using (sub)copula regression. ENGIN A. SUNGUR has a B.A. in City and Regional Planning (Middle East Technical University, METU, Turkey), M.S. in Applied Statistics, METU, M.S. in Statistics (Carnegie-Mellon University, CMU) and Ph.D. in Statistics (CMU). He taught at Carnegie-Mellon University, University of Pittsburg, Middle East Technical University, and University of Iowa. Currently, he is a Morse-Alumni distinguished professor of statistics at University of Minnesota Morris. He is teaching statistics for more than 38 years, 29 years of which is at the University of Minnesota Morris. His research areas are dependence modeling with emphasis on directional dependence, modern multivariate statistics, extreme value theory, and statistical education. ALEXANDER VON EYE is Professor Emeritus of Psychology at Michigan State University (MSU). He received his Ph.D. in Psychology from the University of Trier, Germany. He received his accreditation as Professional Statistician from the American Statistical Association (PSTATTM). His research focuses (1) on the development and testing of statistical methods for the analysis of categorical and longitudinal data, and for the analysis of direction dependence hypotheses. In addition (2), he is member of a research team at MSU (with Bogat, Levendosky, and Lonstein) that investigates the effects of violence on women and their newborn children. His third area of interest (3) concerns theoretical developments and applied analysis of person-orientation in empirical research.

About the Editors xv Notes on Contributors xvii Acknowledgments xxi Preface xxiii Part I Fundamental Concepts of Direction Dependence 1 1 From Correlation to Direction Dependence Analysis 18882018 3 Yadolah Dodge and Valentin Rousson 1.1 Introduction 3 1.2 Correlation as a Symmetrical Concept of X and Y 4 1.3 Correlation as an Asymmetrical Concept of X and Y 5 1.4 Outlook and Conclusions 6 References 6 2 Direction Dependence Analysis: Statistical Foundations and Applications 9 Wolfgang Wiedermann, Xintong Li, and Alexander von Eye 2.1 Some Origins of Direction Dependence Research 11 2.2 Causation and Asymmetry of Dependence 13 2.3 Foundations of Direction Dependence 14 2.3.1 Data Requirements 15 2.3.2 DDA Component I: Distributional Properties of Observed Variables 16 2.3.3 DDA Component II: Distributional Properties of Errors 19 2.3.4 DDA Component III: Independence Properties 20 2.3.5 Presence of Confounding 21 2.3.6 An Integrated Framework 24 2.4 Direction Dependence in Mediation 29 2.5 Direction Dependence in Moderation 32 2.6 Some Applications and Software Implementations 34 2.7 Conclusions and Future Directions 36 References 38 3 The Use of Copulas for Directional Dependence Modeling 47 Engin A. Sungur 3.1 Introduction and Definitions 47 3.1.1 Why Copulas? 48 3.1.2 Defining Directional Dependence 48 3.2 Directional Dependence Between Two Numerical Variables 51 3.2.1 Asymmetric Copulas 52 3.2.2 Regression Setting 59 3.2.3 An Alternative Approach to Directional Dependence 62 3.3 Directional Association Between Two Categorical Variables 70 3.4 Concluding Remarks and Future Directions 74 References 75 Part II Direction Dependence in Continuous Variables 79 4 Asymmetry Properties of the Partial Correlation Coefficient: Foundations for Covariate Adjustment in Distribution-Based Direction Dependence Analysis 81 Wolfgang Wiedermann 4.1 Asymmetry Properties of the Partial Correlation Coefficient 84 4.2 Direction Dependence Measures when Errors Are Non-Normal 86 4.3 Statistical Inference on Direction Dependence 89 4.4 Monte-Carlo Simulations 90 4.4.1 Study I: Parameter Recovery 90 4.4.1.1 Results 91 4.4.2 Study II: CI Coverage and Statistical Power 91 4.4.2.1 Type I Error Coverage 94 4.4.2.2 Statistical Power 94 4.5 Data Example 98 4.6 Discussion 101 4.6.1 Relation to Causal Inference Methods 103 References 105 5 Recent Advances in Semi-Parametric Methods for Causal Discovery 111 Shohei Shimizu and Patrick Blbaum 5.1 Introduction 111 5.2 Linear Non-Gaussian Methods 113 5.2.1 LiNGAM 113 5.2.2 Hidden Common Causes 115 5.2.3 Time Series 118 5.2.4 Multiple Data Sets 119 5.2.5 Other Methodological Issues 119 5.3 Nonlinear Bivariate Methods 119 5.3.1 Additive Noise Models 120 5.3.1.1 Post-Nonlinear Models 121 5.3.1.2 Discrete Additive Noise Models 121 5.3.2 Independence of Mechanism and Input 121 5.3.2.1 Information-Geometric Approach for Causal Inference 122 5.3.2.2 Causal Inference with Unsupervised Inverse Regression 123 5.3.2.3 Approximation of Kolmogorov Complexities via the Minimum Description Length Principle 123 5.3.2.4 Regression Error Based Causal Inference 124 5.3.3 Applications to Multivariate Cases 125 5.4 Conclusion 125 References 126 6 Assumption Checking for Directional Causality Analyses 131 Phillip K. Wood 6.1 Epistemic Causality 135 6.1.1 Example Data Set 136 6.2 Assessment of Functional Form: Loess Regression 137 6.3 Influential and Outlying Observations 140 6.4 Directional Dependence Based on All Available Data 141 6.4.1 Studentized Deleted Residuals 143 6.4.2 Lever 143 6.4.3 DFFITS 144 6.4.4 DFBETA 145 6.4.5 Results from Influence Diagnostics 145 6.4.6 Directional Dependence Based on Factor Scores 148 6.5 Directional Dependence Based on Latent Difference Scores 149 6.6 Direction Dependence Based on State-Trait Models 153 6.7 Discussion 156