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Covers the latest developments in direction dependence research Direction Dependence in Statistical Modeling: Methods of Analysis incorporates the latest research for the statistical analysis of hypotheses that are compatible with the causal direction of dependence of variable relations. Having particular application in the fields of neuroscience, clinical psychology, developmental psychology, educational psychology, and epidemiology, direction dependence methods have attracted growing attention due to their potential to help decide which of two competing statistical models is more likely to reflect the correct causal flow. The book covers several topics in-depth, including: A demonstration of the importance of methods for the analysis of direction dependence hypotheses A presentation of the development of methods for direction dependence analysis together with recent novel, unpublished software implementations A review of methods of direction dependence following the copula-based tradition of Sungur and Kim A presentation of extensions of direction dependence methods to the domain of categorical data An overview of algorithms for causal structure learning The book's fourteen chapters include a discussion of the use of custom dialogs and macros in SPSS to make direction dependence analysis accessible to empirical researchers.

Author Biography

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.

Table of Contents

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 1888–2018 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 Blöbaum 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 References 163 7 Complete Dependence: A Survey 167
Santi Tasena 7.1 Basic Properties 168 7.2 Measure of Complete Dependence 171 7.3 Example Calculation 177 7.4 Future Works and Open Problems 180 References 181 Part III Direction Dependence in Categorical Variables 183 8 Locating Direction Dependence Using Log-Linear Modeling, Configural Frequency Analysis, and Prediction Analysis 185
Alexander von Eye and Wolfgang Wiedermann 8.1 Specifying Directional Hypotheses in Categorical Variables 187 8.2 Types of Directional Hypotheses 192 8.2.1 Multiple Premises and Outcomes 192 8.3 Analyzing Event-Based Directional Hypotheses 193 8.3.1 Log-Linear Models of Direction Dependence 193 8.3.1.1 Identification Issues 197 8.3.2 Confirmatory Configural Frequency Analysis (CFA) of Direction Dependence 198 8.3.3 Prediction Analysis of Cross-Classifications 200 8.3.3.1 Descriptive Measures of Prediction Success 202 8.4 Data Example 203 8.4.1 Log-Linear Analysis 205 8.4.2 Configural Analysis 206 8.4.3 Prediction Analysis 208 8.5 Reversing Direction of Effect 209 8.5.1 Log-Linear Modeling of the Re-Specified Hypotheses 209 8.5.2 CFA of the Re-Specified Hypotheses 210 8.5.3 PA of the Re-Specified Hypotheses 212 8.6 Discussion 212 References 215 9 Recent Developments on Asymmetric Association Measures for Contingency Tables 219
Xiaonan Zhu, Zheng Wei, and Tonghui Wang 9.1 Introduction 219 9.2 Measures on Two-Way Contingency Tables 220 9.2.1 Functional Chi-Square Statistic 220 9.2.2 Measures of Complete Dependence 222 9.2.3 A Measure of Asymmetric Association Using Subcopula-Based Regression 223 9.3 Asymmetric Measures of Three-Way Contingency Tables 225 9.3.1 Measures of Complete Dependence for Three Way Contingency Table 225 9.3.2 Subcopula Based Measure for Three Way Contingency Table 232 9.3.3 Estimation 235 9.4 Simulation of Three-Way Contingency Tables 237 9.5 Real Data of Three-Way Contingency Tables 239 References 240 10 Analysis of Asymmetric Dependence for Three-Way Contingency Tables Using the Subcopula Approach 243
Daeyoung Kim and Zheng Wei 10.1 Introduction 243 10.2 Review on Subcopula Based Asymmetric Association Measure for Ordinal Two-Way Contingency Table 245 10.3 Measure of Asymmetric Association for Ordinal Three-Way Contingency Tables via Subcopula Regression 248 10.3.1 Subcopula Regression-Based Asymmetric Association Measures 248 10.3.2 Estimation 251 10.4 Numerical Examples 253 10.4.1 Sensitivity Analysis 253 10.4.2 Data Analysis 257 10.5 Conclusion 260 10.A Appendix 261 10.A.1 The Proof of Proposition 10.1 261 References 262 Part IV Applications and Software 265 11 Distribution-Based Causal Inference: A Review and Practical Guidance for Epidemiologists 267
Tom Rosenström and Regina García-Velázquez 11.1 Introduction 267 11.2 Direction of Dependence in Linear Regression 268 11.3 Previous Epidemiologic Applications of Distribution-Based Causal Inference 271 11.4 A Running Example: Re-Visiting the Case of Sleep Problems and Depression 273 11.5 Evaluating the Assumptions in Practical Work 274 11.5.1 Testing Linearity 275 11.5.2 Testing Non-Normality 276 11.5.3 Testing Independence 277 11.6 Distribution-Based Causality Estimates for the Running Example 278 11.7 Conducting Sensitivity Analyses 279 11.7.1 Convergent Evidence from Multiple Estimators 279 11.7.2 Simulation-Based Analysis of Robustness to Latent Confounding 279 11.7.2.1 Obtain Data-Based Parameters 281 11.7.2.2 Defining Parameters and Simulation Conditions 281 11.7.2.3 Defining the Simulation Model 282 11.7.2.4 Run Simulation and Interpret Results 283 11.8 Simulation-Based Analysis of Statistical Power 284 11.9 Triangulating Causal Inferences 288 11.10 Conclusion 291 References 292 12 Determining Causality in Relation to Early Risk Factors for ADHD: The Case of Breastfeeding Duration 295
Joel T. Nigg, Diane D. Stadler, Alexander von Eye, and Wolfgang Wiedermann 12.1 Method 298 12.1.1 Participants 298 12.1.1.1 Recruitment and Identification 298 12.1.1.2 Parental Psychopathology 299 12.1.1.3 Ethical Standards 300 12.1.2 Exclusion Criteria 300 12.1.2.1 Assessment of Breastfeeding Duration 300 12.1.3 Covariates 301 12.1.3.1 Parental Education 301 12.1.3.2 Primary Residence and Family Income 301 12.1.3.3 Parental Occupational Status 301 12.1.4 Data Reduction and Data Analysis 301 12.1.4.1 Parental ADHD 301 12.1.4.2 Data Reduction 301 12.1.4.3 Data Analysis 302 12.2 Results 304 12.2.1 Study Participant Demographic and Clinical Characteristics 304 12.3 Discussion 316 12.3.1 Limitations 317 12.3.2 Question of Causality 317 Acknowledgments 318 References 318 13 Direction of Effect Between Intimate Partner Violence and Mood Lability: A Granger Causality Model 325
G. Anne Bogat, Alytia A. Levendosky, Jade E. Kobayashi, and Alexander von Eye 13.1 Introduction 325 13.1.1 Definitions and Frequency of IPV 326 13.1.2 Depression, Mood and IPV 329 13.1.2.1 Depression and IPV 329 13.1.2.2 Mood and IPV 330 13.1.3 Summary 332 13.2 Methods 333 13.2.1 Participants 333 13.2.2 Measures 333 13.2.2.1 Daily Diary Questions 333 13.2.3 Procedures 334 13.3 Results 334 13.3.1 Data Consolidation 334 13.3.2 Descriptive Statistics 335 13.3.3 Model Development 335 13.3.4 Granger Causality Analyses 337 13.4 Discussion 341 References 343 14 On the Causal Relation of Academic Achievement and Intrinsic Motivation: An Application of Direction Dependence Analysis Using SPSS Custom Dialogs 351
Xintong Li and Wolfgang Wiedermann 14.1 Direction of Dependence in Linear Regression 352 14.1.1 Distributional Properties of x and y 353 14.1.2 Distributional Properties of ex and ey 354 14.1.3 Independence of Error Terms with Predictor Variable 355 14.1.4 DDA in Confounded Models 356 14.1.5 DDA in Multiple Linear Regression Models 356 14.2 The Causal Relation of Intrinsic Motivation and Academic Achievement 359 14.2.1 High School Longitudinal Study 2009 360 14.3 Direction Dependence Analysis Using SPSS 363 14.3.1 Variable Distributions and Assumption Checks 363 14.3.2 Residual Distributions 366 14.3.3 Independence Properties 368 14.3.4 Summary of DDA Results 369 14.4 Conclusions 371 14.4.1 Extensions and Future Work 372 References 372 Author Index 379 Subject Index 395

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