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 xvNotes on Contributors xviiAcknowledgments xxiPreface xxiiiPart I Fundamental Concepts of Direction Dependence 11 From Correlation to Direction Dependence Analysis 1888–2018 3Yadolah Dodge and Valentin Rousson1.1 Introduction 31.2 Correlation as a Symmetrical Concept of X and Y 41.3 Correlation as an Asymmetrical Concept of X and Y 51.4 Outlook and Conclusions 6References 62 Direction Dependence Analysis: Statistical Foundations and Applications 9Wolfgang Wiedermann, Xintong Li, and Alexander von Eye2.1 Some Origins of Direction Dependence Research 112.2 Causation and Asymmetry of Dependence 132.3 Foundations of Direction Dependence 142.3.1 Data Requirements 152.3.2 DDA Component I: Distributional Properties of Observed Variables 162.3.3 DDA Component II: Distributional Properties of Errors 192.3.4 DDA Component III: Independence Properties 202.3.5 Presence of Confounding 212.3.6 An Integrated Framework 242.4 Direction Dependence in Mediation 292.5 Direction Dependence in Moderation 322.6 Some Applications and Software Implementations 342.7 Conclusions and Future Directions 36References 383 The Use of Copulas for Directional Dependence Modeling 47Engin A. Sungur3.1 Introduction and Definitions 473.1.1 Why Copulas? 483.1.2 Defining Directional Dependence 483.2 Directional Dependence Between Two Numerical Variables 513.2.1 Asymmetric Copulas 523.2.2 Regression Setting 593.2.3 An Alternative Approach to Directional Dependence 623.3 Directional Association Between Two Categorical Variables 703.4 Concluding Remarks and Future Directions 74References 75Part II Direction Dependence in Continuous Variables 794 Asymmetry Properties of the Partial Correlation Coefficient: Foundations for Covariate Adjustment in Distribution-Based Direction Dependence Analysis 81Wolfgang Wiedermann4.1 Asymmetry Properties of the Partial Correlation Coefficient 844.2 Direction Dependence Measures when Errors Are Non-Normal 864.3 Statistical Inference on Direction Dependence 894.4 Monte-Carlo Simulations 904.4.1 Study I: Parameter Recovery 904.4.1.1 Results 914.4.2 Study II: CI Coverage and Statistical Power 914.4.2.1 Type I Error Coverage 944.4.2.2 Statistical Power 944.5 Data Example 984.6 Discussion 1014.6.1 Relation to Causal Inference Methods 103References 1055 Recent Advances in Semi-Parametric Methods for Causal Discovery 111Shohei Shimizu and Patrick Blöbaum5.1 Introduction 1115.2 Linear Non-Gaussian Methods 1135.2.1 LiNGAM 1135.2.2 Hidden Common Causes 1155.2.3 Time Series 1185.2.4 Multiple Data Sets 1195.2.5 Other Methodological Issues 1195.3 Nonlinear Bivariate Methods 1195.3.1 Additive Noise Models 1205.3.1.1 Post-Nonlinear Models 1215.3.1.2 Discrete Additive Noise Models 1215.3.2 Independence of Mechanism and Input 1215.3.2.1 Information-Geometric Approach for Causal Inference 1225.3.2.2 Causal Inference with Unsupervised Inverse Regression 1235.3.2.3 Approximation of Kolmogorov Complexities via the Minimum Description Length Principle 1235.3.2.4 Regression Error Based Causal Inference 1245.3.3 Applications to Multivariate Cases 1255.4 Conclusion 125References 1266 Assumption Checking for Directional Causality Analyses 131Phillip K. Wood6.1 Epistemic Causality 1356.1.1 Example Data Set 1366.2 Assessment of Functional Form: Loess Regression 1376.3 Influential and Outlying Observations 1406.4 Directional Dependence Based on All Available Data 1416.4.1 Studentized Deleted Residuals 1436.4.2 Lever 1436.4.3 DFFITS 1446.4.4 DFBETA 1456.4.5 Results from Influence Diagnostics 1456.4.6 Directional Dependence Based on Factor Scores 1486.5 Directional Dependence Based on Latent Difference Scores 1496.6 Direction Dependence Based on State-Trait Models 1536.7 Discussion 156References 1637 Complete Dependence: A Survey 167Santi Tasena7.1 Basic Properties 1687.2 Measure of Complete Dependence 1717.3 Example Calculation 1777.4 Future Works and Open Problems 180References 181Part III Direction Dependence in Categorical Variables 1838 Locating Direction Dependence Using Log-Linear Modeling, Configural Frequency Analysis, and Prediction Analysis 185Alexander von Eye and Wolfgang Wiedermann8.1 Specifying Directional Hypotheses in Categorical Variables 1878.2 Types of Directional Hypotheses 1928.2.1 Multiple Premises and Outcomes 1928.3 Analyzing Event-Based Directional Hypotheses 1938.3.1 Log-Linear Models of Direction Dependence 1938.3.1.1 Identification Issues 1978.3.2 Confirmatory Configural Frequency Analysis (CFA) of Direction Dependence 1988.3.3 Prediction Analysis of Cross-Classifications 2008.3.3.1 Descriptive Measures of Prediction Success 2028.4 Data Example 2038.4.1 Log-Linear Analysis 2058.4.2 Configural Analysis 2068.4.3 Prediction Analysis 2088.5 Reversing Direction of Effect 2098.5.1 Log-Linear Modeling of the Re-Specified Hypotheses 2098.5.2 CFA of the Re-Specified Hypotheses 2108.5.3 PA of the Re-Specified Hypotheses 2128.6 Discussion 212References 2159 Recent Developments on Asymmetric Association Measures for Contingency Tables 219Xiaonan Zhu, Zheng Wei, and Tonghui Wang9.1 Introduction 2199.2 Measures on Two-Way Contingency Tables 2209.2.1 Functional Chi-Square Statistic 2209.2.2 Measures of Complete Dependence 2229.2.3 A Measure of Asymmetric Association Using Subcopula-Based Regression 2239.3 Asymmetric Measures of Three-Way Contingency Tables 2259.3.1 Measures of Complete Dependence for Three Way Contingency Table 2259.3.2 Subcopula Based Measure for Three Way Contingency Table 2329.3.3 Estimation 2359.4 Simulation of Three-Way Contingency Tables 2379.5 Real Data of Three-Way Contingency Tables 239References 24010 Analysis of Asymmetric Dependence for Three-Way Contingency Tables Using the Subcopula Approach 243Daeyoung Kim and Zheng Wei10.1 Introduction 24310.2 Review on Subcopula Based Asymmetric Association Measure for Ordinal Two-Way Contingency Table 24510.3 Measure of Asymmetric Association for Ordinal Three-Way Contingency Tables via Subcopula Regression 24810.3.1 Subcopula Regression-Based Asymmetric Association Measures 24810.3.2 Estimation 25110.4 Numerical Examples 25310.4.1 Sensitivity Analysis 25310.4.2 Data Analysis 25710.5 Conclusion 26010.A Appendix 26110.A.1 The Proof of Proposition 10.1 261References 262Part IV Applications and Software 26511 Distribution-Based Causal Inference: A Review and Practical Guidance for Epidemiologists 267Tom Rosenström and Regina García-Velázquez11.1 Introduction 26711.2 Direction of Dependence in Linear Regression 26811.3 Previous Epidemiologic Applications of Distribution-Based Causal Inference 27111.4 A Running Example: Re-Visiting the Case of Sleep Problems and Depression 27311.5 Evaluating the Assumptions in Practical Work 27411.5.1 Testing Linearity 27511.5.2 Testing Non-Normality 27611.5.3 Testing Independence 27711.6 Distribution-Based Causality Estimates for the Running Example 27811.7 Conducting Sensitivity Analyses 27911.7.1 Convergent Evidence from Multiple Estimators 27911.7.2 Simulation-Based Analysis of Robustness to Latent Confounding 27911.7.2.1 Obtain Data-Based Parameters 28111.7.2.2 Defining Parameters and Simulation Conditions 28111.7.2.3 Defining the Simulation Model 28211.7.2.4 Run Simulation and Interpret Results 28311.8 Simulation-Based Analysis of Statistical Power 28411.9 Triangulating Causal Inferences 28811.10 Conclusion 291References 29212 Determining Causality in Relation to Early Risk Factors for ADHD: The Case of Breastfeeding Duration 295Joel T. Nigg, Diane D. Stadler, Alexander von Eye, and Wolfgang Wiedermann12.1 Method 29812.1.1 Participants 29812.1.1.1 Recruitment and Identification 29812.1.1.2 Parental Psychopathology 29912.1.1.3 Ethical Standards 30012.1.2 Exclusion Criteria 30012.1.2.1 Assessment of Breastfeeding Duration 30012.1.3 Covariates 30112.1.3.1 Parental Education 30112.1.3.2 Primary Residence and Family Income 30112.1.3.3 Parental Occupational Status 30112.1.4 Data Reduction and Data Analysis 30112.1.4.1 Parental ADHD 30112.1.4.2 Data Reduction 30112.1.4.3 Data Analysis 30212.2 Results 30412.2.1 Study Participant Demographic and Clinical Characteristics 30412.3 Discussion 31612.3.1 Limitations 31712.3.2 Question of Causality 317Acknowledgments 318References 31813 Direction of Effect Between Intimate Partner Violence and Mood Lability: A Granger Causality Model 325G. Anne Bogat, Alytia A. Levendosky, Jade E. Kobayashi, and Alexander von Eye13.1 Introduction 32513.1.1 Definitions and Frequency of IPV 32613.1.2 Depression, Mood and IPV 32913.1.2.1 Depression and IPV 32913.1.2.2 Mood and IPV 33013.1.3 Summary 33213.2 Methods 33313.2.1 Participants 33313.2.2 Measures 33313.2.2.1 Daily Diary Questions 33313.2.3 Procedures 33413.3 Results 33413.3.1 Data Consolidation 33413.3.2 Descriptive Statistics 33513.3.3 Model Development 33513.3.4 Granger Causality Analyses 33713.4 Discussion 341References 34314 On the Causal Relation of Academic Achievement and Intrinsic Motivation: An Application of Direction Dependence Analysis Using SPSS Custom Dialogs 351Xintong Li and Wolfgang Wiedermann14.1 Direction of Dependence in Linear Regression 35214.1.1 Distributional Properties of x and y 35314.1.2 Distributional Properties of ex and ey 35414.1.3 Independence of Error Terms with Predictor Variable 35514.1.4 DDA in Confounded Models 35614.1.5 DDA in Multiple Linear Regression Models 35614.2 The Causal Relation of Intrinsic Motivation and Academic Achievement 35914.2.1 High School Longitudinal Study 2009 36014.3 Direction Dependence Analysis Using SPSS 36314.3.1 Variable Distributions and Assumption Checks 36314.3.2 Residual Distributions 36614.3.3 Independence Properties 36814.3.4 Summary of DDA Results 36914.4 Conclusions 37114.4.1 Extensions and Future Work 372References 372Author Index 379Subject Index 395