The Mathematics of Marriage Dynamic Nonlinear Models

by Gottman, John M.; Murray, James D.; Swanson, Catherine C.; Tyson, Rebecca; Swanson, Kristin R.

• ISBN13: 9780262572309
• ISBN10: 0262572303
• Format: Paperback
• Copyright: 2005-01-14
• Publisher: Bradford Books

Summary

Divorce rates are at an all-time high. But without a theoretical understanding of the processes related to marital stability and dissolution, it is difficult to design and evaluate new marriage interventions. The Mathematics of Marriageprovides the foundation for a scientific theory of marital relations. The book does not rely on metaphors, but develops and applies a mathematical model using difference equations. The work is the fulfillment of the goal to build a mathematical framework for the general system theory of families first suggested by Ludwig Von Bertalanffy in the 1960s. The book also presents a complete introduction to the mathematics involved in theory building and testing, and details the development of experiments and models. In one "marriage experiment," for example, the authors explored the effects of lowering or raising a couple�s heart rates. Armed with their mathematical model, they were able to do real experiments to determine which processes were affected by their interventions. Applying ideas such as phase space, null clines, influence functions, inertia, and uninfluenced and influenced stable steady states (attractors), the authors show how other researchers can use the methods to weigh their own data with positive and negative weights. While the focus is on modeling marriage, the techniques can be applied to other types of psychological phenomena as well.

Author Biography

John M. Gottman is Professor of Psychology at the University of Washington.

James D. Murray is Professor Emeritus of Applied Mathematics at the University of Washington.

Catherine Swanson is a software engineer at the University of Washington.

Rebecca Tyson is Research Scientist at the University of Arizona.

Kristin R. Swanson is Senior Fellow in Pathology and Applied Mathematics at the University of Washington.

Table of Contents

Preface	xi
What Do We Mean by Theory?	1	(4)
What Phenomena Are We Modeling?	5	(30)
Brief Review of Research on Marriage	6	(22)
The Problem of Power	28	(1)
Review of Gottman and Levenson	29	(5)
What Phenomenon Are We Modeling?	34	(1)
Two More Mysteries in Marital Research	34	(1)
Nonlinear Dynamic Modeling	35	(6)
The General Systems Theory of von Bertalanffy	35	(1)
Two or more equations	36	(2)
Outline of the Subsequent Chapters	38	(3)
Calculus---the Mathematics of Change	41	(24)
Dynamical Motion as Slope	42	(3)
The Concept of a Function	45	(6)
Limits	51	(1)
The Derivative Is a Limit	52	(1)
Common Derivatives in Functional Form	53	(1)
Rules for Differentiation	54	(2)
Integration---the Reverse of Differentiation	56	(1)
Maxima and Minima	56	(2)
Problem	58	(1)
Writing Differential Equations	59	(2)
Taylor's series	61	(4)
Introduction to Dynamic Modeling	65	(16)
Philosophy of Modeling	65	(3)
A Bit of Dynamical Modeling History	68	(1)
One Equation: Malthus Revisited	69	(12)
Appendix 5.1: Stability Results	79	(2)
Modeling Catastrophic Change	81	(18)
The Spruce Budworm Problem	82	(6)
Spruce Budworm Catastrophe	88	(1)
Examples of Catastrophic Change: Zeeman	89	(4)
Catastrophes in Perception	93	(4)
Chapter Summary	97	(2)
Appendix 6.1: Nondimensionalization	98	(1)
Intuitive Discussion of Phase Space Plots	99	(12)
Phase Space Portrait	99	(1)
Force Field Portrait	100	(1)
Null Clines and Steady States	101	(3)
Chapter Summary	104	(7)
Appendix 7.1: Phase Plane Analysis	104	(7)
Interacting Dyadic Systems	111	(16)
A Linear Marriage Model: Romeo and Juliet	111	(2)
Predator and Prey (LotkaVolterra Equations)	113	(6)
Competition Models	119	(3)
Cooperation Models	122	(1)
Chapter Summary	122	(5)
Appendix 8.1: Nonlinear Modeling Concepts	124	(3)
Writing the Equations of Marriage	127	(24)
The Model	129	(5)
Estimating of Parameters	134	(1)
Finding Null Clines	135	(2)
Steady States and Stability	137	(4)
Catastrophes in this Model	141	(10)
Appendix 9.1: The Inertia Parameter	144	(1)
Appendix 9.2: Stability of Steady States	145	(4)
Appendix 9.3: Strength of Attraction	149	(2)
Initial Results of Our Modeling	151	(14)
Power and Affect: Influence Functions	151	(3)
Mismatch Theory of Unstable Marriages	154	(2)
Set Points and Inertia	156	(1)
Validation of the Model Parameters	157	(3)
Discussion of Initial Modeling Results	160	(2)
Prospects for Extending the Model	162	(3)
Who Needs All This Math?	165	(8)
Homeostasis and Feedback	165	(4)
Family Dynamics	169	(1)
Disconfirmability	170	(1)
The Whole Is Greater Than the Sum of Its Parts	171	(1)
Conceptual Power of the Model	171	(2)
Applying the Model to Newlyweds	173	(16)
Developing the SPAFF Coding System	173	(2)
Fitting an Ojive Model to Weighted SPAFF Data	175	(2)
Seattle Newlywed Study	177	(4)
Initial Weighting for the SPAFF	181	(2)
Final Weighting Scheme for the SPAFF	183	(3)
Understanding the Uninfluenced Steady States	186	(1)
Chapter Summary	187	(2)
Repair and Damping	189	(44)
What Are Repair and Damping?	189	(3)
The Repair and Damping Terms	192	(6)
The Bilinear and Ojive Influence Functions	198	(6)
Simulations regarding Repair and Damping	204	(6)
Methods for Evaluating the Two Alternatives	210	(1)
Results	211	(8)
Model Behavior with Repair and Damping	219	(12)
Interpretations and Predictions	231	(2)
Extending the Marriage Model	233	(18)
Gay Male and Lesbian Relationships	233	(8)
Parent-Baby Interaction	241	(9)
Chapter summary	250	(1)
The Core Triad of Balance	251	(40)
What Is the Core Triad of Balance?	251	(1)
On What Data Do We Base the Model?	252	(5)
Brief Review of Behavior-Based Models	257	(1)
Core Triad Model Specification	258	(2)
Core Triad Model Results	260	(29)
Chapter Summary	289	(2)
The Marriage Experiments	291	(46)
Evaluation of Couples Therapy	293	(4)
Gottman's New Theory and Marital Intervention	297	(5)
Goals & Rationale for the Marriage Experiments	302	(8)
Dreams-within-Conflict Intervention	310	(3)
Angie and Dave: Overview	313	(4)
Angie and Dave: Intervention	317	(6)
Angie and Dave: Second Conversation	323	(8)
Angie and Dave: Math Model	331	(2)
Alan and Eve	333	(2)
Chapter Summary	335	(2)
How to prepare data for modeling	337	(6)
Obtaining the Model Program	337	(1)
Basic Concepts in Data Preparation	338	(1)
Example: Deriving Model Data from SPAFF	339	(4)
Bibliography	343	(18)
Appendix A	361	(4)
Appendix B	365	(32)
Own Slope/Own Score Repair	366	(10)
Spouse's Slope/Own Score Repair	376	(5)
Own Slope/Spouse's Score Repair	381	(10)
Spouse's Slope/Spouse's Score Repair	391	(6)
Index	397