The Marriage Equation: A practical theory for predicting divorce & scientifically-based marital therapy

1. The Marriage Equation: A practical theory for predicting divorce & scientifically-based marital therapy James D. Murray University of Oxford & University of Washington Collaborator: Dr. John Gottman,Clinical Psychologist Drs. Julian Cook, Kristin Swanson, Rebecca Tyson, Jane White (Applied Mathematicians) Book: The Mathematics of Marriage MIT Press 2003

2. Gathering a Couple’s Data Video is taken of the couple discussing a topic of contention, such as money, sex, housing, in-laws etc. An accepted scoring system assigns a specific number (positive or negative) to each statement & expression. The scores (positive – negative) for the husband (H) and the wife (W) for each turn of speech (t) are plotted as functions of time giving a kind of “Dow Jones” graph of their discussion. It measures the average positivity of each spouse as a function of time (t).

3. Data Representation: Typical Data for Low Risk Couple Cumulative “Dow Jones” graph of the “positive-negative” scores for each turn of speech for the husband and wife. Stable marriage Examples: affection +4 disgust -3 whining -1 contempt -4 5 to 1 positive to negative ratio

4. Typical High Risk Couple’s Interaction Unstable marriage 0.8 to 1 positive to negative ratio

5. Experimental Data, Model Predictions & Comparison with Statistics of the Control Study 1992: 700 newly married couples in King County in the Washington State participated in this ongoing longitudinal scientific study. We used our model to analyse their 15 minute discussion, and made predictions as to whether they would (i) divorce, or (ii) stay married (a) happily or (b) unhappily. Every one to two years to the present (2004) each couple was asked to complete a questionnaire (one from the wife and one from the husband) assessing their marriage. We compared these with our predictions. Prediction of which couples would get divorced was 94% accurate (and they got divorced within 4 years).Some couples predicted to stay married (unhappily) got divorced.

6. Steady States Intuitive Way to Represent the Data Husband-Wife Space: Evolution of an Interaction Wife (W) + W positive H negative + – Husband (H) W negative W negative H negative H positive –

7. Relationship (H,W) Phase Space H happyW happy H unhappyW happy H happyW unhappy H unhappyW unhappy

8. Example of the Scoring of a Couple’s Discussion

9. A little bit of history to motivate the model Leonardo of Pisa’s arithmetic book 1202 Exercise: Start with a pair (M and F) of immature rabbits at the beginning of a breeding season. After one season they can reproduce and produce 2 pairs of immature rabbits. The parents then stop producing but after another season the immature rabbits produce 2 pairs each and stop. The process continues in exactly the same way. Question: What is the number of pairs of rabbits at each reproductive period? Answer: 1 1 2 3 5 8 13 21 ….. 17th century Leonardo of Pisa was renamed Fibonacci ----- Fibonacci series

10. A little bit of algebra Question: How do we calculate the number of pairs in general?

11. Wife’s influence on Husband Husband’s influence on Wife Wife’s score at time t + 1 Wife’s previous score Husband’s previous score Constant Constant Mathematical Model of the interaction Wt , Htrepresentwife and husband’s scores at time t (turn of speech) in the discussion = + + Wt+1 = a + r1 Wt + IHW (Ht) Husband’s score at time t + 1 = + + H t+1 = b + r2 Ht + IWH (Wt)

12. Mathematical Model: Interpretation Wife’s influence on husband Husband’s influence on wife Wife at time t + 1 Wife’s past behaviour Husband’s past behaviour Constant A Constant Wt , Htrepresentwife and husband’s scores at time t in the discussion = + + IHW (Ht) Wt+1 = a + r1 Wt + IHW (Ht) Husband at time t + 1 = + + IWH (Wt) H t+1 = b + r2 Ht + IWH (Wt)

13. Influence Function Calculated over Complete 15 minute Discussion – Mathematical Best Fit Husband’s Influence on his Wife IHW (H) H linear approximations

14. Couple 1: Influence functions from analysis of the video Wife’s influence function Husband’s influence function Husband’s influence on wife Wife’s influence on husband Wife’s score Husband’s score

15. IHW (H) 0 H positivity Terminology & Basic Interaction Styles Five marriage types – different interaction styles – related to interaction functions (I) in the model. Basic model fits empirical data to a two-slope linear interaction function I (one each for the Husband and Wife) negativity

16. Early/middle marriage profile IHW (H) Long term, post retirement marriage profile 0 H negativity positivity Basic interaction styles change over courtship, early and long term marriage IHW (H) Courtship interaction profile 0 H negativity positivity

17. They have little effect on each other in the negative range negative positive Interaction: Conflict-Avoiding Marriage Conflict-Avoiding husband Conflict-Avoiding wife IHW(H) IWH(W) .3 -.05 .15 W H .01 positive negative positive negative Theoretical influence function: Conflict-Avoidingcouple

18. Basic Marriage Types Observations of couples (RCISS - Rapid Couples Interacting Scoring System) and mathematical model 5 types of marriages: 3 stable: Volatiles, Validators, Avoiders 2 unstable: Hostiles, Hostile-Detached Characteristics (stable and unstable types): Volatile (S) – romantic, passionate, have heated arguments with cycles of fights and sex Validating (S) – calmer, intimate, value companionate marriage, shared experience rather than individuality Avoiders (S) – avoid confrontation and conflict, interact only in positive range of their emotions Hostile (U) – (mixed) conflict-avoiding wife, validating husband Hostile-Detached (U) – (mixed) volatile wife, validating husband

19. Mathematical Model: Interpretation Husband’s influence on wife Wife’s influence on husband Wife at time t + 1 Wife’s past behaviour Husband’s past behaviour Constant Constant = + + r1 Wt Wt+1 = a + r1 Wt + IHW (Ht) Husband at time t + 1 = + + r2 Ht H t+1 = b + r2 Ht + IWH (Wt) Emotional Inertia:r1 and r2 constants, |r| < 1

20. Wt+1 = Wt = Wu Wu = a + rW Wu Wu = a (1- rW ) Ht+1 = Ht = Hu Hu = b + rH Hu Hu = b ( 1- rH ) What are steady states in an interaction? Uninfluenced Steady State (IHW =0, IWH =0): NB: high inertia parameters (r near 1) play crucial role

21. Wife’s influence on husband Husband’s influence on wife Wife at time t + 1 Wife’s past behaviour Husband’s past behaviour Constant Constant Mathematical Model of the interaction Wt , Htrepresentwife and husband’sscores at time t = + + Wt+1 = a + r1 Wt + IHW Ht Husband at time t + 1 = + + H t+1 = b + r2 Ht + IWH Wt

22. Graphical Interpretation of Model Suppose Model becomes 2 straight lines in Wife-Husband plane Wife’s interaction line* Husband’s interaction line Where these lines intersect gives a steady state of the couple’s interaction *Straight lines like Ax+By=C with H as x and W as y

23. Effect of uninfluencedsteady state Husband’s interaction line S H u S – stable U - unstable U W W u Wife’s interaction line H 0 uninfluenced steady states = b/1-r = a/1-r W H u u husband wife

24. Steady state Couple 1: Husband and Wife’s interaction lines Divorce/Therapy? Husband’s interaction line W Stable steady state (H*,W*) = (-.89,-1.06) Strength of stable attractor = 1.32 Wife’s interaction line This marriage has a high probability of divorce H

25. Steady state Couple 2: Husband and Wife’s interaction lines – before workshop therapy Husband’s interaction line Steady state (H*,W*) = (– 2.84,1.6) Strength of stable attractor = 1.11 W This marriage is a good candidate for therapy. Wife’s interaction line H

26. steady state Relationship (H,W) Phase Space H happyW happy H unhappyW happy H unhappyW unhappy H happyW unhappy

27. Wife’s Score Stable Positive Steady State @(3.78, 3.40) Husband’s Score Stable, Happy Marriage Husband’sInteraction Line Wife’s Interaction Line

28. Wife’s Score Husband’s Score Divorced 2 Stable Negative Steady States @(-4.24, -3.58) and (-2.82, -5.28) + 1 UnstableNegative Steady State@(-4.01, -5.09)

29. Wife’s Score Increase uninfluenced steady state new positive influenced steady state Husband’s Score TherapyDesign Husband’s uninfluenced steady state

30. Pre-Marital Therapy HusbandandWifetend toward negativity

31. Post-Marital Therapy HusbandandWifebecome more positive

32. What Have We Gained from the Mathematics?Three practical goals for an empirical based marital therapy & rationale for the marital experiments 1. Reduce emotional inertia 2. Make the uninfluenced steady states of wife and husband (Wu , Hu) more positive Make the influenced steady state (W*,H*) more positive than the uninfluenced steady state (Wu , Hu)

33. What Have We Gained from the Mathematics? New language for describing marital interaction and social influenceand rationale for the marital experiments Concept that marriages can be classified into one of 5 types of marriage depending on the couple’s interaction style: Stable marriages have matched interaction styles. Unstable marriages have mismatched interaction styles Couple’s interaction data suggest specific therapy

34. Giardan Bruno (1548-1600) Se non e' vero, e' ben trovato