1. 1 Distribution Independence Michael H. Birnbaum California State University, Fullerton

2. 2 4-DI is violated by CPT If W(P) is nonlinear, we should be able to predict violations of 4-DI from CPT. RAM satisfies 4-DI TAX violates 4-DI in the opposite way as CPT with its inverse-S weighting function.

3. 3 We manipulate r in both gambles, r ’ > r. This changes where the two equally probable branches fall with respect to the gamble’s distribution.

4. 4 4-Distribution Independence (4-DI)

5. 5 Example Test of 4-DI

6. 6 Generic Configural Model There will be no violation if this ratio is independent of r

7. 7 CPT Analysis of S vs. R

8. 8 CPT Analysis of S ’ vs. R ’

9. 9 Violation of 4-DI in CPT If W(P) has inverse-S shape, the ratios depend on r. CPT implies RS ’.

10. 10 RAM Weights

11. 11 RAM Satisfies 4-DI RAM satisfies 4-DI because the ratio of weights is independent of r.

12. 12 TAX Model

13. 13 TAX Model Implies SR ’ TAX violates 4-DI in the opposite pattern as CPT with inverse-S. Weight ratios: This implies the SR ’ pattern.

14. 14 Summary of Predictions EU and RAM satisfy 4-DI. CPT as fit to previous data violates 4-DI with RS ’ pattern. TAX as fit to previous data predicts the SR ’ pattern of violations.

15. 15 Study of 4-DI Birnbaum, M. H., & Chavez, A. (1997). Tests of Theories of Decision Making: Violations of Branch Independence and Distribution Independence. Organizational Behavior and Human Decision Processes, 71, 161-194. 100 participants, 12 tests with (r, r ’ ) = (.01,.59) and (.05,.55). Study also tested RBI and other properties. Significantly more SR ’ than RS ’ violations.

16. 16 Example Test

17. 17 Results for this Example Choice Pattern SS ’ SR ’ RS ’ RR ’ 4323*628

18. 18 Violations predicted by TAX, not CPT EU and RAM are refuted by systematic violations of 4-DI. TAX, as fit to previous data, correctly predicted the modal choices. CPT, with its inverse-S weighting function predicted opposite pattern.

19. 19 To Rescue CPT: CPT can handle the results if it uses an S-shaped rather than an inverse-S shaped weighting function.

20. 20 Summary PropertyCPTRAMTAX 4-DI RS ’ Viols No Viols SR ’ Viols UDIS ’ R2 ’ Viols No ViolsR ’ S2 ’ Viols RBI RS ’ ViolsSR ’ Viols

21. 21 Summary-Continued PropertyCPTRAMTAX LCINo ViolsViols UCINo ViolsViols UTINo ViolsR ’ S1Viols LDIRS2 ViolsNo Viols 3-2 LDIRS2 ViolsNo Viols

22. 22 Summary-Continued CPT violates RBI, 4-DI, and UDI, but the results show the opposite pattern. It violates 3-LDI and 3-2 LDI, but violations not found. CPT satisfies LCI, UCI, and UTI, but there are systematic violations. TAX correctly predicts all 8 results; RAM correct in 6 cases where it agrees with TAX; RAM disproved by violations of 4-DI and UDI.

23. 23 End of Series on Tests of Independence This presentation concludes the series on Lower and Upper Cumulative Independence, Lower and Upper Distribution Independence, Upper Tail Independence, Restricted Branch Independence, and 4-Distribution Independence. If you have not yet viewed them, the series of programs on Stochastic Dominance Violations and Allais Paradoxes will also be of interest, as will the separate programs on various models of decision making.

24. 24 For More Information: http://psych.fullerton.edu/mbirnbaum/ Download recent papers from this site. Follow links to “brief vita” and then to “in press” for recent papers. mbirnbaum@fullerton.edu