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# Testing Transitivity (and other Properties) Using a True and Error Model - PowerPoint PPT Presentation

Testing Transitivity (and other Properties) Using a True and Error Model. Michael H. Birnbaum. Testing Algebraic Models with Error-Filled Data. Algebraic models assume or imply formal properties such as stochastic dominance, coalescing, transitivity, gain-loss separability, etc.

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### Testing Transitivity (and other Properties) Using a True and Error Model

Michael H. Birnbaum

• Algebraic models assume or imply formal properties such as stochastic dominance, coalescing, transitivity, gain-loss separability, etc.

• But these properties will not hold if data contain “error.”

Some Proposed Solutions Error Model

• Neo-Bayesian approach (Myung, Karabatsos, & Iverson.

• Cognitive process approach (Busemeyer)

• “Error” Theory (“Error Story”) approach (Thurstone, Luce) combined with algebraic models.

Variations of Error Models Error Model

• Thurstone, Luce: errors related to separation between subjective values. Case V: SST (scalability).

• Harless & Camerer: errors assumed to be equal for certain choices.

• Today: Allow each choice to have a different rate of error.

• Advantage: we desire error theory that is both descriptive and neutral.

Basic Assumptions Error Model

• Each choice in an experiment has a true choice probability, p, and an error rate, e.

• The error rate is estimated from (and is the “reason” given for) inconsistency of response to the same choice by same person over repetitions

Solution for e Error Model

• The proportion of preference reversals between repetitions allows an estimate of e.

• Both off-diagonal entries should be equal, and are equal to:

Estimating Error Modele

Estimating Error Modelp

Testing if Error Modelp = 0

Ex: Stochastic Dominance Error Model

122 Undergrads: 59% repeated viols (BB)

28% Preference Reversals (AB or BA)

Estimates: e = 0.19; p = 0.85

170 Experts: 35% repeated violations

31% Reversals

Estimates: e = 0.196; p = 0.50

Chi-Squared test reject H0: p < 0.4

Testing 3-Choice Properties Error Model

• Extending this model to properties using 2, 3, or 4 choices is straightforward.

• Allow a different error rate on each choice.

• Allow a true probability for each choice pattern.

Model for Transitivity Transitive

A similar expression is written for the

other seven probabilities. These can in turn be expanded to predict the probabilities of showing each pattern repeatedly.

Starmer (1999) data Transitive

• A = (\$15, 0.2; \$0, 0.8)

• B = (\$8; 0.3; \$0, 0.7)

• C = (\$8, 0.15; \$7.75; 0.15; \$0, .7)

• Starmer predicted intransitivity from Prospect Theory and the dominance detection (editing) mechanism.

Transitive Solution to Starmer Data Transitive

Full model is underdetermined. One error

Fixed to zero; but other errors not equal.

Most people recognized dominance.

Expand and Simplify Transitive

• There are 8 X 8 data patterns in an experiment with 2 repetitions.

• However, most of these have very small probabilities.

• Examine probabilities of each of 8 repeated patterns.

• Probability of showing each of 8 patterns in one replicate OR the other, but NOT both. Mutually exclusive, exhaustive partition.

New Studies of Transitivity Transitive

• Work currently under way testing transitivity under same conditions as used in tests of other decision properties.

• Participants view choices via the WWW, click button beside the gamble they would prefer to play.

Some Recipes being Tested Transitive

• Tversky’s (1969) 5 gambles.

• LS: Preds of Priority Heuristic

• Starmer’s recipe

• Additive Difference Model

• Birnbaum, Patton, & Lott (1999) recipe.

Tversky Gambles Transitive

• Some Sample Data, using Tversky’s 5 gambles, but formatted with tickets instead of pie charts.

• Data as of May 5, 2005, n = 123.

• No pre-selection of participants.

• Participants served in other studies, prior to testing (~1 hr).

Three of the Gambles Transitive

• A = (\$5.00, 0.29; \$0, 0.79)

• C = (\$4.50, 0.38; \$0, 0.62)

• E = (\$4.00, 0.46; \$0, 0.54)

Results-ACE Transitive

Test of WST Transitive

• Preliminary results were surprisingly transitive.

• Difference: no pre-test, selection

• Probability represented by # of tickets (100 per urn)

• Participants have practice with variety of gambles, & choices.

• Tested via Computer

Test of Gain-Loss Separability Transitive

• Same Structure as Transitivity

• Property implied by CPT, RSDU

• Property violated by TAX.

• Loss Aversion: people do not like fair bets to win or lose.

• CPT: Loss Aversion due to utility function for gains and losses.

Notation Transitive

Birnbaum & Bahra Transitive

Summary GLS Transitive

• Wu & Markle (2004) found evidence of violation of GLS. Modified CPT.

• Birnbaum & Bahra (2005) also find evidence of violation of GLS, violations of modified CPT as well.

• TAX: In mixed gambles, losses get greater weight. Data do not require kink in the utility function at zero.

Summary Transitive

• True & Error model with different error rates seems a reasonable “null” hypothesis for testing transitivity and other properties.

• Requires data with replications so that we can use each person’s self-agreement or reversals to estimate whether response patterns are “real” or due to “error.”