Testing transitivity and other properties using a true and error model
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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 algebraic models with error filled data
Testing Algebraic Models with Error-Filled Data Error Model

  • 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
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
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
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
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 e
Estimating Error Modele

Estimating p
Estimating Error Modelp

Testing if p 0
Testing if Error Modelp = 0

Ex stochastic dominance
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
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
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
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 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
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
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
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
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
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
Results-ACE Transitive

Test of wst
Test of WST Transitive

Comments 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
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
Birnbaum & Bahra Transitive

Summary gls
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.”