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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 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. • But these properties will not hold if data contain “error.”
Some Proposed Solutions • 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 • 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 • 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 • The proportion of preference reversals between repetitions allows an estimate of e. • Both off-diagonal entries should be equal, and are equal to:
Ex: Stochastic Dominance 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 • 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.
WST Can be Violated even when Everyone is Perfectly Transitive
Model for Transitivity 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 • 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 Full model is underdetermined. One error Fixed to zero; but other errors not equal. Most people recognized dominance.
Expand and Simplify • 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 • 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 • Tversky’s (1969) 5 gambles. • LS: Preds of Priority Heuristic • Starmer’s recipe • Additive Difference Model • Birnbaum, Patton, & Lott (1999) recipe.
Tversky Gambles • 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 • A = ($5.00, 0.29; $0, 0.79) • C = ($4.50, 0.38; $0, 0.62) • E = ($4.00, 0.46; $0, 0.54)
Comments • 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 • 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.
Summary GLS • 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 • 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.”
