MARTER: Markov True and Error model of drifting parameters

Abstract

This paper describes a theory of the variability of risky choice that describes empirical properties of choice data, including sequential effects and systematic violations of response independence. The Markov True and Error (MARTER) model represents the formation and fluctuation of true preferences produced by stochastic variation of parameters over time, which produces changing true preference patterns. This model includes a probabilistic association between true preferences and overt responses due to random error. Computer programs have been developed to simulate data according to this model, to fit data to the TE model, and to test and analyze violations of iid (independent and identical distributions) that are predicted by the model. Data simulated from MARTER models show properties that are characteristic of real data, including violations of iid similar to those observed in previous empirical research. This paper also illustrates how methods based on analysis of binary response proportions do not and in many cases cannot correctly diagnose what model was used to generate the data. The MARTER model is extremely general and neutral with respect to models of risky decision making. For example, the transitive transfer of attention exchange (TAX) model and intransitive Lexicographic Semiorder (LS) models can both be represented as special cases of MARTER, and they can be tested against each other, even when binary choice proportions cannot discriminate which model was used to simulate the data. Software to simulate data according to this model, and to fit data to this model, to test this model, and to compare special case theories are included or linked to this article.

Reanalysis of Butler and Pogrebna (2018) using true and error model

“…In this case, an investigator might consider transitivity to be satisfied even though 2/3 of the participants had intransitive preferences. These same problems arise in the analysis of a single individual's data if a person can change true preferences over time (Birnbaum & Wan, 2020).…”

“…Sopher and Giglioti (1993) noted that if different choice problems have different rates of error, then tests of transitivity based on inequality of response patterns (once considered evidence of intransitivity) might easily signal "intransitivity" when the data are actually perfectly transitive. Birnbaum and Schmidt (2008) gave clear examples to illustrate that one cannot properly address the issue of transitivity without measuring error rates in the choice problems used to measure This distinction may be important because the fitting model used here assumes that true preferences are the same in both repetitions (Birnbaum & Wan, 2020). Nevertheless, Birnbaum and Quan (2020) found in a series of simulations that TE analysis was robust with respect to violations of this assumption.…”

“…They do not provide estimates of the incidences of different transitive and intransitive response patterns, which one needs in order to evaluate models of decision making. Birnbaum and Wan (2020) simulated data from transitive or intransitive processes and illustrated that those older methods of analysis simply cannot correctly identify whether a transitive or intransitive model had been used to create the data. They also showed that the true and error model correctly identified whether a transitive or intransitive model had been used to generate the data in these cases where the older methods fail.…”

“…Because the data analyses presented in Butler and Pogrebna (2018) were based on these older methods, a skeptic could remain unconvinced that their data actually contained any real evidence against transitivity, once errors and differences in true preferences are allowed. Fortunately, because their study included two presentations of each choice problem, it is possible to apply the true and error (TE) model to reanalyze their data in order to estimate error rates and the incidence of intransitive and transitive behavior (Birnbaum, 2013;Birnbaum, Navarro-Martinez, Ungemach, Stewart, & Quispe-Torreblanca, 2016;Birnbaum & Wan, 2020)…”

“…A program in R (Birnbaum, et al, 2016), 8 2_ . , is available from this journal's website supplement to Birnbaum and Wan (2020). This program can be used to analyze the TE model when sample sizes are relatively small.…”