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Using Bayesian hierarchical parameter estimation to assess the generalizability of cognitive models of choice

Abstract

To be useful, cognitive models with fitted parameters should show generalizability across time and allow accurate predictions of future observations. It has been proposed that hierarchical procedures yield better estimates of model parameters than do nonhierarchical, independent approaches, because the formers’ estimates for individuals within a group can mutually inform each other. Here, we examine Bayesian hierarchical approaches to evaluating model generalizability in the context of two prominent models of risky choice—cumulative prospect theory (Tversky & Kahneman, 1992) and the transfer-of-attention-exchange model (Birnbaum & Chavez, 1997). Using empirical data of risky choices collected for each individual at two time points, we compared the use of hierarchical versus independent, nonhierarchical Bayesian estimation techniques to assess two aspects of model generalizability: parameter stability (across time) and predictive accuracy. The relative performance of hierarchical versus independent estimation varied across the different measures of generalizability. The hierarchical approach improved parameter stability (in terms of a lower absolute discrepancy of parameter values across time) and predictive accuracy (in terms of deviance; i.e., likelihood). With respect to test–retest correlations and posterior predictive accuracy, however, the hierarchical approach did not outperform the independent approach. Further analyses suggested that this was due to strong correlations between some parameters within both models. Such intercorrelations make it difficult to identify and interpret single parameters and can induce high degrees of shrinkage in hierarchical models. Similar findings may also occur in the context of other cognitive models of choice.

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Notes

  1. Note that Nilsson et al. (2011) used Tversky and Kahneman’s (1992) one-parameter probability-weighting function, which does not disentangle elevation and curvature.

  2. The BUGS programming code for each model implementation is available in the online supplementary materials.

  3. Because Nilsson et al. (2011) employed a different weighting function in their comparison of independent and hierarchical parameter estimations for CPT, our results are not directly comparable with theirs. Nevertheless, note that Nilsson et al. also found the hierarchical approach to yield a lower choice sensitivity. Interestingly, they obtained a pattern of results opposite to ours with regard to loss aversion, with the hierarchical approach yielding a higher λ.

  4. See the online supplementary materials for the BUGS programming code.

  5. See the online supplemental materials for similar plots of the other parameters.

  6. Note, however, that this would not necessarily be the case. It is possible to conceive of situations in which shrinkage reduces the variance but retains the (linear) relationship between the individual parameters; in such cases, the test–retest correlations would not be lower for hierarchically estimated parameters, as they indeed are not for most of the parameters in Fig. 2.

  7. For pragmatic reasons, in the hierarchical case all participants, including those predicted at t2 at any one time, were included in the parameter estimation. This may have yielded a small advantage for the hierarchical approach over the independent approach.

  8. Bayes factor estimates were calculated from conventional t-test outputs on the basis of the template by Rouder, Speckman, Sun, Morey, and Iverson (2009), assuming the Jeffrey–Zellner–Siow prior and r = 1.

  9. The range of the prior distribution has very little impact on the results when taking all 138 choices into account. Presumably, with this amount of data on the individual level, the influence of the prior on the posterior estimates is negligible.

  10. If p is a vector of probabilities for making a correct prediction, the deviance is defined as –2*sum[log(p)], whereas the squared error is defined as sum[(1 – p)2].

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Correspondence to Benjamin Scheibehenne.

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Scheibehenne, B., Pachur, T. Using Bayesian hierarchical parameter estimation to assess the generalizability of cognitive models of choice. Psychon Bull Rev 22, 391–407 (2015). https://doi.org/10.3758/s13423-014-0684-4

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Keywords

  • Bayesian inference
  • Parameter estimation
  • Bayesian modeling
  • Decision making
  • Math modeling
  • Model evaluation