Heterogeneity and Parsimony in Intertemporal Choice

Behavioral theories of intertemporal choice involve many moving parts. Most descriptive theories model how time delays and rewards are perceived, compared, and/or combined into preferences or utilities. Most behavioral studies neglect to spell out how such constructs translate into heterogeneous observable choices. We consider several broad models of transitive intertemporal preference and combine these with several mathematically formal, yet very general, models of heterogeneity. We evaluate 20 probabilistic models of intertemporal choice using binary choice data from two large-scale experiments. Our analysis documents the interplay between heterogeneity and parsimony in accounting for empirical data: We find evidence for heterogeneity across individuals and across stimulus sets that can be accommodated with transitive models of varying complexity. We do not find systematic violations of transitivity in our data. Future work should continue to tackle the complex trade-off between parsimony and heterogeneity

Heterogeneity and Parsimony in Intertemporal Choice

Behavioral theories of intertemporal choice involve many moving parts. Most descriptive theories model how time delays and rewards are perceived, compared, and/or combined into preferences or utilities. Most behavioral studies neglect to spell out how such constructs translate into heterogeneous observable choices. We consider several broad models of transitive intertemporal preference and combine these with several mathematically formal, yet very general, models of heterogeneity. We evaluate 20 probabilistic models of intertemporal choice using binary choice data from two large-scale experiments. Our analysis documents the interplay between heterogeneity and parsimony in accounting for empirical data: We find evidence for heterogeneity across individuals and across stimulus sets that can be accommodated with transitive models of varying complexity. We do not find systematic violations of transitivity in our data. Future work should continue to tackle the complex trade-off between parsimony and heterogeneity

Reply: Birnbaum's (2012) statistical tests of independence have unknown Type-I error rates and do not replicate within participant

Birnbaum (2011, 2012) questioned the iid (independent and identically distributed) sampling assumptions used by state-of-the-art statistical tests in Regenwetter, Dana and Davis-Stober's (2010, 2011) analysis of the ``linear order model''. Birnbaum (2012) cited, but did not use, a test of iid by Smith and Batchelder (2008) with analytically known properties. Instead, he created two new test statistics with unknown sampling distributions. Our rebuttal has five components: 1) We demonstrate that the Regenwetter et al. data pass Smith and Batchelder's test of iid with flying colors. 2) We provide evidence from Monte Carlo simulations that Birnbaum's (2012) proposed tests have unknown Type-I error rates, which depend on the actual choice probabilities and on how data are coded as well as on the null hypothesis of iid sampling. 3) Birnbaum analyzed only a third of Regenwetter et al.'s data. We show that his two new tests fail to replicate on the other two-thirds of the data, within participants. 4) Birnbaum selectively picked data of one respondent to suggest that choice probabilities may have changed partway into the experiment. Such nonstationarity could potentially cause a seemingly good fit to be a Type-II error. We show that the linear order model fits equally well if we allow for warm-up effects. 5) Using hypothetical data, Birnbaum (2012) claimed to show that ``true-and-error'' models for binary pattern probabilities overcome the alleged short-comings of Regenwetter et al.'s approach. We disprove this claim on the same data

Efficient Blockwise Permutation Tests Preserving Exchangeability

In this paper, we present a new blockwise permutation test approach based on the moments of the test statistic. The method is of importance to neuroimaging studies. In order to preserve the exchangeability condition required in permutation tests, we divide the entire set of data into certain exchangeability blocks. In addition, computationally efficient moments-based permutation tests are performed by approximating the permutation distribution of the test statistic with the Pearson distribution series. This involves the calculation of the first four moments of the permutation distribution within each block and then over the entire set of data. The accuracy and efficiency of the proposed method are demonstrated through simulated experiment on the magnetic resonance imaging (MRI) brain data, specifically the multi-site voxel-based morphometry analysis from structural MRI (sMRI)

Efficient Blockwise Permutation Tests Preserving Exchangeability

In this paper, we present a new blockwise permutation test approach based on the moments of the test statistic. The method is of importance to neuroimaging studies. In order to preserve the exchangeability condition required in permutation tests, we divide the entire set of data into certain exchangeability blocks. In addition, computationally efficient moments-based permutation tests are performed by approximating the permutation distribution of the test statistic with the Pearson distribution series. This involves the calculation of the first four moments of the permutation distribution within each block and then over the entire set of data. The accuracy and efficiency of the proposed method are demonstrated through simulated experiment on the magnetic resonance imaging (MRI) brain data, specifically the multi-site voxel-based morphometry analysis from structural MRI (sMRI)

Parahippocampal Cortex Mediates the Relationship between Lutein and Crystallized Intelligence in Healthy, Older Adults

Introduction: Although diet has a substantial influence on the aging brain, the relationship between dietary nutrients and aspects of brain health remains unclear. This study examines the neural mechanisms that mediate the relationship between a carotenoid important for brain health across the lifespan, lutein, and crystallized intelligence in cognitively intact older adults. We hypothesized that higher serum levels of lutein are associated with better performance on a task of crystallized intelligence, and that this relationship is mediated by gray matter structure of regions within the temporal cortex. This investigation aims to contribute to a growing line of evidence, which suggests that particular nutrients may slow or prevent aspects of cognitive decline by targeting specific features of brain aging.Methods: We examined 75 cognitively intact adults between the ages of 65 and 75 to investigate the relationship between serum lutein, tests of crystallized intelligence (measured by the Wechsler Abbreviated Scale of Intelligence), and gray matter volume of regions within the temporal cortex. A three-step mediation analysis was implemented using multivariate linear regressions to control for age, sex, education, income, depression status, and body mass index.Results: The mediation analysis revealed that gray matter thickness of one region within the temporal cortex, the right parahippocampal cortex (Brodmann’s Area 34), partially mediates the relationship between serum lutein and crystallized intelligence. Conclusion: These results suggest that the parahippocampal cortex acts as a mediator of the relationship between serum lutein and crystallized intelligence in cognitively intact older adults. Prior findings substantiate the individual relationships reported within the mediation, specifically the links between (i) serum lutein and temporal cortex structure, (ii) serum lutein and crystallized intelligence, and (iii) parahippocampal cortex structure and crystallized intelligence. This report is the first to demonstrate a specific structural mediation between lutein status and crystallized intelligence, and therefore provides further evidence that specific nutrients may slow or prevent features of cognitive decline by hindering particular aspects of brain aging. Future work should examine the potential mechanisms underlying this mediation, including the antioxidant, anti-inflammatory, and membrane modulating properties of lutein