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Bayesian data analysis

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Abstract Bayesian methods have garnered huge interest in cognitive science as an approach to models of cognition and perception. On the other hand, Bayesian methods for data analysis have not yet made much headway in cognitive science against the institutionalized inertia of 20th century null hypothesis significance testing (NHST). Ironically, specific Bayesian models of cognition and perception may not long endure the ravages of empirical verification, but generic Bayesian methods for data analysis will eventually dominate. It is time that Bayesian data analysis became the norm for empirical methods in cognitive science. This article reviews a fatal flaw of NHST and introduces the reader to some benefits of Bayesian data analysis. The article presents illustrative examples of multiple comparisons in Bayesian analysis of variance and Bayesian approaches to statistical power. Copyright © 2010 John Wiley & Sons, Ltd. This article is categorized under: Psychology > Theory and Methods

Sampling distribution of t for two groups when the null hypothesis is true, when the intention is to fix N = 6 for both groups, regardless of how long that takes (top), or when the intention is to fix the duration of data collection at 2 weeks, when the mean rate is N = 6 per week (bottom).

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Group accuracies, parameterized as µ1 through µ4, for the filtration/condensation experiment. First row: prior distributions on the four group accuracies. The prior is mildly informed by the knowledge that accuracies will tend to be better than chance. Second row: the priors on various differences of group accuracies, implied by the priors in the first row. Third and fourth rows: posterior distributions, corresponding to the first two rows.

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Three types of Bayesian replication probability.

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Posterior estimate for difference of groups in Solari et al.36 The third group is credibly different from the mean of the other eight groups.

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A variety of comparisons of RTs for various test cues. These are the differences of the posterior estimates of parameter values shown in Figure 4, transformed back to the original RT scale of seconds instead of log10(s). Importantly, these differences take into account any correlations in believable parameter values.

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Posterior distributions for the test cue effects (compare with priors in Figure 3). The β values are in units of log10(s); for example, the mean baseline β0 = 0.175 corresponds to 100.175 = 1.50s.

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Prior probability distributions for parameters in Bayesian ANOVA. The left distribution is the prior for the baseline RT, denoted β0. The β0 values are in units of log10(s). The mean of the distribution is at log10(0, s) = 1, s, indicating a prior belief that RTs are about 1 s in duration. Notice that the scale is broad, which indicates that the prior is only mildly informed by the knowledge that human RTs are not on the order of nanoseconds or millennia. The middle distribution is the prior for all seven test cue effects, denoted βj. The right distribution is the prior for all 64 subject effects, denoted βi. Both the middle and right distributions indicate deviations from baseline, hence a prior mean of 0 indicates a mild preference for null effects. The dark bar labeled ‘95% HDI’ indicates the highest density interval, i.e., the interval that contains 95% of the distribution such that parameter values outside the highest density interval (HDI) have less believability than parameter values inside the HDI. These histograms were generated by a large random sample from the continuous and symmetric underlying distribution.

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scale = 0.98 Probability that any of the t values among the comparisons exceeds the critical value, when the null hypothesis is true, and N = 6 is fixed for all five groups. The arrow labeled ‘Grp 1 versus Grp 2’ is at t = 2.23 as shown in the top panel of Figure 1.

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