# IntroductionΒΆ

Sequential sampling models (SSMs) ([TownsendAshby83]) have established themselves as the de-facto standard for modeling reaction-time data from simple two-alternative forced choice decision making tasks ([SmithRatcliff04]). Each decision is modeled as an accumulation of noisy information indicative of one choice or the other, with sequential evaluation of the accumulated evidence at each time step. Once this evidence crosses a threshold, the corresponding response is executed. This simple assumption about the underlying psychological process has the appealing property of reproducing not only choice probabilities, but the full distribution of response times for each of the two choices. Models of this class have been used successfully in mathematical psychology since the 60s and more recently adopted in cognitive neuroscience investigations. These studies are typically interested in neural mechanisms associated with the accumulation process or for regulating the decision threshold (see e.g. [ForstmannDutilhBrownEtAl08], [CavanaghWieckiCohenEtAl11], [RatcliffPhiliastidesSajda09]). One issue in such model-based cognitive neuroscience approaches is that the trial numbers in each condition are often low, making it difficult it difficult to estimate model parameters. For example, studies with patient populations, especially if combined with intraoperative recordings, typically have substantial constraints on the duration of the task. Similarly, model-based fMRI or EEG studies are often interested not in static model parameters, but how these dynamically vary with trial-by-trial variations in recorded brain activity. Efficient and reliable estimation methods that take advantage of the full statistical structure available in the data across subjects and conditions are critical to the success of these endeavors.

Bayesian data analytic methods are quickly gaining popularity in the cognitive sciences because of their many desirable properties ([LeeWagenmakers13], [Kruschke10]). First, Bayesian methods allow inference of the full posterior distribution of each parameter, thus quantifying uncertainty in their estimation, rather than simply provide their most likely value. Second, hierarchical modeling is naturally formulated in a Bayesian framework. Traditionally, psychological models either assume subjects are completely independent of each other, fitting models separately to each individual, or that all subjects are the same, fitting models to the group as if they are all copies of some “average subject”. Both approaches are sub-optimal in that the former fails to capitalize on statistic strength offered by the degree to which subjects are similar in one or more model parameters, whereas the latter approach fails to account for the differences among subjects, and hence could lead to a situation where the estimated model cannot fit any individual subject. The same limitations apply to current DDM software packages such as DMAT [VandekerckhoveTuerlinckx08] and fast-dm [VossVoss07]. Hierarchical Bayesian methods provide a remedy for this problem by allowing group and subject parameters to be estimated simultaneously at different hierarchical levels ([LeeWagenmakers13], [Kruschke10], [VandekerckhoveTuerlinckxLee11]). Subject parameters are assumed to be drawn from a group distribution, and to the degree that subject are similar to each other, the variance in the group distribution will be estimated to be small, which reciprocally has a greater influence on constraining parameter estimates of any individual. Even in this scenario, the method still allows the posterior for any given individual subject to differ substantially from that of the rest of the group given sufficient data to overwhelm the group prior. Thus the method capitalizes on statistical strength shared across the individuals, and can do so to different degrees even within the same sample and model, depending on the extent to which subjects are similar to each other in one parameter vs. another. In the DDM for example, it may be the case that there is relatively little variability across subjects in the perceptual time for stimulus encoding, quantified by the “non-decision time” but more variability in their degree of response caution, quantified by the “decision threshold”. The estimation should be able to capitalize on this structure so that the non-decision time in any given subject is anchored by that of the group, potentially allowing for more efficient estimation of that subjects decision threshold. This approach may be particularly helpful when relatively few trials per condition are available for each subject, and when incorporating noisy trial-by-trial neural data into the estimation of DDM parameters.

HDDM is an open-source software package written in Python which allows (i) the flexible construction of hierarchical Bayesian drift diffusion models and (ii) the estimation of its posterior parameter distributions via PyMC ([PatilHuardFonnesbeck10]). User-defined models can be created via a simple python script or be used interactively via, for example, IPython interpreter shell (:cite:PER-GRA2007). All run-time critical functions are coded in Cython ([BehnelBradshawCitroEtAl11]) and compiled natively for speed which allows estimation of complex models in minutes. HDDM includes many commonly used statistics and plotting functionality generally used to assess model fit. The code is released under the permissive BSD 3-clause license, test-covered to assure correct behavior and well documented. Finally, HDDM allows flexible estimation of trial-by-trial regressions where an external measurement (e.g. brain activity as measured by fMRI) is correlated with one or more decision making parameters.

With HDDM we aim to provide a user-friendly but powerful tool that can be used by experimentalists to construct and fit complex, user-specified models using state-of-the-art estimation methods to test their hypotheses. The purpose of this report is to introduce the toolbox and provide a tutorial for how to employ it; subsequent reports will quantitatively characterize its success in recovering model parameters and advantages relative to non-hierarchical or non-Bayesian methods as a function of the number of subjects and trials (:cite: SoferWieckiFrank).