Hierarchical Drift Diffusion Modeling

SSMs generally fall into one of two classes: (1) diffusion models which assume that relative evidence is accumulated over time and (2) race models which assume independent evidence accumulation and response commitment once the first accumulator crossed a boundary (LaBerge, 1962 (link); Vickers, 1970 (link)). Currently, HDDM includes two of the most commonly used SSMs: the drift diffusion model (DDM) (Ratcliff and Rouder, 1998 (link); Ratcliff and McKoon, 2008 (link)) belonging to the class of diffusion models and the linear ballistic accumulator (LBA) (Brown and Heathcote, 2008 (link)) belonging to the class of race models. In the remainder of this paper we focus on the more commonly used DDM.
As input these methods require trial-by-trial RT and choice data (HDDM currently only supports binary decisions) as illustrated in the below example table:
The DDM models decision-making in two-choice tasks. Each choice is represented as an upper and lower boundary. A drift-process accumulates evidence over time until it crosses one of the two boundaries and initiates the corresponding response (Ratcliff and Rouder, 1998 (link); Smith and Ratcliff, 2004 (link)) (see Figure 1 for an illustration). The speed with which the accumulation process approaches one of the two boundaries is called drift-rate v. Because there is noise in the drift process, the time of the boundary crossing and the selected response will vary between trials. The distance between the two boundaries (i.e., threshold a) influences how much evidence must be accumulated until a response is executed. A lower threshold makes responding faster in general but increases the influence of noise on decision-making and can hence lead to errors or impulsive choice, whereas a higher threshold leads to more cautious responding (slower, more skewed RT distributions, but more accurate). Response time, however, is not solely comprised of the decision-making process—perception, movement initiation and execution all take time and are lumped in the DDM by a single non-decision time parameter t. The model also allows for a prepotent bias z affecting the starting point of the drift process relative to the two boundaries. The termination times of this generative process gives rise to the response time distributions of both choices.
An analytic solution to the resulting probability distribution of the termination times was provided by Wald (1947 ); Feller (1968 ):
$\begin{array}{l}f(x|v,a,z)=\frac{\pi }{a^2}\text{exp}\left(-vaz-\frac{v^{2}x}{2}\right)\\ \times {\displaystyle \sum _{k=1}^{\infty }k}\text{ exp}\left(-\frac{k^2{\pi }^{2}x}{2a^2}\right)\text{sin}\left(k\pi z\right)\end{array}$
Since the formula contains an infinite sum, HDDM uses an approximation provided by Navarro and Fuss (2009 (link)).
Subsequently, the DDM was extended to include additional noise parameters capturing inter-trial variability in the drift-rate, the non-decision time and the starting point in order to account for two phenomena observed in decision-making tasks, most notably cases where errors are faster or slower than correct responses. Models that take this into account are referred to as the full DDM (Ratcliff and Rouder, 1998 (link)). HDDM uses analytic integration of the likelihood function for variability in drift-rate and numerical integration for variability in non-decision time and bias (Ratcliff and Tuerlinckx, 2002 (link)).