Explicit moments of decision times for single- and double-threshold drift-diffusion processes

TL;DR

This paper derives explicit formulas for the first three moments of decision times in drift-diffusion models, enhancing understanding of decision processes and aiding experimental data fitting.

Contribution

It provides the first explicit formulas for higher moments and conditioned moments of decision times in drift-diffusion models, which were previously unavailable.

Findings

Explicit formulas for third and higher moments of decision times.

Analysis of how drift rate, starting point, and non-decision time variability affect moments.

Numerical simulations support the theoretical results.

Abstract

We derive expressions for the first three moments of the decision time (DT) distribution produced via first threshold crossings by sample paths of a drift-diffusion equation. The "pure" and "extended" diffusion processes are widely used to model two-alternative forced choice decisions, and, while simple formulae for accuracy, mean DT and coefficient of variation are readily available, third and higher moments and conditioned moments are not generally available. We provide explicit formulae for these, describe their behaviors as drift rates and starting points approach interesting limits, and, with the support of numerical simulations, discuss how trial-to-trial variability of drift rates, starting points, and non-decision times affect these behaviors in the extended diffusion model. Both unconditioned moments and those conditioned on correct and erroneous responses are treated. We argue that the results will assist in exploring mechanisms of evidence accumulation and in fitting parameters to experimental data.