The first-passage time distribution for the diffusion model with variable drift

- The Ratcliff diffusion model contains normally distributed drift rates.
- Analytic solution for the cumulative distribution function of the diffusion model.
- The analytical solution is highly efficient and ensures high accuracy.
- Implementations in R statistical language and MATLAB included as online Appendix C.

The Ratcliff diffusion model is now arguably the most widely applied model for response time data. Its major advantage is its description of both response times and the probabilities for correct as well as incorrect responses. The model assumes a Wiener process with drift between two constant absorbing barriers. The first-passage times at the upper and lower boundary describe the responses in simple two-choice decision tasks, for example, in experiments with perceptual discrimination or memory search. In applications of the model, a usual assumption is a varying drift of the Wiener process across trials. This extra flexibility allows accounting for slow errors that often occur in response time experiments. So far, the predicted response time distributions were obtained by numerical evaluation as analytical solutions were not available. Here, we present an analytical expression for the cumulative first-passage time distribution in the diffusion model with normally distributed trial-to-trial variability in the drift. The solution is obtained with predefined precision, and its evaluation turns out to be extremely fast.