Fitting Wald and ex-Wald distributions to response time data: An example using functions for the S-PLUS package

Heathcote, Andrew · 2004 · Behavior Research Methods, Instruments, & Computers

DOI: 10.3758/bf03206550

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Summary

This paper addresses the challenge of fitting Wald and ex-Wald distributions to response time (RT) data, a task motivated by Schwarz’s (2001, 2002) proposal of the ex-Wald distribution as a model for simple and go/no-go RT tasks. The ex-Wald distribution is derived from the convolution of a Wald random variable (modeling decision time) and an exponential random variable (modeling nondecision time). Heathcote provides S-PLUS functions to compute maximum likelihood estimates for the ex-Wald, shifted Wald, and ex-Gaussian distributions, aiming to offer robust tools for analyzing RT data within a sequential sampling framework. The study employs a Monte Carlo simulation to evaluate the efficiency and bias of parameter estimates for these distributions. For the shifted Wald distribution, three distributions were tested, equated on mean and standard deviation but varying in skew. For the ex-Wald distribution, five distributions were examined, varying the parameter $k^2$ to cover real, complex, and boundary cases defined by Schwarz. The simulations involved generating samples of varying sizes (from 40 to 50,000) and fitting the distributions using the provided S-PLUS functions, which utilize the `nlminb` optimization routine. The study specifically analyzed the impact of using analytic gradients and Hessians for the shifted Wald versus numerical finite differences for the ex-Wald, as well as the effects of heuristic starting points and parameter scaling. The results indicate that sample size requirements differ significantly between distributions. For the shifted Wald, samples of approximately 100 observations were adequate for obtaining reliable estimates, provided that fits with ill-conditioned Hessians were excluded. Excluding these irregular fits substantially reduced bias and improved efficiency. In contrast, fitting the ex-Wald distribution proved more difficult. Samples of at least 400 were necessary for adequate estimates, but for certain parameter ranges—particularly where $k^2 \le 0$—much larger samples (up to 50,000) were required to achieve unimodal parameter distributions. Even with large samples, the ex-Wald exhibited high variability in estimates for the exponential parameter ($t$) and frequent irregular fits that could not be corrected by exclusion. The study also found that scaling parameters during optimization significantly improved convergence for the ex-Wald. The significance of this work lies in its practical guidance for researchers using distributional models of response time. It demonstrates that while the shifted Wald is relatively robust and requires modest sample sizes, the ex-Wald distribution is statistically challenging to fit, often requiring very large datasets to yield stable parameter estimates. The paper provides open-source S-PLUS functions and detailed implementation strategies, such as handling numerical errors and setting appropriate optimization bounds, facilitating the application of these sophisticated models in behavioral research. The findings caution against the uncritical use of the ex-Wald model in studies with limited data, highlighting the trade-offs between model complexity and estimation reliability.

Key finding

Samples of at least 400 are necessary to obtain adequate estimates for the ex-Wald distribution, whereas smaller samples of around 100 are sufficient for the shifted Wald distribution when ill-conditioned fits are excluded.

Methodology

simulation_modeling

Provenance

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