Fitting distributions using maximum likelihood: methods and packages

The most powerful tests of response time (RT) models often involve the whole shape of the RT distribution, thus avoiding mimicking that can occur at the level of RT means and variances. Nonparametric distribution estimation is, in principle, the most appropriate approach, but such estimators are sometimes difficult to obtain. On the other hand, distribution fitting, given an algebraic function, is both easy and compact. We review the general approach to performing distribution fitting with maximum likelihood (ML) and a method based on quantiles (quantile maximum probability, QMP). We show that QMP has both small bias and good efficiency when used with common distribution functions (the ex-Gaussian, Gumbel, lognormal, Wald, and Weibull distributions). In addition, we review some software packages performing ML (PASTIS, QMPE, DISFIT, and MATHEMATICA) and compare their results. In general, the differences between packages have little influence on the optimal solution found, but the form of the distribution function has: Both the lognormal and the Wald distributions have nonlinear dependencies between the parameter estimates that tend to increase the overall bias in parameter recovery and to decrease efficiency. We conclude by laying out a few pointers on how to relate descriptive models of RT to cognitive models of RT. A program that generated the random deviates used in our studies may be downloaded from www.psychonomic.org/archive/.