A framework for ML estimation of parameters of (mixtures of) common reaction time distributions given optional truncation or censoring

CONOR V. DOLAN 0 HAN L. J. VAN DER MAAS 0 PETER C. M. MOLENAAR 0 0 University of Amsterdam , Amsterdam, The Netherlands We present a framework for distributional reaction time (RT) analysis, based on maximum likelihood (ML) estimation. Given certain information relating to chosen distribution functions, one can estimate the parameters of these distributions and of finite mixtures of these distributions. In addition, left and/or right censoring or truncation may be imposed. Censoring and truncation are useful methods by which to accommodate outlying observations, which are a pervasive problem in RT research. We consider five RT distributions: the Weibull, the ex-Gaussian, the gamma, the log-normal, and the Wald. We employ quasi-Newton optimization to obtain ML estimates. Multicase distributional analyses can be carried out, which enable one to conduct detailed (across or within subjects) comparisons of RT data by means of loglikelihood difference tests. Parameters may be freely estimated, estimated subject to boundary constraints, constrained to be equal (within or over cases), or fixed. To demonstrate the feasibility of ML estimation and to illustrate some of the possibilities offered by the present approach, we present three small simulation studies. In addition, we present three illustrative analyses of real data. - The aim of the present paper is to present a general framework for distributional reaction time (RT) analysis, based on maximum likelihood (ML) estimation. The importance of distributional analyses as a method for characterizing the properties of empirical RT distributions is generally recognized (Heathcote, Popiel, & Mewhort, 1991; Hockley, 1984; Luce, 1986; Ratcliff, 1979, 1993; Ratcliff & Murdock, 1976; Ratcliff, Van Zandt, & McKoon, 1999; Ulrich & Miller, 1994). A distributional analysis involves three steps: (1) determination of a suitable theoretical distribution function to model or approximate the observed RT distribution (e.g., Logan, 1992; Van Zandt & Ratcliff, 1995), (2) estimation of the parameters of this distribution, and (3) evaluation of the fit of the theoretical distribution to the data. The determination of a suitable distribution may be approached from two perspectives. On the one hand, one may want to derive a distribution from first principles concerning the psychological process that generates the RTs (Luce, 1986). On the other hand, and more The research of C.V.D. was made possible by a fellowship from the Royal Dutch Academy of Arts and Sciences. We thank Wulfert van den Brink for free access to his statistics library, Denis Cousineau for his comments and feedback concerning our computer program, Eric-Jan Wagenmakers for his comments on an earlier version of this paper and his general encouragement, and Susanne Hammen and Verena Schmittmann for pointing out several errors. Wery van den Wildenberg kindly contributed data used in the first real-data illustration. Two rounds of detailed reviews by Andrew Heathcote and two anonymous referees resulted in many improvements. Correspondence concerning this paper should be directed to C. V. Dolan, Developmental Psychology, Department of Psychology, University of Amsterdam, Roetersstraat 15, 1018WB Amsterdam, The Netherlands (e-mail: ). pragmatically, one may simply want a distribution that provides an adequate description of the observed distribution. The ex-Gaussian is often used to this end (Heathcote, 1996; Schnipke & Scrams, 1997). The second step, estimation of parameters, is the subject of the present paper. Estimation of the parameters of RT distributions poses several problems. In estimating parameters of a given distribution function, it is important to take account of possible outlying RTs. Within the context of a distributional analysis, these may be accommodated by means of the imposition of censoring (Azzelini, 1996, pp. 26 27; Ulrich & Miller, 1994) or truncation (Ratcliff, 1993). We will use the term right (left) truncation, or censoring, to refer to the treatment of outlyingslow (fast) RTs. Truncation involves discarding RTs beyond chosen thresholds. No assumptions are made concerning the distribution of the discarded observations. The distribution function is renormalized, so that it integrates to unity, and the renormalized distribution is used in ML estimation. Only the retained RTs are used to estimate the parameters. Censoring likewise involves discarding RTs beyond chosen thresholds. However, the number of discarded RTs is supposed to be consistent with the hypothesized distribution of the retained RTs. Ulrich and Miller (1994) have shown that censoring is computationallyfeasible and that it works well in practice. As compared with censoring, truncation requires much larger samples to work well (Ratcliff, 1993; Ulrich & Miller, 1994). In addition to the accommodation of outliers, it may be desirable to model hypothesized heterogeneity, in the psychological process that generates RTs, by specifying a mixture of distributions (Van Zandt & Ratcliff, 1995; Yantis, Meyer, & Smith, 1991). Such heterogeneity may arise when two or more distinct processes (e.g., two different strategies of responding) generate the RTs. Interestingly, Ulrich and Miller (1994) related outlying RTs to mixture distributions by supposing that the outliers (e.g., extremely fast RTs) were generated by an extraneous process (fast guessing), whereas the other RTs were generated by the process in which they were interested. As Ulrich and Miller (1994) pointed out, an actual mixture modeling approach to accommodate outliers does not appear to be feasible, since outliers are small in number and may be generated by a variety of mechanisms, which give rise to unknown distributions. The problem of obtaining estimates in distributionalRT analyses is complicated by the imposition of truncation or censoring in single-component or finite mixture distribution. Van Zandt (2000) has provided an in-depth and clear account of methods of estimation in single-componentRT distributions, without censoring or truncation. She considered both nonparametric estimates of the distributions, such as histograms and kernel density estimates, and the parametric estimations of parameters of selected theoretical distributions. Parametric parameter estimation was achieved by ML estimation and by least-squares fits of the theoretical distributions to various empirical estimates of the distributions. Generally, in terms of efficiency and unbiasedness, ML emerged as the best method of estimating parameters. The main aim of the present paper is to present a framework for ML estimation of RT distribution parameters. We will demonstrate that, with certain information relating to the theoretical distribution functions, one can readily assemble the loglikelihood function for (mixtures of ) common RT distributions, given optional truncation or censoring. The required information relates to distribution function values, function deriv (...truncated)