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

Schwarz (2001, 2002) proposed the ex-Wald distribution, obtained from the convolution of Wald and exponential random variables, as a model of simple and go/no-go response time. This article provides functions for the S-PLUS package that produce maximum likelihood estimates of the parameters for the ex-Wald, as well as for the shifted Wald and ex-Gaussian, distributions. In a Monte Carlo study, the efficiency and bias of parameter estimates were examined. Results indicated that samples of at least 400 are necessary to obtain adequate estimates of the ex-Wald and that, for some parameter ranges, much larger samples may be required. For shifted Wald estimation, smaller samples of around 100 were adequate, at least when fits identified by the software as having ill-conditioned maximums were excluded. The use of all functions is illustrated using data from Schwarz (2001). The S-PLUS functions and Schwarz’s data may be downloaded from the Psychonomic Society’s Web archive, www. psychonomic.org/archive/.     

Assessing the evidence for response time mixture distributions

I describe a technique for comparing two simple accounts of a distribution of response times: A mixture model and a generalized-shift model. In the mixture model, a target distribution is assumed to be a mixture of response times from two other (reference) distributions. In the generalized-shift model, the target distribution is assumed to be a quantile average of the reference distributions. In order to distinguish these two possibilities, quantiles for the target distribution are estimated from the quantiles of the reference distributions assuming either a shift or a mixture, and the predicted quantiles are used to calculate the multinomial likelihood of the obtained data. Monte Carlo simulations reported here demonstrate that the index is relatively unbiased, is effective with moderate sample sizes and modest spreads between the reference distributions, is relatively unaffected by changes in the number of bins or by data trimming, can be used with data aggregated across subjects, and is relatively insensitive to a range of subject variations in distribution shape and in mixture or shift proportion. As an illustration, the index is applied to the interpretation of three effects from distinct paradigms: residual switch costs in the task-switching paradigm, the psychological refractory period effect, and sequential effects in the Simon task. I conclude that the multinomial likelihood index provides a useful and easily applied tool for the interpretation of effects on response time distributions.     

Fitting model-based psychometric functions to simultaneity and temporal-order judgment data: MATLAB and R routines

Research on temporal-order perception uses temporal-order judgment (TOJ) tasks or synchrony judgment (SJ) tasks in their binary SJ2 or ternary SJ3 variants. In all cases, two stimuli are presented with some temporal delay, and observers judge the order of presentation. Arbitrary psychometric functions are typically fitted to obtain performance measures such as sensitivity or the point of subjective simultaneity, but the parameters of these functions are uninterpretable. We describe routines in MATLAB and R that fit model-based functions whose parameters are interpretable in terms of the processes underlying temporal-order and simultaneity judgments and responses. These functions arise from an independent-channels model assuming arrival latencies with exponential distributions and a trichotomous decision space. Different routines fit data separately for SJ2, SJ3, and TOJ tasks, jointly for any two tasks, or also jointly for the three tasks (for common cases in which two or even the three tasks were used with the same stimuli and participants). Additional routines provide bootstrap p-values and confidence intervals for estimated parameters. A further routine is included that obtains performance measures from the fitted functions. An R package for Windows and source code of the MATLAB and R routines are available as Supplementary Files.     

An R package to compute commonality coefficients in the multiple regression case: An introduction to the package and a practical example

Multiple regression is a widely used technique for data analysis in social and behavioral research. The complexity of interpreting such results increases when correlated predictor variables are involved. Commonality analysis provides a method of determining the variance accounted for by respective predictor variables and is especially useful in the presence of correlated predictors. However, computing commonality coefficients is laborious. To make commonality analysis accessible to more researchers, a program was developed to automate the calculation of unique and common elements in commonality analysis, using the statistical package R. The program is described, and a heuristic example using data from the Holzinger and Swineford (1939) study, readily available in the MBESS R package, is presented.     

Estimating response time hazard functions: An exposition and extension

The hazard function of a response time distribution describes the temporal dimension of a response process and completely characterizes the distribution. The function can be estimated by a quadratic spline which is constrained to have one of a number of simple functional forms. The estimator can also be employed when the response time is an unobserved component in a parallel or serial pair of processes.     

Aging and intraindividual variability in performance: analyses of response time distributions

It has been suggested that older adults are more variable in their performance because they are more prone to lapses of either attention or intention. In the present experiment, 9 young and 9 older adults each performed nearly 2000 trials of a same-different judgment task. As expected, older adults were slower and more variable than young adults. When the age-related difference in speed was taken into account, however, the older adults were, if anything, less variable than the young adults. When younger and older adults' RT distributions were analyzed using quantile-quantile plots and by fitting ex-Gaussian and Weibull functions, there was no consistent evidence that older adults' distributions were more skewed than young adults', as would be predicted by age-related increases in lapses of attention or intention. Importantly, there was a positive, linear relation between RT and intraindividual variability, and the same relation was observed both within subjects (practice increased speed and reduced variability) as well as between subjects (regardless of age, slower individuals were more variable). Thus, the present results suggest that there may be a general law governing the relation between average RT and variability, and that the greater performance variability of older adults primarily reflects their greater average RTs.     

Decomposing the problem-size effect: a comparison of response time distributions across cultures

Is the locus of the problem-size effect in mental arithmetic different across cultures? In a novel approach to this question, the ex-Gaussian distributional model was applied to response times for large (e.g., 8 x 9) and small (e.g., 2 x 3) problems obtained from Chinese and Canadian graduate students in a multiplication production task (LeFevre & Liu, 1997). The problem-size effect for the Chinese group occurred in mu (the mean of the normal component), whereas the problem-size effect for the Canadian group occurred in both mu and tau (the mean of the exponential component). The results support the position that the problem-size effect for the Chinese group is purely a memory-retrieval effect, whereas for the Canadian group, it is an effect of both retrieval and the use of nonretrieval solution procedures.     

Dynamics of the attentional control of word retrieval: analyses of response time distributions

Since W. Wundt (1904) and H. J. Watt (1906), researchers have found no agreement on how goals direct word retrieval. A prevailing associative account (E. K. Miller & J. D. Cohen, 2001) holds that goals bias association strength, which determines retrieval latency and whether irrelevant words interfere. A symbolic account (A. Roelofs, 2003) holds that goals enable retrieval rules and predicts no strict dependence of interference on latency. Here, 3 chronometric experiments in which the role of relative retrieval latency was investigated through distributional analyses, following Watt, are reported. Participants verbally categorized picture-word pairs that were semantically related or unrelated, or they categorized single pictures or words. The pairs yielded semantic latency effects in both word and picture categorizing, although single words were categorized slower than single pictures. Semantic effects occurred in word categorizing even when postexposure of the pictures compensated for the difference in categorizing latency. Vincentile and ex-Gaussian analyses revealed that the semantic effects occurred throughout the latency distributions, excluding goal neglect as the cause of the effects. The results were interpreted as most consistent with the symbolic account, which was corroborated by computer simulations.     

The structure of short-term memory scanning: an investigation using response time distribution models

A classic question in cognitive psychology concerns the nature of memory search in short-term recognition. Despite its long history of investigation, however, there is still no consensus on whether memory search takes place serially or in parallel or is based on global access. In the present investigation, we formalize a variety of models designed to account for detailed response time distribution data in the classic Sternberg (Science 153: 652-654, 1966) memory-scanning task. The models vary in their mental architectures (serial exhaustive, parallel self-terminating, and global access). Furthermore, the component processes within the architectures that make match/mismatch decisions are formalized as linear ballistic accumulators (LBAs). In fast presentation rate conditions, the parallel and global access models provide far better accounts of the data than does the serial model. LBA drift rates are found to depend almost solely on the lag between study items and test probes, whereas response thresholds change with memory set size. Under slow presentation rate conditions, even simple versions of the serial-exhaustive model provide accounts of the data that are as good as those of the parallel and global access models. We provide alternative interpretations of the results in our General Discussion.