The baseline for response latency distributions

Response latency – the time taken to initiate or complete an action or task – is one of the principal measures used to investigate the mechanisms subserving human and animal cognitive processes. The right tails of response latency distributions have received little attention in experimental psychology. This is because such very long latencies have traditionally been considered irrelevant for psychological processes, instead, they are expected to reflect ‘contingent’ neural events unrelated to the experimental question. Most current theories predict the right tail of response latency distributions to decrease exponentially [1, 2]. In consequence, current standard practice recommends discarding very long response latencies as ‘outliers’ [3, 4]. Here, I show that the right tails of response latency distributions always follow a powerlaw with a slope of exactly two. This entails that the very late responses cannot be considered outliers. Rather they provide crucial information that falsifies most current theories of cognitive processing with respect to their exponential tail predictions. This exponent constitutes a fundamental constant of the cognitive system that groups behavioral measures with a variety of physical phenomena. A pervading assumption in the literature is that Response Latencies (RLs) follow a distribution whose right tail decreases exponentially [1, 2, 3]. RLs are ultimately by-products of the workings of the brain, and further, of the firing patterns of heavily interconnected neural assemblies. From this perspective, exponential tails would be a rather surprising outcome for RL distributions [5]. They would imply that the RLs were generated by a Poisson process, that is, they would be independent events, despite the interconnections between the neurons that generated them. More in line with the probably correlated origins of behavioral events, two recent theories have predicted that the right tails of RLs distributions should follow a power-law [5, 6]. This is to say that for all times t greater than a certain tmin, their probability density function should be that of a Pareto distribution: p(t) = α− 1 tmin ( t tmin )−α , α > 1, t ≥ tmin, (1) where α is referred to as the scaling parameter, and it corresponds to the slope of the straight line that is formed by the density function when plotted on log-log scale. A more precise theoretical proposal [6] is that RLs arise as the result of the ratio of two correlated normal variables: The excitability of the response effector, and the strength of the signal that excited it. Therefore the distribution of RLs should follow a normal ratio distribution (NRD [7]). This has the further implication that the power-law right tail should have a value of the scaling parameter of exactly two [8, 9]. Such a precise tail behavior would hold irrespective of the properties of the task. It would constitute a complete description of the RL distribution in the far right tail, in the strong sense of having zero degrees of freedom. The scaling parameter value would therefore represent a fundamental constant of the cognitive system. Furthermore, it would group RLs with other well-known natural systems with identical properties, such as Ising models of ferromagnetic materials close to their critical temperature [8] or the intervening times between major earthquakes [10]. Obtaining estimates of the distributions in the far right tail requires very large numbers of ideally untruncated responses. I analyzed six large-scale databases of human responses across experimental tasks and modalities, and at different time ranges. The datasets included ocular fixation and blink durations during reading (The Dundee Corpus [11]), spontaneous ocular fixation durations while participants were inspecting 1 ar X iv :0 90 8. 34 32 v1 [ qbi o. N C ] 2 4 A ug 2 00 9 photographs (DOVES database [12]), and a sample of different web-collected experiments extracted from the PsychExperiments [13] web site. This last set included two-choice decision reaction times to both auditory (tones) and visual (colours) stimuli, reaction times of participants performing a mental rotation task, and the times that participants took to exit a virtual maze. The solid dots in the left panel of Fig. 1 plot (in log-log scale) the histograms of the RLs in each of the datasets, aggregated across participants. Notice that all six distributions show a very similar pattern: The probabilities of the faster latencies rise to a peak, from which they decrease, gradually approaching a straight line, which is the characteristic signature of a power-law distribution. As predicted, the straight line components seem remarkably parallel across the six datasets, with a slope of approximately minus two (black dot-dashed lines). The right panel in Fig. 1 further stresses this apparent invariance. It plots the corresponding distributions when the times have been divided by their medians so as to remove the scale-dependent component of the distributions. One can distinguish three phases in the distributions. The early times rise to a peak, following very different patterns for each dataset. From the mode up to somewhere between five and forty times the median, there is a transition phase where the distributions gradually approach a power-law. The precise speed of convergence to the power-law varies depending on the properties of the participant and the task [6, 5]. From this point onwards – as stressed by the inset panel in the figure – the distribution of latencies is approximately the same, regardless of the particular experimental task. To confirm that this pattern holds when one considers only single-participant data, the figures also plot the histogram of the responses of an individual participant in the Dundee dataset (open red circles; these correspond to participant “sd”, but the pattern also holds for all other participants). Finally, in order to illustrate the theoretical prediction across the whole range of latencies, the figure also includes the theoretical density that would be predicted by an instance of the NRD with arbitrary parameters (black solid lines), and how the histogram from a sample of such would look like (grey open circles). The histograms in Fig. 1 seem consistent with the hypothesis that the right tails of latency distributions follow a power-law with a scaling parameter of two, and most certainly discard the traditional assumption of a light, exponential-type tail. However, other heavy-tailed distributions could also produce histograms with this appearance, and this has given rise to disagreements with respect to the precise nature of heavy tails in some datasets. Therefore, the hypothesis needs to be contrasted with other possible distributions with similarly heavy tails. Both log-normal and stretched-exponential (i.e., Weibull) tailed distributions also give rise to very heavy tails [14, 15, 16], and both have been proposed as plausible theoretical or empirical models for RL distributions [17, 3]. In addition, as I predict that the power-law should have a scaling parameter of exactly two, any other power-law with an arbitrary scaling parameter – not necessarily, but also including two – could be an alternative description [5]. Tab. 1 summarises the posterior evidence supporting the hypothesis that the right tails follow a power law distribution with a scaling parameter of exactly two over each of the other three candidate hypotheses [18]. For four out of the six aggregated datasets, and for the individual participant analysis, the evidence supports the hypothesis over the three competing candidates (i.e., positive values in the table). In the remaining two cases (negative values, highlighted in bold), the best candidate distribution was a power-law with an arbitrary value of the scaling parameter. In both of these cases, it seems like the optimal value of the scaling parameter estimated under a general power-law hypothesis has a value above two (last row in the table). The model comparison method was particularly stringent on the target hypothesis. The implicit truncation (see Supplementary Materials) present in the data could lead to the over-estimation of the scaling parameter that was found for two of the datasets. To investigate this possibility, I fitted an NRD to the RLs in the Maze dataset, as this was the one for which the theory showed the worst performance. From the fitted distribution, I generated a sample of artificial RLs of the same size as the Maze dataset, sampling only points below 50 times the median (this is equivalent to an upper truncation at around eight minutes). The sample was discretised to simulate a measurement resolution of one ms. Fig. 2 compares the original data with the sample and the fitted distribution. Although these simulated data originated from a power-law tailed distribution with a true scaling parameter of exactly two, applying the hypothesis testing procedure revealed a very similar pattern to what was observed in the Maze data (see the last column of Tab. 1).