The race model inequality for censored reaction time distributions

Abstract

In the redundant-signals paradigm for simple reaction time (RT), the observer must initiate a response as quickly as possible following the detection of any stimulus onset. A typical finding is a redundancy gain: Responses are faster, on average, when two or more signals are presented simultaneously than when a single signal appears. This redundant-signals effect (RSE) has often, although not always, been replicated under different experimental settings-for example, comparing uni-versus multimodal stimulation (Diederich, 1995;Diederich & Colonius, 1987;Gielen, Schmidt, & Van den Heuvel, 1983;Miller, 1982Miller, , 1986Molholm, Ritter, Javitt, & Foxe, 2004), single versus multiple stimuli within the same modality (e.g., Schwarz & Ischebeck, 1994), or monocular versus binocular stimulation (Hughes & Townsend, 1998; Westendorf & Blake, 1988)-and for specific populations (see, e.g., Corballis, 1998;Miller, 2004;Reuter-Lorenz, Nozawa, Gazzaniga, & Hughes, 1995;and Savazzi & Marzi, 2004, all for split-brain individuals; and Marzi et al., 1996, for hemianopics).Raab (1962) proposed a race model for simple RT, postulating that (1) each individual stimulus elicits a (normally distributed) detection process performed in parallel to the others and (2) the winner's time determines the observable RT. The race model opens up the possibility that the RSE is generated by statistical facilitation: If detection latencies are interpreted as (nonnegative) random variables, the time to detect the first of several redundant signals is faster, on average, than the detection time for any single signal. Testing the race model amounts to probing whether an observed RT speedup is too large to be attributable to statistical facilitation (viz., probability summation), no matter which distributional assumptions have been made.A test of general race models was developed by Miller (1978Miller ( , 1982, showing thatmust hold for all t 0. This race model inequality (RMI) follows fromfor any pair of random variables (X, Y ) with a joint probability distribution based on Pr XY and with its marginal distributions identical to Pr X and Pr Y . Thus, as was observed in Luce (1986, p. 130), the RMI test requires that the RT distributions in the single-signal conditions are identical to the corresponding (marginal) RT distributions in the redundant-signals condition (cf. Colonius, 1990). Note that, for fixed t, Inequality 2 corresponds to the well-known Boole's inequality (e.g., Billingsley, 1979). Neglecting possible additional components (such as motor time), the inequality stipulates that the RT distribution function for redundant stimuli is never larger than the sum of the RT distributions for the single stimuli. A violation of this inequality is interpreted as an indication of an underlying coactivation mechanism or some other strong form of nonindependence. The race model inequality (RMI) introduced in Miller (1982) puts an upper limit on the amount of reaction time facilitation within the redundant-signals paradigm that is consistent with a race ...

Recalibration of the Multisensory Temporal Window of Integration Results from Changing Task Demands

“…In order to minimize the chances of erroneously rejecting the race model, a conservative approach to the “kill-the-twin” procedure was applied to the observed RT data, and a progressive approach was applied to the simulated RT data generated by the resampling procedure described below [33]. Misses were attributed infinitely long reaction times [35], [36]. The average occurrence of guesses to the catch trials was 3.94% across subjects (range 0–11.67%).…”

A new nonparametric test for the race model inequality

“…In this and in the following simulation study, only non-scaled observations were considered (as the Maris and Maris's KS test as well as the truncated KS test are known to be scale invariant). The simulation was performed by systematically varying two factors in a complete 4 × 5 twofactorial design (see also Rach et al, 2010). In particular, the factors were:…”

“…In this second simulation study, we considered several redundant target conditions in which the distributional patterns were not constructed to reflect a maximal statistical facilitation (and therefore a truncated distribution for F Z ). Also, in this second simulation, the observed responses were modeled according to a Weibull distribution and assumed equal sample size (n = m = s) for F Z and K (Rach et al, 2010). In the simulation design, we varied three factors in a complete 4 × 5 × 10 factorial design.…”

A tutorial on cue combination and Signal Detection Theory: Using changes in sensitivity to evaluate how observers integrate sensory information

“…Thus, researchers often find it necessary to exclude as outliers empirical estimates below and/or above an arbitrary threshold. Such trimming can in turn introduce non-trivial artefacts into the data, which, if not corrected for, can lead to fallacious conclusions 130 . For practical reasons, many authors therefore to prefer measures of sensitivity over response times.…”