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Edited by: Martin Lages, University of Glasgow, United Kingdom

Reviewed by: Juergen Heller, University of Tübingen, Germany; Edgar Erdfelder, University of Mannheim, Germany

*Correspondence: Hans Colonius

This article was submitted to Quantitative Psychology and Measurement, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The race model inequality has become an important testing tool for the analysis of redundant signals tasks. In crossmodal reaction time experiments, the strength of violation of the inequality is taken as measure of multisensory integration occurring beyond probability summation. Here we extend previous results on trimodal race model inequalities and specify the underlying context invariance assumptions required for their validity. Some simulation results comparing the race model and the superposition model for Erlang distributed random variables illustrate the trimodal inequalities.

When stimulus information, perceived via several sensory modalities, indicates the occurrence of some event, an observer is faster detecting and responding to the stimulus compared to receiving only unimodal information, given a background of noisy signals. As a daily-life example consider the warning lights and siren of an ambulance in a traffic environment, a common audio-visual signal that allows, e.g., a driver to initiate an adequate reaction like giving way faster than if only acoustic or only visual information was available. Since the pioneering study by Todd (

A number of different models for the mechanisms underlying the RSE have been suggested. Raab (

for all non-negative time points _{xy}, _{x}, and _{y} denoting the distribution function for the redundant-signals condition and the single-stimulus conditions, respectively. Literature on this race model inequality (RMI) test involving different sensory modalities is huge (for a recent review, see Gondan and Minakata,

Numerous modeling approaches for a coactivation mechanism have been proposed (Diederich,

Sometimes, instead of inequality (1), the race model is tested using inequality

However, this is not generally recommended since it is more restrictive than (1) by assuming stochastic independence between the random latencies. While this assumption is not required by the general race model, there is another, essential assumption hidden in any version of the model, known as “context independence” or “context invariance”: the processing of a stimulus of a given modality does not depend on which and how many stimuli from other modalities are presented concurrently (e.g., Luce,

In the majority of RT studies on the RSE, only the bimodal case has been tested. Exceptions are, without claiming exhaustiveness, (Diederich,

In most cases, the stimuli being tested for multisensory integration are from the visual, auditory, or somatosensory modality. Many notable studies also involve other modalities (e.g., Gu et al.,

Let _{A}, _{V}, and _{S}, denote the distribution function for the unimodal conditions _{AV} is the bivariate distribution function of random vector (_{AVS} stands for the trivariate distribution function of (

for all

The formal definition of “context invariance” is as follows:

and

In other words, complete context invariance holds in the trimodal case if the distributions in the unimodal and bimodal conditions are identical to the corresponding univariate and bivariate marginal distributions of the trivariate distribution. As recently argued in Miller (

The proof of Inequality 1 relies on a simple probability inequality. Rewrite (1) for the

For any

Because of context invariance, _{AV}(_{t} ∪ _{t}), and the inequality follows from

where the last probability inequality is known as a special case of “Boole's inequality” (e.g., Diederich, _{t} and _{t} are defined on the same probability space for all _{AV}(_{A}(_{V}(_{AV}(_{A}(_{AV}(∞, _{V}(

In the following, we assume that trimodal context invariance holds unless indicated otherwise. In order to avoid trivial upper bounds larger than 1, the right-hand side of all inequalities presented may be replaced by, e.g., min{_{A}(_{V}(

For further reference, let us start with a listing of all possible bimodal RMIs:

A straightforward generalization to the trimodal case is

which follows again as special case of Boole's inequality.

As shown in Diederich (

A sharper^{1}_{AVS}(

Given the bimodal inequalities 3–5 hold, the trimodal inequalities 7–9 are sharper than Inequality 6. For example, if Inequalities 3 and 4 are satisfied, Inequality 9 implies Inequality 6:

Next we consider a situation where, for example, data from conditions

for all

The race model then implies

Obviously, with adding the following two, there are three inequalities in total:

with the corresponding, mutually incompatible, restricted context invariance assumptions.

An alternative situation for considering Inequalities 13–15 is when all three univariate distributions are available but violations occur for some of the univariate pairs. For example, there may be one or more values

While the race model for condition

In this section, we (i) illustrate a possible simulation approach and (ii) point to a specific aspect of dependency occurring for trimodal race models.

Simulating the race model and comparing it with a coactivation model requires specifying some RT distributions. Here we select distribution functions derived from the most basic stochastic counting process, i.e., the

A rate parameter λ^{x},

For _{xyz}, or _{xy} for bimodal stimulation, is defined as the minimum of the corresponding single stimulus detection times _{x}, _{y}, _{z} :

For a coactivation model, we choose the _{x}, λ_{y}, λ_{z}:

The prediction of the race model, an exponential distribution of RTs, is of course not consistent with typical data. It is taken here just for illustration; for fitting empirical data, it would be easy to add a Gaussian component, resulting in an ex-Gaussian distribution. For an empirical evaluation of race and superposition models we refer to Diederich (

Figures _{A} = λ_{V} = λ_{S} = 0.01 with sample size _{AV} in bound 13 was obtained from the race model but, in principle, it could be made arbitrarily close to 1 by choosing some coactivation model for condition

It shows the empirical distribution function of the simulated data for condition _{A} = λ_{V} = λ_{S} = 0.01).

It shows the empirical distribution function of the simulated data for condition _{A} = λ_{V} = λ_{S} = 0.01).

One characteristic of the race model is the possibility to increase the size of the redundant signals effect by tweaking the correlations between the involved detection processes. Given a race between two detection processes, consider their random processing times,

However, a situation with three “competing” processes _{xy}, _{xz}, _{yz}. For a fixed set of two-sample Neyman–Pearson correlation coefficients _{xy} and _{xz}, the third coefficient _{yz} can not vary freely between −1 and +1 but is restricted to a narrower range (Stanley and Wang,

Second, values for the correlations generating maximal facilitation are not as trivial to find as in a two-process situation. Limited by the above mentioned constraints, it is not possible to construct a correlation matrix with coefficients _{xy} = _{xz} = _{yz} = −1. Our simulations with a multivariate gamma distribution (not presented here) suggest, for example, that setting _{xy} = _{xz} = _{yz} = −0.5 yields a relatively large redundant signals effect when processing times

We have shown that the race model inequality extends naturally from the bimodal to the trimodal case, as long as essential assumptions about context invariance are specified. Moreover, the trimodal case permits “mixed models,” that is, models (i) where the race assumption is only valid for certain modality combinations but not for others, and (ii) where not all unimodal distributions may be available.

For an application of the generalized race model inequalities presented here, the next step is to extend the current statistical tests developed for the bimodal case to the different trimodal cases (for a recent overview, see Gondan and Minakata,

HC, FW, and AD conceived of the analysis. FW performed the simulations. HC and FW wrote the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

_{12}, given values of

_{13}and

_{23}

^{1}By “sharper” we mean “strictly smaller or equal.”