Let A, V, S be random latencies corresponding to stimulus modalities A,V, and S, respectively. F_A, F_V, and F_S, denote the distribution function for the unimodal conditions A,V, and S respectively. For bimodal condition AV, H_AV is the bivariate distribution function of random vector (A, V), and for trimodal condition AVS, H_AVS stands for the trivariate distribution function of (A, V, S). Thus, for example,

for all s, t, u ≥ 0. Moreover, for marginal distributions we write, e.g.,

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

DEFINITION 1. For all s, t, u ≥ 0, necessary and sufficient conditions for (trimodal) complete context invariance are

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 (2016), context invariance is an essential part of the race model concept.

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

For any t ≥ 0, we define events

Because of context invariance, F_AV(t) = P(B_t ∪ C_t), and the inequality follows from

where the last probability inequality is known as a special case of “Boole's inequality” (e.g., Diederich, 1992b). It is important to recognize the role of the context invariance assumption here: it guarantees that events B_t and C_t are defined on the same probability space for all t or, more generally, that there exists a bivariate distribution H_AV(s, t) with marginals equal to F_A(s) and F_V(t), respectively: H_AV(s, ∞) = F_A(s) and H_AV(∞, t) = F_V(t).

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{F_A(t) + F_V(t), 1}, etc. For simplicity, we do not state this explicitly.

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 (1992b), Bonferroni-type probability inequalities (Worsley, 1982) can be used to derive further trimodal RMIs:

A sharper¹ bound for F_AVS(t) results by taking the minimum (at each value of t) across all three bounds in 7–9:

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 AVS, AV, and S are available but none from V or A. Responses under the bimodal condition AV can then be conceived as representing reaction times to a “combined” visual-auditory modality formally equivalent to a unimodal condition. The underlying distribution function is bivariate,

for all w, z ≥ 0, with AV denoting the RT in condition AV. Context invariance is restricted to the bivariate case, that is:

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 t′ such that

While the race model for condition AV would be ruled out in this case, the joint processing of AV and S may still be consistent with a race.

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 Poisson process. The time between two randomly occurring events follows an exponential distribution with intensity rate λ. Stimulus processing time is defined as the waiting time of the Poisson process for c-th event. Empirically, criterion c may be influenced by the experimental condition. For example, rewarding high detection accuracy would increase the threshold (Luce, 1986) and higher values of c will result in longer detection time, denoted D. The distribution of D is known as Erlang distribution, a special case of the gamma distribution with an integer-valued shape parameter c and rate λ:

A rate parameter λ^x, x ∈ {A, V, S}, has to be specified for each single stimulus condition, auditory (A), visual (V), and somato-sensory (S), respectively, while threshold parameter c is assumed to be constant across the modalities. For simplicity, we neglect residual processes, like motor time, and assume that D equals the observed reaction time.

For x, y, z pairwise different modalities chosen from {A, V, S}, detection time in the race model for trimodal redundant stimulation, D_xyz, or D_xy for bimodal stimulation, is defined as the minimum of the corresponding single stimulus detection times D_x, D_y, D_z :

For a coactivation model, we choose the superposition model proposed in Schwarz (1989) (see also Diederich and Colonius, 1991). While a number of alternative coactivation models are available, choosing one based on the Poisson process has the advantage that, with one and the same set of values for the parameters, detection time D can be simulated either for the race model or for the superposition model. For the latter, detection time for redundant stimuli follows again an Erlang/Gamma distribution with the intensity rate given by the sum of the single stimulus intensity rates λ_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 (1992a).

Figures 1, 2 depict empirical distribution functions obtained from simulations of race and superposition models with parameter values c = 2, λ_A = λ_V = λ_S = 0.01 with sample size n = 2,000. In both figures the trimodal distribution was obtained from the superposition model, i.e., a Gamma (2, 0.03) distribution. Figure 1 compares it to the bounds described by Inequality 6 and 10 (denoted as “sharp RMI”). Both bounds are violated for a large range of percentiles, with the latter bound being violated even for percentiles beyond 80%. Figure 2 compares the trimodal distribution of the superposition model with data from simulating the bounds described by Inequalities 6 and 13 for the race model. Again, there are large-range violations of the bounds. F_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 AV.

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, X and Y, varying from trial to trial. By definition, the shortest processing time min(X, Y) determines the detection time for a trial. Negative correlation means that, when one process is fast, the other tends to be slow in a given trial. Thus, there would only be few trials where both processes are rather slow. This results in decreased average detection times, since long times for one process are replaced by shorter times of the other process. One can show that a race model with two processes and maximal negative correlation yields the maximum redundant signals effect (Miller, 1982; Colonius, 1990, 2016).

However, a situation with three “competing” processes X, Y, Z, e.g., audio, visual, and somato-sensory, is more complicated. First, instead of one there are three correlation coefficients r_xy, r_xz, r_yz. For a fixed set of two-sample Neyman–Pearson correlation coefficients r_xy and r_xz, the third coefficient r_yz can not vary freely between −1 and +1 but is restricted to a narrower range (Stanley and Wang, 1969):

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 r_xy = r_xz = r_yz = −1. Our simulations with a multivariate gamma distribution (not presented here) suggest, for example, that setting r_xy = r_xz = r_yz = −0.5 yields a relatively large redundant signals effect when processing times X, Y, Z have an identical underlying distribution.

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.