RACE MODEL INEQUALITYSUPPLEMETARY MATERIAL1

Supplementary material

The supplementary material for Testing the race model inequality in redundant stimuli with variable onset asynchrony is divided into four sections:

  • How to derive a weighting function from a coactivation model
  • A bootstrapping procedure for comparing violation areas in two sets of conditions
  • Results of the reanalysis of the Miller (1986) data
  • Documentation of the computer program for the test of the race model, written in R statistical language (R Development Core Team, 2008).

Deriving a weight function from the diffusion superposition model

The diffusion superposition model (DSM, Schwarz, 1994) assumes that, upon the presentation of a stimulus, the accumulation of sensory evidence can be described by a diffusion process with drift µ > 0 and variance σ². As soon as a criterion (‘absorbing barrier’, a) is reached for the first time, the stimulus is ‘detected’. In bimodal stimuli, the auditory and the visual diffusion processes are superimposed, resulting in an aggregate process with drift µA + µV and variance σA² + σV² (the covariance term is set to zero, see discussion in Schwarz, 1994). As a consequence of the higher drift, the first passage of the criterion occurs earlier. The distribution functions for unimodal and bimodal stimuli are as follows:

GA(t) = W(t | a, µA, σA²)auditory stimulus
GV(t) = W(t | a, µV, σV²)visual stimulus
GAV(t) = W(t | a, µA + µV, σA² + σV²)synchronous AV stimulus
GAV(τ)(t) = GA(t)t ≤ τ, asynchronous AV
GAV(τ)(t) = GA(τ) + –∞∫a g(x; τ | a, µA, σA²) · W(t – τ | a – x, µA + µV, σA² + σV²) dx,
t > τ, asynchronous AV (S1)

with g(x; τ | a, µA, σA²) denoting the particle density of a diffusion process with drift µA and variance σA² at time τ with an absorbing barrier at a (Schwarz, 1994, Eq. 7). W(t) denotes the inverse Gaussian distribution function (e.g. Schwarz, 2001, Eq.A3). As mentioned in the main article, Miller (1986) suggested using the positive area below FAV(τ)(t) – FA(t) – FV(t – τ) as a measure of the amount to which the race model inequality is violated at a given SOA τ. Using the distribution functions (S1) and assuming a constant time for the motor execution of the response, the violation area predicted by the DSM is as following:

∆τ = 0∫ ∞max[0, GAV(τ)(t) – GA(t) – GV(t – τ)] dt(S2)

This violation area can be readily used as a weighting function. For the test of the coactivation model with exponentially distributed detection times (Ineq.6 of the main article), the second minuend is replaced by the distribution for synchronous presentations:

λ(τ)=0∫∞max[0, GAV(τ)(t) – GA(t) – GAV(t – τ)] dt(S3)

Application using Miller’s (1986) data

The use of the new method can be illustrated using the data from the two participants B.D. and K.Y. of Miller (1986). In the experiment, participants made speeded responses to unimodal and bimodal auditory-visual stimuli. Bimodal stimuli were presented at eleven onset asynchronies (A167V, A133V, A100V, …, AV, …V167A). Inverse cumulative distributions of the response times for each SOA were obtained from Jeff Miller (personal communication). For each SOA condition, I generated 400 response times which closely matched these distributions. In a first analysis, the observed violation areas for the eleven SOAs were added up using the aggregate test (13) with equal weighting, that is, λ(τ) = 1. The violation area ∆τ was calculated according to (12), anticipatory responses were censored by restricting the range of integration to percentiles 5 to 95 of the FAV distribution. The upper bound was chosen to increase the speed of the calculation. The distribution of the aggregate violation area under the race model was simulated 10,000 times as described by Miller (1986), the observed violation area was then compared to the simulated violation areas.

The results are summarized in TableS1. The SOA specific tests roughly reproduce the results of Miller (1986, Table 2). For the aggregate tests, different weighting functions were used, (a) equal weights, (b) an ‘umbrella’, that is, a triangular λ(τ) with maximum at τ=0, (c)a ‘shifted umbrella’ with maximum at E[TV] – E[TA], (d) DSM weights according to (S2). Each of the weighting functions b–d assign high weights to SOAs at which large violation areas are expected, whereas low weights are assigned to SOAs with small expected violation areas. For B.D., both the unweighted and the weighted tests are statistically significant. For K.Y. the P values of the weighted tests (umbrella: P=.094, shifted: P=.005, DSM: P=.019) indicate a significant violation of the race model prediction. Assuming that coactivation has occurred in B.D. as well as in K.Y., this reduction in P value suggests that the choice of an appropriate weighting function increases the power of the test. Alternatively, it is possible to use a test statistic based on the maximally observed violation area (Eq.14), this results in a P value of .040 for K.Y.

Comparison of coactivation effects between conditions

As already mentioned in the main article, Miller (1986) noted that the largest violation of the race model inequality occurred at V67A in both participants. The violation observed in this condition was substantially larger than, for example, in A67V. A statistical test of the difference between two violation areas can be performed using the bootstrap confidence intervals around ∆V67A and ∆A67V (Miller, 1986, p.337f): In each iteration i, samples with replacement are drawn from A, V, and AV(τ), and ∆τ(i) is determined for these samples. This procedure is repeated 1,000 times, the 95% confidence interval then corresponds to percentiles 2.5 and 97.5 of the simulated ∆τ(i). If the confidence intervals for ∆A67V and ∆V67A do not overlap, it is concluded that coactivation effects significantly differ between these two SOAs. Note that this approach is very conservative, because non-overlapping confidence intervals for 95% of two conditions generally imply P values much smaller than 0.05 (e.g. Austin & Hux, 2002).

The comparison of the violation areas observed in two single SOA conditions can again be generalized to weighted sums of violation areas, using two weighting functions with equal weight in total. For example, race model violations observed in vision-first stimuli might be compared to those observed in audition-first stimuli using λ1(τ)=I{τ0} (1 if τ0, otherwise 0) and λ2(τ)=I{τ < 0}. In each iteration, samples are drawn for each SOA and the weighted sum of the violation areas is determined. If the confidence intervals around the aggregate violation areas do not overlap, coactivation effects differ between audition-first and vision-first stimuli.

The SOA-specific P values shown in TableS1 for B.D. and for K.Y. suggest that coactivation effects are higher for stimuli in which the auditory component follows the visual component than for the reverse sequence. To test whether this difference is significant, bootstrap confidence intervals were estimated around the weighted sum of the violation areas obtained by λ1 and λ2 (10,000 simulations). For K.Y., the 95% confidence intervals for the aggregate violation areas in vision-first and audition-first stimuli were [13.1, 32.4], and [0.55, 20.3], respectively. Since the intervals overlap, it cannot be concluded that the stimulus sequence affects the coactivation effect (but see Austin & Hux, 2002). For B.D., coactivation effects differed significantly, with the lower bound of the 95% confidence interval for vision-first violation areas [51.3, 72.6] being substantially above the upper confidence limit for audition-first violation areas [1.54, 19.3].

Table S1.

Results of the different weighting functions applied to the Miller (1986) data

SOA / B.D. / K.Y.
∆obs / S95% / P / B2.5% / B97.5% / ∆obs / S95% / P / B2.5% / B97.5%
A167V / 0.27 / 4.28 / .761 / 0.00 / 3.63 / 1.07 / 6.59 / .563 / 0.00 / 6.94
A133V / 0.25 / 4.31 / .776 / 0.00 / 4.16 / 0.40 / 5.82 / .737 / 0.00 / 5.24
A100V / 1.56 / 4.29 / .359 / 0.21 / 5.38 / 1.87 / 5.23 / .346 / 0.02 / 7.47
A67V / 0.29 / 4.22 / .769 / 0.00 / 2.98 / 0.56 / 4.54 / .619 / 0.00 / 4.26
A33V / 2.78 / 3.55 / .109 / 0.52 / 6.35 / 0.00 / 3.64 / .961 / 0.00 / 1.61
AV / 4.52 / 2.91 / .006 / 1.86 / 7.67 / 2.08 / 2.89 / .121 / 0.24 / 4.85
V33A / 12.9 / 2.70 / .000 / 9.97 / 16.0 / 1.35 / 2.48 / .217 / 0.15 / 3.19
V67A / 14.3 / 2.67 / .000 / 11.6 / 17.2 / 5.66 / 2.29 / .000 / 3.47 / 8.07
V100A / 19.7 / 3.10 / .000 / 16.5 / 23.1 / 7.82 / 3.02 / .000 / 4.83 / 11.1
V133A / 12.2 / 3.73 / .000 / 8.59 / 16.2 / 2.71 / 3.84 / .133 / 0.66 / 6.81
V167A / 2.18 / 3.69 / .185 / 1.04 / 4.43 / 3.04 / 4.56 / .144 / 0.44 / 7.94
∑ / 70.9 / 28.5 / .000 / 57.6 / 94.6 / 26.5 / 32.0 / .109 / 17.8 / 49.9
∑∩ / 255 / 93.2 / .000 / 203 / 337 / 83.4 / 96.9 / .094 / 54.9 / 150
∑→∩ / 311 / 69.2 / .000 / 258 / 374 / 111 / 77.2 / .005 / 75.2 / 171
∑DSM / 398 / 89.5 / .000 / 328 / 483 / 19.6 / 16.0 / .019 / 11.6 / 31.0
∑A...V / 5.15 / 17.1 / .544 / 1.54 / 19.3 / 3.90 / 21.1 / .696 / 0.55 / 20.3
∑V...A / 61.3 / 12.5 / .000 / 51.3 / 72.6 / 20.6 / 12.9 / .004 / 13.1 / 32.4
Max / 19.7 / 5.77 / .000 / 16.5 / 23.1 / 7.82 / 7.53 / .040 / 5.17 / 11.2

Note.

∆obs: observed violation area—S95%: percentile 95 of simulated areas—B2.5%, B97.5%: bootstrap confidence interval around observed violation area—∑: summed violation areas, unweighted —∩: umbrella weighting function— →∩: shifted umbrella—DSM: weight function from superposition model (B.D.: µV=0.53, σ²V=18.1, µA=1.34, σ²A=128; K.Y.: µV=1.02, σ²V=25.1, µA=2.75, σ²A=818)—A...V: Weight function for audition-first stimuli—V...A: Weight function for vision-first stimuli—Max: maximum of the violation areas (Eq.14).

Computer Program

Two scripts in R statistical language (R development core team, 2008) are part of this supplementary material. The first script (rcode1.r) is used for generating response times similar to those reported by Miller (1986), of which only the inverse cumulative distributions are available. The response times for the two participants are generated using the two commands bd.gen() and ky.gen() without further options. The easiest way to do this is to execute the entire script by the source() command or by opening the script and choosing “Edit/Execute all” from the menu of the R environment. Note that two text files bd.dat and ky.dat are generated in the user’s home directory, eventually overwriting existing files with the same filename.

block tauv taua rt

1 Inf 0 265

1 167 0 159

1 0 133 267

1 0 0 338

2 0 Inf Inf

Figure S1. Format of the data used in the computer program.

The data to be analyzed should have the format shown in Figure S1. The header is mandatory. In the first entry (2nd line of the file), tauv = Inf denotes a unimodal auditory stimulus, the following rows denote A167V, V133A, AV (synchronous), V (unimodal visual stimulus), respectively. A response time of Inf denotes an omitted response.

The first column denotes the session number: If the experiment has been split into blocks of, say, 15 min duration, stratified bootstrapping is performed using these blocks as strata. Set the first column to a constant value to run the analysis without stratification. Stratification increases the power of the test, because fluctuations of overall response speed are reflected by sampling a fixed number of trials from each stratum. Note that this requires enough trials per block (e.g. 30 trials per condition and block). Bootstrapping underestimates the variance of the data by (N–1)/N, if stratified sampling is used, N denotes the stratum size (e.g. Smith, 1997, Eq. 5; Smith also discusses stratified bootstrapping techniques for small samples, p. 618f). Table 1 (main article) shows that for 10 trials per condition and block, stratification leads to anti-conservative decisions.

The race model test is performed using the second script, rcode2.r. The easiest way is to execute the entire script first (see above). If the results of Observer 1 are stored in c:/rtdata/obs01.dat, the race model test for this participant is executed by these commands:

d = m86.read(afile='obs01.dat', adir='c:/rtdata')

m86.test(d)

A few options of m86.test(d, qmin, qmax, nsamp, dodsm)might be of interest:

  • qmin, qmax:Range of integration for determining the violation area. Specify as quantiles of the AV synchronous distribution, e.g. qmin=0.05, qmax=0.95
  • nsamp:Number of simulations used for generating the distribution of violation areas under the race model (default is 1000)
  • dodsm:Should the superposition model be fitted? (default is no)

For example, Table S1 was generated using m86.test(d, qmin=0.05, qmax=0.95, nsamp=10000, dodsm=TRUE). For bootstrap confidence intervals around the violation areas, the command m86.ci(d, qmin, qmax, nsamp, dodsm) is used with the same options.

Table 1 of the main text is reproduced by setting the switch TABLE1 to TRUE at the beginning of rcode2.r. Note that the simulations take about one week on a standard 2 GHz computer.

References

Austin, P. C., & Hux, J. E. (2002). A brief note on overlapping confidence intervals. Journal of Vascular Surgery, 36, 194–195.

Miller, J. O. (1986). Timecourse of coactivation in bimodal divided attention. Perception & Psychophysics, 40, 331–343.

R Development Core Team (2008). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.

Schwarz, W. (2001). The ex-Wald distribution as a descriptive model of response times. Behavior Research Methods, Instruments, & Computers, 33, 457-469.

Schwarz, W. (1994). Diffusion, superposition, and the redundant targets effect. Journal of Mathematical Psychology, 38,504–520.

Smith, S. J. (1997). Bootstrap confidence limits for groundfish trawl survey estimates of mean abundance. Canadian Journal of Fisheries and Aquatic Sciences, 54, 616–630.