Testing the race model inequality: An algorithm and computer programs

In divided-attention tasks, responses are faster when two target stimuli are presented, and thus one is redundant, than when only a single target stimulus is presented. Raab (1962) suggested an account of this redundant-targets effect in terms of a race model in which the response to redundant target stimuli is initiated by the faster of two separate target detection processes. Such models make a prediction about the probability distributions of reaction times that is often called the race model inequality, and it is often of interest to test this prediction. In this article, we describe a precise algorithm that can be used to test the race model inequality and present MATLAB routines and a Pascal program that implement this algorithm.     

The race model inequality: interpreting a geometric measure of the amount of violation

An inequality by J. O. Miller (1982) has become the standard tool to test the race model for redundant signals reaction times (RTs), as an alternative to a neural summation mechanism. It stipulates that the RT distribution function to redundant stimuli is never larger than the sum of the distribution functions for 2 single stimuli. When many different experimental conditions are to be compared, a numerical index of violation is very desirable. Widespread practice is to take a certain area with contours defined by the distribution functions for single and redundant stimuli. Here this area is shown to equal the difference between 2 mean RT values. This result provides an intuitive interpretation of the index and makes it amenable to simple statistical testing. An extension of this approach to 3 redundant signals is presented.     

Showing that the race model inequality is not violated

When participants are asked to respond in the same way to stimuli from different sources (e.g., auditory and visual), responses are often observed to be substantially faster when both stimuli are presented simultaneously (redundancy gain). Different models account for this effect, the two most important being race models and coactivation models. Redundancy gains consistent with the race model have an upper limit, however, which is given by the well-known race model inequality (Miller, 1982). A number of statistical tests have been proposed for testing the race model inequality in single participants and groups of participants. All of these tests use the race model as the null hypothesis, and rejection of the null hypothesis is considered evidence in favor of coactivation. We introduce a statistical test in which the race model prediction is the alternative hypothesis. This test controls the Type I error if a theory predicts that the race model prediction holds in a given experimental condition.     

Effects of spatial and selective attention on basic multisensory integration

When participants respond to auditory and visual stimuli, responses to audiovisual stimuli are substantially faster than to unimodal stimuli (redundant signals effect, RSE). In such tasks, the RSE is usually higher than probability summation predicts, suggestive of specific integration mechanisms underlying the RSE. We investigated the role of spatial and selective attention on the RSE in audiovisual redundant signals tasks. In Experiment 1, stimuli were presented either centrally (narrow attentional focus) or at 1 of 3 unpredictable locations (wide focus). The RSE was accurately described by a coactivation model assuming linear superposition of modality-specific activation. Effects of spatial attention were explained by a shift of the evidence criterion. In Experiment 2, stimuli were presented at 3 locations; participants had to respond either to all signals regardless of location (simple response task) or to central stimuli only (selective attention task). The RSE was consistent with task-specific coactivation models; accumulation of evidence, however, differed between the 2 tasks.     

Consequences of Base Time for Redundant Signals Experiments

We report analytical and computational investigations into the effects of base time on the diagnosticity of two popular theoretical tools in the redundant signals literature: (1) the race model inequality and (2) the capacity coefficient. We show analytically and without distributional assumptions that the presence of base time decreases the sensitivity of both of these measures to model violations. We further use simulations to investigate the statistical power model selection tools based on the race model inequality, both with and without base time. Base time decreases statistical power, and biases the race model test toward conservatism. The magnitude of this biasing effect increases as we increase the proportion of total reaction time variance contributed by base time. We marshal empirical evidence to suggest that the proportion of reaction time variance contributed by base time is relatively small, and that the effects of base time on the diagnosticity of our model-selection tools are therefore likely to be minor. However, uncertainty remains concerning the magnitude and even the definition of base time. Experimentalists should continue to be alert to situations in which base time may contribute a large proportion of the total reaction time variance.     

Redundancy gains and coactivation with two different targets: The problem of target preferences and the effects of display frequency

When a visual display contains two targets, both of which require the same response, reaction times (RTs) are faster than when only one target appears. This effect has previously been obtained regardless of whether the redundant targets are the same or different in shape, and in at least one set of two-target experiments, the redundancy gains have been larger for different targets (Grice & Reed, 1992). Experiments with two different targets have also revealed violations of the race-model inequality, suggesting that redundant targets coactivate the response (Miller, 1982). The present paper reexamines both of these findings, because both appear to be inconsistent with the interactive race model (Mordkoff & Yantis, 1991). Experiment 1 shows that the race-model inequality is not violated when the experimental design is free of biased contingencies; Experiment 1 also provides evidence that target preferences may artifactually produce the RT advantage fordifferent- oversame-target trials. Experiment 2, however, shows that the race-model inequality is violated when the frequencies of single- and redundant-target displays are equated (without introducing any biased contingencies), implying that the interactive race model cannot account for the results of experiments involving more than one type of target. Alternative loci for coactivation are briefly discussed.     

Bias produced by fast guessing in distribution-based tests of race models.

A comparison involving cumulative probability distributions of reaction time (RT) has been used to test race models of the redundancy gain observed in certain divided-attention paradigms. It has been pointed out, however, that the presence of fast guesses would interfere with this test, biasing it to accept race models. The present paper reports simulations carried out to determine the size of the bias introduced by fast guesses. In absolute terms, this bias can be quite large--exceeding 175 msec in some conditions. Simulations indicate that the bias increases with the percentage of fast guesses and with the latency difference between the lower tails of guess and nonguess RT distributions. Discarding and rerunning errors reduces bias somewhat, but a more elaborate "kill-the-twin" procedure reduces it much more.     

Age-related differences in interhemispheric visuomotor integration measured by the redundant target effect

Presentation of bilateral redundant visual stimuli produces faster reaction times (RT) than presentation of a single unilateral stimulus; an effect known as the redundant target effect (RTE; Miller, 1982), and is a means of testing interhemispheric visuomotor integration (Ouimet, 2009). RTEs that exceed expectations, based on Miller's race model of inequality (RMI), are referred to as "enhanced RTEs" and imply neural coactivation. Paradoxically, enhanced RTEs are observed in cases of corpus callosum disruption. The Hemispheric Coactivation Hypothesis accounts for this paradox by positing that bihemispheric processing occurs to both unilateral and bilateral stimuli in the normal brain, but occurs only with bilateral stimuli in the disconnected brain. Neuroimaging has revealed decreases in the microstructural integrity of the corpus callosum with age (Ota et al., 2006), but research investigating the bilateral RTE in healthy older individuals is lacking. The present study investigated the bilateral RTE in healthy younger and healthy older adults using simple RT and choice RT tasks. Our prediction that older individuals would show significantly larger RTEs than younger individuals was found to be true for both tasks. Tests of the RMI produced little evidence for coactivation. The crossed-uncrossed difference, generally used as a means of testing visuomotor interhemispheric transfer, was also investigated, but no age effects were found. The observation of greater RTE in age is congruent with the Hemispheric Coactivation hypothesis (Miller, 2004) in which callosal disconnection is associated with increased RTE.     

Further tests of the interactive race model of divided attention: The effects of negative bias and varying stimulus — onset asynchronies

In Go/No-Go detection tasks, responses to redundant targets are typically faster than responses to either of these targets alone. One explanation of this redundant-targets effect is from race models, which assume statistical facilitation due to the activation of more than one processing channel under the redundant-targets condition. J. O. Miller (1982, 1986) has derived an upper boundary for the amount of facilitation these models can predict and has found this boundary to be consistently violated in bimodal divided-attention tasks and in letter-detection tasks. Thus, until recently race models were thought to be unable to predict the amount of facilitation commonly observed.Mordkoff and Yantis (1991) have challenged this conclusion and showed that no facilitation beyond the predictions of race models is observed if certain types of contingency within the experimental design are removed. The present study tries to replicate this basic finding and to generalize it to conditions with (a) nonsimultaneous signal presentation, and (b) negative interstimuius contingency benefit. Several important predictions of Mordkoff and Yantis' interactive race model were found to hold, but for nonsimultaneous signals presentations consistent violations of Miller's upper bound were found under certain conditions. The implications of the present results for models of divided attention are discussed.     

An interactive race model of divided attention.

Two classes of models have been proposed to explain how redundant information extracted from separate sources comes to activate a single response. Each provides a fundamentally different account of why responses to redundant signals are typically faster than those to either signal alone (the redundant-signals effect). Independent race models assume that a race occurs between perceptual codes on independent channels and that only the winner activates the response. Coactivation models assume that there is some form of energy or activation-strength summation, with information being pooled across channels prior to decision. An intermediate class of models is introduced and a specific exemplar, the interactive race model, is tested in a series of redundant-target detection experiments. In particular, we examine the effects on performance of two types of contingency that have previously been overlooked as sources of task-relevant information. The results reveal that response times are significantly influenced by both interstimulus and stimulus-response contingencies. The interactive race model provides a natural account of these findings as well as several otherwise puzzling results in the divided-attention literature.     

Visual search for dimensionally redundant pop-out targets: evidence for parallel-coactive processing of dimensions

In two visual search experiments, the detection of singleton feature targets redundantly defined on multiple dimensions was investigated. Targets differed from the distractors in orientation, color, or both (redundant targets). In Experiment 1, the various target types were presented either in separate blocks or in random order within blocks. Reaction times to redundant targets significantly violated the race model inequality (Miller, 1982), but only when there was constancy of the target-defining dimension(s) within trial blocks. In Experiment 2, there was dimensional variability within blocks. Consistent with Experiment 1, constancy of the target-defining dimension(s), but this time across successive trials (rather than within blocks), was critical for observing violations of the race model inequality. These results provide evidence for parallel-coactive processing of multiple dimensions, consistent with the dimension-weighting account of Müller, Heller, and Ziegler (1995).     

Bias produced by fast guessing in distribution-based tests of race models

A comparison involving cumulative probability distributions of reaction time (RT) has been used to test race models of the redundancy gain observed in certain divided-attention paradigms. It has beenpointed out, however, that the presence of fast guesses wouldinterfera with this test, biasing it to accept race models. The present paper reports simulations carried out to determine the size of the bias introduced by fast guesses. In absolute terms, this bias can be quite large— exceeding 175 msec in some conditions. Simulations indicate that the bias increases with the percentage of fast guesses and with the latency difference between the lower tails of guess and nonguess RT distributions. Discarding and rerunning errors reduces bias somewhat, but a more elaborate “kill-the-twin” procedure reduces it much more.     

Visual search for dimensionally redundant pop-out targets: Redundancy gains in compound tasks

Two visual search experiments investigated the detection of odd-one-out feature targets redundantly defined on multiple dimensions. Targets differed from the distractors in either orientation or colour or both (redundant targets). In Experiment 1, the three types of target were presented either in separate trial blocks or randomized within blocks, and the task involved either a simple target detection response or a “compound” response based on the position of dials inside the target. Mean reaction times (RTs) were faster to redundant targets than to singly defined targets, with greater gains in simple detection than in compound tasks. Further, simple detection RTs to redundant targets were faster than the fastest RTs to singly defined targets, violating Miller's (1982) “race model inequality” (RMI). Experiment 2 showed that, with compound tasks, mean RT redundancy gains (and violations of the RMI) depend on practice. The results suggest that separate colour and orientation feature contrast signals coactivate perceptual mechanisms involved in target detection.     

Redundant visual signals boost saccade execution

The redundant signal effect (RSE) refers to the fact that human beings react more quickly to a pair of stimuli than to only one stimulus. In previous studies of the RSE in the oculomotor system, bimodal signals have been used as the goal of the saccade. In consistency with studies using manual response times (RTs), saccadic RTs have been shown to be shorter for redundant multimodal stimuli than for single unimodal stimuli. In the present experiments, we extended these findings by demonstrating an RSE in the saccadic system elicited only by unimodal visual stimuli. In addition, we found that shorter saccadic RTs were accompanied by an increased saccadic peak velocity. The present results are of relevance for neurophysiological models of saccade execution, since the boost of saccades was elicited by two visual transients (acting as a “go” signal) that were presented not at the goal of the saccade but at various other locations.     

Multisensory processing of redundant information in go/no-go and choice responses

In multisensory research, faster responses are commonly observed when multimodal stimuli are presented, as compared to unimodal target presentations. This so-called redundant-signals effect can be explained by several frameworks, including separate-activation and coactivation models. The redundant-signals effect has been investigated in a large number of studies; however, most of those studies have been limited to the rejection of separate-activation models. Coactivation models have been analyzed in only a few studies, primarily using simple response tasks. Here, we investigated the mechanism of multisensory integration underlying go/no-go and choice responses to redundant auditory–visual stimuli. In the present study, the mean and variance of response times, as well as the accuracy rates of go/no-go and choice responses, were used to test a coactivation model based on the linear superposition of diffusion processes (Schwarz, 1994) within two absorbing barriers. The diffusion superposition model accurately describes the means and variances of response times as well as the proportions of correct responses observed in the two tasks. Linear superposition thus seems to be a general principle in the integration of redundant information provided by different sensory channels, and is not restricted to simple responses. The results connect existing theories of multisensory integration with theories on choice behavior.     

Bimodal and trimodal multisensory enhancement: effects of stimulus onset and intensity on reaction time

Manual reaction times to visual, auditory, and tactile stimuli presented simultaneously, or with a delay, were measured to test for multisensory interaction effects in a simple detection task with redundant signals. Responses to trimodal stimulus combinations were faster than those to bimodal combinations, which in turn were faster than reactions to unimodal stimuli. Response enhancement increased with decreasing auditory and tactile stimulus intensity and was a U-shaped function of stimulus onset asynchrony. Distribution inequality tests indicated that the multisensory interaction effects were larger than predicted by separate activation models, including the difference between bimodal and trimodal response facilitation. The results are discussed with respect to previous findings in a focused attention task and are compared with multisensory integration rules observed in bimodal and trimodal superior colliculus neurons in the cat and monkey.     

Visual search for dimensionally redundant pop-out targets: Evidence for parallel-coactive processing of dimensions

In two visual search experiments, the detection of singleton feature targets redundantly defined on multiple dimensions was investigated. Targets differed from the distractors in orientation, color, or both (redundant targets). In Experiment 1, the various target types were presented either in separate blocks or in random order within blocks. Reaction times to redundant targets significantly violated therace model inequality (Miller, 1982), but only when there was constancy of the target-defining dimension(s) within trial blocks. In Experiment 2, there was dimensional variability within blocks. Consistent with Experiment 1, constancy of the target-defining dimension(s), but this time across successive trials (rather than within blocks), was critical for observing violations of the race model inequality. These results provide evidence for parallel-coactive processing of multiple dimensions, consistent with thedimension-weighting account of Müller, Heller, and Ziegler (1995).     

Redundancy gain in semantic categorisation

Redundancy gain refers to the performance enhancements often associated with the presentation of redundant versus single targets (for example, faster, more accurate, or more forceful responses). Though predominantly observed in relatively simple tasks (e.g., stimulus detection), there have been some efforts to investigate similar phenomena in tasks involving higher level processing. We conducted three experiments aimed at determining (a) whether a redundancy gain would be evident in a task unambiguously requiring higher level processing (the semantic categorisation of visually-presented lexical stimuli), and (b) if so, what accounts might be appropriate to explain such findings. We found that redundancy gains are observed in such tasks, and we conclude that both coactivation and race models can account for these gains.