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.     

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.     

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.     

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.     

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.     

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.     

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).     

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).     

Testing the race inequality: A simple correction procedure for fast guesses

In speeded response tasks with redundant signals, parallel processing of the redundant signals is generally tested using the so-called race inequality. The race inequality states that the distribution of fast responses for a redundant stimulus never exceeds the summed distributions of fast responses for the single stimuli. It has been pointed out that fast guesses (e.g. anticipatory responses) interfere with this test, and a correction procedure (‘kill-the-twin’ procedure) has been suggested. In this note we formally derive this procedure and extend it to the case in which redundant stimuli are presented with onset asynchrony. We demonstrate how the kill-the-twin procedure is used in a statistical test of the race model prediction.     

Multiplicity,directional (type III) errors,and the null hypothesis

L. V. Jones and J. W. Tukey (2000) pointed out that the usual 2-sided, equal-tails null hypothesis test at level alpha can be reinterpreted as simultaneous tests of 2 directional inequality hypotheses, each at level alpha/2, and that the maximum probability of a Type I error is alpha/2 if the truth of the null hypothesis is considered impossible. This article points out that in multiple testing with familywise error rate controlled at alpha, the directional error rate (assuming all null hypotheses are false) is greater than alpha/2 and can be arbitrarily close to alpha. Single-step, step-down, and step-up procedures are analyzed, and other error rates, including the false discovery rate, are discussed. Implications for confidence interval estimation and hypothesis testing practices are considered.     

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.     

Testing models of the redundant-signals effect: A warning concerning the combination-rule regression analysis

The redundant-signals effect is the observed RT advantage for trials presenting two or more targets, as compared with trials with only one target Two general classes of parallel-processing model have been proposed to explain this effect race models (e.g., Raab, 1962) and coactivation models (e.g., Miller, 1982). Various distributional analyses have been used in work aimed at discriminating between these two model classes. The present study reexamined one of these tests—the combination-rule regression analysis based on variable-criterion theory (Grice, Canham, & Boroughs, 1984)—by applying it to the data from two sets of simulated experiments. One set of simulations assumed coactivation; the other set assumed an independent race on redundant-target trials. Nearly identical combination-rule values were observed in the two sets of simulations. This finding shows that the combination rule of variable-criterion theory does not discriminate between models capable of explaining the redundant-signals effect The implications of this finding are briefly discussed     

Heterogeneous heterogeneity by default: Testing categorical moderators in mixed-effects meta-analysis

Categorical moderators are often included in mixed-effects meta-analysis to explain heterogeneity in effect sizes. An assumption in tests of categorical moderator effects is that of a constant between-study variance across all levels of the moderator. Although it rarely receives serious thought, there can be statistical ramifications to upholding this assumption. We propose that researchers should instead default to assuming unequal between-study variances when analysing categorical moderators. To achieve this, we suggest using a mixed-effects location-scale model (MELSM) to allow group-specific estimates for the between-study variance. In two extensive simulation studies, we show that in terms of Type I error and statistical power, little is lost by using the MELSM for moderator tests, but there can be serious costs when an equal variance mixed-effects model (MEM) is used. Most notably, in scenarios with balanced sample sizes or equal between-study variance, the Type I error and power rates are nearly identical between the MEM and the MELSM. On the other hand, with imbalanced sample sizes and unequal variances, the Type I error rate under the MEM can be grossly inflated or overly conservative, whereas the MELSM does comparatively well in controlling the Type I error across the majority of cases. A notable exception where the MELSM did not clearly outperform the MEM was in the case of few studies (e.g., 5). With respect to power, the MELSM had similar or higher power than the MEM in conditions where the latter produced non-inflated Type 1 error rates. Together, our results support the idea that assuming unequal between-study variances is preferred as a default strategy when testing categorical moderators.     

Comparison of a two-stage and three-stage interim-analysis procedure.

A statistical model for combining p values from multiple tests of significance is used to define rejection and acceptance regions for two-stage and three-stage sampling plans. Type I error rates, power, frequencies of early termination decisions, and expected sample sizes are compared. Both the two-stage and three-stage procedures provide appropriate protection against Type I errors. The two-stage sampling plan with its single interim analysis entails minimal loss in power and provides substantial reduction in expected sample size as compared with a conventional single end-of-study test of significance for which power is in the adequate range. The three-stage sampling plan with its two interim analyses introduces somewhat greater reduction in power, but it compensates with greater reduction in expected sample size. Either interim-analysis strategy is more efficient than a single end-of-study analysis in terms of power per unit of sample size.     

Effects of Covariance Heterogeneity on Three Procedures for Analyzing Multivariate Repeated Measures Designs

Empirical Type I error and power rates were estimated for (a) the doubly multivariate model, (b) the Welch-James multivariate solution developed by Keselman, Carriere and Lix (1993) using Johansen's results (1980), and for (c) the multivariate version of the modified Brown-Forsythe (1974) procedure. The performance of these procedures was investigated by testing within- blocks sources of variation in a multivariate split-plot design containing unequal covariance matrices. The results indicate that the doubly multivariate model did not provide effective Type I error control while the Welch-James procedure provided robust and powerful tests of the within-subjects main effect, however, this approach provided liberal tests of the interaction effect. The results also indicate that the modified Brown-Forsythe procedure provided robust tests of within-subjects main and interaction effects, especially when the design was balanced or when group sizes and covariance matrices were positively paired.     

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.     

Effects of redundant visual stimuli on temporal order judgments

Four experiments were conducted in order to compare the effects of stimulus redundancy on temporal order judgments (TOJs) and reaction times (RTs). In Experiments 1 and 2, participants were presented in each trial with a tone and either a single visual stimulus or two redundant visual stimuli. They were asked to judge whether the tone or the visual display was presented first. Judgments of the relative onset times of the visual and the auditory stimuli were virtually unaffected by the presentation of redundant, rather than single, visual stimuli. Experiments 3 and 4 used simple RT tasks with the same stimuli, and responses were much faster to redundant than to single visual stimuli. It appears that the traditional speedup of RT associated with redundant visual stimuli arises after the stimulus detection processes to which TOJs are sensitive.     

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.     

Testing two variances for superiority/non-inferiority and equivalence: Using the exhaustion algorithm for sample size allocation with cost

The equality of two group variances is frequently tested in experiments. However, criticisms of null hypothesis statistical testing on means have recently arisen and there is interest in other types of statistical tests of hypotheses, such as superiority/non-inferiority and equivalence. Although these tests have become more common in psychology and social sciences, the corresponding sample size estimation for these tests is rarely discussed, especially when the sampling unit costs are unequal or group sizes are unequal for two groups. Thus, for finding optimal sample size, the present study derived an initial allocation by approximating the percentiles of an F distribution with the percentiles of the standard normal distribution and used the exhaustion algorithm to select the best combination of group sizes, thereby ensuring the resulting power reaches the designated level and is maximal with a minimal total cost. In this manner, optimization of sample size planning is achieved. The proposed sample size determination has a wide range of applications and is efficient in terms of Type I errors and statistical power in simulations. Finally, an illustrative example from a report by the Health Survey for England, 1995–1997, is presented using hypertension data. For ease of application, four R Shiny apps are provided and benchmarks for setting equivalence margins are suggested.