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.     

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.     

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 detection of feature singletons defined in two dimensions is based on salience summation, rather than on serial exhaustive or interactive race architectures

Influential models of visual search assume that dimension-specific feature contrast signals are summed into a master saliency map in a coactive fashion. The main source of evidence for coactivation models, and against parallel race models, is violations of the race model inequality (RMI; Miller, 1982) by redundantly defined singleton feature targets. However, RMI violations do not rule out serial exhaustive (Townsend & Nozawa, 1997) or interactive race (Mordkoff & Yantis, 1991) architectures. These alternatives were tested in two experiments. In Experiment 1, we used a double-factorial design with singleton targets defined in two dimensions and at two levels of intensity, to distinguish between serial versus parallel models and self-terminating versus exhaustive stopping rules. In Experiment 2, we manipulated contingency benefits that are expected to affect the magnitude of redundancy gains and/or RMI violations on the assumption of an interactive race. The results of both experiments revealed redundancy gains as well as violations of the RMI, but the data pattern excluded serial-exhaustive and interactive race models as possible explanations for RMI violations. This result supports saliency summation (coactivation) models of search for singleton feature targets.     

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.     

Redundancy gain with static versus moving hands: a test of the hemispheric coactivation model

A simple reaction time (RT) experiment was conducted to test the hypothesis that redundancy gain arises partly because of hemispheric coactivation. Stimuli were presented to the left or right of fixation, or redundantly to both, and the participants had to make keypress responses as rapidly as possible to stimulus onset. A "static" condition, in which the participants held their hands at rest, was compared with a "dynamic" condition, in which they moved their hands back and forth in an oscillating motion prior to stimulus onset. As predicted from the hypothesis of hemispheric coactivation, redundancy gain, measured as the decrease in mean RT to redundant stimuli as compared with single stimuli, was smaller in the dynamic condition than in the static one.     

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     

Exact procedures for the analysis of multidimensional contingency tables

The log-linear model for contingency tables expresses the logarithm of a cell frequency as an additive function of main effects, interactions, etc., in a way formally identical with an analysis of variance model. Exact statistical tests are developed to test hypotheses that specific effects or sets of effects are zero, yielding procedures for exploring relationships among qualitative variables which are suitable for small samples. The tests are analogous to Fisher's exact test for a 2 × 2 contingency table. Given a hypothesis, the exact probability of the obtained table is determined, conditional on fixed marginals or other functions of the cell frequencies. The sum of the probabilities of the obtained table and of all less probable ones is the exact probability to be considered in testing the null hypothesis. Procedures for obtaining exact probabilities are explained in detail, with examples given.     

Identification of own-race and other-race faces: Implications for the representation of race in face space

Own-race faces are recognized more easily than faces of a different, unfamiliar race. According to the multidimensional space (MDS) framework, the poor discriminability of other-race faces is due to their being more densely clustered in face space than own-race faces. Multidimensional scaling analyses of similarity ratings (Caucasian participants, n = 22) showed that other-race (Chinese) faces are more densely clustered in face space. We applied a formal model to test whether the spatial location of face stimuli could account for identification accuracy of another group of Caucasian participants (n = 30). As expected, own-race (Caucasian) faces were identified more accurately (higher hit rate, lower false alarms, and higher A) than other-race faces, which were more densely clustered than ownrace faces. A quantitative model successfully predicted identification performance from the spatial locations of the stimuli. The results are discussed in relation to the standard MDS account of race effects and also an alternative “race-feature” hypothesis.     

Testing differences between nested covariance structure models: Power analysis and null hypotheses

For comparing nested covariance structure models, the standard procedure is the likelihood ratio test of the difference in fit, where the null hypothesis is that the models fit identically in the population. A procedure for determining statistical power of this test is presented where effect size is based on a specified difference in overall fit of the models. A modification of the standard null hypothesis of zero difference in fit is proposed allowing for testing an interval hypothesis that the difference in fit between models is small, rather than zero. These developments are combined yielding a procedure for estimating power of a test of a null hypothesis of small difference in fit versus an alternative hypothesis of larger difference.     

Fisher与Neyman–Pearson的分歧与心理统计中的假设检验争议

假设检验思想的提出者Fisher与Neyman–Pearson在统计模型的方法论基础、两类错误的性质、显著性水平的理解、以及假设检验的功能等方面存在诸多分歧, 使得心理统计中最常用的原假设显著性检验模式呈现出隐含的各种矛盾, 从而引发了应用上的争议。心理统计不仅需要检讨现有检验模型的模糊之处和提出其他补充性的统计推论方式,更应注重反思心理统计的教育传统, 以建立更加开放和多元的统计应用视野, 使心理统计为更好地心理学研究服务。     

Increasing stimulus intensity does not affect sensorimotor synchronization

When people synchronize taps with isochronously presented stimuli, taps usually precede the pacing stimuli [negative mean asynchrony (NMA)]. One explanation of NMA [sensory accumulation model (SAM), Aschersleben in Brain Cogn 48:66–79, 2002] is that more time is needed to generate a central code for kinesthetic-tactile information than for auditory or visual stimuli. The SAM predicts that raising the intensity of the pacing stimuli shortens the time for their sensory accumulation, thereby increasing NMA. This prediction was tested by asking participants to synchronize finger force pulses with target isochronous stimuli with various intensities. In addition, participants performed a simple reaction-time task, for comparison. Higher intensity led to shorter reaction times. However, intensity manipulation did not affect NMA in the synchronization task. This finding is not consistent with the predictions based on the SAM. Discrepancies in sensitivity to stimulus intensity between sensorimotor synchronization and reaction-time tasks point to the involvement of different timing mechanisms in these two tasks.     

Information Seeking in a Non-Hypothesis Testing Task

The information gain model (Oaksford and Chater, Psychological Review 101, 608–631, 1994) advocates that participants attempt to achieve a larger expected information gain when they have to test an if-then rule or hypothesis. However, acquisition of larger expected information gain could also be operational when participants do not have to test a hypothesis. This study devised a new task to investigate whether participants would seek larger expected information gain when they were not required to test a hypothesis. The task required participants to select one out of two balance scales for weighing coins in order to detect an underweight coin. We discovered that participants more frequently selected the balance scale that provided smaller expected information gain. This finding suggests that the preference for larger expected information gain may not apply to non-hypothesis testing settings.     

Dividing attention between color and shape: Evidence of coactivation

When attention is divided between spatially distinct objects, the time-to detect a target decreases when two or more targets are present. This redundancy gain can be accounted for by an interactive race model (Mordkoff & Yantis, 1991) in which separate decisions are made about each object, but environmental contingencies among the objects can influence decision times. In the present study, we examined whether the model also accounts for performance when attention must be divided between stimulus attributes other than spatial location. Subjects made target-present responses when displays included a prespecified color, a prespecified letter, or both target features. The data violated the predictions of all separate-activations models, including the interactive race model. Two control experiments ruled out an alternative account based on task complexity. We conclude that coactivation occurs when target attributes from two separable dimensions are simultaneously present, but not when target attributes come from the same dimension. A modular hybrid of race and coactivation models is tentatively proposed.     

Statistical Learning of Unfamiliar Sounds as Trajectories Through a Perceptual Similarity Space

In typical statistical learning studies, researchers define sequences in terms of the probability of the next item in the sequence given the current item (or items), and they show that high probability sequences are treated as more familiar than low probability sequences. Existing accounts of these phenomena all assume that participants represent statistical regularities more or less as they are defined by the experimenters—as sequential probabilities of symbols in a string. Here we offer an alternative, or possibly supplementary, hypothesis. Specifically, rather than identifying or labeling individual stimuli discretely in order to predict the next item in a sequence, we need only assume that the participant is able to represent the stimuli as evincing particular similarity relations to one another, with sequences represented as trajectories through this similarity space. We present experiments in which this hypothesis makes sharply different predictions from hypotheses based on the assumption that sequences are learned over discrete, labeled stimuli. We also present a series of simulation models that encode stimuli as positions in a continuous two‐dimensional space, and predict the next location from the current location. Although no model captures all of the data presented here, the results of three critical experiments are more consistent with the view that participants represent trajectories through similarity space rather than sequences of discrete labels under particular conditions.     

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.     

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.     

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.