Models for bimodai signal detection: A reply to Fideil

Fidell (1982)has raised questions about the validity of our conclusions regarding performance in a bimodal detection task (Mulligan & Shaw, 1980). This reply explains why his critique is in error. Our paper is concerned with how subjects combine auditory and visual sensory information to decide whether either an auditory or a visual signal is present. Under conditions of continuous auditory and visual noise, subjects were presented with four types of stimuli: (1) auditory signal, (2) visual signal, (3) auditory plus visual signal, and (4) neither auditory nor visual signal. The subject's task was to decide whether at least one signal (visual or auditory) was presented. We presented and tested the predictions of two decision models. According to the independent decisions model, a subject makes a separate, covert, binary (yes-no) decision for each modality by comparing a measure of signal strength in that modality to the relevant decision criterion. The total number of covert "yes" decisions is then used to make an overt final decision: "yes" if one or more covert "yes" decisions have been made and "no" otherwise. In contrast, according to the integration model, a singledecision variable is formed by taking a weighted sum of the signal strength measures from the separate modalities. If this decision variable exceeds its criterion, the subject responds "yes." The data from our experiment supported a version of the independent decisions model and rejected the integration model. In his critique, Fidell emphasizes a formulation of the integration model in which ad' value for bimodal detection is predicted from two separate d' values estimated from unimodal detection data. He shows that our data conform quite well to this prediction, and from this he infers that our conclusion favoring the independent decisions model is erroneous; indeed, he suggests that it must be based exclusivelyon "aesthetic preference." We claim that this inference is faulty.' We will substantiate our position below by arguing that a prediction in terms of d' is not appropriate for discriminating between the two models, and that the predictions we use provide a better and more general test of the models. Earlier comparisons of independent-decisions integration models (cf. Green & Swets, 1973) have been restricted to a threshold (i.e., two-state) independentdecisions model with a nonzero probability of a detect state given noise (hence "low" threshold). In our paper, we consider an independent-decisions model in which sensory strength measures are treated as continuous random variables, and the internal strength measure of each modality is compared with its own criterion. The criteria selected for the auditory and visual modalities are independently adjustable, and can differ between unimodal and bimodal tasks. A consequence of these two properties is that the continuous independent decisions model can produce hit and false alarm rates that mimic the hit and false alarm rates of the integration model. This means that these models are not identifiable on the basis of the relation between bimodal and unimodal d' values. Thus, Fidell's observation that our data conform to the integration model predictions for d' is meaningless-because this conformity also applies to predictions from the independent decisions model, given appropriate choices of criteria. In our paper, we do not rely exclusively on the prediction of one model to discriminate the two theoretical alternatives. Instead, we derive predictions from each model in terms of the same statistic (probability of a "no" response), and these two predictions are pitted against each other. (See Green & Swets, 1973, and Sternberg, 1963, for a discussion.) A separate analysis (Shaw, 1980) shows that these and the other predictions we use differ enough between the models to provide reliable discrimination. The data we collected clearly reject the integration model and support the independent decisions model. Only if it can be argued that thereliable discrimination of one model from another confers a certain beauty on that model can we agree with Fidell that our choice of the independent decisions model is "aesthetic. " We claim that our tests of these models are more general in several ways than the test proposed by Fidell (and others; cf. Green & Swets, 1973). First, our predictionsapply to both threshold and continuousstate variants of the independent-decisions model (see Shaw, in press). Second, for none of the models we considered do the critical predictions depend upon subjects' maintaining fixed criteria from unimodal to bimodal detection tasks. Third, for none of our critical predictions is it required that attentional demands be invariant between unimodal and bimodal