When less equals more: Probability summation without sensitivity improvement.

Abstract

Many perceptual and cognitive tasks permit or require the integrated cooperation of specialized sensory channels, detectors, or other functionally separate units. In compound detection or discrimination tasks, 1 prominent general mechanism to model the combination of the output of different processing channels is probability summation. The classical example is the binocular summation model of Pirenne (1943), according to which a weak visual stimulus is detected if at least 1 of the 2 eyes detects this stimulus; as we review briefly, exactly the same reasoning is applied in numerous other fields. It is generally accepted that this mechanism necessarily predicts performance based on 2 (or more) channels to be superior to single channel performance, because 2 separate channels provide "2 chances" to succeed with the task. We argue that this reasoning is misleading because it neglects the increased opportunity with 2 channels not just for hits but also for false alarms and that there may well be no redundancy gain at all when performance is measured in terms of receiver operating characteristic curves. We illustrate and support these arguments with a visual detection experiment involving different spatial uncertainty conditions. Our arguments and findings have important implications for all models that, in one way or another, rest on, or incorporate, the notion of probability summation for the analysis of detection tasks, 2-alternative forced-choice tasks, and psychometric functions.

When two heads are better than one: Interactive versus independent benefits of collaborative cognition

“…Fortunately, research on perceptual decision-making in individuals has already laid that groundwork. For example, Schwarz and Miller (2014) discuss the interactive versus independent effects of redundant signals on response accuracy, recommending that both hits and false alarms be considered with equal rigor in any analysis. In the realm of perceptual sensitivity at threshold, researchers have applied the statistical logic of Miller's RMI to study the perception of facial expressions based on single features and combinations of features (Gold, 2014).…”

GSDT: An integrative model of visual search.

“…One third of them were shown at each of three eccentricities along imaginary circles, of radius 1.8 deg, 3.8 deg, and 6.2 deg, jittered slightly in both dimensions so as to give the display an irregular appearance but to avoid item overlap. As illustrated in Figure 2, all crosses were upright and consisted of two orthogonal lines of 15 pixels length each (cf., Schwarz & Miller, 2014). Distractor crosses intersected at pixel number 8, so that all four arms of the cross were formed by seven pixels, whereas targets were crosses with arms of unequal lengths.…”

One model fits all: Explaining many aspects of number comparison within a single coherent model—A random walk account.

“…As an alternative to symmetric step size densities in the present study we consider the skew double exponential distribution (for background on this distribution in the context of perceptual decision making, see Foley & Schwarz, 1998; Schwarz & Miller, 2014), which is shown in Figure 1 and described formally in the Appendix. Critically, under the usual mirror-image property g ij ( x ) = g ji (– x ), this skew distribution does not predict errors for a given digit pair to be on average as fast as correct responses to its complementary pair.…”

“…The double exponential model shown in Figure 1 assumes that for i < j the step size density associated with the pair ( i , j ) has a double exponential distribution (Figures 1 and 7, see Kotz & Nadarajah, 2000; for background on this model in the context of perceptual decision making, see Foley & Schwarz, 1998; Schwarz & Miller, 2014), with associated cumulative distribution function where μ ij is the net drift rate of the induced walk, and γ=0.5772 . .…”