Modeling probability and additive summation for detection across multiple mechanisms under the assumptions of signal detection theory

Abstract

Many studies have investigated how multiple stimuli combine to reach threshold. There are broadly speaking two ways this can occur: additive summation (AS) where inputs from the different stimuli add together in a single mechanism, or probability summation (PS) where different stimuli are detected independently by separate mechanisms. PS is traditionally modeled under high threshold theory (HTT); however, tests have shown that HTT is incorrect and that signal detection theory (SDT) is the better framework for modeling summation. Modeling the equivalent of PS under SDT is, however, relatively complicated, leading many investigators to use Monte Carlo simulations for the predictions. We derive formulas that employ numerical integration to predict the proportion correct for detecting multiple stimuli assuming PS under SDT, for the situations in which stimuli are either equal or unequal in strength. Both formulas are general purpose, calculating performance for forced-choice tasks with M alternatives, n stimuli, in Q monitored mechanisms, each subject to a non-linear transducer with exponent τ. We show how the probability (and additive) summation formulas can be used to simulate psychometric functions, which when fitted with Weibull functions make signature predictions for how thresholds and psychometric function slopes vary as a function of τ, n, and Q. We also show how one can fit the formulas directly to real psychometric functions using data from a binocular summation experiment, and show how one can obtain estimates of τ and test whether binocular summation conforms more to PS or AS. The methods described here can be readily applied using software functions newly added to the Palamedes toolbox.

Length summation in noise

“…Conversely, the detection threshold decreased with target size with a log-log slope of approximately À1/2 (thick dashed line) at all noise levels. Such À1/2 slope summation is commonly observed in spatial summation with periodic patterns (Polat & Tyler, 1999;Tyler & Chen, 2006;Kingdom et al, 2015), faces (Tyler & Chen, 2006) and texts (Kao & Chen, 2012). There were telltale signs of the function asymptoting to a À1 slope (dotted line in Figure 2) for small target sizes and a trend of flattening more than the slope of À1/2 for the largest sizes.…”

“…Recently, multiple stage models have been proposed to explain various spatial summation phenomena (Baker & Meese, 2011;Meese & Summers, 2012;Kingdom et al, 2015). Although there are variations among the different models, they generally start with a band of local filters followed by a nonlinear transform, and an array of summation mechanisms pooling the nonlinear responses of a number of local filters.…”

“…This kind of mechanism is an implementation of the divisive transducer function formalism of Legge and Foley (1980) or Foley (1994), although the numerator of their response functions (Meese & Summers, 2007, Equation B1) would imply the target and the pedestal being processed by different local mechanisms. However, those models were designed to account for spatial summation for the target alone (Kingdom et al, 2015) or the target on a patterned pedestal (Meese & Summers, 2012). It is unclear how these models can explain the effect of a noise mask.…”

“…It is suggested that such matched filters can be implemented as second-order filters 1 combining numbers of local filters in proportion to target size (Tyler & Chen, 2000). Some formulations thus simply use the square root of the number of local channels as the dominator of d 0 calculation (see Kingdom, Baldwin, & Schmidtmann, 2015). Such matched-filter behavior may also be called ''ideal summation'' behavior (Tanner & Jones, 1960;Tyler & Chen, 2000).…”

“…Another type of spatial summation is often termed ''probability summation'' (Pelli, 1985;Meese & Summers, 2012;Kingdom et al, 2015), which is governed by a selection or attention mechanism that identifies the channels with the maximum signal-to-noise ratio across the array of stimulated local channels (rather than summing their outputs linearly). It is thus better termed ''max-rule summation.''…”