**Abstract**:

**Abstract**
**Studies investigating the neural bases of cognitive phenomena increasingly employ multialternative detection tasks that seek to measure the ability to detect a target stimulus or changes in some target feature (e.g., orientation or direction of motion) that could occur at one of many locations. In such tasks, it is essential to distinguish the behavioral and neural correlates of enhanced perceptual sensitivity from those of increased bias for a particular location or choice (choice bias). However, making such a distinction is not possible with established approaches. We present a new signal detection model that decouples the behavioral effects of choice bias from those of perceptual sensitivity in multialternative (change) detection tasks. By formulating the perceptual decision in a multidimensional decision space, our model quantifies the respective contributions of bias and sensitivity to multialternative behavioral choices. With a combination of analytical and numerical approaches, we demonstrate an optimal, one-to-one mapping between model parameters and choice probabilities even for tasks involving arbitrarily large numbers of alternatives. We validated the model with published data from two ternary choice experiments: a target-detection experiment and a length-discrimination experiment. The results of this validation provided novel insights into perceptual processes (sensory noise and competitive interactions) that can accurately and parsimoniously account for observers' behavior in each task. The model will find important application in identifying and interpreting the effects of behavioral manipulations (e.g., cueing attention) or neural perturbations (e.g., stimulation or inactivation) in a variety of multialternative tasks of perception, attention, and decision-making.**

*change*detection tasks that require the observer to detect and report the location at which a

*change*occurred in a stimulus feature, such as a change in orientation from a standard value (Figure 1B), and (b) feature-based detection tasks that require the observer to detect and identify the occurrence of stimuli with particular features (e.g., colors, directions of motion, tones of a particular pitch). The former task has been commonly employed in studies of visual attention (Cavanaugh & Wurtz, 2004; Cohen & Maunsell, 2009; Ray & Maunsell, 2010). For brevity, we will refer to such tasks as

*m-ADC*tasks (the acronym stands for

*m*ulti

*a*lternative

*d*etection/

*c*hange-detection tasks).

*N*: No stimulus or

*S*: Stimulus present) based on noisy sensory evidence. The decision variable (Ψ) that encodes this sensory evidence is modeled as a Gaussian random variable with unit variance. The mean of the decision variable is specified as zero when no stimulus is presented and takes on a nonzero value,

*d*, when a non-null stimulus is presented (Figure 2A).

*d*, also termed the “perceptual sensitivity,” is determined by, and increases with the strength of the presented stimulus. In a given trial, the observer chooses

*S*if the decision variable exceeds a particular cutoff value, the “criterion” or

*c*; such a specification permits optimizing a variety of objective functions, including maximizing success (proportion correct) in such tasks (for a detailed discussion, see Green & Swets, 1966, section 1.7). The well-known 2 × 2 stimulus–response contingency table for this type of task is shown in Table S1A (Supplemental Data).

*d*and

*c*as, respectively,

*d̂*= Φ

^{−1}(

*HR*) − Φ

^{−1}(

*FA*) and

*ĉ*= −Φ

^{−1}(

*FA*) (where Φ

^{−1}represents the

*probit*function, the inverse cumulative distribution function associated with the standard normal distribution). As mentioned in the Introduction,

*c*is a measure of the observer's bias for reporting a Yes versus a No response.

*S*

_{1}, stimulus at location 1;

*S*

_{2}, stimulus at location 2; or

*N*, no stimulus at either location.

*N*versus

*S*

_{1}or

*S*

_{2}(Figure 2B, top left) as in a conventional Yes/No task. In the second stage, the observer decides whether a stimulus was presented at location 1 or location 2, i.e., between giving a Go response to location 1 versus 2 based on the relative strengths of evidence for the hypotheses

*S*

_{1}versus

*S*

_{2}(Figure 2B, top right) as in a conventional 2-AFC task. The two binary-choice one-dimensional models that capture this decision process are shown in Figure 2B (top).

*N*) trials, the observer is free to give Go responses (false alarms) to either location 1 or location 2 (Table S1B, last row). However, this model does not specify these response contingencies individually; rather, it only specifies the aggregate of the observer's false alarms (Go responses) to both locations during catch trials (Figure 2B, top left, hatched). Conversely, the observer may give different proportions of “miss” (NoGo) responses when stimuli are presented at location 1 versus at location 2 (Table S1B, last column). Again, this model does not specify responses to these contingencies individually but only specifies the aggregate of the observer's miss rates to stimuli presented at either location (Figure 2B, top left, gray shading).

*d*and

*c*) at each location (Figure 2B, bottom row; Yeshurun, Carrasco, & Maloney, 2008). This model does indeed specify, separately, the FAs (during catch trials) for each location (Figure 2B, bottom row, hatched) as well as individual miss rates for each stimulus event (Figure 2B, bottom row, gray shading). However, such a model is not sufficient to model behavior in this task. For example, the model specifies that, in each trial, the observer gives a Go response to a location at which the decision variable (Ψ

_{1}or Ψ

_{2}) exceeds the criterion. But what if the decision variables (Ψ

_{1}and Ψ

_{2}) were to exceed their respective criteria (

*c*

_{1}and

*c*

_{2}) at both locations in a particular trial? Go responses cannot be made to more than one location in a given trial. It is possible that, under these conditions, observers respond with a random “guess” at one of the two locations. Whatever the case, this model is insufficient, and a more elaborate framework is required, for instance, to model the observer's guessing strategy.

*m*> 2). We illustrate the model with a stimulus-detection task, such as the one shown in Figure 1A. However, the model is applicable, with a simple translation of the origin of the coordinate axes (see next), to change detection tasks, such as the one shown in Figure 1B. We describe the model verbally below and then provide an analytical formulation.

**Ψ**, whose components encode sensory evidence at each location,

*k*, along orthogonal decision variable axes (also called “perceptual dimensions”) in a two-dimensional decision space (also called a “perceptual space”; Figure 2C). When no stimulus is presented (catch trials), the distribution of

**Ψ**, given by the joint distribution of the two decision variable components, is centered at the origin with equal variance along each axis (Figure 2C, black; noise distribution). A stimulus presented at a particular location results in a “signal” distribution whose mean lies along the decision axis for that location (Figure 2C, red or blue). The value of this mean of the signal distribution at each location,

*k*, determined by the strength of the stimulus at that location and measured in units of noise standard deviation along the corresponding dimension, is defined as the perceptual sensitivity (

*d*).

_{k}*c*) for each location: In each trial, a response is made to the location at which the decision variable component exceeds the (respective) choice criterion. A difference in criteria between the two locations gives rise to a choice bias (relative preference) for one location over the other. If decision variable components at both locations exceed their respective criteria, the response is made to the location at which the difference between the decision variable component and the corresponding choice criterion was the greatest. If no decision variable component exceeds its respective criterion, the observer gives a NoGo response (Figure 2C, gray shaded region). In this model, the response probability for each stimulus event is the proportion (integral) of the corresponding joint distribution within the respective region. Thus, this two-dimensional 2-ADC model fully specifies each stimulus-response contingency for the 2-ADC task (Table S1B, Supplemental Data).

_{k}*structural model*of the observer's perceptual sensitivity for detecting the presented stimulus and a

*decision rule*that models the effect of choice bias on the observer's response. In the Discussion, we analyze the assumptions inherent in this formulation and discuss potential extensions.

*Y*:

*Y*=

*i*indicates that the observer chose to respond at location

*i*(Go response) whereas

*Y*= 0 indicates that the observer gave a NoGo response. Similarly, we denote the stimulus event with the variable

**X**whose components

*X*denote where the event occurred:

_{i}*X*= 1 indicates that a stimulus was presented at location

_{i}*i*. We further stipulate that no more than one stimulus be presented in a given trial, a common practice in psychophysics tasks of perception and attention (see Discussion). Thus, ||

*X*||

_{1}=

*for each of the two locations and specifies how these distributions change with each stimulus event: where Ψ*

_{i}*denotes the decision variable that encodes sensory evidence at location*

_{i}*i*(

*i*∈ {1, 2}),

*ε*is a random variable that represents the distribution of Ψ

_{i}*when*

_{i}*X*= 0, and

_{i}*d*is the perceptual sensitivity, an indicator of the strength of the perceived signal when a stimulus was presented at location

_{i}*i*(elaborated below).

_{1}and Ψ

_{2}when a stimulus was presented (stimulus trials, ||

*X*||

_{1}= 1) is termed a “signal” distribution whereas the joint distribution of the Ψ

*when no stimulus was presented (catch trials, ||*

_{i}*X*||

_{1}= 0) is termed the “noise” distribution; the latter distribution is identical with the joint distribution of the

*ε*. In line with conventional SDT for a binary choice stimulus-detection task, we assume that the noise distribution along each dimension is unit normal, i.e.,

_{i}*ε*εi ~ 𝒩(0,1). We note that the assumption of Gaussian distributions is not necessary for the model and the results presented here (except for the demonstration of model optimality).

_{i}*d*represents the change in the expected value of Ψ

_{i}*when a stimulus is presented at location*

_{i}*i*versus when no stimulus is presented; in other words,

*d*=

_{i}*E*(Ψ

*|*

_{i}*X*= 1) –

_{i}*E*(Ψ

*|*

_{i}*X*= 0).

_{i}*d*, measured in the units of noise standard deviation (unity in conventional SDT), determines the amount of the overlap (or lack thereof) between the “signal” distribution when a stimulus was present at location

_{i}*i*(

*X*= 1), and the noise distribution. Hence,

_{i}*d*is termed the perceptual sensitivity associated with detecting a stimulus at location

_{i}*i*and is determined by the strength of the stimulus at that location.

*(additively) without altering its variance or higher moments. Thus, the distribution of each Ψ*

_{i}*is Gaussian with unit variance. If the Ψ*

_{i}*distributions have unequal variances across the different locations, the 2-ADC structural model with unit normal distributions (Equation 1) can be readily recovered by scaling each Ψ*

_{i}*by its respective standard deviation.*

_{i}*can be considered a component of a bivariate random variable (*

_{i}**Ψ**) represented in a two-dimensional, Cartesian decision space (such as that shown in Figure 2C). Henceforth, in describing the model we will interchangeably refer to the Ψ

*as “decision variables” or “decision variable components” with the understanding that in either case these represent the univariate (scalar) component variables that constitute the bivariate (vector) decision variable (*

_{i}**Ψ**).

*i*when the value of the decision variable Ψ

*exceeds choice criterion*

_{i}*c*. If the values of Ψ

_{i}*exceed the criterion at both locations, then the observer responds to the location with the larger difference between the decision variable and the (respective) criterion value (larger Ψ*

_{i}*–*

_{i}*c*). On the other hand, if Ψ

_{i}*values fall below the choice criterion at every location, then the observer makes a NoGo response.*

_{i}*c*(Figure 2C) constitute an SDT measure of bias. The relative value of the criteria between locations indicates the magnitude of the bias: A lower choice criterion at a location corresponds to a greater choice bias for that location. The analytical formulation of the 2-ADC decision rule is, arguably, more complex than that of related models that incorporate NoGo responses (Ashby & Townsend, 1986; García-Pérez & Alcalá-Quintana, 2010). Consequently, the partitioning of decision space and the analytical treatment of bias represent fundamentally novel aspects of the 2-ADC model.

_{i}*p*(

*Y*=

*i*|

**X**) represents the conditional probability of a Go response to location

*i*(

*i*∈ {1, 2}) for each stimulus event

**X**;

*p*(

*Y*= 0|

**X**) represents the conditional probability of a NoGo response for each stimulus event; and

*ϕ*and Φ represent, respectively, the probability density and the cumulative distribution functions of the unit normal distribution.

*d*

_{1},

*d*

_{2},

*c*

_{1},

*c*

_{2}}) with two degrees of freedom to test the goodness of fit of the model.

*d*,

_{i}*c*},

_{i}*i*∈ {1, 2}). To facilitate representation, we examined a pair of two-dimensional subspaces by varying the criteria holding the sensitivities constant (parameter values in Table S2A, Supplemental Data) and vice versa. In line with conventional SDT, noise was assumed to be normally distributed with zero-mean and unit variance. The task specification requires that no more than one stimulus be presented in a given trial. This permits us to employ the following notational shorthand for the response probabilities:

*p*(

*Y*=

*i*|

*X*= 1) =

_{j}*c*at each location (

_{i}*i*∈ {1, 2}) on the response probabilities at a particular location, say, location 1. The following general trends are apparent from the figure: A higher choice criterion at a location

*i*(lower bias toward location

*i*) reduces the probability of response at that location (

*j*≠

*i*) regardless of where the stimulus was presented (

*i*,

*j*,

*k*∈ {1, 2}). Also apparent is the effect of sensitivity (

*d*) on response probabilities: Greater sensitivity to a stimulus at a location enhances the HR at that location (Figure S1B, red) and reduces the probability of a false alarm (incorrect response) at the opposite location (Figure S1B, blue).

_{i}*E*(Ψ

*|*

_{i}*X*= 0) = 0 so that

_{i}*d*is simply equal to

_{i}*E*(Ψ

*|*

_{i}*X*= 1), in a change-detection task

_{i}*E*(Ψ

*|*

_{i}*X*= 0) =

_{i}*i*so that

*d*=

_{i}*E*(Ψ

*|*

_{i}*X*= 1) –

_{i}*d*and

_{i}*c*must be measured with the origin of the coordinates at the center of the decision variable distribution for the standard stimulus (Figure 2C, black distribution). For simplicity of illustration, the formulation henceforth will be based solely on the stimulus-detection task (e.g., Figure 1A) with the understanding that the same logic and analogous equations are readily applied to change-detection tasks with an appropriate translation of the origin of the coordinate axes.

_{i}*d*is now defined as a function of stimulus strength: Stronger, more salient stimuli are more reliably detected because the respective signal distribution is further removed from the noise distribution (higher

*d*), resulting in less overlap between the signal and noise distributions. The

*psychophysical function*describes the variation of perceptual sensitivity,

*d*, with stimulus strength. Here, we relate the psychophysical function to the

*psychometric function*, which describes the variation in the observer's response proportions

*p*with stimulus strength in the m-ADC model.

*d*) with stimulus strength, we specify the m-ADC structural model as follows: where

*ξ*(

_{i}*i*∈ {1, 2, …

*m*}) represents the stimulus strength at location

*i*(e.g., contrast in Figure 1A or orientation change magnitude in Figure 1B), the psychophysical function

*d*(

_{i}*ξ*) describes variation of sensitivity at location

_{i}*i*with the stimulus strength at that location, and

*ε*~ 𝒩(0,1). For ease of illustration, we choose

_{i}*ξ*to represent the contrast of the stimulus. In this exemplar case, our theory relates response probabilities to the well-known psychophysical function of stimulus contrast.

_{i}*in the m-ADC model can be considered an independent component of a multivariate (random) decision variable (*

_{i}**Ψ**) represented in a multidimensional decision space. In addition, the assumption of orthogonality (independence) among the Ψ

*implies that the covariance matrix of this decision variable is a diagonal matrix.*

_{i}*c*, at each location

_{i}*i*that is independent of (does not vary with) stimulus strength. Such an assumption is plausible for task designs in which stimulus strength is varied pseudorandomly across trials so that the observer cannot adjust her/his criterion systematically based on foreknowledge of stimulus strength.

*ξ*

_{1},

*ξ*

_{2}, …

*ξ*) denotes a stimulus event with its

_{m}*i*

^{th}component representing the contrast of the stimulus presented at location

*i*,

*p*(

*Y*=

*i*|) represents the psychometric function, the conditional probability of Go responses to location

*i*for each stimulus event (), and

*p*(

*Y*= 0|) represents the psychometric function of a NoGo response for each stimulus event. For the m-ADC task, as for the 2-ADC task, we specify that the stimulus is presented at no more than one location in a given trial so that, at most, one

*ξ*is nonzero.

_{i}*n*

_{S}m^{2}+

*m*independent observations (assuming

*n*stimulus levels at each location) from which the

_{S}*m*+

*n*parameters, corresponding to the

_{S}m*m*criteria, and

*m*sensitivities for each of the

*n*stimulus strengths must be estimated. Even in the case of only a single stimulus level at each location (

_{S}*n*= 1), there are at least

_{S}*m*

^{2}–

*m*degrees of freedom to evaluate goodness of fit for all

*m*≥ 2.

*d*(

*ξ*)): Does the manipulation scale, shift, or change the slope of the psychophysical function (Herrmann, Montaser-Kouhsari, Carrasco, & Heeger, 2010; Lee & Maunsell, 2009; Reynolds & Heeger, 2009)? A parametric form of the psychophysical function, which provides an analytical relationship between sensitivity

*d*, and stimulus contrast,

*ξ*, facilitates such an analysis. Sigmoidal functions, such as the hyperbolic ratio function, as well as linear or power functions are all candidate psychophysical functions.

*d*(

*ξ*) =

*d*(

_{max}*ξ*) / (

^{n}*ξ*+ (

^{n}*ξ*

_{50})

*). The parameters of this function,*

^{n}*d*,

_{max}*ξ*

_{50}, and

*n*(which we call

*psychophysical parameters*) correspond to the asymptotic value, contrast at 50% of asymptotic value, and slope of the psychophysical function, respectively. Altering each parameter in turn scales (

*d*), shifts (

_{max}*ξ*

_{50}), or changes the slope (

*n*) of the psychophysical function.

*m*corresponding to the three psychophysical parameters and one criterion at each of the

*m*locations. Thus, the number of degrees of freedom are

*n*

_{S}m^{2}–

*n*where

_{p}m*n*is the number of parameters characterizing the psychophysical function (

_{p}*n*= 3 for the hyperbolic ratio function).

_{p}*p*(

*Y*=

*i*|

*ξ*

_{1}) (parameter values in Table S3A, Supplemental Data). For values of the parameters that do not saturate the response probabilities, the effect of varying the psychophysical parameters

*d*,

_{max}*ξ*

_{50}, and

*n*on the psychometric functions (Figure S2A through C) is similar to the effect of the respective parameter on the psychophysical function,

*d*(

*ξ*), viz. scaling, shift, and slope change (Figure S2A through C, insets). On the other hand, altering each response criterion (

*c*), which, by definition, has no impact on the psychophysical function, alters the psychometric function in complex ways: The effects include apparent scaling, shifting, and/or slope changes (Figure S2D through E). However, the proportion of responses increases (across all

_{i}*ξ*) with decreasing criterion at that location and with increasing criterion at the opposite location, consistent with the monotonic trends noted before (Figure S1A, Supplemental Data).

*m*> 2) and catch trials renders the m-ADC model multidimensional and raises several challenges. First, in this multidimensional SDT model, the system of Equation 6 is not readily inverted (analytically) to yield model parameters. Thus, given a set of experimentally observed m-ADC response probabilities (e.g., contingency table, Table S1B), is it possible to obtain estimates of the underlying perceptual sensitivities and choice criteria that generate these response probabilities? Second, can one guarantee model identifiability so that a given set of response probabilities can be produced by only one set of parameters in the model? Finally, can one show that the specification of independent criteria at each location (linear, intersecting decision surfaces; Figure 2C) constitutes an optimal decision rule? We addressed the first of these challenges (parameter estimation) by developing and extending numerical approaches noted in a recent study (DeCarlo, 2012), described next. We addressed the remaining two challenges (demonstrating model identifiability and optimality) with analytical approaches, described subsequently.

*c*

_{1}<

*c*

_{2}).

*c*=

_{i}*c*), we plotted the reconstructed psychometric functions both separately for each location as well as with the responses pooled (as “correct” vs. “incorrect”) across locations. When data were plotted separately for each location, the reconstructed psychometric functions (Figure 3C and D, dashed curves) deviated systematically from the original data (Figure 3C and D, circles). In addition, the model systematically overestimated the psychophysical function at location 1, the location of greater bias (Figure S4B, dashed red curve), and systematically underestimated it at the other location (Figure S4B, dashed blue curve). On the other hand, when data were pooled across locations as the proportion of correct (hit) and incorrect (misidentification) responses, the reconstructed psychometric functions closely fit the data (Figure 3E). The significance of these observations is discussed later (see Discussion).

*θ*(sensitivities and criteria) that produce a set of response probabilities

*θ** that also produces the same probabilities? In the previous section, we demonstrated that, for various initial values of parameter guesses, numerical approaches reliably recover an identical set of 2-ADC model parameters, suggesting that the 2-ADC model is identifiable. However, the model contains nonlinear integral equations, and we must entertain the possibility that multiple parameter configurations may be consistent with a given set of response probabilities, especially in the m-alternative case (m > 2).

*d*and 2

_{i}*c*). We depict this four-dimensional likelihood function in a pair of two-dimensional subspaces by holding

_{i}*c*constant and varying

_{i}*d*or vice versa (Figure 4A and B, parameter values in Table S2A, Supplemental Data). In the domain of parameter values shown in the figure, the likelihood function appears to be concave (Figure 4A through D) indicating a single, global minimum corresponding to a unique set of underlying parameters. However, demonstrating, analytically, the concavity of the likelihood function appears to be intractable even for the 2-ADC model (see Appendix C, Supplemental Data).

_{i}*i*∈ {0, 1, 2} are produced by exactly one pair of criterion values (

*c*

_{1},

*c*

_{2}). Next, we demonstrate that a given set of response probabilities during stimulus trials

*i*,

*j*∈ {1, 2} are produced by no more than one set of sensitivity values (

*d*

_{1},

*d*

_{2}). We develop a geometric intuition by varying the criterion and sensitivity parameters and examining the effects on the probabilities of each response (Figure 5A through C).

*c*

_{1},

*c*

_{2}) (Figure 5A, solid lines) during catch trials (note that in catch trials,

*d*

_{1}=

*d*

_{2}= 0, by definition). These response probabilities correspond to the area under the joint distribution of Ψ

_{1}and Ψ

_{2}in each of the three (shaded) regions (Figure 5A). Let us assume that another set of (distinct) criteria (

*c*

_{1}and

*c*

_{2}.

*c*

_{1}. As is apparent from Figure 5A and B, the smaller criterion at location 1 would result in a smaller NoGo response probability (

*c*

_{2}. However, a smaller criterion at location 1 and a larger criterion at location 2 would result in an increase in the response probability to location 1 (

*c*

_{1}, would result in the opposite scenario: to maintain the probability of a NoGo response,

*c*

_{2}, resulting in a decrease in the response probability to location 1 and an increase to location 2. Thus, the only way the alternate set of criteria could produce the same response probabilities is if the two sets of criteria were identical, i.e.,

*c*

_{1}and

*c*

_{2}.

*d*

_{1}=

*d*

_{2}= 0). The sets of all possible pairs of choice criteria that could determine the probability of each type of response during catch trials (locus of variation of

*c*

_{1}and

*c*

_{2}for specific values of

*i*= 1, red;

*i*= 2, blue;

*i*= 0, green). Note that the three contours intersect at exactly one point in the

*c*

_{1}–

*c*

_{2}plane (Figure 4F), indicating that exactly one pair of criteria is consistent with these response probabilities.

*d*

_{1},

*d*

_{2}).

_{1}-axis). Note that the two distributions cross over at the point Ψ

_{1}=

*c*

_{1}(dashed vertical line). In a given trial, let the decision variable take some value

**Ψ**

*. In order to maximize success, it is intuitively clear that the optimal (Bayesian) strategy would be to report the event corresponding to the distribution that is most likely (greatest likelihood) to have produced this value*

_{t}**Ψ**

*. The ideal observer would choose to report a stimulus at location 1 (Go response) if the component of*

_{t}**Ψ**,

^{t}*c*

_{1}because the marginal probability for the stimulus event at location 1 (p(Ψ

_{1}| Stim 1)) exceeds that for the catch event (p(Ψ

_{1}| Catch)) for all Ψ

_{1}>

*c*

_{1}. Similarly, the observer would choose to report a catch event (NoGo response) if

*c*

_{1}. Thus, the linear decision boundary Ψ

_{1}=

*c*

_{1}(dashed, thick vertical line) forms an optimal decision surface for distinguishing between these two events.

_{2}=

*c*

_{2}) and a stimulus at location 1 versus at location 2 (dashed, thick oblique line, Ψ

_{1}– Ψ

_{2}=

*C*), where

*C*is a constant. The value of

*C*can be determined by noting that the three decision boundaries intersect at a point (see Appendix D.2 for proof). Thus,

*C*=

*c*

_{1}–

*c*

_{2}, and the decision boundary is given by Ψ

_{1}– Ψ

_{2}=

*c*

_{1}–

*c*

_{2}or (with a slight rearrangement) Ψ

_{1}–

*c*

_{1}= Ψ

_{2}–

*c*

_{2}.

_{1}to

*c*

_{1}(dashed, thick vertical line), Ψ

_{2}to

*c*

_{2}(dashed, thick horizontal line), and Ψ

_{1}–

*c*

_{1}to Ψ

_{2}–

*c*

_{2}(dashed, thick oblique line). These are identical with the decision boundaries specified in the 2-ADC model.

_{1=}

*indecision*model is a ternary choice model that has been widely applied to explain behaviors in two-alternative nonforced choice (2-ANFC) tasks that incorporate NoGo (or “undecided”) responses and, optionally, catch trials (García-Pérez & Alcalá-Quintana, 2010, 2013).

*c*

_{1},

*c*

_{2}), one for detection in each temporal interval, and one sensitivity parameter (

*d*). We assumed that detection sensitivities for the two intervals were identical. We also fit the data with the indecision model (which also assumes equal detection sensitivities, as in the original study) with the following three parameters (Figure 6A): the sensitivity (

*μ*) of detection during either target interval, a criterion (

*δ*) that delineates an indifference zone such that the observer indicates a guess response if –

*δ*≤ Ψ

_{2}– Ψ

_{1}≤

*δ*, and a finger error term (

*λ*). The finger-error term models unintentional response (motor) errors (“hitting an unintended response key by mistake,” García-Pérez & Alcalá-Quintana, 2010, p. 880). The indecision model in the original study assumed a noise standard deviation of

*p*value = 0.60, randomization test; the model failed for observers #4 and #15). On the other hand, the three-parameter indecision model fit performance for 14 of the 18 observers (the model failed for observers #4, #6, #11, and #13), replicating the findings of the original study. The goodness-of-fit G-statistic distribution across observers was not significantly different between the two models (median ± std: 1.67 ± 1.58 for the 2-ADC model and 1.54 ± 1.53 for the indecision model;

*p*= 0.31, Wilcoxon signed rank test,

*n*= 17, excluding observer #4's data, which were not well fit by either model). Finally, the estimates of sensitivity, and its standard error, derived from the 2-ADC model were similar to those derived from the indecision model for each observer (Figure 6B;

*p*= 0.80, paired Wilcoxon signed rank test).

*μ*) for the two intervals, the criteria for the two intervals must have been unequal in order to explain this differential pattern of guess (or NoGo) responses. Hence, we extended the indecision model to incorporate different criteria

*δ*

_{1}and

*δ*

_{2}for each interval (Figure 6C); a similar extension has been proposed recently (García-Pérez & Alcalá-Quintana, 2013). Such an indecision model “with bias” is described by four parameters:

*μ*,

*δ*

_{1},

*δ*

_{2},

*λ*.

Observer #1 | Observer #2 | |||||

Model | 2-ADC | 2-ADCX | Indecision | 2-ADC | 2-ADCX | Indecision |

# parameters | 4 | 5 | 5 | 4 | 5 | 5 |

β (horizontal)_{s} | 0.482 | 0.399 | 0.554 | 0.348 | 0.305 | 0.388 |

β (vertical)_{t} | 0.489 | 0.404 | 0.561 | 0.366 | 0.321 | 0.409 |

PSE (mm) | 102.62 | 102.64 | 102.61 | 98.74 | 98.82 | 98.71 |

c or _{A}δ_{A} | 1.307 | 1.218 | 1.272 | 1.198 | 1.146 | 1.189 |

c or _{B}δ_{B} | 0.961 | 0.866 | 0.608 | 0.671 | 0.611 | 0.155 |

α | n/a | −0.385 | n/a | n/a | −0.227 | n/a |

λ | n/a | n/a | 0.007 | n/a | n/a | 0.012 |

AICc | 10,676 | 10,649 | 10,646 | 11,010 | 11,001 | 11,002 |

BIC | 10,697 | 10,676 | 10,672 | 11,031 | 11,028 | 11,029 |

ΔAICc_{indecision} | 30 | 3 | 0 | 8 | −1 | 0 |

ΔBIC_{indecision} | 25 | 4 | 0 | 2 | −1 | 0 |

*δ*

_{1}–

*δ*

_{2}) closely correlated with 2-ADC model estimates (

*c*

_{1}–

*c*

_{2}) for each observer (Figure 6D; correlation

*R*

^{2}= 0.89,

*p*< 0.001; as before, data from observer #4 were excluded from this, and subsequent, analyses).

*K*) units for one model represent an exponential increase in the relative likelihood (

*e*

^{K/}^{2}) of that model.

_{2-ADC−indecision}= 1) was not significantly different from zero (

*p*= 0.18, paired test). On the other hand, BIC scores were significantly lower for the 2-ADC model compared to the indecision model (ΔBIC

_{2-ADC−indecision}= −3,

*p*= 0.016, paired test), favoring the 2-ADC model over the indecision model for explaining these data.

*d*,

*c*

_{1},

*c*

_{2}vs.

*μ*,

*δ*

_{1},

*δ*

_{2}), the indecision model contains an extra parameter to model finger errors,

*λ. λ*has been referred to as a “non-sensory” parameter (García-Pérez & Alcalá-Quintana, 2010; their supporting information, p. 4) that, as mentioned previously, is thought to reflect inadvertent motor errors when reporting responses. The inclusion of this parameter in the model is usually justified, not by its relevance for explaining perceptual confusion but for controlling for motor/finger errors in order to obtain more accurate estimates of the sensitivity and criterion parameters (García-Pérez & Alcalá-Quintana, 2010, 2013).

*μ*,

*δ*

_{1},

*δ*

_{2}) with the fit of the three-parameter 2-ADC model (

*d*,

*c*

_{1},

*c*

_{2}). Because the number of parameters were identical in the two models, any differences in AICc or BIC scores must reflect, specifically, differences in goodness of fit. As finger errors were estimated to be less than ∼5% for most of the observers (García-Pérez & Alcalá-Quintana, 2010, their table 1), we expected to obtain marginally poorer fits and perhaps slightly different values for the sensitivities and criteria when this parameter was excluded from the model.

*p*value = 0.001, randomization test). Sensitivity and bias parameter estimates from the indecision model differed significantly depending on whether finger errors were or were not included in the fit (Figure 6E and F, gray data;

*p*< 0.001, Wilcoxon signed rank test). In particular, sensitivity was systematically underestimated across subjects when the finger-error parameter was excluded. In contrast, parameter estimates from the 2-ADC model were virtually identical regardless of whether the finger-error parameter was included or not (Figure 6E and F, white data;

*p*> 0.9).

_{2-ADC−indecision}= −13; ΔBIC

_{2-ADC−indecision}= −13;

*p*< 0.001, paired Wilcoxon signed rank test). Conversely, incorporating a finger-error parameter into the 2-ADC model did not improve the goodness of fit (increase less than 0.05%):

*λ*estimates were vanishingly small (less than 0.1%) across nearly all (16/18) observers and were less than 2% for the other two observers.

*λ*(Discussion).

*d*and

_{i}*c*, respectively, whereas the analogous parameters in the indecision model are referred to as

_{i}*μ*and

_{i}*δ*, respectively.

_{i}*c*and

_{A}*c*(in the conventional 2-ADC model, the NoGo response domain is unbounded on two sides). Such a modification was necessary for this task because the observer must not only indicate whether the test stimulus is of a different length from the standard, but must also indicate whether it is longer or shorter. Thus, the decision rule for the NoGo response is –

_{B}*c*≤ Ψ

_{B}*≤*

_{A}*c*∩ –

_{A}*c*≤ Ψ

_{A}*≤*

_{B}*c*, where Ψ

_{B}*and Ψ*

_{A}*denote the decision variable for each location, above and below, respectively (Figure 7A). Note that this decision rule is different from that of the conventional detection model (which would be Ψ*

_{B}*≤*

_{A}*c*∩ Ψ

_{A}*≤*

_{B}*c*) and permits modeling data from two-alternative discrimination tasks that incorporate a NoGo response. The model equations relating sensitivity and criteria to response probabilities are derived in Appendix E (Supplemental Data).

_{B}*d*(

_{z}*x*) =

*β*for the 2-ADC model, or

_{z}x*μ*(

_{z}*x*) =

*β*for the indecision model (

_{z}x*z*= {

*s*,

*t*}), where

*s*and

*t*refer to the standard (horizontal) and test (vertical) stimuli, respectively (García-Pérez & Alcalá-Quintana, 2013). Due to the linearity of the psychophysical function for both test and standard stimuli, the point of subjective equality (PSE) of the test to the standard was calculated as

*PSE*=

*β*/

_{s}x_{s}*β*, where

_{t}*x*is the length of the standard stimulus (104 pixels). Thus, the 2-ADC model incorporated four parameters (

_{s}*β*,

_{s}*β*,

_{t}*c*,

_{A}*c*) whereas the indecision model incorporated five parameters (

_{B}*β*,

_{s}*β*,

_{t}*δ*,

_{A}*δ*,

_{B}*λ*),

*λ*being the finger-error term discussed previously.

*β*>

_{t}*β*, the vertical-horizontal illusion), in line with results from the indecision model. Although the estimated values of

_{s}*β*and

_{s}*β*were different between models, their relative magnitudes (ratios) and, hence, the PSE for each observer were nearly identical across models. In addition, both models indicated that the observers exhibited a bias for reporting the vertical line as longer when it was below compared to when it was above the horizontal line (

_{t}*c*<

_{B}*c*;

_{A}*δ*<

_{B}*δ*). Despite these similarities, the indecision model fared substantially better than the 2-ADC model in fitting data for both observers (Figure 7B, dashed lines vs. solid lines). In addition, AICc and BIC scores were markedly lower for the indecision model; the differences in AICc or BIC scores relative to the indecision model were much larger for observer #1 compared to observer #2 (Table 1). What might explain the poorer fits of the 2-ADC model to these data?

_{A}*α*as shown in Figure 7C (model equations derived in Appendix E, Supplemental Data). In the estimation process, we did not specify the sign of

*α*so that a positive value would indicate a facilitatory interaction whereas a negative value would indicate a competitive interaction between test and standard stimuli. We refer to this 2-ADC model, which incorporates an interaction term, as the

*2-ADCX*model. This model, like the indecision model, incorporates five parameters (

*β*,

_{s}*β*,

_{t}*c*,

_{A}*c*,

_{B}*α*).

*α*< 0) in this task. Such interactions could not be readily identified with the indecision model (Discussion). In the Discussion, we elaborate on the relative advantages of each model for analyzing behavior in nonforced choice detection and discrimination tasks.

*− s are not independent, and perceptual sensitivities do not vary along orthogonal dimensions: Signal covariation may arise from facilitative or competitive interactions that operate across locations. Thus, decision variable distributions at different locations could be correlated, or, equivalently, decision variable axes could be separated by angles different from 90°. In this case, the covariance matrix of*

_{i}**Ψ**is no longer diagonal. The 2-ADCX model (Figure 7C) incorporates such interactions for the two-alternative task. This model could be extended to the multialternative case as well.

*– Ψ*

_{i}*=*

_{j}*c*–

_{i}*c*) is optimal only if the values of sensitivity (

_{j}*d*) are identical across locations (

*d*=

_{i}*d*∀

*, Figure 8A, left). In certain experiments, such as when a particular spatial location is cued for attention, it is possible that the sensitivities at different locations (e.g., cued vs. uncued) could be significantly different. The model may then be extended with a modified decision rule to capture optimal decision-making in this more general scenario of unequal sensitivities (Figure 8A, right).*

_{i}*X*,

_{i}X_{j}*X*, …) in the structural model vanish automatically (as at least one

_{i}X_{j}X_{k}*X*= 0). Tasks that violate this requirement and incorporate compound stimuli (e.g., stimuli presented at more than one location or more than one stimulus feature presented in a given trial) fall under the purview of the General Recognition Theory (GRT, Ashby, 1992) framework (discussed next).

_{i}*most apparent*at the weaker stimulus strengths (deviation between data and bias-free model for low

*ξ*). At higher strengths, the psychometric functions of models with and without bias approach each other closely (e.g., Figure 3C and D solid vs. dashed curves at

*ξ*> 0.5). Thus, it may be difficult, in practice, to identify the occurrence of choice bias with stimuli of high strengths alone. Catch trials, which are essentially zero stimulus strength trials, provide an elegant and efficient means to identify and account for choice bias in such multialternative tasks. In addition, the inclusion of catch trials reduces the standard error of parameter estimates (García-Pérez & Alcalá-Quintana, 2011b).

*= Ψ*

_{S}_{1}– Ψ

_{2}, Figure 8C). Indeed, previous studies have exclusively employed such a one-dimensional formulation of the indecision model (e.g., García-Pérez & Alcalá-Quintana, 2010, 2011b, 2013), and the same one-dimensional formulation was employed in our analyses that replicated the results of previous studies (Figures 6 and 7). Our m-ADC model is the first and only model, to our knowledge, that can be applied to data from unforced choice tasks with any number (three or more) of alternatives, based on a multidimensional formulation.

*λ*as “nonsensory.”

*λ*has been described as being “irrelevant” to the core indecision model (García-Pérez & Alcalá-Quintana, 2010, their supporting information, p. 4) but is useful for obtaining more accurate estimates of the other model parameters (García-Pérez & Alcalá-Quintana, 2010). Yet we found that behavioral performance for more than two thirds of the observers could not be adequately fit without including this parameter in the indecision model. In contrast, a finger-error term was not necessary for the 2-ADC model: Incorporating this parameter did not result in any improvements in model fits, and the term was uniformly truncated to vanishingly small values for most observers in both tasks.

*without*finger errors fit the data as well as or, often, even marginally better than the indecision model

*with*finger errors, either explanation is possible, and the explanations are not mutually exclusive. To resolve this issue, it would be worthwhile to obtain an independent measure of finger errors from each observer (e.g., with a postexperiment questionnaire) in future experiments. Additional independent evidence (e.g., with neural recordings) could also help resolve which factor contributed predominantly to these erroneous responses.

*β*(vice versa for facilitative interactions) as can be inferred from Figure 7C. Indeed we noticed that estimates of

_{t}*β*were consistently higher for the indecision model compared to the 2-ADC or 2-ADCX models (Table 1). This result highlights a key advantage of the two-dimensional formulation of the 2-ADC model: It readily enables modeling interactions that occur among the decision variable components.

_{t}*Y*=

*i*) for each stimulus event (

**X**) can be derived from the structural model (Equation 1) and decision rule (Equation 2). We illustrate the case for

*p*(

*Y*= 1|

**X**). The other cases may be similarly derived.

*f*

_{1}=

*f*

_{2}=

*ϕ*and

*F*

_{1}=

*F*

_{2}= Φ where

*ϕ*and Φ are respectively the probability density and cumulative distribution functions of the unit normal distribution.

*p*(

*Y*= 0|

**X**), can be calculated by observing that the NoGo decision region in Figure 2C is simply a quadrant of two-dimensional decision space. Because Ψ

_{1}and Ψ

_{2}are independent, this can be readily shown to be

*p*(

*Y*= 0|

**X**) = 1 –

*p*(

*Y*= 1|

**X**) –

*p*(

*Y*= 2|

**X**).

*e*

_{1}and

*e*

_{2}with the variable

*e*because

*e*

_{1}and

*e*

_{2}are simply dummy variables of integration.

*i*(probabilities of response at location

*Y*=

*i*, as a function of stimulus strength

*ξ*at each location

_{k}*k*), based on the structural model (Equation 4) and decision rule (Equation 5) in the m-ADC model:

*ε*=

_{i}*e*. where 𝓗(

_{i}*x*) is the Heaviside function, and

*F*represents the cumulative distribution function of the decision variable distribution at location

_{k}*k*, Ψ

*. In deriving this expression, we have used the fact that the Ψ*

_{k}*distributions are mutually independent, such that their joint probability density factors into the product of the individual densities.*

_{k}*p*(

*Y*= 0|), can be calculated by observing that the NoGo decision region in the m-ADC model is simply a orthant (2

*-tant) in m-dimensional decision space. Again, based on the independence of the Ψ*

^{m}*-s this can be shown to be*

_{i}*d*(

_{j}*ξ*)) and criterion

_{j}*c*at each location,

_{j}*j*(reproduced in the results as Equation system 6).

*d*,

_{i}*c*},

_{i}*i*∈ {1, 2}) for various initial guesses (Table S2C, Supplemental Data). In these figures, the search trajectory in four-dimensional parameter space is depicted as two two-dimensional trajectories, one for each pair of criterion and sensitivity parameters.

*burn-in*period (about 500 iterations, Figure S3C) to converge to a stable parameter set; the chi-square error value reduced and the log-likelihood value increased systematically over successive iterations (Figure S3D). The posterior distribution was generated with the parameter values from the last 1,000 iterations, well after the burn-in period of the MCMC algorithm (Figure S3E, Methods). Error estimates of the parameters were also highly similar between the two estimation approaches (Table S2C).

*ξ*∈ [0, 100]) with 50% catch trials and 25% stimulus trials at each of the two locations; this process was repeated for 100 simulated experimental blocks (1,000 trials per contrast value for each simulation). As before, we denote these by

_{k}*ξ*) (Figure 3C and D circles, error bars denote standard deviations across simulation blocks), corresponding to the probability of response at location

_{k}*r*when a stimulus is presented at location

*s*with contrast

*ξ*(

_{k}*k*= 1–6).

*quadgk*function in Matlab) in order to evaluate these integrals.

*fminunc*, in Matlab's Optimization Toolbox). The optimization algorithm also yields a numerical approximation to the Hessian matrix. Standard errors based on ML-LS estimation were derived as the square root of the diagonal elements of the inverse of this Hessian matrix.

*r*for a stimulus at location

*s*). In the following,

*N*denotes the total number of trials for each stimulus event

_{s}*s*, and the symbol

**d**

*is used as a general notation either for sensitivity*

_{i}*d*when estimation was performed at a single value of stimulus strength or for the collection of psychophysical parameters (

_{i}*d*,

_{max}*n*,

*ξ*

_{50})

*when estimation was performed with the entire psychometric function.*

_{i}*r*denotes a reference set). Designate this as the reference parameter set. Determine response probabilities from Equation system 3 based on this set. We denote these probabilities by

*, assuming that responses*

^{r}*N*,

_{s}*based on the new guess. (e) Compute a likelihood ratio based on the older and newer guesses: 𝓛*

^{n}*= 𝓛*

_{R}*/ 𝓛*

^{n}*. (f) Accept the new guess for the parameters with a probability*

^{r}*a*, which depends on the magnitude of the likelihood ratio,

*a*= min(𝓛

*,1). Once accepted, the new set of parameters becomes the reference set, and the likelihood value based on the last set of accepted parameters is used as the reference value (𝓛*

_{R}*). (g) Repeat steps (c) through (f) until convergence.*

^{r}*σ*= 0.02 in each dimension). The MCMC simulation proceeded until the algorithm converged on a specific set of parameters

**d**

*,*

_{i}*c*,

_{i}*i*∈ {1, 2} in four-dimensional space. The algorithm was determined to have converged when the value of 𝓛 and the chi-square error function changed by less than 2% over at least 100 consecutive iterations. The burn-in period was generally achieved within about 500 iterations (e.g., Figure S3C and D). Posterior distributions were computed based on parameter values between iterations 1,000 and 2,000. Standard errors for the parameters and 95% credible intervals reported (Table S2C, Supplemental Data) were based on the standard deviation and the [2.5–97.5] percentile of the posterior distributions.

*d*,

_{i}*c*} were permitted to take both positive and negative values (unconstrained optimization); no constraint was placed on their sign or magnitude. However, negative values of sensitivity parameters (

_{i}*d*) lack physical meaning. We repeated the estimation by constraining sensitivity parameters to take only positive values (with the constrained optimization function

_{i}*fmincon*in Matlab or with a custom-implemented MCMC Metropolis-Hastings algorithm); this analysis yielded sensitivity estimates that matched those obtained with the unconstrained optimization approaches.

**Ψ**= [Ψ

_{1},Ψ

_{2},…Ψ

*] with a diagonal (identity) covariance matrix. The equation of such a multivariate Gaussian variable*

_{m}**Ψ**with mean () = [

*d*(

*ξ*

_{1}),

*d*(

*ξ*

_{2}), …

*d*(

*ξ*)] and covariance matrix = (

_{m}*= 1,*

_{ii}*= 0,*

_{ij}*i*,

*j*∈ {1, …,

*m*}, i ≠

*j*), can be written as where 𝒩

*is the m-dimensional Gaussian density function, and*

_{m}*A*is a normalization constant in order for 𝒩

*to be a probability density (*

_{m}*A*= 1/

*d*and posit that the psychophysical function is the same at all locations, although the results hold even without this assumption.

_{i}*j*with strength

*ξ*versus no stimulus is denoted by

_{j}*p*/

_{ξj}*p*

_{0}=

*p*(

*ξ*= 1,

_{j}*ξ*= 0 ∀

_{k}*k*≠

*j*)/

*p*(||||

_{1}= 0)). Thus, the log-posterior odds is obtained by adding log(

*p*

_{0}) to the log-likelihood ratio.

*ξ*at location

_{j}*j*versus no stimulus are surfaces of constant Λ

_{j}_{0}(see Appendix D.1, Supplemental Data):

*. The specification of a cutoff criterion at Ψ*

_{j}*=*

_{j}*c*, as in the m-ADC model, corresponds to the observer employing a decision boundary from among this family of optimal decision surfaces. The precise choice of

_{j}*c*would depend on the cost/utility of choosing each alternative (

_{j}*β*

_{j}_{0}, see Appendix D.1) and the prior odds ratio as well as the perceptual sensitivity to that stimulus (

*d*(

*ξ*)). Specifically, when the prior odds, relative costs, and stimulus strength at each location remain constant across trials, the optimal value of

_{j}*c*also remains constant across trials.

_{j}*i*versus a stimulus at location

*j*are surfaces of constant Λ

*(see Appendix D.1, Supplemental Data):*

_{ij}*(*

_{i}d*ξ*) – Ψ

_{i}*(*

_{j}d*ξ*) =

_{j}*B*.

_{ij}*, Ψ*

_{i}*) = (*

_{j}*c*,

_{i}*c*). Hence, the constant

_{j}*B*=

_{ij}*c*(

_{i}d*ξ*) –

_{i}*c*(

_{j}d*ξ*) and the optimal decision hyperplane are given by Ψ

_{j}*(*

_{i}d*ξ*) – Ψ

_{i}*(*

_{j}d*ξ*) =

_{j}*c*(

_{i}d*ξ*) –

_{i}*c*(

_{j}d*ξ*).

_{j}*d*(

*ξ*) =

_{i}*d*(

*ξ*) =

_{j}*d*, i.e., the perceptual sensitivities at the two locations are equal (and constant), these decision surfaces are planes of constant Ψ

*– Ψ*

_{i}*=*

_{j}*c*–

_{i}*c*. Thus, in this case, the decision surfaces in the m-ADC model (constant Ψ

_{j}*– Ψ*

_{i}*) belong to the family of optimal decision surfaces for detecting a stimulus at location*

_{j}*i*versus at location

*j*.

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