ABSTRACT
The contribution of sensory and decisional processes to perceptual decision making is still unclear, even in simple perceptual tasks. When a decision maker needs to select an action from a set of balanced alternatives, any tendency to choose one alternative more often− choice bias−is consistent with a bias in the sensory evidence, but also with a preference to select that alternative independently of the sensory evidence. To decouple sensory from decisional biases, here we asked humans to perform a simple perceptual discrimination task with two symmetric alternatives under two different task instructions. The instructions varied the response mapping between perception and the category of the alternatives. In two experiments, we found that 30 out of 32 participants exhibited sensory biases and 15 decisional biases. The decisional biases were consistent with a criterion change in a simple signal detection theory model. Perceptual decision making, thus, even in simple scenarios, is affected by sensory and decisional choice biases.
INTRODUCTION
You ask a friend about the tilt of a canvas that is perfectly horizontal and she says that the top right corner is up. What causes her inaccuracy? One possibility is that her sensory representation is biased and she perceives the canvas tilted. Another possibility, however, is that under uncertainty about the orientation of the canvas, she prefers to choose up over down. This situation exemplifies a major problem in the study of perception: perceptual decisions not only depend on the sensory evidence, but also on decisional components (Green and Swets 1966; Gold and Ding 2013).
When a decision maker needs to select an action from a set of alternatives, the tendency to choose one alternative over the others is known as choice bias (Gold and Ding 2013). Choice biases occur in perceptual tasks even in simple scenarios like the one described above, in which the stimuli carry similar levels of signal relative to a neutral point (Newsome and Paré 1988; Mareschal and Clifford 2012; Jazayeri and Movshon 2007; Milner, Brechmann, and Pagliarini 1992; Tadin et al. 2003). Choice biases are extensively studied in relation to how current decisions are influenced by previous decisions, what is known as choice history biases (Abrahamyan et al. 2016; Akaishi et al. 2014; Fründ, Wichmann, and Macke 2014; Urai, Braun, and Donner 2017; Fischer and Whitney 2014; Braun, Urai, and Donner 2018; St John-Saaltink et al. 2016; Fritsche, Mostert, and de Lange 2017). Depending on the perceptual scenario, people tend to repeat their previous choice (Akaishi et al. 2014; Braun, Urai, and Donner 2018), alternate between choices (Fritsche, Mostert, and de Lange 2017) or idiosyncratically repeat or alternate (Urai, Braun, and Donner 2017; Abrahamyan et al. 2016). Whether choice history biases reflect a bias in the sensory evidence or in the decision process is under debate (Akaishi et al. 2014; Fischer and Whitney 2014; St John-Saaltink et al. 2016; Fritsche, Mostert, and de Lange 2017).
Another form of choice bias refers to the overall tendency to choose one alternative over the others−global choice biases. The existence of these biases is acknowledged (Gold and Ding 2013; Kingdom and Prins 2016; Michael J. Morgan et al. 2012; García-Pérez and Alcalá-Quintana 2013; Peters, Ro, and Lau 2016) and they are included in current models of perceptual decision making (Abrahamyan et al. 2016; Akaishi et al. 2014; Urai, Braun, and Donner 2017; Braun, Urai, and Donner 2018), but whether they reflect sensory or decisional processes is unknown. The problem to identify their origin is that, in standard perceptual paradigms, the tendency to choose one alternative more often is consistent with a biased sensory representation, but also with a bias in the decision process (García-Pérez and Alcalá-Quintana 2013; Gold and Ding 2013). Here, to disentangle the contribution of sensory and decisional processes to global choice bias, we measured choice behavior in a simple common perceptual discrimination task with two symmetric alternatives, but under two different task instructions. The instructions varied the response mapping between perception and the category of the alternatives.
RESULTS
In Experiment 1, the orientation of a grating centered on a fixation point was chosen randomly on each trial from a range centered around the horizontal orientation. Seventeen participants judged whether the grating was pointing down or up (by pressing the down or up arrow keys on a keyboard) relative to a reference that we asked participants to imagine placed on the right at the same height of the fixation point (Figure 1A). Down and up choices correspond to clockwise and counterclockwise orientations relative to horizontal.
We first describe the results of four representative participants that illustrate the four types of choice behaviors that we found (Figure 1C-F). The circles in Figure 1C-F show the probability of clockwise responses (pressing the down key) as a function of the orientation of the grating and the curves show the psychometric fits (see Models; figure supplement 1 shows the results for all participants). Participant 12 (Figure 1C) did not show asymmetries in choice behavior−no global choice bias was present (to see how we assessed that the bias was not significant, see Materials and Methods). Participant 2 (Figure 1D) and 4 (Figure 1F) had a tendency to give a response consistent with clockwise orientation (pressing the down key)−that shifts the psychometric curve leftwards− and participant 9 (Figure 1F) had a tendency to give a response consistent with counterclockwise orientation (pressing the up key)−that shifts the psychometric curve rightwards.
The biases could reflect that the perceived horizontality of the grating corresponds to different orientations for different participants (sensory bias) or that under uncertainty participants had a tendency to select either the down or up alternative (decisional bias). It could also be that biases include a sensory and a decisional component.
To decouple these possibilities, in other trials intermixed with the trials just described we presented the same stimuli but asked participants whether the grating was pointing down or up relative to a reference that now we asked them to imagine on the left (Figure 1B). This variation in the instructions was easy for the participants to understand and effectively reversed the mapping between perception and the category of the alternatives. We did not display the reference to keep the stimuli identical across locations of the reference. For the left reference, down choices correspond to counterclockwise orientations relative to horizontal and up choices to clockwise. If the bias is of sensory origin, the probability of choosing an alternative consistent with clockwise orientation (triangles in Figure C-F) should not depend on where the reference was imagined (right or left). This is the choice behavior that participant 2 exhibited (Figure 1D). Notice that the similar pattern of clockwise responses for the two locations of the reference indicates that the participant reversed the frequency of pressing the down and up keys. If the bias is decisional, the probability of choosing an alternative consistent with clockwise orientation for the left reference should be shifted symmetrically relative to 0 deg (see Models). This is the choice behavior of participant 9 (Figure 1E). In this case, the participant had a tendency to press more often the up key independently of the location of the reference. If sensorial and decisional biases contribute, the probability of choosing an alternative consistent with clockwise orientation should be shifted, but not symmetrically. This is the choice behavior of participant 4 (Figure 1F).
For each participant, we quantified the magnitude of the sensory and decisional biases (illustrated for participant 4 using arrows, Figure 1F, see Models). Figure 1G shows, for each participant, the magnitude of the decisional bias against the magnitude of the sensory bias and which is the bias model that better describes the data (see Models and Materials and Methods). For each participant, two data points are plotted. The circles correspond to the trials just described, in which the grating was presented around the horizontal orientation. The triangles correspond to other trials−intermixed with the previous ones−in which the grating was presented around the vertical orientation (in this case, participants judged the orientation using the right and left keys relative to a reference that we asked participants to image on top or at the bottom). Most participants showed significant biases. About half of the participants showed biases consistent with only a sensory origin and about half showed decisional biases mostly combined with sensory biases (although two participants showed biases consistent with only a decisional origin). Across participants, the absolute value of the magnitude biases was larger for the sensory biases than for the decisional biases, indicating that the sensory biases dominated (Figure 1H).
The task described could be considered symmetric because the evidence supporting clockwise or counterclockwise choices should be similar when the stimulus has the same magnitude but different sign relative to the neutral point. Given this symmetry, it could be argued that what we have described as sensory biases might be a more complex form of decisional biases than the one we tested: participants might not prefer the down or up alternative (or right and left), but instead biased to choose the alternative consistent with clockwise or counterclockwise orientation. In this case, the sensory biases that we found should not remain when the same stimuli are presented, but the task has two choices unrelated to the clockwise and counterclockwise alternatives. To assess this prediction, in some other trials−intermixed with the previous trials−we asked participants to imagine a reference on the right or on the left (or bottom and top) and perform a two-alternative asymmetric choice task that consisted in indicating whether the grating was aligned or not with the imagined reference (Figure 2A-B).
To illustrate the choice behaviors on this task, we first describe the results of four representative participants (Figure 2C-F). The symbols in Figure 2C-F show the probability of responding aligned as a function of the orientation of the grating and the curves show the psychometric fits for the two possible locations of the reference (see Supplementary Models; figure supplement 2 shows the results for all participants). The arrows in Figure 2F illustrates the estimation of the magnitude of the sensory and decisional bias (see Supplementary Models). For participant 14 (Figure 2C), the probability of responding aligned was centered around 0 and did not depend on the location of the reference (no bias). For participant 3 (Figure 2D), the probability of responding aligned was not centered around 0, but did not depend on the location of the reference (sensory bias). For participant 1 (Figure 1E), the probability of responding aligned was not centered around 0 and depended on the location of the reference, but the tendency to respond aligned more often for one location did not affect where the psychometric fit was centered (sensory bias with a symmetric decisional bias, see Supplementary Models). Finally, for participant 16, the probability of responding aligned was not centered around 0 and depended on the location of the reference, but in this case this tendency affected where the psychometric fits were centered (sensory bias with an asymmetric decisional bias, see Supplementary Models).
Figure 2G shows, for each participant and overall orientation of the grating (horizontal or vertical), the magnitude of the decisional bias against the magnitude of the sensory bias and which is the bias model that better describes the data (see Materials and Methods). Most participants showed significant biases that include sensory and decisional biases. It has been suggested that asymmetric tasks are less prone to decisional biases than symmetric tasks (Schneider and Komlos 2008; but see Anton-Erxleben, Abrams, and Carrasco 2010). We did find less groups with significant decisional biases (the asymmetric ones, Figure 2F) for the asymmetric task (5 fits; figure 2G and figure supplemental 2) than for the symmetric task (8 fits; figure 1G and figure supplemental 1), but the difference was not significant (x2 (1) = 0.52, p = 0.47). Across participants, the absolute value of the magnitude biases was larger for the sensory biases than for the decisional biases, indicating that the sensory biases dominated (Figure 1H).
Figure 3A shows that, within participants, the sensory biases estimated from the asymmetric task are very similar to those estimated from the symmetric task. This shows that what we described as sensory biases for the symmetric task had indeed a sensory origin, and not a complex form of decisional biases. We also found a good agreement, within participants, between the sensory biases for the horizontal grating and the vertical grating for the two tasks (Figure 3B). This suggests that the sensory biases are consistent with a clockwise or counterclockwise global perceptual rotation of the stimulus.
Across trials we intermixed four locations of the reference and two tasks (symmetric and asymmetric) that required pressing different keys. These high demands on participants might have minimized decisional biases. To test whether the task demands influenced the proliferation of decisional biases, in Experiment 2, for 16 new participants, we repeated the symmetric task using two references (Top and Bottom) that could be blocked in two ways. In some blocks of trials the two references were mixed across trials and participants, thus, needed to establish a mapping between perception and the category of the alternatives on each trial. In some other blocks, only one reference was presented on each block (on different blocks, the reference changed) and participants did not need to update the response mapping on each trial. Consistent with Experiment 1, we found that participants exhibited significant sensory and decisional biases, but we did not find significant differences in decisional biases between the two blocking conditions (figure supplementary 3). We found, however, that the decisional biases in Experiment 2 were larger than in Experiment 1 (figure supplementary 3), which suggests that very high demands on the task like those in Experiment 1 can reduce the proliferation of decisional biases.
Taking into account Experiment 1 and 2, we have found that from the 32 participants, 30 (94%) had significant global choice biases. From these 30, the sensory contribution was significant in all of them (that is, 94% of participants had sensory bias) and the decisional contribution was significant in 15 (that is, 47% of participants had decisional biases).
MODELS
On a given trial of the symmetric task (the model for the asymmetric task is described in Supplementary Models) a grating with orientation θi that could be clockwise or counterclockwise, is presented and the participant decides whether its orientation is consistent with clockwise acw or counterclockwise accw orientation. A simple standard signal detection theory (SDT) model assumes that: 1) the sensory evidence s associated to the stimulus is a random variable normally distributed with variance σ2 centered on μcw when θi is clockwise and centered on Hccw when θi is counterclockwise; 2) to make choices, the participant sets a criterion β, computes the log likelihood ratio log Λ of s under the hypothesis that μ=μcw and the hypothesis that μ=μccw and chooses the action acw when log Λ(s) > β
In standard SDT models the origin and scale of the sensory evidence is arbitrary (Wickens 2001; Green and Swets 1966). A typical choice is (Wickens 2001; Green and Swets 1966) which results in log Λ (s)=d’s That is, one can use directly the evidence as a decision variable and choose acw when s > c = β/d’ (Wickens 2001; Green and Swets 1966).
The probability of choosing clockwise when θi is clockwise is
Where Φ is the standard cumulative normal function. To relate this probability to the magnitude of the stimulus, this standard SDT could be extended to include how the stimulus is transduced into sensory evidence (Garcia-Perez and Alcala-Quintana 2013; Schneider and Bavelier 2003; Garcia-Perez and Peli 2014; Schneider and Komlos 2008). Adding the transduction provides a meaningful origin and scale to the sensory evidence. Assuming that the transduction is linear the probability of choosing clockwise as a function of θi becomes where which corresponds to a psychometric function centered on δD – δs with slope given by σ.
Consider the trials in which the grating is presented in orientations around the horizontal orientation and the reference is on the right (the reasoning that follows also holds for vertical orientations). If the orientation of the grating can take M different values, each value is presented n times and ki is the number of times that the participant pressed the down key (clockwise), then the probability density function that defines statistical model for these trials is
Given that ƒ depends on δD – δs, it is not possible to distinguish sensory from decisional biases.
Consider now the trials for the left reference. A criterion c associated to a bias to choose responses consistent with clockwise orientation for the right reference corresponds to a criterion − c for the left reference. The statistical model that incorporates responses for the two locations of the reference is
Where and the index i larger than M refers to the trials for the left reference. Now, by fitting two psychometric functions conjointly, δD and δs are not degenerated and could be estimated. This is the exact model that we fitted for 17 of the 34 groups (17 participants that discriminate orientations around the horizontal and the vertical orientation in Experiment 1). For the rest of the groups, we also fitted this basic model, but adding some extra features that significantly improved the quality of the fit (using likelihood ratio tests, see Materials and Methods).
The first feature is related to the variability of the sensory evidence σ The basic model considers that this variability does not depend on where the participants imagined the reference, which results in the two psychometric functions having the same slope. We found this to be the case in 28 of the 34 fits, but in 6 groups we found that considering different variabilities (σ1 and σ2) for the two locations of the reference was better. It might be that orienting attention to different parts of the visual field might change the uncertainty in the sensory evidence for some participants.
The other feature that we added consisted in adding lapses, which are responses that are incorrect independently of the sensory evidence (Kingdom and Prins 2016; Gold and Ding 2013). This might occur, for example, when the participant misses the stimulus because of blinking or a loss of attention. To incorporate lapses into the basic model, ΦR need to be replaced by λ1+ (1 − λ1 − λ2).ΦR and ΦL by (Kingdom and Prins 2016). From the 34 groups, adding lapses improved the fit in 14 cases. From those, 3 fits required the 4 lapse parameters, 4 fits required 2 lapse parameters (one for each psychometric function, ) and 7 fits required 1 lapse rate . Importantly, including a lapse rate parameter Λ * placed on the right asymptote for the right reference and placed on the left asymptote for the left reference did not improve significantly the fit for any group, which indicates that decisional biases cannot be explained by a tendency of participants to select one alternative completely independently of the sensory evidence.
Perceptual tasks with two symmetric alternatives have been also modeled using a high threshold model, called the indecision model (García-Pérez and Alcalá-Quintana 2013; García-Pérez and Peli 2014). This model is similar to the SDT model described, but divides the sensory axis in 3 regions delimited by c1 and c2. When the sensory evidence is lower than c1 the participant chooses accw and when is larger than c2 chooses acw Critically, when the sensory evidence lies between c1 and c2−called interval of uncertainty−the model assumes that the observer is undecided and guesses the response (choosing accw with probability ξ). The probability of choosing clockwise when θi is clockwise and the reference is on the right is
For the left reference, assuming that c1 and c2 do not change, ξ should be replaced by (1 −ξ). We fitted this model to the 19 groups with significant decisional biases and found that for all of them (except for participant 8 for the vertical condition in Experiment 1) the SDT model was better using the Akaike information criterion (Burnham and Anderson 2004). Given that the indecision model has 2 parameters more than the SDT model (c is replaced by c1 and c2 and ξ is introduced), we also compared the SDT model to a simplified version of the indecision model with symmetric boundaries c1= −c2 (García-Pérez and Alcalá-Quintana 2013; García-Pérez and Peli 2014). In this case, the SDT model was better for all the groups.
DISCUSSION
We showed that, in a simple discrimination task with two symmetric alternatives, most people exhibit global choice biases. Changing the stimulus-response mapping and testing a task with two asymmetric alternatives, we found that these biases reflect biases in the sensory evidence and in the decisional process. Specifically, about half of the participants showed biases consistent with only a sensory origin and about half of the participant biases with a contribution of sensory and decisional biases. We also found that a very few participants exhibited biases consistent with only a decisional origin.
A SDT model, in which the sensory biases were included as an intercept term in a linear transduction of the stimuli (García-Pérez and Alcalá-Quintana 2013; Schneider and Bavelier 2003; García-Pérez and Peli 2014; Schneider and Komlos 2008) and decisional biases correspond to a shift in the criterion, fitted well the choice behavior of the participants. A model in which participants have a tendency to choose one alternative independently of the magnitude of the sensory evidence did not fit well the data; it rather predicts an asymmetry on the lapse rate across locations of the reference that was not observed. Finally, a model (García-Pérez and Alcalá-Quintana 2013; García-Pérez and Peli 2014) in which participants guess when the sensory evidence lies within some uncertainty range was not parsimonious: first, to fit the data for the participants with pure sensory biases, the model needs that participants, when uncertain, guess the two alternatives equally often; second, for the participants with decisional biases, we found that the SDT model provided a better fit.
Perceptual discrimination tasks with two symmetric alternatives are often regarded as performance tasks (Carandini and Churchland 2013; Dayan and Abbott 2001). If a stimulus has positive signal (e.g. rightward motion) relative to a neutral point (no net motion), but the decision--maker chooses the alternative consistent with negative signal (leftward motion), the response is considered an error (false alarm). Our results, however, suggest that in some cases a positive signal might be perceived consistently as a negative signal (sensory bias). Consequently, it might be inappropriate to consider these responses as errors and, in case feedback is given, provide a negative reward. Our results suggest that perceptual discrimination tasks with two symmetric alternatives should be regarded as appearance tasks (Kingdom and Prins 2016; Gold and Ding 2013).
Discrimination tasks with two symmetric alternatives are commonly used to assess how perception is affected by contextual cues (Carrasco, Ling, and Read 2004; Schwartz, Hsu, and Dayan 2007), but in some scenarios it is unclear whether the context influences perception or biases decisions (Carrasco 2011; García-Pérez and Alcalá-Quintana 2013; Michael J. Morgan et al. 2012; Stormer, McDonald, and Hillyard 2009; Schneider 2011; Anton-Erxleben, Abrams, and Carrasco 2010; Jogan and Stocker 2012; Mather and Sharman 2015). Our results indicate that, even when the symmetry of the task is not broken by the context, half of the participants exhibit decisional biases. This suggests that to reliably estimate sensory biases in the presence of contextual cues, it might be better to use manipulations of the stimulus-response mapping like the one that we used or tasks for which the potential contribution of decisional biases is reduced or eliminated (Jogan and Stocker 2012; Patten and Clifford 2015; M. J. Morgan 2014).
To facilitate that participants internalize two different associations between perception and the responses, we asked them to compare the orientation of the stimulus to a reference imagined in two different locations. Given that the reference was not physically present, we think, however, that it is likely that participants were performing judgments of absolute orientation relative to an internal point of horizontality (or verticality). As we found evidence that the sensory biases were consistent with a global perceptual rotation, if participants were performing judgments of relative orientation between the stimulus and the reference− instead of judgments of absolute orientation−then sensory biases should not be expected, as both the stimulus and the reference should be biased in the same direction.
We found a good agreement between the sensory biases estimated from the symmetric and the asymmetric task. We also found a good agreement between tasks when the biases were estimated taking into account only one localization of the reference, and thus for which it was not possible to assess the contribution of sensory and decisional biases (see legend for Figure 3A). This is expected given the large contribution of sensory biases to the total magnitude of the biases. In the temporal domain, however, the biases estimated from temporal order judgments (symmetric task) and simultaneity judgments (asymmetric task) do not agree (Love et al. 2013; Linares and Holcombe 2014; van Eijk et al. 2008), which suggests that decision making for time perception might be even more affected by decisional biases (Linares and Holcombe 2014; Shore, Spence, and Klein 2001; Schneider and Bavelier 2003).
MATERIALS AND METHODS
Participants
The study was approved by the ethical committee of the University of Barcelona and followed the requirements of the Helsinki convention. The participants, who did not know the hypothesis of the experiments, provided written consent to perform the experiments. Twenty-one participants were recruited for Experiment 1 and sixteen for Experiment 2.
Stimuli and tasks
Stimuli were generated using PsychoPy (Peirce 2007), displayed on a Sony G520 CRT screen (40 cm width and 30 cm height; 60 Hz refresh rate) and viewed from a distance of 57 cm in a dark room. They consisted in a Gabor patch (standard deviation (sd) of the Gaussian envelope: 1.33 degrees of visual angle (dva); maximum luminance: 79 cd/m2) and a red Gaussian blob (sd: 0.1 dva; maximum luminance: 19 cd/m2) on top of it that participants were asked to fixate during the experiment. Stimuli were presented against a uniform circular grey background (diameter: 25 dva; luminance: 43 cd/m2) that was displayed in a black background (luminance: 0.2 cd/m2). The verticality of the Gabor was calibrated using a pendulum.
Experiment 1
Participants performed 6 blocks of 360 trials. In each block, 8 conditions were randomly intermixed across trials. On each trial, before the Gabor was presented, a text message informed participants about the condition. A right: up or down? message instructed participants to imagine a reference on the right (at the same height of the fixation point) and respond whether the Gabor was pointing down (clockwise) or up (counterclockwise) relative to it. A left: up or down? message instructed participants to imagine a reference on the left and respond whether the Gabor was pointing down (counter-clockwise) or up (clockwise). For these conditions, the orientation of the Gabor was chosen randomly from a range centred around horizontal orientation (from -2 to 2 deg in steps of 0.5 deg) according to the method of constant stimuli (Kingdom and Prins 2016). The up: right or left? and the down: right or left? messages provided parallel instructions for imaginary references on top and at the bottom. For these conditions the orientation was centered around vertical orientation. Participants used the arrow keys to respond. A right? message instructed participants to imagine a reference on the right and respond whether the Gabor was aligned with it (pressing m key) or not (pressing n key). A left?, up? and down? provided parallel instructions for references in other locations. The messages were available until participants pressed the spacebar. The Gabor was presented for 100 ms, 500 ms after the keypress.
Experiment 2
Experiment 2 was like Experiment 1, but included only the symmetric task and orientations around vertical. Participants performed 4 blocks of 270 trials. Half of the participants performed first two blocks in which they were asked to imagined the reference always in the same location within a block and, then, two blocks in which they were asked to imagine the reference at one location that changed randomly across trials. For the other half of participants, the block order was inverted.
Instructions
Before the experiments, to facilitate the understanding of the instructions, a reference (a green Gabor patch; sd: 0.1 dva; maximum luminance: 29 cd/m2) was displayed for 5 to 10 trials at the corresponding cardinal location at a distance of 6 dva from the center of the fixation point.
Analysis
The data and the code to do the statistical analysis and create the figures is available at https://github.com/danilinares/2018LinaresAguilarLopezmoliner. The model fitting, goodness of fit and model selection (likelihood ratio test and Akaike information criterion) was performed using the R package quickspy (Linares and López i Moliner 2016), which under the development version allows fitting several psychometric functions conjointly.
Experiment 1 (symmetric task)
For four of the twenty-one participants, a preliminary analysis revealed that the responses were inverted or not modulated by the orientation of the grating and were not analyzed further. For each participant and condition of overall orientation (horizontal or vertical), we fitted the model in equation (8) with all the extra features (2 slopes and 4 lapses, see Models; parameters:) using maximum likelihood estimation. Then, using consecutive likelihood ratio tests performed using the χ2 distribution and a significance level of 0.05 (Prins and Kingdom 2018), we reduced model complexity to select for each participant and condition of orientation the simplest model that captured the data and incorporated at least the parameters δs, δD and σ. The first two parameters correspond to the sensory and decisional biases plotted in Figure 1G. To assess, for each participant and condition of orientation, which was the best bias model, we performed further likelihood ratio tests with simpler models. The best bias model was described as Sensory bias if a simpler model with δD = 0 provided a better fit; Decisional bias if a model with δs = 0 was better; No bias if a model with δD = δS = 0 was better; and Sensory + decisional bias if the complexity of the original model could not be further reduced. When we compared the models with lapse parameters with the models without lapses, we used a lapse rate of 0.01 instead of zero because this resulted in better fits (Prins 2012).
Experiment 1 (asymmetric task)
For one of the participants that could perform the symmetric task, the responses were not modulated by the orientation of the grating for the asymmetric task and the data was not analyzed. To estimate sensory and decisional biases and assess which bias model was better, we used the same procedure described for the symmetric task.
Experiment 2: The data were analyzed as it is described for the symmetric task in Experiment 1.
SUPPLEMENTARY MODELS
A simple SDT model for the asymmetric task divides the sensory axis in 3 regions delimited by c1 and c2. When the sensory evidence is lower than c1 or larger than c2 the participant responds no aligned. When the sensory evidence lies between c1 and c2 the participant responds aligned−in contrast to the indecision model (García-Pérez and Alcalá-Quintana 2013; García-Pérez and Peli 2014), there is not guesses in this case. The probability of choosing aligned when θi is clockwise is
Where
The model assumes that for one of the references (right) the criteria are symmetric δD2=−δD1, but not for the other reference (left). The statistical model that incorporates the two locations of the reference is thus where ki is the number of times that the participant responds aligned and
This was the best model for 5 or the 32 groups (likelihood ratio tests, see Materials and Methods). For 2 groups, assuming symmetric criteria also for the left reference provided a better fit. For 24 groups, assuming the same symmetric criteria for the two locations of the reference was better. Finally, for 1 group, the best fit resulted from replacing σ by σ1 and σ2.
SUPPLEMENTARY FIGURES
ACKNOWLEDGMENTS
The research was funded by the Departament de Salut of the Generalitat de Catalunya (PERIS-ICT Ref: SLT002/16/00338), the Catalan government (Ref: 2017SGR-48), the Fundación Alicia Koplowitz and project Ref: PSI2017-83493-R, AEI/Feder, UE. We thank Joâo Barbosa, Albert Compte and Genís Prat for comments on the manuscript.