22.01.2020
DIFFERENTIATING
PERCEPTUAL FROM
POST-PERCEPTUAL BIASES
Author
Laura Van den Akker
11171464
Lauraa.vdakker@gmail.com
Supervisors
Dr. Simon van Gaal
Ms. Nicolás Sanchez-Fuenzalida
Word count
Abstract:
257
Differentiating perceptual from
post-perceptual biases
Laura Van den Akker
Department of Psychology, University of Amsterdam Received: 22.01.2020
Abstract: When people are making decisions, their responses can be biased through a shift of the
internal threshold (criterion). However, it is not always clear whether this shift results from a change in perception, or from a post-perceptual shift in their response criterion. This paper aims to have developed a two-task method that experimentally separates decision from perception. Participants had to judge whether the presented line was shorter or longer than the reference line (length comparison task), followed by an estimation of the last five presented lines (average length estimation task). The Müller-Lyer illusion was assumed to bias participants’ responses in both tasks (perceptual bias) while a category-cost contingency was assumed to bias participants’ responses only in the length comparison task (post-perceptual bias). The results revealed a large bias effect of the Müller-Lyer manipulation in both tasks, that is, the Müller-Lyer being a perceptual manipulation. There was also a bias effect of the category-cost contingency in both tasks, suggesting being a perceptual illusion. However, considering that the category-cost contingency bias effect was much smaller than the Müller-Lyer bias effect and even decreased by half between tasks, this is not believed to be the case. Furthermore, the small category-cost contingency bias effect in the average length estimation task could have been a carry-over effect from the length comparison answers, because the tasks were not completely independent. Meaning that the category-cost contingency is indeed a post-perceptual manipulation. Overall, future research involving independent tasks or equal bias effects of the bias sources in the length comparison task is needed to clarify.
Keywords: perceptual bias, post-perceptual bias, Müller-Lyer illusion, category-cost contingency, decision-making, Signal Detection Theory, drift-diffusion model
1 Introduction
Humans are frequently subjected to decision-making in their everyday life. Prior to perceptual decision-making, an integration process of sensory information followed by implementation of the internal threshold occurs in the brain (Jin & Glickfeld, 2018). This internal decision threshold, often referred to as the criterion, can be influenced (or ‘biased’) by perceptual inputs or by post-perceptual mechanisms (Wickens, 2002).
Biases can take place at different levels, i.e. perceptually or post-perceptually. Some biases can be intuitively assumed to be perceptual, like the Müller-Lyer illusion. In this illusion lines are perceived shorter or longer despite the fact that the length of the line did not altered (see fig. 1). However, for many other bias sources it remains unclear if they take place on a perceptual or post-perceptual level (Wickens, 2002; Gold & Ding, 2013; Linares, Aguilar-Lleyda & pez-Moliner, 2019).
Imagine a task where participants are asked to determine whether a presented line is shorter or longer compared to the reference line. After one block of trials, the participants are punished every time they incorrectly answer ‘short’. As a result, the participants are likely to become biased towards reporting a greater number of long lines, i.e. correctly identifying more long lines as being longer than the reference line, but also incorrectly reporting more short lines as being longer than the reference line. That is, although the sensitivity of the participants has not changed, the participants’ criterion will be biased to prefer long-line answers. In Signal Detection Theory (SDT), such a bias can be modelled by a shift in the participants’ criterion parameter (c) (Wickens, 2002; Sanabria, Spence & Soto-Faraco, 2007; Cardoso-Leite, Mamassian, Schütz-Bosbach & Waszak, 2010; Rahnev et al., 2011; Liu, Mercado III & Church, 2011). However, SDT does not clarify the nature of the criterion shift.
Are participants consciously perceiving a greater proportion of lines as being longer than the reference line (involving a perceptual bias when processing the stimuli) and therefore reporting more lines as being longer than the reference line, or are they simply reporting more long lines without actually consciously perceiving a greater number of long lines (involving a post-perceptual bias or strategic change to increase their reward) Put differently: does the decision shift take place at a perceptual
or post-perceptual level (Wei & Stocker, 2012; Grove, Ashton, Kawachi & Sakurai, 2012; White & Poldrack, 2014)? Witt, Taylor, Sugovic & Wixted (2015) showed that both perceptual and post-perceptual induced biases modelled in SDT result in non-distinct criterion shifts. Therefore, SDT cannot dissociate between these types of biases and remains agnostic about their nature.
A more recent approach to understand the differences between perceptual and post-perceptual bias is to use drift-diffusion models (DDMs). DDMs are used to model bias by a shift in the starting point parameter, or by a change in the direction of the evidence accumulation rate (drift bias) (Ratcliff, 1978; Leite & Ratcliff, 2011: White & Poldrack, 2014; Urai, de Gee & Donner, 2018; Kloosterman et al., 2018). Previous research of White & Poldrack (2014) showed that a shift in starting point can be induced by increasing the proportion or reward of stimuli, while a shift in drift rates can be induced by shifting the stimuli using illusory cues. However, although the starting point and evidence accumulation rate parameters seem reasonably well-established, it is incorrectly assumed that perceptual bias reflects changes in drift rate and post-perceptual bias reflects a shift in starting point because this has not been experimentally showed. Therefore, DMM did not provide information about the nature of perceptual and post-perceptual biases based on experimental evidence.
Overall, the question about the nature of criterion shifts as perceptual or post-perceptual bias remains unanswered. To address this question, a two-task setup was devised that aims to dissociate the perceptual component from the post-perceptual processes involved in a decision bias. Firstly, the response bias had been tested in the length comparison task by asking participants to compare a series of lines, one by one, as being shorter or longer than the reference. Secondly, for every five lines seen in the task just described,
Figure 1. The Müller-Lyer illusion. Two lines of the same
length appear to be different in length due to outward or inward flanking arrowheads. A: The length of a line with inward-pointing arrowheads is perceived longer than the actual length.
B: The length of a line with outward-pointing arrowheads is
they were asked to estimate the average length of those lines. This average length estimation was used as a proxy of the length they actually perceived while doing the length comparison task. In this setting, the bias manipulation is expected to be perceptual in nature if participants’ responses are biased in both the length comparison task and the average length estimation task. The bias manipulation is expected to be post-perceptual if only participants’ responses in the length comparison task are biased but not in the average length estimation task.
To prevent that a smaller bias effect of the category-cost contingency condition compared to the Müller-Lyer condition in the average length estimation task can be attributed to an overall lower-efficacy of the bias source in that condition, the bias effect of the category-cost contingency manipulation in the length comparison task needs to be at least as big as the bias effect of the Müller-Lyer manipulation. To accomplish this, first an experiment to determine effect sizes for the Müller-Lyer and category-cost contingency for various parameters was performed (experiment 1). Then, this parameter setting was used in experiment 2 for which the effect size of the category-cost contingency condition in the length comparison task is as big (or bigger) compared to the effect size of the Müller-Lyer condition in the length comparison task.
2 Experiment 1
2.1 Material & Methods 2.1.1 Participants
Participants were recruited through an ad published at the University of Amsterdam research pool website composed exclusively by college students. Only a minimum/maximum age of 18 and 35 years old was set. Also, participation
was not possible for participants that already completed one of the two experiments, or any of the previous pilots.
2.1.2 Ethics
The experiment design, type of data collected from participants, the information brochure and the informed consent were approved by the Ethics Review Board of the University of Amsterdam.
2.1.3 General procedure
Participants registered voluntarily to be part of the experiment for monetary compensation. They read a brochure with information about the tasks, the expected duration of the experiment, the payment and the performance discount scheme before starting the experiment. The researcher was present while participants read the brochure to answer any doubts they could have about it. Participants were naive to the actual objectives of the experiment but were debriefed afterwards.
After reading the information brochure
participants signed an informed consent.
2.1.4 Apparatus
The instructions and the experiment were completed on a desktop computer in an isolated behavioural cubicle. The computer monitor was 23" (58.4cm) and had a resolution of 1920x1080. The size of each pixel was 0.265mm. At 75 cm from the monitor each pixel is approximately 0.02 visual angle degrees. The refresh rate of the monitor was set to 120 Hz.
2.1.5 Independent variable
This experiment employed a between-participants factor with two bias sources (the Müller-Lyer illusion and a category-cost contingency) x a within-participants factor with two bias directions (short and long) nested in each bias source (see fig. 2A). Additionally, there are four different Müller-Lyer arrowhead lengths nested in each bias direction of the Müller-Lyer
bias source (see fig. 2B). The two biases sources were independently used in this experiment. In the Müller-Lyer condition target lines flanked by inward-pointing arrowheads (four arrowhead lengths) were biased towards long. While target lines flanked by outward-pointing arrowheads (four arrowhead lengths) were biased towards short. Different Müller-Lyer arrowhead lengths were used because Bulatov, Bertulis, Bulatova & Loginovich (2009) showed that among other parameters, the bias effect size of the Müller-Lyer increases as the arrowhead length increases. An asymmetrical discount scheme was used in the category-cost contingency condition. If the bias was directed to long, there was a bigger monetary discount for incorrectly answering short (five cents) compared to the monetary discount for incorrectly answering long (one cent). If the bias was directed to short, there was a bigger monetary discount for incorrectly answering long (five cents) compared to the monetary discount for incorrectly answering short (one cent). To make the conditions similar, horizontal lines flanked by vertical lines were presented in the category-cost contingency and a symmetrical discount scheme (three cents) were added to the Müller-Lyer condition (see fig. 2A).
2.1.6 Dependent variable
In the length comparison task participants had to classify, one by one, a series of target lines as being shorter or longer than the reference line.
Using the participants’ responses in the length comparison task the decision bias was calculated (length comparison bias).
2.1.7 Experimental procedure
Each participant completed 1200 trials in the length comparison task in one of the bias source manipulations. For each participant the answers in the length comparison task resulted in a proportion of correctly classified short and long lines, and incorrectly classified short and long lines. This was used to calculate the SDT centered criterion and sensitivity (see Analysis section of Supplementary methods S.3 for detailed information about how the variables were calculated). The reference line had the same length throughout the entire experiment (300 pixels) and was presented every five trials. The presentation of the reference line and the discount scheme, followed by five trials of this task were considered a mini-block (for the mini-block layout see fig. 3).
First, only the category-cost contingency manipulation and both bias directions were tested to estimate its bias effect. Participants completed 600 trials of each bias direction in the length comparison task. Then the Müller-Lyer manipulation was tested using four different arrowhead lengths on each bias direction. This resulted in eight possible conditions; two bias directions (short x long) and four arrowhead lengths (30, 40, 50 and 60 pixels, see fig. 2B).
Figure 2. The independent variables and Müller-Lyer arrowhead lengths example. A: The Müller-Lyer illusion and a category-cost
contingency were used to bias participants’ responses. The bias could be directed to either ‘short’ or ‘long’ answers. The length of the reference line was always 300 pixels. B: Four arrowhead lengths (30, 40, 50 and 60 pixels) inward and outward pointing were used to test for different levels of bias strength.
Participants completed 150 trials of the length comparison task in each of the eight conditions.
The experiment was divided in blocks of 50 trials, or 10 mini-blocks. For the duration of 50 trials the bias direction (and the arrowhead length when the bias source was the Müller-Lyer illusion) was kept the same. Then, on the next 50 trials the bias direction (and the arrowhead length when the bias source was the Müller-Lyer illusion) could change. The actual order was randomized. Also, every 50 trials participants received feedback about the number of mistakes and the money lost during the last 50 trials. In the category-cost contingency condition the feedback was specified by type of mistake (incorrectly answering long and incorrectly answering short), while in the Müller-Lyer condition there was a general feedback about total number of mistakes and total money lost during the last 50 trials.
Participants were able to take a voluntary break during the feedback screens. However, every 300 trials there was a forced break. Participants were asked to go out of the room and to take a break of three minutes.
2.2 Results
2.2.1 Participants
90 participants (20 males and 70 females; mean = 20.5, SD = 2.62) completed experiment 1.
2.2.2 General bias measure
In the Müller-Lyer conditions, when the arrowhead length was 30 pixels, the mean bias effect was 0.49 (SD = 0.41) for the long bias direction and -0.03 (SD = 0.46) for the short bias direction. When the arrowhead length was 40 pixels, the mean was 0.59 (SD = 0.44) for the long bias direction and -0.04 (SD = 0.46) for the short bias direction. When the arrowhead length was 50 pixels, the mean was 0.61 (SD = 0.49) for the long
Condition Arrowhead length mean SD
ML-long 30 pixels 0.49 0.41 ML-short -0.03 0.46 ML-long 40 pixels 0.59 0.44 ML-short -0.04 0.46 ML-long 50 pixels 0.61 0.49 ML-short -0.05 0.49 ML-long 60 pixels 0.64 0.49 ML-short -0.16 0.50 CCC-long 0.64 0.50 CCC-short -0.26 0.50
Figure 3. Mini-block layout. The experiment was divided into mini-blocks. Each mini-block started with the reference line and the
discount scheme (left-end column) followed by five target lines separated by a fixation cross. When participants memorized the reference line they had to press ‘space’ to continue and answered if the target line was shorter or longer compared to the reference line respectively by clicking left or right using the computer mouse (length comparison task). Only in experiment 2, participants had to estimate the average length of the previous five target lines seen in that mini-block (average length estimation task). On the top row, there is a category-cost contingency manipulation biased towards ‘long’ mini-block. On the bottom row, there is a Müller-Lyer manipulation biased towards ‘short’ mini-block. The solid horizontal arrow indicates the order of the sequence.
Table 1.
The mean centered criterion and SD for each bias source (ML = Müller-Lyer and CCC= category-cost contingency), bias direction (long and short) and arrowhead length (30, 40, 50 and 60 pixels, this variable only in the Müller-Lyer condition).
bias direction and -0.05 (SD = 0.49) for the short bias direction. When the arrowhead length was 60 pixels, the was 0.64 (SD = 0.49) for the long bias direction and -0.16 (SD = 0.50) for the short bias direction. In the category-cost contingency conditions, the mean was 0.64 (SD = 0.50) for the long bias direction and -0.26 (SD = 0.50) for the short bias direction (see table 1 and fig. 4).
2.3 Discussion
The aim of experiment 1 was to determine a Müller-Lyer arrowhead length that would render a bias effect size as similar as possible, without being bigger than the category-cost contingency bias effect size. An arrowhead length of 60 pixels was chosen (see fig. 4). Moreover, increasing the length of the arrowheads resulted in increasing levels of bias size (Bulatov et al., 2009).
3 Experiment 2
3.1 Hypotheses
The following hypotheses were tested in experiment 2. Müller-Lyer condition: (H1) The
Müller-Lyer manipulation will bias participants’ responses in both the length comparison task and the average length estimation task (see fig. 5). For each task: (H1.1) In the length comparison task, the Müller-Lyer manipulation will result in a smaller criterion-value for the outward-pointing
Figure 4. Results general bias measurement experiment 1. 50 participants completed the task with the Müller-Lyer manipulation
and 40 participants with the category-cost contingency manipulation. A: The mean centered criterion and SD is grouped by bias source (Müller-Lyer and category-cost contingency), bias direction (long and short) and arrowhead length (30, 40, 50 and 60 pixels, only in the Müller-Lyer condition). B: For each bias source, and arrowhead length of the Müller-Lyer condition, the bias was calculated as the difference between the mean centered criterion of the long and short bias direction (showed are mean and SD). See table C2 in the Analysis section of Supplementary methods S.3 for more detailed information. An arrowhead length of 60 pixels was chosen which rendered a bias effect close to but still smaller than the category-cost contingency bias effect (black parenthesis).
Figure 5. Predicted pattern of results experiment 2. The left
panel corresponds to the length comparison task and the right panel to the average length estimation task. The distance between the dark and light color, respectively within blue and yellow, represents the strength of the bias within the bias source condition, e.g. the closer the means within a color, the weaker the effect. In the length comparison panel, the bias for both conditions are the same, while in the average length estimation error there is an effect in the Müller-Lyer condition but not in the category-cost contingency condition.
arrowheads compared to the criterion-value for the inward-pointing arrowheads; (H1.2) In the length comparison task, the sensitivity of the
Müller-Lyer outward-pointing arrowheads
condition and the sensitivity of the Müller-Lyer inward-pointing arrowheads condition will be the same; (H1.3) In the average length estimation task, the average length estimation of the Müller-Lyer manipulation under outward-pointing arrowheads will be smaller than the average length estimation of the inward-pointing arrowheads condition.
Category-cost contingency condition:
(H2) The category-cost contingency
manipulation will bias participants’ responses in the length comparison task, but in the average length estimation task, the average length estimation will be the same for the short and long bias direction condition. Or, if there is a difference, it will be smaller than the difference in the average length estimation of the Müller- Lyer condition (see fig. 5). For each task: (H2.1) In the length comparison task, the category-cost contingency manipulation will result in a smaller criterion-value for the punish-long condition compared to the criterion value for the punish-short condition; (H2.2) In the length comparison task, the sensitivity of the category-cost contingency punish-short condition and the sensitivity of the category-cost contingency punish-long condition will be the same; (H2.3) In the average length estimation task, there will be no difference between the average length estimations in the punish-short compared to the punish-long condition, or the difference in average length estimation will be smaller than the difference in average length estimation of the Müller-Lyer condition.
As an alternative, when the category-cost contingency manipulation results in a perceptual shift: (H2.3.A) The difference in average length estimation for punish-short and punish-long will
be as large or larger compared to the difference in average length estimation in the Müller-Lyer condition, given that the Müller-Lyer and category-cost contingency resulted in a bias of equal size in the length comparison task.
3.2 Materials & Methods 3.2.1 Participants
Same as stated in the Participants section of previous experiment (see 2.2.1).
3.2.2 Ethics
Same as stated in the Ethics section of previous experiment (see 2.2.2).
3.2.3 General procedure
Same as stated in the General procedure section of previous experiment (see 2.2.3).
3.2.4 Apparatus
Same as stated in the Apparatus section of previous experiment (see 2.2.4).
3.2.5 Planned sample
Since all the analyses were conducted in a Bayesian framework, there was tested until a BF > 30 for model A, B or C (for a detailed description of the models see the Statistical analysis section 3.3) compared against each other in each task. In this framework optional stopping or data peaking was not considered problematic (Rouder, 2014).
3.2.6 Independent variable
A within-participants factor with 2 bias sources (Müller-Lyer and category-cost contingency) x within-participants factor with 2 bias directions (short and long) nested in each bias source was employed to create four within-participants conditions (see fig. 3). The arrowhead length resulting in a category-cost contingency bias
effect at least as big as the Müller-Lyer bias effect was used in experiment 2 (bias sources and bias directions are same as stated in the independent variable section of previous experiment, 2.2.5). 3.2.7 Dependent variable
Participants completed the length comparison task as stated in the dependent variable section of previous experiment (2.2.6). They were also asked to estimate the average length of the last five seen lines in the length comparison task, referred to as the average length estimation task. Using the participants’ responses in the length comparison task the decision bias and decision sensitivity was calculated (length comparison bias and sensitivity). How much participants deviated from the actual length average of the presented mini-block was calculated as the difference between the actual length average and the participants’ average length estimation (average length estimation error). This resulted in positive estimation error if there was an overestimation and a negative estimation error if there was an underestimation.
3.2.8 Experimental procedure
In this experiment, the presentation of the reference line and the discount scheme, followed by five trials of the length comparison task (as described in the Experimental procedure section of previous experiment 2.1.7) and then 1 trial of the average length estimation task was considered a mini-block (for the mini-block layout see fig. 3). Each participant completed 30 mini-blocks (150 length comparison task trials and 30 trials of the average length estimation task) in each of the four conditions.
As in experiment 1, the entire experiment was divided in blocks of 50 trials, or 10 mini-blocks. For the duration of 50 trials the bias source and bias direction combination was kept the same. Then, on the next 50 trials the bias source and bias direction combination could change. The actual
order was randomized. Also, every 50 trials participants received feedback about the number of mistakes and the money lost during the last 50 trials. In the category-cost contingency condition the feedback was specified by type of mistake (incorrectly answering long and incorrectly answering short), while in the Müller-Lyer condition there was a general feedback about total number of mistakes and total money lost during the last 50 trials.
Participants were able to take a voluntary break during the feedback screens. However, there was a forced break every 200 trials. Participants were asked to go out of the room and to take a break of 3 minutes.
3.3 Statistical Analysis
The Bayesian model comparison framework was adopted to test ordinally-constrained models described in Haaf, Klaassen & Rouder (2019) and based on Klugkist, Laudy & Hoijtink (2005) encompassing estimation method for Bayes factors. For the unconstrained models, a g-prior approach as described in Rouder, Morey, Speckman & Province (2012) with a default setting on the scale of effect, r = √2/2 was used. The other models are restricted versions of the unconstrained model using ordinal and equality constraints. For the analysis, the BayesFactor package in Rstudio Version 1.1.463© was used (Morey & Rouder, 2015). The evidential threshold toward making a conclusion is described in the Planned sample section of this experiment (see 3.2.5). See the Analysis section of Supplementary methods S.3 for more detailed information.
3.3.1 Length comparison task models
On model A, the mean bias size of the Müller-Lyer condition is equal to the mean bias size of the category-cost contingency condition. On model B, the mean bias size of the category-cost contingency condition is bigger than the mean
bias size of the Müller-Lyer condition. These two models capture the situation were the category-cost contingency bias effect is at least as big as the Müller-Lyer bias effect. The manipulation check was considered successful if model A or B were better than the other models at explaining data. Further, model C explains the mean bias size of the Müller-Lyer condition being bigger than the mean bias size of the category-cost contingency condition. Finally, a null model (D), where the means of all conditions are the same, and an unconstrained model (E), where all means are different but there are no ordinal relations established between the conditions, were tested (see fig. 6).
3.3.2 Average length estimation task models There are three main models, first model A describes the situation where there is only an estimation error in the Müller-Lyer condition. However, if there is an estimation error on both conditions, model B captures the situation where the Müller-Lyer estimation error is bigger than the category-cost contingency estimation error
and model C captures the situation where the estimation error on both conditions are the same. As before, a null model (D), where the estimation
errors of all conditions are the same, and an unconstrained model (E), where all estimation
errors are different but there are no ordinal relations established between the conditions, were tested (see fig. 7).
3.4 Results
3.4.1 Participants
50 participants (13 males, 35 females, 2 others; mean age = 21.1, SD = 3.33) completed experiment 2.
3.4.2 Sensitivity
In the Müller-Lyer conditions, the mean sensitivity (SDT d’) was 1.14 (SD = 0.60) for the long bias direction and 1.39 (SD = 0.58) for the short bias direction. In the category-cost contingency conditions, the mean sensitivity was 1.31 (SD = 0.59) for the long bias direction and 1.38 (SD = 0.57) for the short bias direction (see
Figure 7. Ordinal models for average length estimation task. The
four nodes represent the mean centered criterion value of that condition in the length comparison task. Cells labeled “ml-l” refer to the Müller-Lyer – biased towards long condition, cells labeled “ml-s” refer to the Müller-Lyer – biased towards short condition, cells labeled “cc-l” refer to the category-cost contingency – biased towards long condition and cells labeled “cc-s” refer to the category-cost contingency – biased towards short condition. A: Müller-Lyer bias effect only. B: the Müller-Lyer bias effect is bigger than the category-cost contingency bias effect. C: the Müller-Lyer bias and category-cost contingency bias effect are the same size. D: null model, no effect of the bias manipulations.
E: unconstrained model where all orderings of cell means are
possible.
Figure 6. Ordinal models for the length comparison task. The four
nodes represent the mean centered criterion value of that condition in the length comparison task. Cells labeled “ml-l” refer to the Müller-Lyer – biased towards long condition, cells labeled “ml-s” refer to the Müller-Lyer – biased towards short condition, cells labeled “cc-l” refer to the category-cost contingency – biased towards long condition and cells labeled “cc-s” refer to the category-cost contingency – biased towards short condition. A: the Müller-Lyer and category-cost contingency manipulation have the same bias effect. B: the Müller-Lyer bias effect is smaller than the category-cost contingency bias effect. C: the Müller-Lyer bias effect is bigger than the category-cost contingency bias effect. D: null model, no effect of the bias manipulations. E: unconstrained model where all orderings of cell means are possible.
table 2 and fig. 8). Bayesian model comparison showed that the null model was 0.58 (BF01) more
likely than the unconstrained model (E).
3.4.3 Bayesian model comparison length comparison task short bias direction.
In the Müller-Lyer conditions, the mean bias was 0.93 (SD = 0.62) for the long bias direction and -0.26 (SD = 0.54) for the short bias direction. In the category-cost contingency conditions, the mean bias was 0.29 (SD = 0.54) for the long bias direction and -0.26 (SD = 0.50) for the short bias direction (see top rows of table 3). The Müller-Lyer manipulation (Cohen’s d = 2.06, calculated
as long – short condition) and the category-cost contingency manipulation (Cohen’s d = 1.07, calculated as long – short condition) had both a large effect (Cohen, 1992; see fig. 9A). Then, in de model-based analysis model C (BF = 52.7) was preferred over the other models with a Bayes factor of 7.6 x 1022 -to-one over model A and a
Bayes factor of 2.2 x 104-to-one over model B.
3.4.4 Bayesian model comparison average length estimation task
In the Müller-Lyer conditions, the mean bias was 8.03 (SD = 10.89) for the long bias direction and -15.74 (SD = 10.81) for the short bias
Condition mean SD
ML-long 1.14 0.60 ML-short 1.39 0.58 CCC-long 1.31 0.59 CCC-short 1.38 0.57
Figure 9. Results effect sizes (Cohen’s d) for each bias manipulation in the length comparison task (A) and in the average length
estimation task (B). Showed are the mean centered criterion grouped by bias source (Müller-Lyer and category-cost contingency) and bias direction (long and short). Cohen’s d was calculated for each manipulation within a task.
A B
Figure 8. Results sensitivity measurement. The mean d’ is
grouped by bias source (Müller-Lyer and category-cost contingency) and bias direction (long and short).
Table 2.
The mean sensitivity and SD for each bias source (ML = Müller-Lyer and CCC= category-cost contingency) and bias direction (long and short).
direction. In the category-cost contingency conditions, the mean bias was -4.04 (SD = 13.67) for the long bias direction and -10.69 (SD = 12.12) for the short bias direction (see bottom rows of table 3). The Müller-Lyer manipulation (Cohen’s d = 2.19, calculated as long – short) had a large effect, while the category-cost contingency manipulation (Cohen’s d = 0.48, calculated as long – short) had a medium effect (Cohen, 1992; see fig. 9B). In the average length estimation task, there was a preference for model B (BF = 38.7) over the other models, with a Bayes factor of 23.8-to-one over model A and a Bayes factor of 6.0 x 1016-to-one over model C. The
second preferred model was model A (BF = 36.7).
3.5 Discussion
The aim of experiment 2 was to experimentally dissociate between the perceptual component and post-perceptual processes. Participants had to judge whether a presented line was shorter or longer than the reference line in the length comparison task, then in the average length estimation task they had to estimate the average length of the last five lines seen.
3.5.1 Sensitivity
According to Bayesian model comparison, there is anecdotal evidence for the null model (BF < 3; Wagenmakers, Wetzels, Borsboom & van der
Maas, 2011). This means that the performance among conditions was likely not the same. However, this seems to concern the small difference between the Müller-Lyer bias towards long condition compared to the other conditions (see fig. 8 and table 2) and therefore is not problematic. Hence, more data collection should address this finding.
3.5.2 Length comparison task
Model C was preferred over the other models, the Müller-Lyer bias effect size being bigger than the category-cost contingency bias effect size. Hence, the control for the bias effect of the bias sources to be equal in the length comparison task did not work, in fact, the Müller-Lyer bias effect even increased from experiment 1 to experiment 2. Although, there is not a clear explanation for this, it is suggested to be a result of a not very consistent Müller-Lyer manipulation, or of an inconsistent criterion measurement through time.
3.5.3 Average length estimation task
Model B was preferred over the other models, the category-cost contingency bias effect size being smaller than the Müller-Lyer bias effect size. It should be noticed that the difference in Bayes factor with model A was very small. If the bias effect sizes of both bias sources were equal in the length comparison task, model A and B would capture the situation were the category-cost contingency is a post-perceptual manipulation. However, since this was not the case the category-cost contingency manipulation can be a perceptual illusion but underpowered in the length comparison task. Contrariwise, the small bias effect of the category-cost contingency manipulation in the average length estimation task could also be a carry-over effect from the answers in the length comparison task because the tasks were not completely independent. That is, the category-cost contingency being a post-perceptual manipulation. Overall, it is necessary
Task Condition mean SD
Length estimation task ML-long 0.93 0.62 ML-short -0.26 0.54 CCC-long 0.29 0.54 CCC-short -0.26 0.50 Average length estimation task ML-long 8.03 10.89 ML-short -15.74 10.81 CCC-long -4.04 13.67 CCC-short -10.69 12.12 Table 3.
The mean bias and SD for each bias source (ML = Müller-Lyer and CCC = category-cost contingency) and bias direction (long and short) in the length comparison task (four top rows) and average length estimation task (four bottom rows).
to control better for the effect sizes, or to completely dissociate the to tasks.
4 General discussion
As mentioned before, Witt et al. (2015) showed that both perceptual and post-perceptual bias modelled in SDT result in a non-distinct criterion shift. Therefore, SDT remained agnostic about the nature of these biases. To address this, a two-task experimental setting was developed that separates decision from perception. The Müller-Lyer manipulation was expected to bias participants’ responses in the length comparison and the average length estimation task, assuming being a perceptual manipulation (Howe & Purves, 2005; White & Poldrack, 2014; Witt, Taylor, Sugovic & Wixted, 2015; Knotts & Shams, 2016). Whereas the category-cost contingency manipulation was expected to bias participants’ responses only in the length comparison task but not in the average length estimation task, assuming being a post-perceptual manipulation (Connine & Titone, 1990; Diederich & Busemeyer, 2006; Windmann, 2007; Leite & Ratcliff, 2011; Witt, Taylor, Sugovic & Wixted, 2015). Although, there was also an alternative hypothesis proposed were the category-cost contingency is a perceptual manipulation.
In experiment 2, the results revealed a bias effect of the Müller-Lyer and category-cost contingency manipulation in both tasks. The Müller-Lyer manipulation had a large bias effect and was of almost equal size in both tasks, that is, the Müller-Lyer being a perceptual manipulation. Also, the category-cost contingency manipulation resulted in a bias effect in both tasks, suggesting being a perceptual illusion but underpowered due to a smaller bias effect compared to the Müller-Lyer bias effect in the length comparison task. However, considering that the bias effect of the category-cost contingency was much smaller than
the Müller-Lyer bias effect and even decreased by half from the length comparison task to the average length estimation task, this is not believed to be the case. If the category-cost contingency would be a perceptual illusion the bias effect should have been of equal size in both tasks, as is the case in the Müller-Lyer manipulation. Furthermore, the category-cost contingency answers in the average length estimation task could have been influenced by the length comparison task answers because the tasks were not completely independent. This means that the small category-cost contingency bias effect in the average length estimation task was a carry-over effect from the length comparison answers, hence, the category-cost contingency indeed being a post-perceptual manipulation.
Taken together, this research provided a method to dissociate perceptual from post-perceptual biases, but there are still some limitations. Therefore, in future research it is recommended to collect more data and to equalize the bias effect sizes of both manipulations in the length comparison task or to use a setting with completely independent tasks.
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Supplementary Methods
S.1 Staircase procedure
In both experiments the length deviation of the target line was titrated to achieve a 75% of accuracy. The initial value of the deviation is 20 pixels. This value was updated trial-by-trial using a weighted up-down method as described by Kaernbach (1991). The step size was 2 pixels and the up/down update was 2/1 pixels. There was a minimal value correction so the titrated value could not go lower than 2 pixels. Participants completed 25 reversals but only the last 20 were used to calculate the final threshold value.
S.2 Target line distribution
For each experiment, a normally-shaped distribution of target lines is drawn using the titrated deviation value mentioned in the staircase section. The value cannot be less than 3 pixels so no target value could have the same length of the reference line. First an integer-only distribution around zero ranged between -2 and 2 is drawn. In experiment 1 the distribution has 75 observations in the Müller-Lyer condition, and 300 observations in the category-cost contingency condition. This is because there are 8 conditions (2 bias directions x 4 arrowhead lengths) under the Müller-Lyer manipulation but only 2 under the category-cost manipulation (2 bias directions). In experiment 2 the distribution has 75 observations. The distribution is composed by 38.67% of 0 values, 42.66% of -1 and 1 values, and 18.66% of -2 and 2 values. Then the reference line length value is added to each value of the distribution. To create the shorter- and longer-than-the-reference distributions the titrated deviation value is subtracted and added to each value of the distribution respectively. In experiment 1, the final distribution has 150 observations in the Müller-Lyer condition and 600 observations in the category-cost contingency condition. Each of these distributions is presented in each condition under the correspondent bias source manipulation. In experiment 2, there is only one final distribution of 150 observations that is presented in each condition.
Each distribution is randomized and then is divided in mini-blocks composed of 5 target lines. The order within a mini-block is also randomized. In experiment 1, under each bias source manipulation the composition of each mini-block is maintained across conditions. In experiment 2 there is only one distribution so each mini-block is tested in all the conditions.
S.3 Analyses
To measure the bias in the length comparison task the centered criterion c (Green & Swets, 1966) of each participant were calculated across the trials in each condition. Using the followed labeled trials:
Bias direction Participant answer Confusion matrix
Short Short Long True short False long Long Long Short True long False short Table C1.
To correct for extreme values, 0.5 is added to the frequencies of every cell in the confusion matrix if any of the cells is zero (Hautus, 1995).
Then the hit and false alarm were calculated using:
𝐻𝑖𝑡 − 𝑟𝑎𝑡𝑒 = True long trials
True long trials + False short trials
𝐹𝐴 − 𝑟𝑎𝑡𝑒 = False long trials
False long trials + True short trials
And the centered criterion c was calculated using:
𝑐 = −1
2[Z(Hit − rate) + Z(FA − rate)
In experiment 1, the mean centered criterion is grouped by bias source (Müller-Lyer and category-cost contingency), bias direction (short and long) and arrowhead length (30, 40, 50 and 60 pixels, this variable only in the Müller-Lyer condition). Then for each bias source (and arrowhead length in the case of the Müller-Lyer condition) the bias is calculated as the difference between the mean centered criterion of the short and long bias direction (see table C2).
Bias source Bias direction Bias direction bias Bias source bias Arrowhead lengths
Müller-Lyer Short μml short μml = μml long – μml short μ
ml short 30px, 40px, 50px and 60px
Long μml long μml long 30px, 40px, 50px and 60px
Category-cost
Short μcc short
μcc = μcc long – μcc short
Long μcc long
In experiment 2, the mean centered criterion is grouped only by bias source (Müller-Lyer and category-cost contingency) and bias direction (short and long). Also, the mean centered criterion for each bias source is calculated as the difference between the mean centered criterion of the long and short bias direction within each bias source (see table C2).
Each participants’ d’ (Wickens, 2002) for each combination of bias source and bias direction (and arrowhead length in experiment 1) was calculated as followed:
𝑑G = Z(Hit − rate) − Z(FA − rate)
To measure the estimation error in the average length estimation task, the average length of the presented target lines for each mini-block were calculated and then the participant’s average length estimation was subtracted to it:
Mini-blockLMNOPQNORS LTTRT = Mini-blockQULTQVL LMNOPQNORS− Mini-blockQULTQVL
Additionally, the mean average estimation error for each bias direction combination and bias source was calculated. As described for the average length estimation bias, the mean estimation error for each
Table C2.
bias source and bias direction combination was calculated and then the bias source bias was calculated as the difference between the mean estimation error of the short and long bias direction (see table C2).
