A mathematical description of theSimon task

ByTessa Bouzidi (10348212)

supervisor: Gilles de Hollander

A thesis submitted in partial fulfillment (18 EC) for the degree of

Bachelor of Science

in

Artificial IntelligenceFaculty of Science

University of Amsterdam

Science Park 904

1098 XH Amsterdam

June 22, 2016

A B S T R A C T

In this project two mathematical models have been studied on theirability to predict results in the Simon task. The first model is theDiffusion Model for Conflict (DMC), from which it was unknown tobe identifiable. First it is showed that the model is able to find thecorrect parameter value when all but one parameter (from in totaleight parameters) is fixed. The DMC has then also been tested if itis able to find the correct parameters when all eight are variable indifferent simulations, and showed to be unable to do so. The estima-tions and real values had a correlation almost equal to zero, meaningthat this model is unidentifiable, and thus unable to find the correctparameters belonging to the data. The DMC does shows the ability offitting to the real data of individual participants, but the general ac-curacy of its quantile predictions are much lower. The second model,the Fast-Guess Linear Ballistic Accumulator (F-G LBA) is afterwardsalso used to estimate the results in the same two real data sets, andshowed that it is more accurate than the DMC.

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C O N T E N T S

1 introduction 4

2 theoretical overview 6

2.1 Mathematical models for the Simon task 6

2.1.1 Identifiability 6

2.1.2 Validity 6

2.1.3 Chi-square test 9

2.1.4 SIMPLEX optimization 9

3 method 11

4 results 13

5 evaluation 16

6 conclusion 18

7 discussion and future work 19

Appendix 22

8 appendix a 23

9 appendix b 24

10 appendix c 25

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1I N T R O D U C T I O N

Human decision making occurs continuously and in a wide varietyof contexts. Many different factors can influence decision making,which complicates predictions about human decisions. Mathematicalmodels are helpful tools which facilitate and improve such predic-tions, giving a better understanding of the processes involved in hu-man decision making[1]. An improvement of the current knowledgeon this topic can have a great impact in the field of Artificial Intel-ligence. Artificially intelligent agents themselves, for example, arebound to make decisions. Improving their decisions to become moresimilar to human decisions will not only make them more intelligentin a human perspective, but could ultimately enhance the interactionbetween the agent and human beings.

This thesis describes a study on one specific decision making task,the Simon task, in which participants have to quickly push a left ora right button based on the colour of a given visual stimulus suchas a circle. The relevant information in the task is the specific colorof the circle, which indicates the correct button to push. The place-ment of the circle on the screen contains no information critical forthis endeavour. A trial is called incongruent when the place of thecircle is on the opposite side of the correct button to push. Researchhas shown that in certain instances, an irrelevant stimulus feature, itslocation, influences performance in the Simon task, as is the case inincongruent trials.[2]. In such trials, reaction times become slowerand mistakes are made more frequently in comparison to congruenttrials[2]. This difference in congruent and incongruent trials is calledthe congruency effect, which implies that irrelevant information af-fects humans in their decision making[3].

In an attempt to predict the congruency effect, different mathemat-ical models have been developed, such as the Diffusion Model forConflict (DMC)[4]. Ulrich et. al (2015) fitted this model on theirobserved data, and tested the accuracy of its predictions. However,evidence is lacking in the assumption that the DMC is identifiable,which implies that it is uncertain if the parameter values of a datasetcan be predicted. In other words, identifiability means the abilityof the model to retrieve a unique set of parameter values, given a

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result[5]. Ulrich et. al (2015) only checked this ability on aggregateddata, and not on the individual results. Only using the averages cancreate a biased result, which makes the identifiability of the DMC stillunknown.[6]A second mathematical model is the Fast-Guess Linear ballistic accu-mulator (F-G LBA) model, used by Noorbaloochi et. al (2015), whichhas been shown to be identifiable[7]. A comparison between the twomodels could yield the creation of an improved or consolidated ver-sion, and provide better insight in the human decision making pro-cess.

The aim of this project is to first investigate the identifiability of theDMC, after which observed data is used to test the accuracy of itsprediction on individual subjects. This will then be compared withthe accuracy of the predictions of the F-G LBA model on the samedata. The results will be evaluated in order to provide possible im-provements of both models, and knowledge about human decisionmaking. The main research questions that will be answered in thisthesis is: Is the diffusion model for conflict (DMC) identifiable, and howaccurate are its predictions compared to the Fast-Guess Linear Ballistic Ac-cumulator (F-G LBA) model?

2T H E O R E T I C A L O V E RV I E W

2.1 mathematical models for the simon task

The DMC dissects the congruency effect observed in the Simon task,assuming that the reaction time is built up of two different parts:the automatic and the decision making process. Further, the DMCstates that the congruency effect only appears in the first part, ie. thedecision making process[4]. In their study, Ulrich et. al (2015) fit themathematical DMC to their observed data, varying some parametersof the mathematical model, such that the observed behaviour is wellpredicted.

2.1.1 Identifiability

Identifiability entails the ability to find the specific set of variables ina model belonging to a given dataset[5]. When a model is not identi-fiable, there are thus multiple, and mostly even infinite, possibilitiesfor the parameter values that can result in the given dataset.[1] A sim-ple example is the model that returns the sum of two parameters: a +b = y. When the result, the value of y, is known, an infinite numberof possible values for a and b can match this result. Therefor, thissimple example model is not identifiable.If a model is not identifiable, the parameter values belonging to thedata are unsure, which makes the model less informative, or evenless useful. Since the DMC is not yet shown to be identifiable, theremight be no unique mapping between the results of an experimentand a set of parameter values. When researching the reliability of thismodel, it is thus of high importance to assure its identifiability. If themodel turns out to be identifiable, the results can be compared withneural data, for so called model-based neuroscience, to see how thefound parameter values correlate with the neural activity.[8]

2.1.2 Validity

A valid model is capable of accurate predicting the observations inreality, which in this research include specific patterns such as manyerrors in fast responses, or significant differences in reaction times

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2.1 mathematical models for the simon task 7

between the two types of trials. These patterns exist in the individualdata of participants, and should therefore be predicted on an individ-ual basis. So far, research has only shown that the DMC is able topredict those observations when the mean of several participants isused, so not only individual results.[4] This makes the validity of thismodel never completely proven, causing that it is unknown if individ-ual differences between participants can be predicted. Furthermore,this makes the usefulness of the model on predicting interesting pat-terns in real data uncertain.

In the diffusion model, eight parameters are important, and will beused to test its identifiability and validity[4]:

• The diffusion constant (σ), which determines the dispersion rateinside a trial.

• The decision boundary (b), which when reached means that thedecision has been made.

• The mean of non-decision component (µ r), that is the meanamount of time needed of the non-decision processes.

• The standard deviation of non-decision component (σ r), thestandard deviation of the time needed by the non-decision pro-cess.

• The amplitude of automatic activation (A), indicating the im-pact of the automatic reaction, caused by the irrelevant informa-tion, thus the placement of the stimulus.

• The drift rate of controlled processes(µ c), that shows how theaverage in the decision making process varies.

• The time to peak automatic activation (τ), determining the timeneeded to reach the top of the amplitude of the automatic reac-tion (A).

• The shape parameter of the starting point distribution (α), giv-ing the start point of the participant in the decision makingprocess.

Figure 1 shows how the two parts, the automatic and the decision pro-cess, influence the decision to make, according to the DMC. Reachingthe upper horizontal black line, at the height of parameter b, meansthat the correct button is chosen to push, whereas the lower horizon-tal line (-b) is the boundary for the incorrect button. The solid blackline is similar in the two graphs, showing that the influence of therelevant information, the colour of the circle, does not differ betweenboth trials. The grey lines visualize the difference in the decision pro-cess, having a bump towards the correct boundary in the left graph,

2.1 mathematical models for the simon task 8

Figure 1: The DMC model, showing the difference between congruent and incongru-ent trials against the reaction time in milliseconds[4].

whereas the bump in the right one gives a trigger to choose for theincorrect button. Parameter A represents the height of this bump,and the time the bump takes to peak is given by tau. The parametervalues with the half of their 95% confidence interval (HW) used byUlrich et. al (2015) can be found in table 1.

Table 1: The parameter values with the half of their 95% confidence interval, and theroot mean squared error found by Ulrich et. al (2015)

The ranges used in this project are these parameter values minus andplus the HW, since previous research showed that these values areable to fit the model to real data, so the sigma values in the simula-tions for example lie between (4.25-0.40 =) 3.85 and (4.25+0.40 =) 4.65.

0.0 0.2 0.4 0.6 0.8 1.00

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Figure 2: The F-G LBA model, showing how different options reach for the deci-sion bound, illustrated by the black horizontal line. The dotted lines showthe accumulation of the irrelevant information, for the correct (green) andincorrect (red) button. The solid line reflects the relevant information.

2.1 mathematical models for the simon task 9

The F-G LBA model has only an upper decision bound, and the accu-mulator that reaches that horizontal bound first, belongs to the choiceof the participant. In the Simon task, this model takes the irrelevantand relevant information separately into account, as is shown by thedotted (irrelevant) and solid (relevant) line(s) in figure 2. The placingof the stimulus in congruent trials, is called the fast accumulator forthe correct button (the green dotted line), whereas in the incongruenttrials it encourages the incorrect button (red line). The relevant infor-mation is enclosed in the green solid line, that could still cause theparticipant to make the correct choice in an incongruent trial if thissolid line reaches the upper bound before the red line (as is the casein figure 2).

2.1.3 Chi-square test

The chi-square test of Pearson is a test statistic to measure the simi-larity of two distributions[9]. This test is used in this project to showhow similar the expected results are to the observed ones. The ideais to split the expected reaction times, given a set of parameters, intosix different quantiles: the fastest first 10%, 10-30%, 30-50%, 50-70%,70-90%, and the slowest 10%. The reaction times on the borders ofthese quantiles are used to split the observed data also in six parts.To clarify this concept, let the fastest 10% of the expected data be de-fined as all the reaction times faster than 240ms. The percentage ofthe reaction times in the observed data, faster than the 240ms borderis, for example, 9%. Both percentages will be compared; in this case,the expected quantile was 10%, and the observed quantile was 9%,and the closer those percentages are, the lower the chi-square valuewill be. The parameter values of the expected data belonging to thelowest (best) chi-square value, are in the end returned to be the mostoptimal parameter set.If the parameter set of the observed data is known, the chi squareerror can be used in the optimization function to find the lowest errorvalue. The parameter values belonging to this optimal simulation arethen returned. This optimal parameter set can in the end be com-pared to the real values, in order to investigate the accuracy of themathematical model.

2.1.4 SIMPLEX optimization

To estimate the parameter values of the real data the Simplex opti-mization method is used, which optimizes the chi square error. Thischi square calculation is thus used as the cost function in the opti-mization. The Simplex optimization takes a set of start values as firstestimates for the variables, which can be represented as the vertexes

2.1 mathematical models for the simon task 10

Figure 3: Vertexes in the Simplex optimization and their reflection, where A has 2

variables, B 3, and C 4 situated in a 3-D factor space[10].

in figure 3. To optimize the values, so decreasing the chi square error,the geometric figure will flip, as is shown with the dotted lines, tocreate new vertexes (so new parameter values). It continues to doso until an optimal result, so a significantly low chi square error, isfound[10].

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M E T H O D

The DMC has first been implemented in python, for which iPythonnotebook is used - an online working environment for python pro-gramming - and checked with simulations on its correctness andefficiency[11]. The chi square error between the expected and ob-served simulated data are also implemented to be calculated correctly,taken into account the four different outcomes: congruent-correct,congruent-error, incongruent-correct, and incongruent-error.Observed data is then first simulated, using the parameter values oftable 1, after which different simulations of expected data are created,using the ranges of the values minus and plus the HW of table 1. Thenumber of trials in this observed simulation should start from 10.000

to make it prepared for outliers. However, the number of trials inthe expected simulation is preferred to keep smaller to make it moresimilar to the data, since the available real time data consists of ap-proximately 200 trials.These chi square error calculations will be done for simulations onlyvarying each parameter separately, but also for the combination oftwo parameters as the variables. If the values with the lowest chisquare error not correspond to the ones in the expected data simula-tion, the DMC can be assumed to be unidentifiable, but these resultscan not make sure yet if the model is identifiable.

To assure the identifiability, the correlation between expected and ob-served parameter values in a simulation having all eight parametersas variables will be calculated. If all expected and observed parame-ters show a high correlation, the model is proven to be identifiable.To check if the DMC is able to predict patterns in real data, and thus isvalid, the ability of the model to estimate all parameter values givenobserved data, should be tested. Even if the model turned out to beunidentifiable, it can still be valid. Simplex is an optimization methodthat will be used to test the validity. Starting points for all parametervalues have to be given, for which again a range of the values of table1 are used.

The reaction times and decision (correct or incorrect button) of thetwo real data sets will be compared to a simulation that uses the

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estimated parameter values by the optimize function, using graphsplotting the quantiles of both results. For the model to be valid, theestimated plots should be significantly similar to the real data plots.All results so far can provide an answer on the first part of the re-search question: is the diffusion model for conflict identifiable?

Now the real time data can be used to compare the DMC with theF-G LBA model. The F-G LBA model will also be estimating the pa-rameters of the same two data sets, after which the correctness of theestimations of both models can be compared. The second part of theresearch question: how accurate are the predictions of the DMC comparedto the Fast-Guess Linear Ballistic Accumulator (F-G LBA) model, will beanswered upon.

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R E S U LT S

Figure 7 in Appendix A contains the scatter plots of six parameters ofthe DMC, visualizing the chi square error when using each parametervalue, whereby the simulated correct values are the ones in table 1.The mean and standard deviation of the non-decisional componentare the only two variables that are not used in this part, since thesetwo are not part of the decision process that influences the magnitudeof the congruency effect. These scatter plots are made to check theability of finding the correct parameter given some data. The numberof trials of the expected data was 10.000, for the observed data thisis set at 200. The experiment has been done many times, to examinethe certainty of the ability of recovering the parameter values, andto make sure the amount of observations was not the cause of theinability of finding the correct parameters.

Two outcomes of the chi square error between observed and expecteddata, that uses two parameters as variables is visualized in heatmapsin figure 8 in Appendix A. The lighter the colour in the heatmap,the lower the error. So the lightest point is the most probable set ofparameter values. An unidentifiable model would show a light line,probably tubular shaped, containing all the most optimal value com-binations on the line, whereas an identifiable model would have onelowest point, so having a more conical shape.

Figure 4: The estimated paramater values plotted against the simulated ones.

In order to assure the model’s identifiability, the correlations betweenthe estimated parameter values and the simulated observed ones havebeen researched. These observed ones belong to a simulation, be-

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results 14

cause the exact values for the real data are unknown. This correlationis visible in figure 4. A perfect model would again show a diagonal.The correlation value can range from 0 to 1, and the higher it is, themore accurate the models’ estimates are, and thus the more likely themodel is to be identifiable. The exact values for different simulationsare shown in figure 5. Perfect estimations would amount to a cor-relation of 1, however, only finite data is available which makes thisperfect correlation very unlikely to occur.

Figure 5: The correlation between the observed and expected value for each parame-ter in the simulation with a range of different parameter values. Simulationis done using 100, 200, 1000 and 5000 number of trials.

The validity of the model has been studied with figure 9 in AppendixB, where the quantiles of the estimated parameter values by the Sim-plex optimization are plotted against the ones of the reaction times inthe real data. The more similar both plots are, the better the estima-tion is, and thus the more likely the DMC is to be valid.

For each quantile the expected reaction time boundary is plottedagainst the observed one in a simulation, see figure 6. This showsagain how accurate the model is in estimating results, and if thisimplementation is able to predict results in a simulation. A perfectmodel would have all results on the diagonal, as these points havethe exact same expected reaction times as observed.

After this simulation, the expected quantiles are also compared withthe observed ones in the real data sets. Two different datasets havebeen used, the first one, the speed stress data, had a time limit tomake the participants respond faster, and the second one was donein an fmri scanner, and is thus called the open fmri data. (obtainedby Gilles de Hollander, 2016). Figure 10 through 13 in appendix Cshow the outcomes for both models.

results 15

Figure 6: The observed reaction times of each quantile against the expected ones ina simulation using sampled observed parameter values, by the DMC

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E VA L U AT I O N

The scatter plots in figure 7 in appendix A are all parabolic shaped, inwhich the minimum is or tends to be the value used for the observeddata simulation. This implies that it is possible for the DMC to beidentifiable.Both heatmaps (figure 8) indicate a conical shape with the lowestpoint at the correct parameter values. This shape is not very clear,as there are many points with an (almost) equal low or even lowerchi-square value as/than the correct point. Also, the conical shapeseems, especially in right heatmap, rather stretched. This shows thatparameter recovery becomes more difficult and has a higher variancewhen only a small sample size is used.

From the correlation results in figure 4 and 5 can be concluded thatthe DMC is not identifiable. In these simulations all parameters areused as variables, a identifiable model would in this case still be ableto find all estimated and observed parameters to be correlated, whichis not the case for the DMC. Combining the results of both figures,only parameter t0 seem to show a correlation, as this is the onlyslightly diagonal shaped graph in figure 4, and has an increasingand quite high value in figure 5.

The parameter estimation by the DMC for the simulated data, as wellas the real data, figure 9 in appendix B, show an estimation with aquite high accuracy. Compared with the F-G LBA model the DMCseems slightly less accurate in its predictions.This difference is much more visible in the quantile plots in figure10 through 13 in appendix C, where the estimated quantiles of theF-G LBA are indicating a diagonal shape, whereas the ones from theDMC seem more randomly distributed. Especially interesting is fig-ure 13, indicating that the mean off all estimates would lie close to thediagonal, since all points, both above and below, seem to be equallyfar off the diagonal.In previous research (Ulrich et. al 2015), only the mean of all par-ticipants was taken in account, and not the individual data as donein this project. This figure shows that the mean would indeed esti-

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mate the correct values, as has been claimed[4], but would have anextremely high variance in individual estimates.

6C O N C L U S I O N

As figure 6 shows, the model is able to predict the simulated data, butas it is visible in figure 4, the correlation in the simulation betweenthe estimated and observed parameters are nearly zero. This showsthat it is unable to estimate the the parameters belonging to the data.Fitting the data could thus be done by different combinations of pa-rameter values, which means that the DMC is unidentifiable. Thisconclusion is also clear in the correlations of figure 5, where the val-ues are extremely low, and even not increase when more simulationsare used.

However, when all parameters except one are fixed, the correct valueof the one variable can be predicted, as figure 7 showed. Therefor,the use of a smaller set of the current parameters could make this anidentifiable and more accurate model.

The DMC seems to be valid when looking at the results in figure9, in which the predictions of the model are very similar to the realdata. However, figure 13 takes this conclusion in doubt, where thepredicted reaction times seem very inaccurate, and thus indicates thatthe DMC is not valid.

Comparing the quantile plots of the two models (figures 10 and 11

for dataset 1, and figures 12 and 13 for dataset 2), the F-G LBA modelseems more accurate in its predictions on the real data.

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D I S C U S S I O N A N D F U T U R E W O R K

The Simplex parameter optimization method started in every case inthis project with the mean values of table 1. This could have causedthe parameters to always estimate close to these values. Future workcould vary these start values to make sure the optimization workscorrectly and is not only estimating the simulation well because thesevalues always lie close to the start values.Also could one or more parameters be fixed, to research if this makesthe DMC identifiable. When doing so, it is also highly importantto assure the model is still valid, thus able to predict the real data.Specifically parameter sigma could first be fixed, since this is oftenthe case in other models.

The two models are currently only compared on their accuracy of thequantile predictions in two datasets. This might firstly be expandedto more data, to reassure the conclusions. Secondly, the influence ofthe different parameters of both models could be compared, in or-der to investigate if parameters cause the same effect. This means,study the effect on the results when making a specific parameter ofthe one model bigger or smaller, and compare this with the effect ofthe parameters in the other model. This parameter comparison couldthen lead to a new model, as an improved version of these currenttwo, and furthermore give a better insight in how the human deci-sion making works.

One final possibility for future research is to combine the results withmri data, as is available for dataset 2. The parameter knowledge canbe compared with the neural activity, but only if the parameter re-covery by the model is done correctly. This is not yet proven, so touse the DMC for model-based neuroscience, further research in itsparameter recovery is required.

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B I B L I O G R A P H Y

[1] Stephan Lewandowsky and Simon Farrel. Computational Mod-elling in Cognition. SAGE Publications, Inc, Thousand Oaks, Cal-ifornia, 2011.

[2] Bernhard Hommel. The simon effect as tool and heuristic. ActaPsychologica, 136(2011):189–202, 2010.

[3] B.U. Forstmann, R. Ratcliff, and E.-J. Wagenmakers. Sequentiasampling models in cognitive neuroscience: Advantages, appli-cations, and extensions. Annual Review of Psychology, 67:641–666,2016.

[4] Rolf Ulrich, Hannes Schrter, Hartmut Leuthold, and Teresa Birn-gruber. Automatic and controlled stimulus processing in conflicttasks: Superimposed diffusion processes and delta functions.Cognitive Psychology, 78:148174, 2015.

[5] L. Ljung and T. Glad. On global identifiability for arbitrarymodel parametrizations. Automatica, 30(2):265276, 1994.

[6] A. Heathcote, S. Brown, and D. J. K. Mewhort. The power law re-pealed: The case for an exponential law of practice. Psychonomicbulletin review, 7(2):185–207, 2000.

[7] S. Noorbaloochi, S. Sharon, and J. L. McClelland. Payoff infor-mation biases a fast guess process in perceptual decision makingunder deadline pressure: Evidence from behavior, evoked poten-tials, and quantitative model comparison. Journal of Neuroscience,35(31):1098911011, 2015.

[8] G. de Hollander, B. U. Forstmann, and S. D. Brown. Differentways of linking behavioral and neural data via computationalcognitive models. Biological Psychiatry: Cognitive Neuroscience andNeuroimaging, 1(2):101109, 2016.

[9] V. Bagdonavicius, R. Levuliene, M.S. Nikulin, and Q.X. Tran. Onchi-squared type tests and their applications in survival analysisand reliability. Journal of Mathematical Sciences, 199(2):8899, 2014.

[10] F. H. Walters, R. Lloyd, S. L. Morgan, and S. N. Deming. Se-quential simplex optimization: a technique for improving qual-ity and productivity in research, development, and manufactur-ing. Chemometrics series, 1991.

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[11] F. Perez and B. E. Granger. Ipython: a system for interac-tive scientific computing. computing in science & engineering.9(3):2129, 2007.

A P P E N D I X

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8A P P E N D I X A

The 2-D plots of the estimations of the DMC in a simulation for eachparameter separately, and the 3-D heatmaps for the combination ofµ c & b and σ & b.

Figure 7: From top down on the left side the 2D plots of sigma, mu c, and b, withthe correct simulated value on the green line, and on the right side A(10 executions), tau (10 executions), and alpha (100 executions), with thecorrect value at the exact middle point on the x-axis

Figure 8: The heatmaps of on the left mu c against b, and on the right sigma againstb. Both have the correct simulated value in the exact middle point of thegraph.

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A P P E N D I X B

The estimations of both models on the two real data sets.

Figure 9: The observed reaction times of each quantile against the expected ones, indata set 1 (’speed stress data’) for the visual data, by on the left side theDMC and on the right the F-G LBA model

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10A P P E N D I X C

The quantile-quantile plots of both models on the two datasets.

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Figure 10: The observed reaction times of each quantile against the expected ones,in data set 1 (’speed stress data’) by the F-G LBA model

Figure 11: The observed reaction times of each quantile against the expected ones,in data set 1 (’speed stress data’) by the DMC

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Figure 12: The observed reaction times of each quantile against the expected ones,in data set 2 (’open fmri data’) by the F-G LBA model

Figure 13: The observed reaction times of each quantile against the expected ones,in data set 2 (’open fmri data’) by the DMC