Interpreting the parameters of the diffusion model: An ... · method of estimating the parameters of the diffusion model based on the KolmogorovÐSmirnov (KS) test (Kolmogorov, 1941)

Copyright 2004 Psychonomic Society Inc 1206

Memory amp Cognition2004 32 (7) 1206-1220

Ratcliff (1978 Ratcliff amp Tuerlinckx 2002 RatcliffVan Zandt amp McKoon 1999) has proposed the diffusionmodel as a form of data analysis for speeded binary de-cisions The diffusion model assumes that binary deci-sions are based on a continuous process that fluctuatesbetween two possible outcomes (Figure 1) As soon asthe process reaches a critical upper or lower value a de-cision is made and the corresponding response is exe-cuted One main advantage of the diffusion model is thatdifferent components of the decision process (rate of in-formation uptake bias conservatism and motor compo-nent) are represented by different parameters of the modelTheoretically distinct aspects of the decision process canbe separated statistically In a sense the model allows in-ferences regarding hidden cognitive processes Anotheradvantage of the diffusion model data analysis is the highdegree of information utilization In contrast to conven-tional forms of data analysis the diffusion model incor-porates response times (RTs) for correct responses and er-rors as well as the ratio of correct and erroneous responses

The diffusion model thus provides a powerful statisti-cal tool that allows a very fine-grained analysis of the

cognitive processes underlying simple response tasksPrevious applications of the diffusion model have in-cluded for example the analysis of retrieval processes(Ratcliff 1978 1988) and the identification of the fac-tors underlying age-related slowing (Ratcliff Spieler ampMcKoon 2000 Ratcliff Thapar amp McKoon 2001 2003)These examples reveal that the diffusion model has awide range of possible applications and can be used todecide between competing theoretical interpretations ofexperimental findings However although interpreta-tions of the different parameters of the diffusion modelseem fairly straightforward it should be kept in mindthat these interpretations are based mainly on theoreticalreflections and presuppose the validity of the diffusionmodel as an adequate representation of the underlyingcognitive processes1 The main aim of the present articletherefore is to provide an empirical validation of the the-oretical interpretation of the parameters of the diffusionmodel To this end we investigated the effects of differ-ent experimental manipulations on the parameter set in adiffusion model data analysis The experimental manip-ulations were intended to specifically affect the differentprocessing components of the model (rate of informationuptake setting of response criteria duration of the motorresponse and bias see Ratcliff 2002 Ratcliff amp Rouder1998 Ratcliff et al 2001) Analyzing whether experi-mental manipulations of the different processing com-ponents that are specified in the model map uniquelyonto the corresponding parameters of the model (and not

Correspondence concerning this article should be addressed to A VossInstitut fuumlr Psychologie Albert-Ludwigs-Universistaumlt Freiburg Engel-bergstr 41 D-79085 Freiburg im Breisgau Germany (e-mail andreasvosspsychologieuni-freiburgde) or to K Rothermund Institut fuumlr Psy-chologie der Universitaumlt Jena Am Steiger 3 Haus 1 07743 Jena Ger-many (e-mail klausrothermunduni-jenade)

Interpreting the parameters of the diffusionmodel An empirical validation

ANDREAS VOSSAlbert-Ludwigs-Universitaumlt Freiburg Germany

KLAUS ROTHERMUNDFriedrich-Schiller-Universitaumlt Jena Germany

and

JOCHEN VOSSUniversity of Warwick Coventry England

The diffusion model (Ratcliff 1978) allows for the statistical separation of different components ofa speeded binary decision process (decision threshold bias information uptake and motor response)These components are represented by different parameters of the model Two experiments were con-ducted to test the interpretational validity of the parameters Using a color discrimination task we in-vestigated whether experimental manipulations of specific aspects of the decision process had specificeffects on the corresponding parameters in a diffusion model data analysis (see Ratcliff 2002 Ratcliffamp Rouder 1998 Ratcliff Thapar amp McKoon 2001 2003) In support of the model we found that (1) de-cision thresholds were higher when we induced accuracy motivation (2) drift rates (ie informationuptake) were lower when stimuli were harder to discriminate (3) the motor components were in-creased when a more difficult form of response was required and (4) the process was biased towardrewarded responses

DIFFUSION MODEL DATA ANALYSIS 1207

onto others) is a strong empirical test of the modelrsquos va-lidity Some of these manipulations have been used in asimilar manner by others (eg Ratcliff 2002 Ratcliff ampRouder 1998 Ratcliff et al 2001 2003) In the presentarticle however a somewhat different focus is adoptedby addressing two fundamental points First all the im-portant parameters of the diffusion model are validatedwithin the same paradigm and second the question ofvalidation is separated from other research questions(eg age-related changes in information processing)this separation allows for clear interpretation of the ex-perimental results

A second objective of this articlemdashnext to the valida-tion of the model parametersmdashis to introduce a newmethod of estimating the parameters of the diffusionmodel based on the KolmogorovndashSmirnov (KS) test(Kolmogorov 1941) This approach is an alternative tothe more established methods (ie maximum likelihoodchi-square and weighted least square see Ratcliff ampTuerlinckx 2002 for a detailed discussion of the differ-ent algorithms) The KS statistic also provides a test ofthe model fit

In the following we first will give a brief overview ofthe rationale of the diffusion model In a second step wewill describe our estimating algorithm and the new modeltest Finally we will report two experiments with a sim-ple color classification task in which the effects of dif-ferent experimental manipulations on the parameter setof the diffusion model were investigated Mathematicaldetails of the diffusion model are given in the Appendix

THE RATIONALE OF THEDIFFUSION MODEL

The diffusion model assumes that binary decision pro-cesses are driven by systematic and random influencesThe systematic influences are called the drift rate Theyldquodriverdquo the process continuously in one direction while therandom influences add an erratic fluctuation to this con-stant path The process is terminated immediately whenone of two thresholds is reached In a psychological appli-cation this means that the decision process is finished andthe response system is being activated which will initiatethe corresponding response (eg a keypress) With a givenset of parameters the model predicts distributions of pro-cess durations (ie RTs) for the two possible outcomes ofthe process Figure 1 provides a schematic illustration ofthe model The process begins at the starting point z andmoves over time until it reaches the lower threshold at zeroor the upper threshold at a Then the corresponding re-sponse is executed immediately In Figure 1 a sample pathof the process is presented As can be seen this path is notstraight but oscillates between the two boundaries due torandom influences (ie the intratrial variability of thepath) on the process

Applied to a perceptual situation in which participantshave to categorize ambiguous stimuli the process de-scribes the ratio of information gathered that fosters eachof the two possible stimulus interpretations Informationis collected over time until evidence points with suffi-cient clearness to one interpretation then a response is

a

z

d (t )i

it

Sample Path

Drift Rate (v )

Response A

Response B

0

Figure 1 Schematic illustration of the diffusion model The process starts at point z andmoves over time until it reaches the lower threshold at 0 or the upper threshold at a De-pending on which threshold is reached different responses are initiated The erratic line givesan example of one process that reaches the upper threshold at time ti The mean deviance ofsingle paths from the mean path is defined by an additional (scaling) parameter of the model(ie the diffusion constant s) Outside the thresholds the distributions of process durationsfor a constant set of parameters are sketched Note that the lower limit can be reached withrandom influences although the distributions shown are based on a positive drift rate

1208 VOSS ROTHERMUND AND VOSS

initiated As was mentioned above there are random in-fluences that cause variance in the RT distributions andalso generate erroneous responses in some trials For ex-ample the different sizes of the shaded areas that areconstituted by the time distributions in Figure 1 denotethat in most cases the process reaches the upper thresh-old indicating a positive drift rate Nonetheless in sometrials the random influences outweigh the drift and theprocess terminates at the lower threshold

Parameters of the Diffusion ModelThe model is defined by a set of parameters (see

Table 1) which will be described in the following sec-tion The upper threshold (a) which is also the distancebetween both limits has already been introduced Thisparameter is interpreted as a measure of conservatism Alarge value of a indicates that the process takes moretime to reach one limit and that it will reach the limit op-posite to the drift less frequently The second parameteris the starting point (z) or bias This parameter reflectsdifferences in the amount of information that is requiredbefore the alternative responses are initiated If z exceedsa2 a smaller (relative) amount of information support-ing Interpretation A is needed to give Response A ascompared with a larger (relative) amount of informationsupporting Interpretation B that is required before Re-sponse B is executed In other words participants aremore liberal in giving the answer connected to the upperthreshold (Response A) and more conservative in givingthe opposite response (Response B) If z is smaller thana2 the contrary is true If z diverges from a2 the aver-age decision times will differ for the different responsesThe smaller the distance between the starting point andthe limit the shorter the process durations will be for thecorresponding responses

The third parameter of the diffusion model is the driftrate (v) which stands for the mean rate of approach tothe upper threshold (negative values indicate an approachto the lower threshold) The drift rate indicates the rela-tive amount of information per time unit that is absorbedTherefore the drift can be interpreted as a measure ofperceptual sensitivity (in a between-person comparison)or as a measure of task difficulty (in a between-conditioncomparison) The path of the process is also influencedby (normally distributed) random influences (ie thenoise in the model) The amount of the random influ-ences is given by the diffusion constant (s) which is a

scaling parameter This means that s is not a free param-eter to be estimated but has to be fixed to any (positive)constant value In the Appendix the mathematical defi-nition of the diffusion model is presented

If the diffusion model is used to analyze RT data it isimportant to note that the decision process constitutesonly one component of the RTs measured The othercomponent consists of motor processes of the responsesystem and of encoding processes preceding the deci-sional phase These processes are not mapped by the dif-fusion process Nonetheless the model allows for esti-mating their total duration The parameter t0 indicatesthe duration of all extradecisional processes

Finally it might be necessary to allow for intertrialvariability of the different parameters in the model Rat-cliff (eg Ratcliff amp Rouder 1998) uses a model withintertrial variability of the drift rate (h) the startingpoint (sz ) and the time constant of the motor response(st ) The intertrial variability parameters of drift andstarting point are important because they allow for mod-eling of RT distributions in which error responses areslower or faster than correct responses (see Ratcliff ampRouder 1998) Therefore we included variability ofdrift and starting point For the sake of simplicity how-ever we did not allow for intertrial variance of the RTconstant Following Ratcliff we assumed that the actualdrift in one trial is normally distributed around the aver-age drift (v) with the standard deviation h the actualstarting point is supposed to be equally distributedwithin the interval from z sz to z sz

Estimating the ParametersThe aim of the estimation of the parameters is to opti-

mize the fit of the empirical and the predicted RT distri-butions for both types of responses simultaneously Forthe optimization process a criterion is needed that spec-ifies the degree of correspondence between the theoret-ical and the empirical distributions (goodness of fit)Ratcliff and Tuerlinckx (2002) compared three differentparameter estimation procedures (ie maximum likeli-hood chi-square and weighted least square) From ourpoint of view all these methods have specific problemsThe maximum likelihood method seeks to maximize thepredicted probability for each empirical RT This algo-rithm might be especially vulnerable to the contamina-tion of data (Ratcliff amp Tuerlinckx 2002) Especiallyoutlier times may strongly bias the parameter estimationhowever Ratcliff and Tuerlinckx are able to reduce thisvulnerability by introducing a new parameter that con-trols for contamination The chi-square and the weightedleast-square methods are not based on single RTs but onthe number of times found in certain ranges on the timeaxis These methods are quite coarse (because they arenot based on individual responses) and have the disad-vantage that the specification of the time segments is al-ways somewhat arbitrary2

An alternative to these three procedures for parameterestimation is presented in the following section The opti-

Table 1Parameters of the Diffusion Model

Parameter Description

Upper threshold (a) Response criterion (conservative vsliberal)

Starting point (z) Asymmetries in the response criterionDrift (v) Information uptake per timeResponse time constant (t0) Nondecisional portion of the response

time

NotemdashSee text for further explanations


mization criterion for the algorithm is provided by the KStest (Kolmogorov 1941 see also Conover 1999) This testis used to check whether an empirical distribution of val-ues matches a given theoretical distribution The test sta-tistic (T ) is the maximum vertical distance between thetheoretical and the empirical cumulative distributions(see Figure 2) For a given value of T and a given num-ber of instances (eg response times) the test providesthe exact probability p that the empirical times fit to thetheoretical distribution (Conover 1999 Miller 1956) Asignificant result (eg one below a 05) indicates thatcorrespondence between both distributions is unlikely

The statistic of the KS test (T ) can be used as an opti-mization criterion for the parameter estimation proce-dure Parameters of the model are varied in such a waythat T (ie the maximum vertical distance between theempirical RT distribution and the predicted time distrib-ution) becomes minimal Because T is directlymdashandmonotonicallymdashrelated to the probability of match be-tween the empirical and the predicted distributions it al-lows for an identification of the optimal set of parametervalues It might also evade the disadvantages of the othermethods mentioned above First in contrast to the max-imum likelihood method times with a predicted proba-bility nearmdashor even equal tomdashzero have no inflated in-fluence Second the procedure is fine graded because itis based on single RTs

The application of the KS test for evaluating themodel fit requires the comparison of two predicted dis-tributions with two empirical distributions (ie the dis-tributions at the upper and the lower thresholds) becausethe model is based on separate distributions for the RTsfor the two alternative responses To test the fit for bothdistributions simultaneously we have to associate thetwo predicted distributions and the two empirical distri-butions respectively This can be done by mirroring thedistribution of RTs for one type of response (eg the dis-tribution at the lower threshold) at the zero point of the

time dimension so that all the times in this distributionreceive a negative sign Please note that in most casesone threshold is linked to correct responses and the otherto erroneous responses In this case all error responseshave to be mirrored (or vice versa all correct responses)Composing a single RT distribution from the two RT dis-tributions for the different response alternativesmdashwithshort times lying at the inner edges (near zero) and thetails with the longer times at the outer edgesmdashallows fora single KS test3 Figure 3 illustrates this procedure Ifthere are different types of stimuli different drift ratescan be expected It might be promising not to estimatecompletely independent models for each type of stimu-lus but to determine that all parameters except for thedrift take the same value in both models In this case theproduct of the probabilities of the different KS statis-ticsmdashthat is the compound probability that the differentmodels fit the datamdashare used as the criterion in the op-timization procedure The T statistic of the KS test is tobe minimized in an iterative search process To search forthe minimum in the parameter space the simplex method(Nelder amp Mead 1965) was used In the Appendix themathematical definition of the diffusion model as wellas the formula for the KS test is provided

Testing the ModelThe rationale for the appropriate test of model fit is

based on the optimization criterion that has been de-scribed in the preceding section Again the KS statisticis used to evaluate the degree of match between the em-pirically observed RT distributions and the cumulativedensity function that is predicted on the basis of the pa-rameter values of the diffusion model However in a typ-ical experiment there are different participants andorexperimental conditions for which different models areestimated This implies that a KS test has to be con-ducted for each model (ie for participants and condi-tions different KS tests have to be conducted) To get an

10

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4

2

00 025 050 075 100 125 150

Response Time

p cum

T = 009

Figure 2 The statistic of the KolmogorovndashSmirnov test (T ) is the maximum vertical distance between two cumulativedistribution functions Each stage in the stepped line represents one response at the time represented on the abscissa


estimate of the overall fit of the models with the datathe number of significant KS tests (a 05) can itself besubjected to a binomial test with parameters of n (num-ber of tests) and p (a) A significant result of this overallbinominal test indicates that the diffusion model in thetested form does not fit the data from the total sampleThis method provides an exact test applicable to a wholesample of distributions which we consider to be a valu-able complement to graphical demonstrations of modelfits (ie quantile probability functions Ratcliff 2001Ratcliff amp Tuerlinckx 2002)

EXPERIMENT 1

In a f irst experiment we manipulated the relativespeed of information uptake conservatism and the du-ration of the motor response in a color discriminationtask These manipulations should specifically affect thedrift rate (v) the distance between thresholds (a) and themotor component (t0) respectively in a diffusion modeldata analysis If the predicted pattern emerges this isstrong support for the validity of the model

Altogether four different experimental conditions arereported First a standard condition which provided base-line values for all the parameters is presented In a sec-ond more difficult condition the stimuli were harder todiscriminate Therefore the information uptake shouldbe reduced which should result in a lower drift rate (v)In a third block an accuracy motivation was inducedThis manipulation should lead to more conservative re-sponse criteria and should increase the distance betweenthresholds (a) Finally a more difficult form of response

had to be executed In this condition an increase in thevalue of the motor component (t0) was predicted

MethodParticipants Thirty-six students (27 females and 9 males) at

the University of Trier participated in partial fulfillment of courserequirements

Design Essentially the design consisted of the within-subjectsfactor condition (standard difficult accuracy and response handi-cap) The four blocks were presented in random order

Apparatus and Stimuli The experiment was implemented onan IBM-compatible Pentium computer connected to a 17-in colormonitor using a Turbo Pascal 70 program in graphics mode Asstimuli we used colored squares (10 10 cm) consisting of 250 200 screen pixels Each pixel was either orange (RGB color values63 25 0) or light blue (0 25 63) The proportion of colors withineach stimulus was 5347 In the difficult condition the ratio was515485 Pixels were randomly intermixed For half of the stimuliorange was the dominant color for the other half of the stimuli bluewas dominant The stimuli were presented on a dark background

Procedure The participants started with a training block of 42trials Thereafter the four experimental blocks followed in randomorder Each block consisted of 2 warm-up trials and 40 experimen-tal trials Each trial had the following sequence First a gray squareof the same size as the stimuli was presented This square was re-placed by the colored stimulus 750 msec later The stimulus re-mained on the screen until a response was registered or the time limitof 1500 msec (3000 msec in the accuracy condition) was reachedwhichever happened first The remaining time was visually repre-sented by a vertical time bar that decreased in length below the stim-ulus The participantrsquos task was to decide which of the two colorswas dominantmdashthat is which color composed the major part of thestimulus Two marked keys on the computer keyboard (ldquoCrdquo andldquoMrdquo) were used for the responses The assignment of keys to colorswas counterbalanced across participants Directly after the responsethe word ldquocorrectrdquo or ldquowrongrdquo was presented below the color field

Figure 3 Comparison of empirical and predicted response time distributions Times for one of the two responses (eg allleft keypresses in a dual-response task) have been mirrored on the zero point of the time axis The mirrored times are rep-resented by the left part of the graph For example a response time of 0850 is recoded as 0850 The steep rise of the dis-tribution functions at about 06 indicates that many responses occurred with a latency of 06 sec The level of the functionin the middle of the graph shows that about 60 of all the reactions have been of the mirrored type The ascending blackline is the accumulated probability function computed according to the diffusion model The gray line shows the cumula-tive probability of empirical response times Note that both cumulative functions must converge to 1

10

8

6

4

2

0ndash25 ndash20 ndash15 ndash10 ndash05 00 05 10 15 20 25

(Mirrored) Response Time

p cum


depending on whether or not the correct answer had been given Si-multaneously the stimulus was replaced by a monochromatic fieldshowing the formerly dominating color Feedback was removedfrom the screen 1000 msec later and a new trial started

Experimental manipulations As was outlined above the ex-periment consisted of a standard condition and three variations Inthe difficult condition the dominant color occupied only 515 (in-stead of 53 in the standard condition) of the pixels of a stimulusSecond there was an accuracy condition in which the RT limit wasdoubled (3 sec instead of 15 sec) In addition the participants wereinstructed to work especially carefully and to avoid mistakes in thisblock The third variation refers to the form of response In the re-sponse handicap condition the participants were told to use thesame finger for both responses To ensure that the finger was posi-tioned between both response keys before the stimulus was shownthe participants had to press a key that lay between the two responsekeys (ie the ldquoBrdquo key) to start each trial

ResultsParameter estimation For each participant four in-

dependent models were calculated one for each experi-mental condition Each of these models allowed for twodifferent drift rates (for the two stimulus types) and com-prised two different diffusion models (see the Estimatingthe Parameters section)

Therefore there were 36 4 144 independentmodels based on 40 trials each For the interpretation ofparameters average values were computed across par-ticipants All responses below 300 msec were discardedprior to parameter estimation (1 of the trials) This isimportant because low outliers may have a strong influ-ence on the complete model (see Ratcliff amp Tuerlinckx2002 for a discussion of the influence of outliers)

For all blocks of the experiment a model was chosenin which the upper threshold was associated with the or-ange response (ie the press of the orange key) and thelower threshold was associated with the blue responseFor the two types of stimuli (orange dominant vs bluedominant) two different drift rates were estimated Strictlyspeaking our model is composed of two diffusion modelswith the constraint that all the parameters except drift rate(a z t0) are identical for both models Please note thatthese two diffusion models were always estimated si-multaneously in a single estimation process Togetherf ive parameters were estimated a z v1 v2 and t0Table 2 shows the mean parameter values across partic-ipants for all blocks in the experiment To facilitate theinterpretation za is presented as a measure of bias in-stead of z Values greater than 05 on za indicate that thestarting point lies closer to the upper threshold (a)whereas lower values indicate a starting point closer tothe lower threshold

Within-block analyses Table 2 shows that the pro-cesses always started roughly at the midpoint betweenboth thresholds One-group t tests revealed that only inthe response handicap condition did z diverge signifi-cantly from a2 [t(35) 239 p 05]

The v1 parameter is the drift rate for orange-dominatedstimuli In all blocks v1 had a positive value [all ts(35) 384 p 001] indicating a positive mean gradient for

the process Contrarily the drift in trials with blue-dominated stimuli (v2) was negative in all conditions [allts(35) 535 p 001] To check for symmetry theabsolute values of the drift parameters were comparedbetween orange-dominated and blue-dominated stimuliin four paired t tests These analyses revealed a signifi-cantly larger drift rate for blue-dominated stimuli in theresponse handicap condition [t(35) 209 p 05]No other comparison reached significance [all ts(35) 132 p 19]

Between-block analyses To test the influence of theexperimental manipulations all the parameters in thethree variation blocks were contrasted with the param-eters in the standard condition All significant effects aremarked in Table 2 The absolute values of the drift pa-rameters in the difficult condition were lower than thosein the standard condition both for v1 [t(35) 244 p 05] and for v2 [t(35) 319 p 01] The distance be-tween thresholds (a) was larger in the accuracy conditionthan in the standard condition [t(35) 463 p 001]In addition we found an increased value for t0 in this con-dition [t(35) 371 p 01] Finally in the responsehandicap condition the RT constant (t0) significantly ex-ceeded the value in the standard condition t(35) 827p 001] In this condition the relative starting point (zr)also was somewhat increased [t(35) 239 p 05]and the drift parameter v2 was decreased [t(35) 232p 05] No other contrasts reached significance

Model fit As was outlined above we performed in-dividual tests of fit separately for each diffusion model(ie for each participant and for each condition) Therewere 2 (stimulus types) 4 (condition) models for eachof the 36 participants resulting in a total of 288 modelsThe KS tests revealed no significant result at the 05level It can be concluded that the individual models de-scribe the individual RT distributions well

To demonstrate model fit in a more descriptive wayempirical and predicted values of the medians of the RTdistributions for correct and erroneous responses and of

Table 2Average Parameter Values for Different Experimental

Conditions of Experiment 1

Condition

ResponseParameter Standard Difficult Accuracy Handicap

a 118 119 137 116za 049 047 052 056v1 089 046 091 102v2 103 057 118 141t0 047 047 052 073h 080 090 085 089sz 009 009 009 008

NotemdashThe parameters describe diffusion models with the orange re-sponse linked to the upper threshold and the blue response linked to thelower threshold The parameter v1 represents the drift for orange-dominated stimuli and v2 represents the drift for blue-dominated stim-uli All tests refer to a comparison with the standard condition p 05 p 01


proportions of correct responses for all the conditions inExperiment 1 are presented in Figure 4 As can be seenfrom the high concordance of the empirical and the pre-dicted values there is a good fit for most of the partici-pants However there are some points with rather large

deviations of predicted and empirical medians Theselarge deviations are found only for distributions of errorresponses and especially where very few errors weremade (in most cases fewer than 10) This indicates thatreliable predictions require a somewhat higher number

1500

1250

1000

750

500

250

100

80

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40

250 500 750 1000 1250 1500

40 60 80 100

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median

Proportion Correct (Empirical)

ResponseCorrect

Error

ConditionStandard

Difficult

Accuracy

Resp Handicap

ConditionStandard

Difficult

Accuracy

Resp Handicap

Figure 4 Individual model fit (Experiment 1) The top panel showsthe concordance of the medians of the empirical and the predicted re-sponse time (RT) distributions For each participant and experimentalcondition the distributions of correct responses and errors are repre-sented by single symbols In case of large misfit the number of RTsbuilding the empirical distribution is also shown The bottom panelshows the concordance of the proportions of correct responses


of RTs within each empirical distribution This is also il-lustrated by the fact that there is a significant negativecorrelation of the absolute deviations between the medi-ans of the empirical and the predicted RT distributionsand the number of RTs making up the empirical distrib-ution (r 42 p 001]

DiscussionThe results provide evidence that the diffusion model ad-

equately represents the different process components thatare involved in speeded binary color dominance decisionsThe within-block analyses showed that the decision pro-cess is roughly symmetric The process starts at the mid-point between thresholds and is driven upward by orange-dominated stimuli or downward by blue-dominated stimuliMore interesting are the between-block comparisons Thedifferent conditions had been implemented to investigatethe validity of the interpretation of the parameters Itshould be demonstrated that each parameter reflects spe-cific components of the decision process and is not in-fluenced by other processes Only if the validity of the pa-rameters is demonstrated can diffusion model results beinterpreted clearly In detail we found decreased drift rateswhen we made stimuli harder to discriminate an increaseddistance between thresholds when we induced an accu-racy motivation and a substantial increase in the motorcomponent when we introduced a more time-consumingform of response Some unexpected effects emerged aswell There was an increase of t0 in the accuracy condi-tion This finding can be easily accommodated becauseit is plausible that participants execute their responsesmore slowly if time pressure is reduced which was thecase in this condition The changes in drift rate and start-ing point in the difficult condition however cannot eas-ily be explained

The KS tests revealed a good model f it for all themodels (ie for all participants and all conditions) Thisis an additional clue that the diffusion model was appro-priate for describing the data without losing much infor-mation Descriptive data on the model fit showed that fitwas worst for the error distributions of the accuracy con-dition This was probably due to the low numbers of er-rors in this condition

EXPERIMENT 2

In Experiment 1 the parameter z which reflects thestarting point of the diffusion process was not manipu-lated Therefore Experiment 2 focused on the empiricalvalidation of the interpretation of the z parameter Againa color discrimination task was used This time howeverthe two responses were not equivalent anymore The par-ticipants collected points during the experiment and anasymmetric payoff matrix promoted one response overthe other We expected that this manipulation would biasthe starting point of the diffusion process toward thethreshold that was linked with the rewarded responsemdashthat is less information would be needed to reach thecorresponding threshold as compared with the opposite

threshold As a second variation the stimuli were pre-sented in two different color ratios

MethodParticipants Twenty-four students (17 females and 7 males) at


Design One response was fostered by an asymmetric payoff ma-trix Which of the two responses was selected as the privileged re-sponse (orange vs blue) was counterbalanced across participants

Apparatus and Stimuli The procedure was the same as that inthe previous experiment with the following exception The stimuliwere presented in two different color ratios 5248 and 51494

Procedure The experiment consisted of 12 training trials and160 experimental trials which were presented without intermissionin one block each of the four stimulus types (orange dominant vsblue dominant easy vs difficult) was presented 40 times Trialswere presented in random order The participants received pointsfor correct answers and lost points for errors These point creditswere exchanged into monetary payoffs later For each participantone color response had a higher expected value because of an asym-metric payoff matrix (Table 3) If the promoted response was givencorrectly 10 points were credited if it was given mistakenly how-ever only 5 points were lost Opposite rules held for the nonpro-moted response Only 5 points were gained when the color responsewas given correctly but 10 points were subtracted when an erroroccurred These rules were explained carefully in the instructionsDuring the experiment the current amount of points was visible atthe bottom of the screen In addition gains of 5 points were sig-naled by one high-frequency beep and gains of 10 points by twohigh-frequency beeps Accordingly losses of 5 or 10 points wereaccompanied by one or two low-frequency beeps respectively Atthe end of the experiment the participants received 20 Euro Cent(about $018 at that time) for each 100 points they reached The pro-cedure resulted in a mean benefit of 128 Euro (about $115)

ResultsParameter estimation RTs below 300 msec (03 of

all the times) were excluded from all analyses In Exper-iment 2 a model with four drift parameters one for eachstimulus type was estimated (ie this model consistedof four connected diffusion models see the Estimatingthe Parameters section for further explanations) The pa-rameters v1 to v4 described the drift rate for a descendingproportion of orange pixels (52 51 49 and 48see the Apparatus and Stimuli section) Furthermorethere were the same parameters as above a z t0 h andsz resulting in a total of nine free parameters for eachmodel For each participant one model was computedon the basis of all 160 experimental trials The averageparameter values for both groups of participants areshown in Table 4 We expected that the manipulation

Table 3Payoff Matrix Used in Experiment 2

Stimulus Color

Response Color 1 Color 2 S (Response)

Color 1 10 5 5Color 2 10 5 5

S (stimulus) 0 0

NotemdashSelection of the promoted response (Color 1) was counterbal-anced across participants


should bias the relative starting point of the processmdashthat is the position of z in relation to a Therefore therelative starting point (za) is presented in Table 4 Avalue of 50 for za indicates that the process starts at themidpoint between both thresholds larger values indicate

that the starting point is closer to the orange thresholdthan to the blue threshold and vice versa

Statistical analyses All the parameters were testedfor between-group differences (see Table 4) A signifi-cant difference emerged only for the relative starting

Figure 5 Individual model fit (Experiment 2) The top panel showsthe concordance of the medians of the empirical and the predicted re-sponse time distributions For each participant and each of the two dif-ficulty levels the distributions of correct responses and errors are rep-resented by single symbols The bottom panel shows the concordance ofthe proportions of correct responses

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400 600 800 1000

40 50 60 70 80 90

Pre

dict

ed M

edia

nP

ropo

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orre

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Pre

dict

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Empirical Median


Response

Correct

Error

StimulusEasy

Difficult

StimulusEasy

Difficult


point za [t(22) 408 p 01] The effect reveals thatthe starting point was biased toward the promoted re-sponsemdashthat is the participants were more liberal ingiving the promoted answer and more conservative re-garding the nonpromoted answer In the between-groupcomparison no other effects reached significance

The drift parameters were analyzed additionally in a 2(group) 4 (color ratio) analysis of variance There wasa strong effect of color ratio [F(320) 8955 p 001] indicating larger (ie more positive) drift rates forhigher proportions of orange in the stimuli No other ef-fects reached significance

Model fit In this experiment 4 (stimulus types) 24(participants) 96 different probability distributions werepredicted by the models Six of the KS tests detected a sig-nificant (a 05) deviation of the empirical from the pre-dicted distribution The probability of finding six or moremismatches with a 05 in 96 tests was p 35 indicat-ing that the models describe the empirical RT distributionswell Again model fit should also be demonstrated de-scriptively The coherence model predictions and the em-pirical distributions can be seen on an individual level inFigure 5 Again the medians of the empirical RT distribu-tions are plotted against the medians of the predicted dis-tributions both for errors and for correct responses

DiscussionThe second experiment was carried out to validate the

interpretation of the bias parameter (za) in the diffusionmodel Specifically we investigated whether biasing re-sponse tendencies in a binary decision task by means ofan asymmetric payoff manipulation is mapped specifi-cally onto the bias parameter in a diffusion model dataanalysis The parameter z indicates the starting point ofthe diffusion process and thus definesmdashin combinationwith the total distance between thresholdsmdashthe neces-sary amount of information that is needed to draw a spe-cific conclusion In an unbiased model z will equal a2mdash

that is the same amount of information is required foreach of both decisions The closer z moves to one of theboundaries the more biased is the process In this casethe preferred response is initiated quickly and often

In Experiment 2 we successfully manipulated z byldquorewardingrdquo the participants for executing one of the tworesponses The starting point moved toward the thresh-old that was connected with the rewarded answer Noother parameter was affected by this manipulation Asrevealed by the KS tests model fit was good

GENERAL DISCUSSION

Diffusion models were introduced in psychology byRatcliff (1978) two and a half decades ago These mod-els describe a theoretical process of decision making In-terestingly recent neurobiological studies based on single-cell recording have shown activation patterns prior toaction and decision making that resemble the path of thediffusion process (see Schall 2001 for an overview)

Only the recent developments in computing powerallow for efficient parameter estimation Our procedurediffers from the methods employed by Ratcliff in a cou-ple of ways (1) We use distributions of single RTs in theestimation process instead of using only certain per-centiles of the RT distribution (2) We introduced a newoptimization criterion to estimate the parameters thatwas deduced from the KS test (Kolmogorov 1941 seealso Conover 1999) (3) The KS test is also used as a testof the model fit This method allows a clear-cut statisti-cal test of whether or not a model fits the data

The estimation procedure presented in this article hasbeen developed on the basis of theoretical considera-tions Because the KS statistic provides an exact test forthe match of an empirical and a predicted distributionthis measure for the probability of match seems to be anoptimal criterion for parameter estimation So far noempirical comparisons with other estimation procedureshave been made (eg concerning the influence of con-taminants see Ratcliff amp Tuerlinckx 2002) althoughthis is clearly an interesting challenge for the procedureAnother important question for the application of themodel is how reliably the different procedures (ie KSmaximum likelihood and chi-square) can estimate theparameters in cases in which there are the relativelysmall sets of data (ie few RTs per participant) typicallyused in cognitive research It is plausible that proceduresbased on single RTs (such as the KS-based procedure orthe maximum likelihood procedure) may be better at thisbecause they use the total amount of information To em-pirically answer such questions more simulation studieshave to be conducted

The diffusion model provides plausible interpretationsfor the estimated parameters The distance betweenthresholds (a) is interpreted as a measure of conser-vatismmdashthat is the larger a is the more information iscollected prior to a response being given The startingpoint (z) is an estimate of relative conservatism or bias

Table 4Average Values of the Diffusion Model Parameters for Both

Groups of Experiment 2

Promoted Response

Parameter Orange Blue Difference

a 113 119 006za 054 044 010v1 070 069 001v2 046 003 042v3 047 069 022v4 070 102 032t0 044 049 005h 102 111 009sz 013 012 001

NotemdashThe parameters describe diffusion models with the orange re-sponse linked to the upper threshold and the blue response linked to thelower threshold The drift parameters v1 to v4 are linked to the four dif-ferent stimulus types with a descending proportion of orange pixels Inaddition to the estimated parameter the relative starting point (za) isshown (see the text for further details) p 01


If the process starts above or below the midpoint be-tween the two thresholds different amounts of informa-tion are required for both responses that is a more con-servative decision criterion is applied for one responseand a more liberal criterion for the opposite responseThe drift rate (v) indicates the (relative) amount of in-formation gathered per time denoting either perceptualsensitivity or task difficulty The parameter t0 finallyprovides a measure of the duration of nondecisionalcomponents of the response processmdashthat is first of allof the motor processes

A possible criticism of the diffusion model is that ituses a fairly large number of parameters to model RTdistributions Given that it is always easy to fit any pos-sible data set provided that enough independent param-eters are available this leads to the question of whetherunambiguous interpretations of the parameters of themodel are warranted In a recent set of simulation stud-ies Ratcliff (2002 Ratcliff amp Tuerlinckx 2002) demon-strated that diffusion models impose a number of impor-tant restrictions on data and that violations of theserestrictions can be detected by a low fit of the modelHowever this still leaves open the question of whetherthe parameters of the model yield unique estimates ofdistinct components of real decision processes For ex-ample the model might be insensitive in disentanglingdifferent components of the decision process and mightshow a strong tendency to represent different compo-nents of the process on only one parameter Alterna-tively the model might have difficulty in uniquely iden-tifying a certain component of the decision process withonly one parameter Variations in different parameters ofthe model might be equifunctional in representing dif-ferences in the same component of the decision processso that only slight variations in the data set can lead todramatic changes of the parameter estimates Ratcliffand Tuerlinckx have shown that parameters can be re-produced reliably from a set of simulated data even ifcontaminated times are added after the simulation How-ever this does not demonstrate whether the different pa-rameters represent specific cognitive processes An eval-uation of the empirical validity of the parameter estimatescannot be decided in theory but requires empirical testingThe experiments reported in this article are a step in thisdirection We used simple straightforward experimentalmanipulations that should uniquely influence certaincomponents of the decision process The face validity ofthe manipulations is of utmost importance because it isa prerequisite for taking the observed effects as a test ofthe modelrsquos validity rather than as a test of what wasbeing manipulated In contrast to other studies based onsimilar paradigms (eg Ratcliff et al 2001 2003) inthe present article systematic manipulations within oneparadigm have been used to check for cognitive pro-cesses related to the different model parameters in theabsence of any other research question

Two experiments were carried out to test whether sim-ple experimental manipulations aimed at influencing spe-

cific components of the decision process in a straightfor-ward way map onto corresponding parameters of themodel while leaving others unaffected In Experiment 1stimuli that were difficult to categorize produced smallerdrift rates than did easy stimuli an induction of accuracymotivation increased the conservatism parameter and aresponse handicap did not affect the decision-related pa-rameters but led to increments in the motor componentIn Experiment 2 an asymmetric payoff matrix promot-ing one of the two responses moved the starting point to-ward the threshold that was linked to the promoted re-sponse In sum our findings clearly support the validityof the previous interpretations of the parameters of thediffusion model5

CONCLUSION

RT data from binary decision tasks are a major sourceof information in experimental psychological researchThe diffusion model (Ratcliff 1978) offers a powerfulapproach for analyzing these data The separation andidentification of different components of the decisionprocess allow fine-grained interpretations of the find-ings that also provide interesting information for theo-retical debates regarding the processes that underlie ormediate experimental effects To give just a few examplesa diffusion model data analysis might help to clarify thenature of the processes involved in cognitive aging (Rat-cliff et al 2000 Ratcliff et al 2001 2003) of the under-lying processes in associative or affective priming (Wen-tura 2000) or of what drives effects in the IAT (BrendlMarkman amp Messner 2001 Rothermund amp Wentura2001) Another advantage of this form of data analysis isthat it is able to integrate a large amount of the availableinformation in only one analysis (eg RTs for correct re-sponses and errors and percentages of errors) These ex-amples reveal that the diffusion model is a highly promis-ing statistical tool with a very wide range of potentialapplications In the present article we have reported somenew strategies that further improve the application of themodel (new techniques of parameter estimation andmodel testing) The main focus of the article howeverwas two validation experiments that yielded further evi-dence in support of the modelrsquos capacity to identify hiddencognitive processes in binary decision tasks

REFERENCES

Brendl C M Markman A B amp Messner C (2001) How do in-direct measures of evaluation work Evaluating the inference of prej-udice in the Implicit Association Test Journal of Personality amp So-cial Psychology 81 760-773

Conover W J (1999) Practical nonparametric statistics (3rd ed)New York Wiley

Feller W (1971) An introduction to probability theory and its appli-cations (Vol 2) New York Wiley

Kolmogorov A (1941) Confidence limits for an unknown distribu-tion function Annals of Mathematical Statistics 12 461-463

Miller L H (1956) Table of percentage points of Kolmogorov sta-tistics Journal of the American Statistical Association 51 111-121


Nelder J A amp Mead R (1965) A simplex method for function min-imization Computer Journal 7 308-313

Ratcliff R (1978) A theory of memory retrieval Psychological Re-view 85 59-108

Ratcliff R (1988) Continuous versus discrete information process-ing Modeling accumulation of partial information PsychologicalReview 95 238-255

Ratcliff R (2001) Diffusion and random walk processes In Interna-tional encyclopedia of the social and behavioral sciences (Vol 6pp 3668-3673) Oxford Elsevier

Ratcliff R (2002) A diffusion model account of response time andaccuracy in a brightness discrimination task Fitting real data andfailing to fit fake but plausible data Psychonomic Bulletin amp Review9 278-291

Ratcliff R amp Rouder J N (1998) Modeling response times fortwo-choice decisions Psychological Science 9 347-356

Ratcliff R Spieler D amp McKoon G (2000) Explicitly modelingthe effects of aging on response time Psychonomic Bulletin amp Re-view 7 1-25

Ratcliff R Thapar A amp McKoon G (2001) The effects of agingon reaction time in a signal detection task Psychology amp Aging 16323-341

Ratcliff R Thapar A amp McKoon G (2003) A diffusion modelanalysis of the effects of aging on brightness discrimination Percep-tion amp Psychophysics 65 523-535

Ratcliff R amp Tuerlinckx F (2002) Estimating parameters of thediffusion model Approaches to dealing with contaminant reactiontimes and parameter variability Psychonomic Bulletin amp Review 9438-481

Ratcliff R Van Zandt T amp McKoon G (1999) Connectionist anddiffusion models of reaction time Psychological Review 106 261-300

Rothermund K amp Wentura D (2001) Figurendashground asymme-tries in the Implicit Association Test (IAT) Zeitschrift fuumlr Experi-mentelle Psychologie 48 94-106

Schall J D (2001) Neural basis of deciding choosing and actingNature Reviews Neuroscience 2 33-42

Wentura D (2000) Dissociative affective and associative priming ef-fects in the lexical decision task Yes versus no responses to word tar-gets reveal evaluative judgment tendencies Journal of ExperimentalPsychology Learning Memory amp Cognition 26 456-469

NOTES

1 Whether the model adequately represents the underlying cognitiveprocesses in a task remains an open question even when the model fitsthe data well because there are always infinitely many theoretically dis-tinct models that fit the data equally well

2 The number (and the width) of the time segments may influencethe result of the estimation process (Slightly) different parameter val-ues may for example emerge in estimation processes based on five orsix segments It is hard to pronounce upon what would be an optimalnumber of segments If few large segments are chosen much informa-tion is lost (concerning the exact shape of the distribution) if manysmall segments are chosen the number of empirical RTs in each islargely chance dependent

3 We thank Heinrich von Weizsaumlcker for this ingenious hint4 A third stimulus condition with color ratios of 5347 was also in-

cluded in the experiment Trials in this condition were not entered intothe analyses because model fit was not especially good for this condi-tion The low model fit is probably due to the extremely low error ratesin this condition (median of the number of errors 8) Nonetheless theresults regarding the starting point of the diffusion process do notchange substantially if this condition is included in the analysis

5 With only two data points for each manipulation it is not possibleto describe the exact functions relating the parameter values to physicalproperties of the stimuli However it is clear that monotonic relation-ships will emerge (Ratcliff amp Rouder 1998 point out a probability-matching explanation of the drift rate)

APPENDIXMathematical Details

Mathematical Definition of the Diffusion ModelThe probability that a diffusion process with constant drift v and start at z 0 will hit a z before time t

and before the first visit at 0 has a density which we shall denote by g (t z a v) We have two different rep-resentations for this density (all formulas are derived from Feller 1971)

(A1)

and

(A2)

By symmetry we get the corresponding densities for the probability that a diffusion process with constantdrift v and start at z 0 will hit 0 before time t and before the first visit at a z as g(t z a v) g (t aza v) This gives

(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=

bull= loz -( ) loz Ecirc

Eumlˆmacr loz - loz +( ) loz[ ]Acirc( ) exp sin exp p p p

21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=

bull= loz -[ ] loz -Egrave

IcircIacute˘˚

loz - loz +( ) loz[ ]Acirc( ) exp ( ) sin ( ) exp p p p2

1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ

macr˜ loz + -[ ]Acirc( )

exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p


APPENDIX (Continued)

The case of a diffusion constant s other than 1 can be reduced to the case above If we scale down every-thing by the factor s we get an ordinary diffusion with diffusion constant 1 with constant drift vs start at zsand the upper threshold as So the probability that a diffusion process with diffusion constant s drift v andstart at z 0 will hit a z before time t and before the first visit at 0 has the density g (t zs as vs)

We need to evaluate these series numerically The speed of convergence in these series is determined by theexponential terms in the sum So the series in Formula A1 will converge quickly when t is small in compar-ison with 2a2 and the series in Formula A2 will converge quickly when t is large in comparison with p 22a2An advantage of Formula A2 is that we can calculate the distribution function The probability that a diffu-sion process with constant drift v and start at z will hit a z before time t and before the first visit at 0 is

(A5)

For our parameter estimation we use Formula A5 Note that t is not the RT but the duration of the decisionprocess Therefore t rt t0 has to be substituted in the equation

Necessary Number of Steps in the Calculation of the Infinite Series of Formula A5Here we derive an upper bound for the difference between the exact value of the infinite sum from For-

mula A5 and the finite sum which we will calculate using our program We can then choose the required num-ber of steps in order to keep the error below a given bound e

1 For the calculation below we will need a bound for

that does not depend on n Since

we will use

(A6)

for this bound Because the set

is a circle in the complex plane the possible values of the fraction in Formula A6 still form a (scaled and trans-lated) circle in the plane

Using a Re(1eic) 1cos(c) b Im(leic) sin(c) and r |1eic| dividea2 b2 divide22cos(c)the center of this circle has the real part ar 2 and the imaginary part br 2 The radius of the circle is lr sowe conclude that

(A7)

2 The next step is to obtain a general result about the tails of oscillating series Let an be a sequence mo-notonically decreasing to 0 Analogous to partial integration we find

and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M

sin( ) sin( ) sin( )

( ) ( )

=+

= =+

=

+ + loz

Acirc Acirc Acirc Acircpound -( ) loz + loz

pound -( ) +

1 1

1 1

a a ck a cn a ckn na N

M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =

Acirc Acirc Acirc Acirc1 1sin( ) sin( ) sin( )

S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc

=loz -[ ] loz -Egrave

IcircIacute˘˚

loz - loz +( ) loz[ ]=

loz -[ ] loz -EgraveIcircIacute

˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +

= loz -[ ] loz -EgraveIcircIacute

˘˚

loz- - loz +( ) loz[ ]

+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p



Letting M AElig bull proves

This helps to estimate the tail of the series which is omitted from the calculation3 In Formula A5 we have

and

From c we can calculate S(c) with Formula A7 and then use the following estimate

(A8)

Formula A8 gives an upper bound for the difference between the theoretical value from Formula A5 andthe value that our program calculates (summing up to N) To achieve a prescribed precision e we simply in-clude enough terms in the sum to make an + 1S(c) smaller than e

4 A simplification is possible because for large n the exponential term and the term a2v22 in the defini-tion of an make almost no difference So we get

If we want to estimate the integral up to accuracy e we can use

(A9)

to get

Formula A9 gives the required number of steps to keep the error smaller than e5 Examples For the results from Experiment 1 (see Table 2) we get the following required numbers of

steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )

g r z a dr a cn a cn a cn

a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+

Uacute Acirc Acirc Acirc

Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30



The Probability of the KS StatisticA formula for the probability of a given value of the KS statistic depending on the number of observations

is provided by Conover (1999)

(A10)

where T is the observed KS statistic n is the number of observations and [n(1T )] is the maximum integerless than or equal to n(1T ) Note that Formula A10 is valid for the two-sided test

(Manuscript received July 30 2003revision accepted for publication February 21 2004)

p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )


onto others) is a strong empirical test of the modelrsquos va-lidity Some of these manipulations have been used in asimilar manner by others (eg Ratcliff 2002 Ratcliff ampRouder 1998 Ratcliff et al 2001 2003) In the presentarticle however a somewhat different focus is adoptedby addressing two fundamental points First all the im-portant parameters of the diffusion model are validatedwithin the same paradigm and second the question ofvalidation is separated from other research questions(eg age-related changes in information processing)this separation allows for clear interpretation of the ex-perimental results

A second objective of this articlemdashnext to the valida-tion of the model parametersmdashis to introduce a newmethod of estimating the parameters of the diffusionmodel based on the KolmogorovndashSmirnov (KS) test(Kolmogorov 1941) This approach is an alternative tothe more established methods (ie maximum likelihoodchi-square and weighted least square see Ratcliff ampTuerlinckx 2002 for a detailed discussion of the differ-ent algorithms) The KS statistic also provides a test ofthe model fit

In the following we first will give a brief overview ofthe rationale of the diffusion model In a second step wewill describe our estimating algorithm and the new modeltest Finally we will report two experiments with a sim-ple color classification task in which the effects of dif-ferent experimental manipulations on the parameter setof the diffusion model were investigated Mathematicaldetails of the diffusion model are given in the Appendix

THE RATIONALE OF THEDIFFUSION MODEL

The diffusion model assumes that binary decision pro-cesses are driven by systematic and random influencesThe systematic influences are called the drift rate Theyldquodriverdquo the process continuously in one direction while therandom influences add an erratic fluctuation to this con-stant path The process is terminated immediately whenone of two thresholds is reached In a psychological appli-cation this means that the decision process is finished andthe response system is being activated which will initiatethe corresponding response (eg a keypress) With a givenset of parameters the model predicts distributions of pro-cess durations (ie RTs) for the two possible outcomes ofthe process Figure 1 provides a schematic illustration ofthe model The process begins at the starting point z andmoves over time until it reaches the lower threshold at zeroor the upper threshold at a Then the corresponding re-sponse is executed immediately In Figure 1 a sample pathof the process is presented As can be seen this path is notstraight but oscillates between the two boundaries due torandom influences (ie the intratrial variability of thepath) on the process

Applied to a perceptual situation in which participantshave to categorize ambiguous stimuli the process de-scribes the ratio of information gathered that fosters eachof the two possible stimulus interpretations Informationis collected over time until evidence points with suffi-cient clearness to one interpretation then a response is

a

z

d (t )i

it

Sample Path

Drift Rate (v )

Response A

Response B

0

Figure 1 Schematic illustration of the diffusion model The process starts at point z andmoves over time until it reaches the lower threshold at 0 or the upper threshold at a De-pending on which threshold is reached different responses are initiated The erratic line givesan example of one process that reaches the upper threshold at time ti The mean deviance ofsingle paths from the mean path is defined by an additional (scaling) parameter of the model(ie the diffusion constant s) Outside the thresholds the distributions of process durationsfor a constant set of parameters are sketched Note that the lower limit can be reached withrandom influences although the distributions shown are based on a positive drift rate
















time









10

8

6

4

2

00 025 050 075 100 125 150

Response Time

p cum

T = 009




EXPERIMENT 1










10

8

6

4

2



p cum
















Condition


a 118 119 137 116za 049 047 052 056v1 089 046 091 102v2 103 057 118 141t0 047 047 052 073h 080 090 085 089sz 009 009 009 008





1500

1250

1000

750

500

250

100

80

60

40

250 500 750 1000 1250 1500

40 60 80 100

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


ResponseCorrect

Error

ConditionStandard

Difficult

Accuracy

Resp Handicap

ConditionStandard

Difficult

Accuracy

Resp Handicap







EXPERIMENT 2











Stimulus Color


Color 1 10 5 5Color 2 10 5 5

S (stimulus) 0 0







1000

800

600

400

90

80

70

60

50

40

400 600 800 1000

40 50 60 70 80 90

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


Response

Correct

Error

StimulusEasy

Difficult

StimulusEasy

Difficult









GENERAL DISCUSSION







Promoted Response


a 113 119 006za 054 044 010v1 070 069 001v2 046 003 042v3 047 069 022v4 070 102 032t0 044 049 005h 102 111 009sz 013 012 001







CONCLUSION


REFERENCES





















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )
















time









10

8

6

4

2

00 025 050 075 100 125 150

Response Time

p cum

T = 009




EXPERIMENT 1










10

8

6

4

2



p cum
















Condition


a 118 119 137 116za 049 047 052 056v1 089 046 091 102v2 103 057 118 141t0 047 047 052 073h 080 090 085 089sz 009 009 009 008





1500

1250

1000

750

500

250

100

80

60

40

250 500 750 1000 1250 1500

40 60 80 100

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


ResponseCorrect

Error

ConditionStandard

Difficult

Accuracy

Resp Handicap

ConditionStandard

Difficult

Accuracy

Resp Handicap







EXPERIMENT 2











Stimulus Color


Color 1 10 5 5Color 2 10 5 5

S (stimulus) 0 0







1000

800

600

400

90

80

70

60

50

40

400 600 800 1000

40 50 60 70 80 90

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


Response

Correct

Error

StimulusEasy

Difficult

StimulusEasy

Difficult









GENERAL DISCUSSION







Promoted Response


a 113 119 006za 054 044 010v1 070 069 001v2 046 003 042v3 047 069 022v4 070 102 032t0 044 049 005h 102 111 009sz 013 012 001







CONCLUSION


REFERENCES





















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )








10

8

6

4

2

00 025 050 075 100 125 150

Response Time

p cum

T = 009




EXPERIMENT 1










10

8

6

4

2



p cum
















Condition


a 118 119 137 116za 049 047 052 056v1 089 046 091 102v2 103 057 118 141t0 047 047 052 073h 080 090 085 089sz 009 009 009 008





1500

1250

1000

750

500

250

100

80

60

40

250 500 750 1000 1250 1500

40 60 80 100

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


ResponseCorrect

Error

ConditionStandard

Difficult

Accuracy

Resp Handicap

ConditionStandard

Difficult

Accuracy

Resp Handicap







EXPERIMENT 2











Stimulus Color


Color 1 10 5 5Color 2 10 5 5

S (stimulus) 0 0







1000

800

600

400

90

80

70

60

50

40

400 600 800 1000

40 50 60 70 80 90

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


Response

Correct

Error

StimulusEasy

Difficult

StimulusEasy

Difficult









GENERAL DISCUSSION







Promoted Response


a 113 119 006za 054 044 010v1 070 069 001v2 046 003 042v3 047 069 022v4 070 102 032t0 044 049 005h 102 111 009sz 013 012 001







CONCLUSION


REFERENCES





















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )



EXPERIMENT 1










10

8

6

4

2



p cum
















Condition


a 118 119 137 116za 049 047 052 056v1 089 046 091 102v2 103 057 118 141t0 047 047 052 073h 080 090 085 089sz 009 009 009 008





1500

1250

1000

750

500

250

100

80

60

40

250 500 750 1000 1250 1500

40 60 80 100

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


ResponseCorrect

Error

ConditionStandard

Difficult

Accuracy

Resp Handicap

ConditionStandard

Difficult

Accuracy

Resp Handicap







EXPERIMENT 2











Stimulus Color


Color 1 10 5 5Color 2 10 5 5

S (stimulus) 0 0







1000

800

600

400

90

80

70

60

50

40

400 600 800 1000

40 50 60 70 80 90

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


Response

Correct

Error

StimulusEasy

Difficult

StimulusEasy

Difficult









GENERAL DISCUSSION







Promoted Response


a 113 119 006za 054 044 010v1 070 069 001v2 046 003 042v3 047 069 022v4 070 102 032t0 044 049 005h 102 111 009sz 013 012 001







CONCLUSION


REFERENCES





















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )
















Condition


a 118 119 137 116za 049 047 052 056v1 089 046 091 102v2 103 057 118 141t0 047 047 052 073h 080 090 085 089sz 009 009 009 008





1500

1250

1000

750

500

250

100

80

60

40

250 500 750 1000 1250 1500

40 60 80 100

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


ResponseCorrect

Error

ConditionStandard

Difficult

Accuracy

Resp Handicap

ConditionStandard

Difficult

Accuracy

Resp Handicap







EXPERIMENT 2











Stimulus Color


Color 1 10 5 5Color 2 10 5 5

S (stimulus) 0 0







1000

800

600

400

90

80

70

60

50

40

400 600 800 1000

40 50 60 70 80 90

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


Response

Correct

Error

StimulusEasy

Difficult

StimulusEasy

Difficult









GENERAL DISCUSSION







Promoted Response


a 113 119 006za 054 044 010v1 070 069 001v2 046 003 042v3 047 069 022v4 070 102 032t0 044 049 005h 102 111 009sz 013 012 001







CONCLUSION


REFERENCES





















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )




1500

1250

1000

750

500

250

100

80

60

40

250 500 750 1000 1250 1500

40 60 80 100

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


ResponseCorrect

Error

ConditionStandard

Difficult

Accuracy

Resp Handicap

ConditionStandard

Difficult

Accuracy

Resp Handicap







EXPERIMENT 2











Stimulus Color


Color 1 10 5 5Color 2 10 5 5

S (stimulus) 0 0







1000

800

600

400

90

80

70

60

50

40

400 600 800 1000

40 50 60 70 80 90

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


Response

Correct

Error

StimulusEasy

Difficult

StimulusEasy

Difficult









GENERAL DISCUSSION







Promoted Response


a 113 119 006za 054 044 010v1 070 069 001v2 046 003 042v3 047 069 022v4 070 102 032t0 044 049 005h 102 111 009sz 013 012 001







CONCLUSION


REFERENCES





















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )






EXPERIMENT 2











Stimulus Color


Color 1 10 5 5Color 2 10 5 5

S (stimulus) 0 0







1000

800

600

400

90

80

70

60

50

40

400 600 800 1000

40 50 60 70 80 90

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


Response

Correct

Error

StimulusEasy

Difficult

StimulusEasy

Difficult









GENERAL DISCUSSION







Promoted Response


a 113 119 006za 054 044 010v1 070 069 001v2 046 003 042v3 047 069 022v4 070 102 032t0 044 049 005h 102 111 009sz 013 012 001







CONCLUSION


REFERENCES





















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )






1000

800

600

400

90

80

70

60

50

40

400 600 800 1000

40 50 60 70 80 90

Pre

dict

ed M

edia

nP

ropo

rtio

n C

orre

ct (

Pre

dict

ed)

Empirical Median


Response

Correct

Error

StimulusEasy

Difficult

StimulusEasy

Difficult









GENERAL DISCUSSION







Promoted Response


a 113 119 006za 054 044 010v1 070 069 001v2 046 003 042v3 047 069 022v4 070 102 032t0 044 049 005h 102 111 009sz 013 012 001







CONCLUSION


REFERENCES





















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )









GENERAL DISCUSSION







Promoted Response


a 113 119 006za 054 044 010v1 070 069 001v2 046 003 042v3 047 069 022v4 070 102 032t0 044 049 005h 102 111 009sz 013 012 001







CONCLUSION


REFERENCES





















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )






CONCLUSION


REFERENCES





















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )
















NOTES









(A1)

and

(A2)


(A3)

and

(A4)g t z a va

zv n zna

v n a tn

-=



21

2 2 2 20 5

g t z a vzv v t

t

na z

tna z

n-

=-bull

bull=

- -( )-

+( )Egrave

IcircIacuteIacute

˘

˚˙˙

loz +( )Acirc( )exp

exp0 5

2

2

22

2

3

2

p

g t z a va

a z v n a z na

v n a tn

+=


IcircIacute˘˚


1

2 2 2 20 5

g t z a va z v v t

t

n a z

tn a z

n+

=-bull

bull=

- -[ ]-

+ -[ ]Ecirc

EumlAacuteAacute

ˆ


exp ( ) exp

( )( )

0 5

2

1 2

21 2

2

3

2

p





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )





(A5)






we will use

(A6)




(A7)


and thus

a cn a a ck a ck

a a S c a S c

nn N

M

n nn N

M

k N

n

Mk N

M

N M M


( ) ( )

=+

= =+

=

+ + loz


pound -( ) +

1 1

1 1


M

k N

n

n Ma N

M

k N

M

-( ) = loz -+= =

+= =


S c br r

b r

r

c c

c( )

sin( ) cos( )

cos( )= + =

+=

+ --2 2

1 2 2

2 2

1 - Œ e xix

S c ee

xix

ic( ) sup Im| |= -

-ŒIuml

IgraveOacute

cedil˝˛

11

sin( ) Im Im ( )

ck e ee

ick

k

n ic n

ick

n

=EcircEumlAacute

ˆmacr

= --

EcircEumlAacute

ˆmacr=

+

=AcircAcirc

0

1

0

11

sin( )ckk

n

=Acirc

0

g r z a v dr

a z v

an a z n

av n a r dr

a z v

an a z n

a

t

n

t

n

+

=

bull

=

bull

Uacute

Acirc Uacute

Acirc


IcircIacute˘˚



˘˚

loz

( )

exp ( )sin ( ) exp

exp ( )sin ( )

0

21

2 2 2 20

21

0 5

2

p p p

p p 11 0 5

21 0 5

2 2 2 2

2 2 2 2

1

2 2 2 2

2 2 2 2

- - loz +( ) loz[ ] +


˘˚


+=

bull

Acirc

exp

exp ( ) sin ( ) exp

v n a t

v n a

a z v n a z na

v n a t

v a nn

p

p

p p p

p





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )





and


(A8)




(A9)

to get


steps

a S c eN

S c ee S c

S cN

a z a z

a z+

- -

-pound

+pound =1

2 11

22

( ) ( )( )

( ) ( ) ( )

( )

n n

np ppe e

N e S ca z

=Egrave

IcircIacute

˘

˚˙

-2 ( ) ( )n

pe

a en

enn

a za z

pound =- -2 1 2 1

2pp p

n n( )

( )


a cn

a S c

tn

n

N

nn

nn

N

nn N

N

+= =

bull

=

= +

bull

+


Acirc

- = -

=

pound

( ) sin( ) sin( ) sin( )

sin( )

( )

n0

1 1 1

1

1

a e n

a na

t

a n

e

a na

t

a n n

na z

a z

=- - +Ecirc

EumlAacuteˆmacr

+

=- - +Ecirc

EumlAacuteˆmacr

+

-

-

2

12

2

12

2 2 2 2

2

2 2 2 2

2 2 2 2

2

2 2 2

p

n p

n p

p

n p

n p

n

n

( )

( )

exp

exp

c a za= -p ( )

a cn a S cnn N

Nsin( ) ( )=

bull

Acirc pound

a z v e N

118 049 a 089 001 108118 049 a 103 001 34119 047 a 046 001 85119 047 a 057 001 44137 052 a 091 001 115137 052 a 118 001 29116 056 a 102 001 107116 056 a 141 001 30





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )





(A10)



p Tn

jt

jn

tjn

j

n T n j j

= EcircEumlAacute

ˆmacr

- -EcircEuml

ˆmacr +Ecirc

Eumlˆmacr=

-[ ] - -

Acirc2 11

1 1( )

Documents

Interpreting the parameters of the diffusion model: An ... · method of estimating the parameters of the diffusion model based on the KolmogorovÐSmirnov (KS) test (Kolmogorov, 1941)