quantitative methods in psychology correcting response measures for...

QUANTITATIVE METHODS IN PSYCHOLOGY

Correcting Response Measuresfor Guessing and Partial Information

S. W. LinkMcMaster University, Hamilton, Ontario, Canada

An experiment or test measuring response choice and response time generatesa large pool of correct and error responses. These indicators of mental processingare often more revealing when corrected for guessing. An example drawn fromlexical decision making motivates the principles behind correcting proportions,response time moments, distributions and histograms, and response confidencewhen only a single stimulus category, such as words, is used. A complete set ofparameter estimates is obtained by analyzing only responses to word stimuli;therefore, false alarms (such as responses to nonwords) are not a necessarycomponent of a correction for guessing.

Nearly everyone is familiar with themethod of correcting dichotomous or serialrecall tests for guessing (Lord & Novick,1968; Sperling & Melchner, 1976). Tomany, it comes as a surprise that in a similarcontext mean response times, response timedistributions, confidence, and other mea-sures of performance may also be "cor-rected." Like any analysis of data, these cor-rections require that assumptions be met bycareful attention to experimental conditions.But before elaborating on these conditions,assumptions, and methods of correction, letus examine a rather typical case.

A Numerical Example:Lexical Decision Making

Reports that lexical decision times andcorrect response probabilities covary suggestthat an experiment be conducted to deter-mine the basis for this often discovered find-ing. An entirely hypothetical experimentproduced the commonly expected resultsshown in Table 1. The results are averages

The National Research Council of Canada, the Nat-ural Sciences and Engineering Research Board of Can-ada, and the U.S. Office of Naval Research providedsupport for this project. I thank David E. Rumelhartfor his generous support during my tenure as VisitingScholar at the University of California, San Diego.

Requests for reprints should be sent to S. W. Link,Department of Psychology, McMaster University,Hamilton, Ontario, Canada L8S 4K1.

for a large number of trials and of subjectswho were to identify a letter string either asa word or as a nonword. To initiate a trial,each subject depressed two response keys,and while both keys were depressed a letterstring appeared on a visual display. The sub-jects indicated whether the letter string wasa word or a nonword by releasing one or theother of the two depressed response keys.

Three groups of subjects provided data forTable 1. The first group were Korsakoff syn-drome patients selected from a local hospi-tal. The second group were speakers of a rareOriental language who learned English afterimmigrating to North America. The thirdgroup were readers of a well-known psycho-logical journal. Table 1 presents generalmeasures of performance derived for eachgroup from the subjects' responses to thepresentations of words. These measures arethe relative frequency of correct identifica-tions, mean response time, mean correct re-sponse time, and mean error response time.The table shows that the relative frequencyof a correct response and the mean correctresponse time tend to covary and that a fastmean response time corresponds to a lowrelative frequency of a correct response.

Korsakoff syndrome patients apparentlyrecognize words very quickly, in 540 msec.It makes no difference whether the responseis correct or incorrect; 540 msec is sufficient.For the nonnative speakers the mean re-sponse time increases, and, somewhat sur-

Psychologicai Bulletin. 1982, Vol. 92, No. 2, 469-486Copyright 1982 by Ihe American Psychological Association, Inc. 0033-2909/S2/9202-0469S00.75

469

470 S. W. LINK

Table 1Probability and Response Time of Subjects'Recognizing a Letter String as a Word

Response time (in msec)

GroupPC

M Mc

(1) Korsakoff(2) Nonnatives(3) Readers

607595

540570616

540560580

540600

1,300

Note. Pc = probability of correct response; M = meanresponse time; Mc = mean correct response time; andME = mean error response time.

prisingly, a small difference between meancorrect and mean error time emerges. Forthe readers of the well-known psychologicaljournal, the mean response time is yet longerand the difference between mean correct andmean error response time is even more dra-matic. In order to ensure that these resultswere credible, the performances were ana-lyzed by the application of a thoroughly in-appropriate analysis of variance to the cor-rect and error response time measures. Theanalysis revealed a significant main effectdue to group, a significant main effect dueto category of response (correct or error),and a significant interaction. Further dis-cussion of this analysis comes later.

Turning now to the examination of re-sponse probability, we see that the groupsdiffered in their ability to recognize words.Group 1 seemed to have great difficulty inrecognizing a word; only 60% of the pre-sented words were identified correctly. ForGroup 2 this percentage increased to 75, andfor Group 3, nearly perfect performance,95% occurred.

One interpretation of these response pro-portions is that they reflect the effects of twodistinct phenomena. First, readers of psy-chological journals (Group 3) are known tobe highly verbal and to have large lexicons.The subjects in Group 2, who learned En-glish as a second language, apparently pos-sess a lexicon that is somewhat smaller thanthat of Group 3. The Korsakoff syndromepatients (Group 1) are special because theyare native speakers of English, but (and herewe invoke the second phenomenon, which isa common observation concerning these pa-

tients) they apparently forgot the instruc-tions given to them and seemed to respondat nearly a chance level with correct anderror responses requiring only a short lengthof time.

A curious result is that as correct responseproportions increase, there is a concomitantincrease in both mean response time andmean correct response time. These increasesin response time are consistent with the hy-pothesis that the size of an effective lexiconvaries across groups. For the Korsakoff pa-tient, the size of the lexicon is effectivelysmall because of the frequent inability togain access to it. For the foreign languagespeaker, the size of the lexicon is also small,and thus little time is consumed in seekingout and identifying a letter string. For thereader of the psychological journal, the lex-icon is vast, and comparing a letter stringwith many internal word representationsconsumes appreciable time. We conclude,therefore, that in recognizing a word, sub-jects search an internal stored representationof words, called a lexicon, and that the timetaken to identify a word correctly dependson the number of stored words. Or do we?

In considering all of the results reportedin Table 1 we are left to wonder why errorresponse times change from group to groupas dramatically as they do. (Fortunately, thisexperimenter chose to provide us with afairly complete report of data by includingmean error response time.) The increase inerror response time appears to be consistentwith a search of an increasingly large lexiconand a failure to locate successfully the pre-sented letter string. Yet the fact that errorsdo occur suggests that the measures of cor-rect response performance may not be directmeasures of the subjects' word-recognizingability. Perhaps some correct responses aredue to a subject's guessing that a letter stringis a word. Other subjects, also failing to rec-ognize this same letter string, respond inerror. Under such circumstances the correctresponse measures may be contaminated byguessing, and although the percentage oferrors may be small, as for Group 3, theremay still be a sizable increase in mean cor-rect response time due to these correctguesses.

The data reported in Table 1 are manu-

CORRECTING FOR GUESSING 471

factured for the purpose of illustration, butthe conclusions drawn from these data arequite like conclusions drawn from similardata reported in contemporary researchjournals. We turn now to a method of cor-recting response times for guessing, a methodthat is not altogether unknown but that re-quires some extensive discussion of its prop-erties and application.

Correction of Response ProbabilitiesUnder Equal Bias

The method of correction derives from asimple two-state psychological theory remi-niscent of models put forth by Herbart, Inaddition to a simple theory of competencethere is a necessary rule of correspondencethat relates psychological states of compe-tence to observable responses by specifyingwhich responses can result from a particularstate. The psychological states representmental processes that turn a stimulus intoa psychological variable. Subsequently, arule of correspondence transforms the outputof the process into a response.

In Figure 1 we see that after a stimulusis presented, the experimental subject mayoccupy either of two mutually exclusive psy-chological states. State A represents a men-tal process that is totally distinct from a sec-ond mental process that is identified as StateB. Although State B represents a processthat is qualitatively different from State A;State B may also characterize a temporalsequence that includes State A, which is then

followed by a subsequent process. For manyfinite state theories State A is associatedwith the mental process of theoretical inter-est, whereas State B characterizes perfor-mance features such as guessing. This theo-retical development contrasts sharply withthe additive error theory that dominated ear-lier psychological theorizing.

The rule of correspondence specifies thatthe output from State A leads to a correctresponse with probability 1. The output ofState B leads to a correct response with prob-ability b and an error response with proba-bility 1 - b. Psychological processes are be-lieved to consume time, and therefore eachbranch of the model shown in Figure 1 hasa distribution of response time that char-acterizes the total time taken by both thepsychological process and the making of theresponse. Each of these distributions has amean response time /ttA and JUB> respectively.The distributions FA(t) and FK(t) are thecumulative distributions of response timeand depend on the subject's psychologicalstate.

Let us now examine the application of thismodel to the experiment reported above. Ourfir'st assumptions are directed toward deriv-ing a model based on a simple hypothesis.Given the nature of the experiment, one rea-sonable hypothesis suggests that on presen-tation of a letter string the subject entersState A if and only if the letter string isidentified as a word. Thus, State A repre-sents a word recognition state from which

Rule ofProcess Correspondence

RTResponse Mean Distribution

State A

«/Present /Word /Latter \^String \A,

State B

Word

Word HB

FA(tl

FB(tl

Non-word \IQ Fgltl

Figure I, A two-state representation of distinct psychological processes. (Of major interest is State A,which results in perfect performance when a word is presented. Associated with each State is a uniqueresponse time [RT] distribution.)

472 S. W. LINK

the subject emits a correct response withprobability 1.

In all other cases the letter string, al-though it is in fact a word, is not perceivedto be a word. The subsequent response isbased on factors other than the fact that aword was recognized. It may be argued thatbecause a word was not recognized, the sub-ject will definitely respond incorrectly andthat the probability b, shown in Figure 1,should be zero. Yet it is equally well arguedthat because the subject recognized bothwords and nonwords, the subject may guesswhen unsure of the proper classification fora letter string. If the subject is led to believethat there are as many words as nonwords,then whenever a letter string is not identi-fiable the subject may guess that the letterstring is a word with probability b = .50.State B is much like State A except thatwhen the process engaged in State A runsto completion without recognition, a guessoccurs. The mean response time in State Bis MB regardless of which response results.

Of course the experimenter does not wit-ness this internal basis for performance.Rather, only correct and error responses areobserved together with their response times.The two upper branches of Figure 1 corre-spond to those cases where a word is pre-sented and the experimenter observes a cor-rect response. The probability of such a re-sponse equals the probabilities obtained byadding together probabilities multiplied alongthe two branches involved. Thus, we find thatPC, the probability of a correct response, canbe written as

PC = ~p)b

or, equivalently,

PC = p + KPE, where

PE = 1 - PC and AT = b/(\ - b). (1)

The value p determines the frequency of oc-currence of the mental process that leads toword recognition. The difficulty in correctingfor guessing derives from the undisputed factthat both p and K are unknown and thatfrom a single equation two free parameterscannot be determined. Yet, when responsetimes are considered, this long standingproblem may be solved.

For these experimental conditions subjectswere encouraged to believe that, and respondas if, word and nonword letter strings wereequally likely. Therefore, we assume that thesubject guesses without bias, and we set bequal to .50. Equation 1 then leads to anestimate of the unknown value of p. To cal-culate p, notice that the standard correctionfor guessing for a true/false test may beapplied. Because b = .50, the value of Kequals 1, and

p = Pc - PE

provides a simple correction for guessing andan estimator of the probability that State Aoccurred. Applying this method of estimat-ing p to each group proves especially re-vealing. The values of p obtained for Groups1, 2, and 3 turns out to be .20, .50, and .90.These values, given our model, suggest majordifferences among the three groups. We nowdetermine whether these differences are cor-roborated by estimates of ^A and ^B againobtained by a correction for guessing.

Correction of Mean Response TimesUnder Equal Bias

The mean correct response time can alsobe determined by referring to the twobranches of Figure 1 that lead to a correctresponse. A more elaborate computation isinvolved, but the essence remains the sameas in the response probability calculation. Inorder to compute the mean correct responsetime the two mean response times for eachbranch are mixed together according to theirconditional probabilities of occurrence. Giventhat a correct response occurs, the upperbranch is involved with probability p/(Pc)and the lower branch is involved with prob-ability (1 - p)b/Pc. These are the condi-tional probabilities associated with the twopossible ways in which a correct responsecan occur. For each branch leading to a cor-rect response, the conditional probability ismultiplied by the branch's mean responsetime. The mean correct response time is thenthe sum of these two products. We find thenthat Mc, the mean correct response time, is


Estimates of p and MB are used to estimatethe value of MA- The value of p must be de-termined, but MB is estimated directly fromthe mean error response time because theonly way an error can occur is through StateB. Either response from State B requiresmean response time MB, and therefore themean error time given the presentation ofa word estimates MB- We may therefore useas an estimate MB = ME. Each group providesa different estimate of MB that is used to de-termine the value of MA according to Equa-tion 2 by using the following equivalence,

PCMC = MnU - (3)

Assuming, as above, that the response biasis identical for both responses, b = .50, and,therefore,

PCMC =

Because p = Pc - PE, the estimator ofis

PCMC- PEME =

PC-PP. M A '(4)

For Group 1 the correct and error meanresponse times are equal to each other andtherefore equal to the overall mean responsetime, M, so that Equation 3 can be rewrittenas

(Pc - PE)M

= 540.

For Group 2 we have

(.75)(S60) - (.2S)(600) _.50

Finally, Group 3 yields

- (.95)(580)-(,05)1300.95 - .05

= 540.

= 540.

As unlikely as it at first appears, all groupsrecognize a word as a word equally fast, onthe average requiring 540 msec to read theletter string, to decide, and to respond.

The interpretation of these findings is

quite distinct from the conclusions drawnfrom the uncorrected response measures ofTable 1. First, the notion of searchingthrough a lexicon of various sizes is no longertenable. If such a search occurs, in anythingother than an unlimited-capacity parallelmanner in which every stored representationis examined simultaneously, then the meantime to locate a word should increase withthe size of the lexicon. Yet the mean re-sponse time for State A, for each group, is540 msec. Perhaps unlimited-capacity par-allel processing is necessary.

Yet changes in error response time appearto argue against fully parallel processing. Ifall representations in the lexicon are simul-taneously searched, then the failure to locatea representation requires no longer than thesuccessful location of a representation. Therather sizable changes in mean error re-sponse time certainly argue against this fullyparallel notion as the only explanation. Infact, it appears that two processes are nec-essary to provide a good account of thesedata. The theory presented below illustrateshow these processes and our assumed psy-chological states are related.

A Stochastic Guessing ModelWith Equal Bias

The Diligent Guessing Model illustratedin Figure 2 proposes the existence of twopsychological states. The first, State A, rep-resents a fully parallel recognition process.If recognition occurs, then a word is iden-tified with probability 1. Failure to recognizeleads to two possibilities. Either guessing willoccur at this time, and a correct or errorresponse will occur with probability .50, orthe subject begins another fully parallelsearch of memory. The theory makes thisadditional assumption: After the first searchof memory the image of the stimulus is sodecayed that subsequent searches can neverlead to identification. Repeated searches,based only on the noisy remains of the earlierstimulus, result in guessing. The more dili-gent the subject is in processing the stimulus,the longer the eventual response time. In thisway increasingly longer error times occur.1

1 A second process based on parallel processing is sug-gested by Townsend (Note 1). In his view the KorsakofT

474 S. W. LINK

Stimulus

Recognition 1 CorrectSuccess

RecognitionFailure

,<V./Corrects

Guess

NoisySearch >/g>

Correct

Error

Noisy ,_Search

Figure 2. A Diligent Guessing theory compatible with the two-process theory shown in Figure I.

The difference between correct and errorresponse probabilities measures the proba-bility that the subject recognizes a letterstring to be a word during the first parallelprocessing of the lexicon. At first it appearsthat this cannot be so, but a simple calcu-lation reveals its truth. With probability p,parallel analysis resulting in a correct re-sponse occurs. With probability 1 — p, thesubject enters State B, in which case a guessactually occurs with probability, let us say,g; or with probability I — g; another parallelsearch is made. This parallel search contin-ues, although the icon has vanished, there-fore, State A may never again be entered.The only future possibilities are that the sub-ject guesses with probability g, or with prob-ability 1 - g the subject once again searches

patients can search a portion of their lexicon both inparallel and exhaustively. The probability that the tar-get item is in the scanned portion of the lexicon equalsp, and because the scanning is exhaustive, a correctresponse occurs with probability 1.0. If the effective lex-icon does not contain the target, with probability 1 -p, then the patient guesses. Either correct or error re-sponses require identical response times because the ef-fective lexicon is always exhaustively scanned. The re-maining two groups process the target in a similar man-ner except that a target is discovered with probabilityp, the processing is then terminated, and a correct re-sponse is emitted. With probability 1 - p the target isnot located, even with an exhaustive scan, and a guesswith a long response time occurs. The size of the meanerror response time depends, then, on the size of thelexicon—the size that is smallest for Group 1, larger forGroup 2, and largest for Group 3.

the lexicon without an icon. The name Dil-igent Guessing derives from the fact thatsubjects may persist in this pointless searchof the lexicon, thereby impressing the ex-perimenter with their earnest approach tothe experimental task.

We may now calculate both the proba-bility of an error and the mean error time.From the relevant branches of Figure 2 theprobability of an error can be written as

k=o

The probability of a correct response Pc

equals 1 - PE or p + (1 — p)/2, and thedifference between correct and error re-sponse probabilities is Pc — PE ~ P-

To calculate the mean error response timelet us impose an easily relaxed condition onthe mean time to parallel search the lexicongiven a well-defined versus a noisy icon. Forthe first search only, the mean time is <5. Forall searches subsequent to the first the iconis essentially noise, and the mean time is-y. To calculate the mean error responsetime, compute first the value of P^M^ whichis obtained by multiplying the probability of


every branch leading to an error by the meantime for the error to occur. The beginningfew terms are

540

(\ CO

f Z ( i -^/ k=0

Or, by dividing both sides by PE =

we have

In a similar fashion

PCMC = p5 + (1 -/>)(- IS + (1 - p)

(6)

= P5 + PnME .

Dividing both sides by Pc = p +

yields

(7)

- —2s

- «>The important value 6, which measures themean parallel processing time, is in fact ftA.The value of 5, may be computed by notingfrom Equation 7 that

PCMC - PEME = pb

= (Pc ~ PB)«.

This leads to the general correction forguessing, which applies to a large numberof models and which is identical to Equation4,

PCMC - PEME

01

LU0.

270o

2oa.

1.0

Figure 3. The relation between PCMC - Pt.ME andPC ~ PE must be linear for a variety of finite-statemodels.

When MA is a constant we should discoveracross subjects, experimental conditions, orother factors that a graph of the denomi-nator of Equation 9 versus the numeratorwill yield a linear function with interceptzero and slope ^A. Figure 3 illustrates thisgraph for the data reported in Table 1 andindicates, of course, remarkable correspon-dence between the Diligent Guessing Modeland the prediction in Equation 9. Othermodels leading to an identical predictionwere examined in extenso by Oilman (1966),Yellott (1967, 1971), and Link and Tindall(1971).

To return to the analysis of Table 1, anexplanation for the changes in mean re-sponse time may be found in the subject'stendency to guess. For Group 1 the proba-bility of continued processing following thefirst parallel analysis of the stimulus is zero,that is 1 — g = 0. Thus, any response, corrector error, occurs in mean time 5 = JUA. Theestimate of the (corrected) relative fre-quency of recognizing a word equals .20.For Group 2 this value increases to .50, thefirst pass parallel search time requires 540msec = /iA, but error and correct responsetimes increase due to diligent guessing. Thisexplanation of Group 2's performance alsoapplies to Group 3 where the probability ofrecognition is .90, /UA = 540 msec, and ap-parently the error and correct times are con-

476 S. W. LINK

siderably longer due to the quite small prob-ability of chancing a guess.

It is tempting to determine the value ofg, which presumably changes from group togroup; yet without some additional structureits value will remain unknown. As an ex-ploratory matter, the mean time to searchthe lexicon and respond may be supposedconstant regardless of when the search oc-curs. Then 7 equals 5, and we see in Equation

5 that g = — . For the three groups we findME

that g = 1.0, .90, and .415, illustrating thepoint that extended guessing rather thana serial search of a large lexicon can ac-count quite nicely for the data reported inTable 1.

Corrections for Unequal Response Bias

The examples above illustrate how re-sponse proportions and means are correctedfor guessing when the value of the bias orpartial information parameter equals one-half. In most common experimental proce-dures it is desirable to test the assumptionthat b = .50 to correct for both guessing andpartial information. By a correction forguessing we mean the removal of those ob-servations that result from State B. By alsocorrecting for partial information we meanadjusting for the tendency to produce moreof one or the other response when guessingoccurs. That is, adjusting the observed cor-rect responses for biased guessing.

The first method of correcting for bias orpartial information assumes the model pro-posed in Figure 1 and thus requires that bothresponse proportions and response times aremeasured. A slight modification of Equation3 provides the relation

PCMC =and

PCMC -

PC ~ KPP

(10)

where K = . The parameter K mea-

sures the relative amount of response bias.For large values of K, correct guesses pre-dominate. When K = 1 the probabilities ofguessing correctly or in error are equal and

as K approaches zero the proportion of errorsapproaches 1 - p. The unknown parametersare /iA, /tB, p, and K. We want to determineK to correct for partial information (bias),p to correct for guessing, and /UA and HB todetermine the differences in processing timesfor the two psychological states. Given themodel, the unknown parameter /XB can beestimated from the mean error time ME, butestimation of K requires more sophisticationand a new result concerning response timehistograms.

Estimating the Relative Response Bias, K

Imagine the time axis to extend from mi-nus infinity to zero to infinity. The correctresponse time histogram is defined on thepositive axis by the selection of a set of times,t} <(2< • • • < tN-\, that divide the responsetimes into TV categories of size «,-. The re-sponse time distribution for errors stretchesto minus infinity from zero, and the values-t\ > — /2 > • • • > — f/v-i divide this dis-tribution into TV categories each of size «_,-.As a further definition let the number ofcorrect responses be «c, the number of er-rors, HE, and the total number of responses,« = nc + «E.

The responses from State A fall into bin;' with a probability that depends in part onthe response time distribution FA(t) and thevalues /,- and /,_,. The value FA(t) is the prob-ability that a response from State A occursbefore time /. We let a,- be the probabilitythat a State A response falls into bin i, thatis, a,- = F^(t,) - FA(tj-i). Corresponding toa, is the probability c,- that a response fromState B falls into bin /, that is, c, = FB(ti) -FB(t,--}). When these definitions are kept inmind, the expected number of correct re-sponses in bin i is

n, = npctj + n( 1 - p)bct ,

and the expected number of errors is

«_,- = H( 1 -

Notice that the probability c, remains fixedregardless of which guessing response, cor-rect or error, occurs. This assumption con-forms to the idea that although there maybe a bias toward one response, the act ofguessing requires the same time regardless


of the eventual response. F^(t) is the dis-tribution of response time regardless of theguessing response made. The fundamentalequation for the correct response time his-togram is written to include the error re-sponse histogram and the unknown param-eter K as

n, = npa, + Kn~, . (11)

When only guessing occurs, the value ofp equals zero, and there are no State A re-sponses. When there is no bias, the measureof relative bias K equals 1, and then Equa-tion 1 1 becomes

«,- = n-i .

With the two histograms stretched out toplus and minus infinity from zero, the his-tograms must appear to be (mirror) sym-metric with respect to time zero. When onlybiased guessing occurs, p is zero, and thereare no State A responses; but K + 1, andthus

«, = *«_,. (12)

The responses in corresponding (i, -i) his-togram bins bear a constant proportion. Agraph of n~, versus «,- must yield a straightline with slope K and intercept zero. Fur-thermore, this result is independent of thenumber of bins, N, and the particular set oftimes selected by the experimenter. There-fore, when performance is at chance levelsa perfectly satisfactory estimate of K is nc/HE, which is the ratio of the number of cor-rect to the number of error responses.

For responses generated by State A tooccur, the value of p must be greater thanzero. In this instance estimates of all param-eters are made available from response dif-ferences. On the basis of Equation 1 1 thedifference

«,- - «,• = np(a, - dj) + ( 1 3)exists for any bins i andj. The values a\, a2,. . . , aN, which are the probabilities that aState A response enters histogram bin '/, aredetermined entirely by the experimenter'schoice of the values of t\, t2) . . . , tN-{. Theset of such values may be selected free fromany constraint other than order and used toensure that, for example, a{ = a2.

The estimation of K can be an operationof great delicacy requiring mathematical

techniques far beyond the scope of this ar-ticle. Certain simple procedures used by theexperimenter in the design of experiments,however, can tame a formidable estimationproblem. The methods suggested below takeadvantage of the fact that the experimentercan include in the experimental design cer-tain conditions that provide data necessaryin estimating unknown parameters.

One goal of experimental design is to pro-vide direct measurement of as many un-known parameters as possible. An estimateof the amount of response bias may be ob-tained by including in the stimulus sequence,as is done in psychophysical studies, stimulithat provide no discriminative informationat all. Any response to such a stimulus pro-vides a measure of response bias, and bothmeasures of response choice and time arereadily available. Data such as these arecommonly obtained when the experimentaldesign provides for various levels of stimulusdiscriminability, as in experiments produc-ing psychometric functions. In this caseEquation 12 provides an estimate of K.

In many instances, however, there is nostimulus that corresponds to the indiscrim-inable stimulus found in psychophysicalstudies. Instead, the subject is presented withone or another stimulus, and a choice re-sponse is required. The response may be dueto guessing, but the amount of bias towardthe response alternatives is unknown. A sim-ple technique for correcting unknown biasis to create conditions with symmetric bias.In particular, let the bias value in experi-mental condition j = 1, 2 be bjt whereb2 = 1 — bi. The probability of an error re-sponse in Condition 1 is PEl = (1 — p)bi,and the probability of an error response inCondition 2 is (1 - p)b2 = (1 - p)(l - &,).Thus, the ratio PEI/^EZ estimates K. Thetechnique used by Link (1975) to bias a sub-ject prior to each experimental trial appearsto provide symmetric biasing effects andmay be applied in the present context.

An implicit assumption used to determineK is that the distributions of response timesmust exist over common time intervals. Also,the observed correct and error response timedistributions must exist over common timeintervals or the assumptions concerningguessing responses being identical except for

478 S. W. LINK

choice of response are violated. The distri-bution of time to recognize, however, maybe shifted significantly lower or higher thanthe distribution for guessing. If this fortui-tous event occurs, a strong test of the theoryis possible, for within the range in which onlyguesses occur, «//«_, = K. Having estimatedK from pure guessing, Equation 1 1 can beconfirmed for the range where both recog-nition and guessing occur jointly.

Once the parameter b can be estimatedfrom data for a single experimental sessionor test, the unknown estimates of all otherunknowns such as alt a2, . . , , aN used forthe response categories are available. Inshort, the data needed to correct for guessingare from the responses to the stimuli underinvestigation. The use of alternative stimuli,such as nonwords, to generate errors to beused in correcting for guessing is unneces-sary.

When the unknown distribution FA is hy-pothesized to be of a particular form or fam-ily, still a third procedure offers a new ap-proach to estimating K. Neither p nor K isknown, but p may be removed by using thefollowing device. Let o,- = FA(f,-) - FA(?,-_i) bethe proportion of State A-generated respon-ses between /,- and ?,-_,, and let PQi and Peirepresent the proportion of correct and errorresponses found within this interval. Ac-cording to the theory,

PCPC1 = pa, + KPEPei .

The unknown parameter p may be removedfrom this equation by comparing two dif-ferent response time intervals. The differ-ence between correct response proportionsfor two arbitrarily selected intervals / andj will be

= P(a, el - PeJ).

When as — a} = 0 an estimate of K may beobtained from

-TV CM)fE^fol ~ Pej)

where in all cases the estimators are substi-tuted for values on the equation's right-handside.

Many such estimates of K are obtainable,and to reduce the influence of sampling error

due to the selection of intervals a regressionanalysis of independent intervals should beused. Then, a graph of Pc(Pa - Pcj) versusPz(Pei ~~ PV) will yield a slope with value Kand intercept 0.

Of course the values of a,- and a,- must bedetermined by evaluation of the distributionfunction FA(r). At first this seems obvious;select a set of parameter values for FA(f),compute various values of /, and tj so thata, = dj, from the data estimate values for theright-hand side of Equation 14, and thencontinue computing estimates of K until sta-bility is reached. Thereafter, obtain new pa-rameter values, and repeat the procedure.

This method is useful only if some infor-mation concerning FA(f) is known. Supposenothing is known about FA(f) except that itobeys the usual conditions for distributionfunctions. Suppose also that values of p andb are unknown but that all other quantitiesFC) FG = FE) omitting the understood (0,and Pc are known. Then, as Rumelhart(Note 2) suggests, for any t 0 < t < I, themodel M = (FA) FG> />, b) is identical to themodel M* = (F£, F& p*, b*), where

b* = eb,

** = />//>*,

and-7r*)FG.

Therefore, unless some additional infor-mation concerning the distribution of FA isforthcoming, it will always be possible tochoose a value of t to satisfy the constraintson the data imposed by the model. Such in-formation may, in fact, be quite minimal.Three cases below illustrate how new infor-mation about FA. can be used to determinean estimate of K.

As an example suppose that FA and FGare Gamma distributions differing by a sin-gle independent exponential component.Note first that two different probability den-sity functions defined on the same intervalmust cross each other, or otherwise the in-tegral of their difference would exceed zero,and one density function would integrate tomore than one, thereby contradicting thehypothesis that it is a density function. Let


such a crossover point be /*, and for theGamma distributions we see that this cross-over occurs at ?* = ( / • — 1)/a, the mean ofthe unknown distribution /A.

The value of t* can be estimated by ob-serving that the mode of the distribution oferrors, /G, occurs at / = (r — \ )/a. The valuet* is a fixed point for which /A = /G, and asa consequence of the model, fc = fa = /Aso that at the point t* the correct and errordensity functions must intersect. Thus, t*can be estimated from either the crossoverpoint or the position of the mode for /E. Fi-nally, the mean of the observed error distri-bution is known, the mean of the unobserveddistribution is known, the mean of the cor-rect responses is known, and consequentlyby Equation 1-0 the unknown value of K canbe determined.

As a second example suppose that only themodes of the distributions are known. Themodel's predicted mixture of density func-tions for correct responses, corresponding toEquation 3, Pcfc = p/A + (1 - p)bfn, canbe differentiated to provide an obvious re-lation between the derivatives of the threedensity functions fc, /A. and /G; that is,pcf'c = P/'A + KPEf'E. The derivative of anydensity function is zero at a mode. Assuming/A to be unimodal and setting the value of/A to zero, we see that at the mode of /A thevalue K can be estimated by K = Pcf'c/PE/'E- That is, at the mode of /A the de-rivatives of the correct and error densityfunctions provide enough information to es-timate K.

The third example draws on the fact thattwo density functions defined on the samedomain must intersect at a point t*. At thispoint the densities /A and /B must be equal.The value of t* can be determined by notingwhere /c(0 equals /E(/), for at this point acrossover of /A and /G necessarily occurs.For a single-parameter density function suchas the exponential distribution, the value oft* can provide a method for estimating p andthen K. Let /A and /G be exponentially dis-tributed with parameters a and ft. At t*these two densities are identical, and there-fore t* = (a ~ /3)-1 In(a/j3). The value of18 can be determined directly from the ob-served errors because /0 = /E. Then, becausethe value of t* is known, the value of a. can

be estimated. With /A and /G known, valuesof p and K can be obtained.

To summarize these examples, there areoften hypothesized or known features of den-sity functions that provide significant infor-mation about the expected behavior of abinary mixture. This additional informationaids in the estimation of the mixing param-eter p and the relative bias parameter K.Finally, when assumptions concerning /A arestrong there is no reason not to move im-mediately to maximum-likelihood methodsand expectation-maximize techniques dis-cussed by Dempster, Laird, and Rubin(1977) and by Murray and Titterington(1978).

The point is that although problems ofidentifiability exist, assumptions concerninggeneral characteristics of the unknown dis-tribution can provide handy solutions to theestimation problem,

Correction of Response Time Distributionsand Higher Moments

Correcting response probabilities and, byuse of Equation 10, mean response times iseasily accomplished. In a manner strictlyanalogous to previous corrections, estimatesof response time distributions are obtained.Let Gc(t) and GE(t) be the observed responsetime distributions for correct and error re-sponses, and from Figure 1 note that FA(t)and Fu(t) are (cumulative) response timedistribution functions associated with StatesA and B. The distribution of responses fromState B gives the estimate

FB(t) = CE(<)

because all errors occur in State B. The dis-tribution of responses from State A followsthe development given above for the cor-rection of mean response times (Equation10). Replacing /UA with FB(0 and MB withFB(I) = GE(t), we obtain the estimate

PcGc(t) - . (15)PC - KPE

Of course Equation 15 may be differentiatedwith respect to t to yield the probability den-sity function

(16)Pc -

480 S. W. LINK

NON-WORD

Low_

WORD

Low

Medium i • Medium

High, .High

CHome Key

Figure 4, An arrangement of response keys for rsuring response choice, response confidence, andsponse time.

mea-re-

Estimates of the unknown distribution andprobability density functions are now pos-sible.

As a matter of completeness all momentsof responding in State A are available froma minor extension of either Equation 15 or16. When both sides of Equation 16 aremultiplied by t' and integrated with respectto t, the result is

(1?)

Pc - KPE '

where ̂ is the rth raw moment of f ^ ( t ) ,and Me and ME are the rih raw momentsof the observed correct and error responsetime. Again, the method of correction is bysubtraction of common paths leaving onlypaths that contribute to the desired result.Higher order central moments follow fromthe usual relations among raw moments.

Correction of Response Confidence

A typical method for measuring confi-dence provides the subject with a semi-circleof response buttons divided into two responsegroups. Those keys on, say, the right-handside are designated to correspond to responseRw (word), whereas those on the left-handside correspond to RNW (nonword). From themidline to the extremes, the keys are labeledlow-high confidence in the judgment made,that is low-high confidence that the stimuluspresented was a word or a nonword. Figure4 illustrates this arrangement, including thefeature of a home key used to initiate ex-perimental trials and to facilitate measuring

response time. Whenever a word is presentedand a nonword response is produced in error,a simultaneous measure of response choiceand confidence is obtained.

Various theories suggest the form of theseconfidence judgments; for example, entirelyuninformed guessing might generate a uni-form distribution over confidence responses.We do not distinguish among the many com-peting theoretical possibilities but concen-trate instead on how to use the distributionof confidence given an error to correct forguessing the distribution of correct-responseconfidence judgments. The important differ-ence between this response measure and theresponse time distributions and momentsexamined before rests in the small numberof categories of confidence typically avail-able to the subject. This small number ofcategories introduces some difficulties re-sulting in a method of analysis that is dif-ferent from the correction of response time.Were confidence measured on a continuousdimension, the response time distributionmethods would directly apply.

As a matter of convenience let us label theconfidence responses in Figure 4 as C_3, C_2,C_h C], C2, C3, from left to right and ingeneral from C-M to CM when there are 2Mkeys symmetrically placed. The elaborationof the two-state model diagrammed in Fig-ure 1 defines the probabilities of occurrenceof confidence responses given the states oc-cupied. Let q, (i = 1, 2, . . . , M) be theprobability of responding with confidenceresponse C, given State A. State B, whichis taken to represent guessing, generatesconfidence response C/ or C-,- with proba-bility d(i = 1, 2, . . . , M). This last as-sumption captures the spirit of symmetri-cally distributed guessing by supposing thedistribution of confidence responses to beidentical for the two response choices.

Correction of Response ConfidenceUnder Equal Bias

The distribution of confidence correctedfor guessing is easily obtained when the re-sponse bias value b equals .50. Even whenb does not equal one-half, clear predictionsfollow from these assumptions and permita test of the finite state assumptions made


above. To examine one such prediction, de-fine the total number of responses to be mand the number of Q confidence responsesto be m,(i ± 1, . , . , ±M). From Figure 1and the rule of correspondence for confi-dence, the values of m-t are

m, = mpqt - p)bc{

m-t = m(\ -/>)(! - b)c,. (18)

The sum of these two numbers of symmetricconfidence responses is

m, + m-i = m[pq, + ( 1 - p)c,] (19)

and does not depend on the unknown re-sponse bias value b.

According to Equation 19, experimentalconditions producing changes in responsebias should lead to invariance in the sum ofsymmetric confidence values. Let j and j'index J experimental conditions generatingchanges in response bias. The number of C,plus the number of C_, confidence responsesfor condition j should equal the number forCondition / for all i = 1, 2, . . . , M. Thatis, letting mtj refer to the number of C,- re-sponses in Condition j,

mtj (20)

A graph of the left-hand side of Equation20 versus the right-hand side should producea straight line with slope 1.

Because the equality of Equation 20 holdsfor all values of z, a convenient, althoughweak, statistical test can be based on thefollowing argument. Across all / and j ¥= j'the differences between the left- and right-hand sides of Equation 20 exceed zero withthe same probability as not exceeding zero(ties excluded). Therefore, the distributionof excesses is binomially distributed withmean .5 (M X J) and variance .25 (M X J).

When response bias b = .50, an immediatecorrection for confidence responses is avail-able. From Equation 18,

m,- m-i = mpqi, (21)

which is the number of confidence responsesC, generated in State A and the desiredcorrection for guessing. Because <?i + <?2 4-1 • • + QM - 1-0, the sum of all such dif-ferences defined by Equation 21 equals mp.

Therefore, the value of q, is estimated fromthe relation

<?/ = (22)2 (mk - m-k

k=l

In a similar manner estimates of the confi-dence response probabilities for State B canbe obtained. Noting that

M

m(l — p) = m — 2 (mk - m~k)

leads to the estimator

2m^,m(\-pY

(23)

In summary of this special case of b = .50,the parameters qh ch and/? can be estimated.The correction for guessing amounts to asimple subtraction of the number of confi-dence responses C_; from the number of cor-rect confidence responses C,. The extensionof these corrections to the response time dis-tributions associated with each confidenceresponse follows the principles set forth inthe section on correcting latency distribu-tions for guessing and provides students withpractice in applying the finite state principlesof correcting for guessing.

Correcting Response ConfidenceUnder Unequal Bias

The correction of response confidencewhen there is bias toward a response followsprecisely the correction for guessing princi-ples used for the analysis of response timehistograms. From Equation 18 and its ob-vious correspondence to Equation 1 1 we findthat

m, = +

When K is determined using the method ofhistogram analysis, an estimate of q, isreadily obtained from

m, -*' m(Pc - KPE) '

Estimates of the probability that a guessproduces confidence response C,-, that is ch

482 S. W. LINK

follow directly from the confidences for er-rors,

where nE is the total number of error re-sponses. Finally, the value of p is determinedby application of Equation 1.

Advanced Techniques

If states may be used to characterize theunderlying psychological phenomena, thenare two states sufficient to describe empiricalresults such as those shown in Table 1?There are potentially as many states as ob-servations, but a less ambitious number ispreferable. In order to demonstrate the ex-istence of two states, the empirical paradigmmust promote sufficient variability in per-formance for the states to be revealed. If, forexample, performance can be biased towardone or the other of the two responses, thenerror and correct response probabilities willvary. The distribution of errors is unaffectedby the amount of bias, so the response timedistribution of errors should remain constantacross bias conditions.

More important is the fact that the mar-ginal distribution of the times of all respon-ses should exhibit the properties of a binarymixture of the two distributions, FA(/) andFn(t). The existence of a mixture can betested by an extension of Falmagne's (1968)fixed-point property (cf. Townsend & Ashby,in press, who also summarized additionaltests). This test is most easily performed on(cumulative) distribution functions of re-sponse times because these functions, whichare composed of probabilities, are easier toestimate than the corresponding densityfunctions.

Changes in experimental conditions arehypothesized to have no influence on thedistributions of responses within the twostates, A and B, that is on FA(f) and FB(r).The marginal response time distribution forexperimental condition j(j = 1, 2, . . . , / )

PjFA(t) + (I - Pj)FB(t).

of statistical and exploratory analysis. Twocases are distinguished below.

In the first case, the probabilities p} maybe hypothesized to remain constant acrossexperimental manipulations. If so, all G(t)must be identical but for statistical fluctu-ation. This is precisely the null hypothesisof simultaneous Kolmorogov-Smirnov teststreated by Hajek and Sidak (1967).

The second case hypothesizes that the val-ues PJ change from experimental conditionto condition. This assumption meets so manycommonly found experimental methods thatit merits attention. When the values of p,change, the values of G/0 also vary, yet thefollowing calculation demonstrates an in-variance property of mixtures of distribu-tions that is easily verified. Let G(<) be theaverage distribution function, that is,

where

G(t) = - 2 G/0J j=i

= pFA(t) + (I -

(24)

VP - ~T 2, PJ•.J =i

The difference between G(t) and any G//)is

- Gj(t)

= (p - Pj)[FA(t) - FB(0]. (25)

The values on the right-hand side are en-tirely unknown, but the values on the left areeasily calculated. This suggests that we ex-amine one more difference and then comparethe two left-hand sides, that is, compute fora different experimental condition, m,

G(/) - Gm(/)

Whether G/f) is a mixture of two fixed butunknown distribution functions is a matter

(26)<p-pjr "The ratio of Equation 25 to Equation 26 isa constant independent of t and depends onlyon observable differences between responsetime distribution functions. Therefore, agraph of Equation 25 versus Equation 26should produce a straight line with zero in-tercept and slope (p — pm)/(p ~ PJ).


A convenient method of performing thistest is to choose values of t that increment(7(0 by steps of .05 (in which case there are19 values of t to provide 20 response timebins). Then each successive point of G(t) isbased on a fixed number of additional ac-cumulated observations. Extreme values ofp, however, may lead to highly variable ob-servations.

Only two experimental conditions are nec-essary to examine the prediction of constantdistributions F^ and FB across experimentalconditions. In this instance only a single linewith slope -1 and intercept zero must occur.When three experimental conditions exist,a likely outcome is that p is approximatelyequal to a value of pj resulting from one ofthe three conditions. Then the denominatorof Equation 26 may be so small that thiscomparison with G(t) is proper but useless.Rather than use G(t), a direct comparisonof G,(0 - G2(t) against G,(f) - G3(0 willprovide a method of evaluation yielding astraight line with slope (/>) — p2)/(/>i - p3).Whenever the slope in Equation 26 becomesunstable, the method of comparing all dis-tributions against a fixed standard conditionis to be preferred. For any two conditions mand n and a standard s,

Gn(0

-Pi.

Ps -- Gm(0]. (27)

This technique may be applied to densityfunctions and moments. For the sake of com-pleteness, Equations 26 and 27 are now gen-eralized to these cases. For density functions

- gm(t)}. (28)Ps ~ P

Multiplying both sides of Equation 28 by fand integrating gives a result for the rawmoments

En(r)

Ps - In (29)

Estimates of the slope (ps — pn)/(ps - pm)may first be obtained from the moments,especially for r = 1, where the means are

easily calculated, and compared with theslope obtained from a best fit (with zero in-tercept) using Equation 27. Severe devia-tions put to question the mixture hypothesis.

The important feature of mixtures of dis-tributions is that observable properties maybe deduced even when the probability valuesdetermining the frequencies of occurrenceof the states are unknown. For example, theresponse time distributions depend on statesand not on the different responses occurringwithin a state. Therefore, experimental ma-nipulations designed to affect only responsebias, which occurs only within a state, mustyield marginal response time distributionsfor all responses that are identical for eachbias condition. This result is theoreticallyclear but mildly counterintuitive.

These methods are easily applied to dataobtained from the hypothetical experimentreported in Table 1 . Under the null hypoth-esis that the distributions of responses inStates A and B remain invariant across thethree experimental conditions, the linearityproperty of the (cumulative) distributionfunctions (Equation 27) will find support.Alternatively, changes in the distribution ofFB(t) across groups will result in nonlinear-ity. Suppose that FB was not observed. Tak-ing as a standard for comparison the Kor-sakoff syndrome patients, Group 1, and com-paring against this both Group 2 and Group3, the unknown slope is (p{ — p2)/(pi ~ Pa)estimated by

p. - p2 ^ £,(;') -E2Q')

= 540 - 570"540-616

= .395.

Notice that this value is quite near the valueof .429 obtained by substitution of the es-timates of pi = .20, p2 = .50, and p3 = .90in Equation 29 to determine the unknownslope. If the null hypothesis is true, a graphof G,(r) - G2(f) versus G{(t) - G3(<) willproduce a straight line with slope of about.40 and intercept 0. If the hypothesis is false,as suspected, then deviations from this func-tion should be revealed.

484 S. W. LINK

Using False Alarms to Measure Guessing

The previous analyses focus on a singleclass of stimuli used to probe mental pro-cesses such as recognition of a word. The useof stimuli that are, for example, nonwordsto obtain an estimate of response bias canfacilitate the analysis when the introductionof this alternative recognition task does notintroduce more unknown probabilities. Acommon assumption stipulates that the pre-sentation of a nonword letter string leads toa recognition (State A) with probability pand to guessing with probability 1 — p. Thenthe probability of an error to a nonwordstimulus is (1 — p)b. This probability equalsthe probability of a correct guess when aword is presented. An estimate of p is easilyobtained by subtracting the error probabilitywhen a nonword is presented (1 — p)b fromthe correct response probability when a wordis presented, p + (1 — p)b.

The very great difficulty with this proce-dure involves the a priori assumption thatwords and nonwords lead to State B withidentical probabilities, that is, each withprobability 1 — p. This assumption is con-venient and provides a simple correction forguessing, yet it always requires empiricalsupport. In order to validate this assumptionthe following technique, based on methodsderived from Oilman (1966), Yellott (1967,1971), Link and Tindall (1971) and Link,Ascher, and Heath (Note 3) together withextensions of Equations 27, 28, and 29, issuggested. Experimental conditions mustpromote only changes in response bias, pro-duce J bias parameters, &,, b2, . . . , bj and,across both word and nonword stimulus pre-sentations as well as the J bias conditions,ensure that the probability of entering StateB equals a constant 1 - p while the prob-ability of entering State A equals p. Underthese conditions, and regardless of theamount of bias, Equations 27, 28, and 29hold with suitable substitutions of word andnonword data. In particular, let s, n, and mindex three different bias conditions, and leti = W (word), NW (nonword) indicate thestimulus presented. Then identical values of(Ps ~ Pn)/(ps ~~ Pm) must be obtained for theratios

i = W.NW. (30)

This implies that a comparison of word tononword ratios will give equality. That is,omitting the understood (t) in all cases,

_N W

or, equivalently,

,w

Precisely the same conclusion holds for theprobability density functions gj,j(/)(j = s, n,m) and for the moments Ej/fj.

When empirical support for a constantvalue of p is obtained, then estimates of theunknown response time means and distri-butions follow immediately. The value ( 1 -p)b, for example, unknown for word-onlypresentations, is now known from errorsmade in response to nonwords. The methodof analysis used in Equation 3 and leadingto Equation 4 applies, with the value PE,NWsubstituted for PE. This substitution yieldsan estimate of ^A, which is

(31)E,NW

By analogous argument to those yielding thecorrections contained in Equations 15, 16,and 17 we find

and

/OT.(32)" cw -

p _ p" C,W -< E,NW

c,w ~ P\

, _ .,• (34)

E.NSV

The word and nonword subscripts may beexchanged to obtain estimates of the meanresponse time for nonwords recognized inState A (not necessarily equal to /*A). Thedistribution of nonword recognition times in


State A follows from Equations 32 and 33,and estimates of higher response time rawmoments in State A follow from the appli-cation of Equation 34. In each instance thesubscripts are changed from word to non-word or from nonword to word. For example,to estimate the rth raw moment of nonwordrecognitions in State A we find

MA.NW — .(35)C.NW E,W

Conclusion

The example in Table 1 illustrates twoimportant principles. First, and most impor-tant, is that a suitably designed experimentpermits the subject to exhibit instances oferrors. These errors are not thought of asadditive random error but may be treatedas the result of imperfect mental processing.Furthermore, even small error rates of onlya few percent can generate corrections thatproduce far more orderly data than the orig-inal uncorrectcd results. Attempts to reducethe need for correction by rerunning errortrials or deletion of errors from the dataare simply unacceptable capitalizations onchance.

Second, theories described in terms ofstates apply to what at first appear to bequalitatively different modes of processing.For the two-state model illustrated in Figure2, State A is invariant, but State B repre-sents the result of a sequence of mental actsconveniently collapsed across time into a sin-gle state. State B is as invariant as State A,only the probability associated with the men-tal act contained in State B varies. Thus,both nonstochastic as well as stochastic phe-nomena can be cast in the finite state format.This is one basis for the finite state theory'sempirical and theoretical appeal.

Traditional statistical methods suppose asa rule of correspondence that error is an ad-ditive component of the measured responseand that this error is approximately nor-mally distributed. The application of theadditive normally distributed error model tothe data summarized by Table 1 would sug-gest some form of analysis of variance basedon the null hypothesis of invariance of means

and error terms. This hypothesis would nodoubt be rejected and the conclusion of dif-ference between the groups adopted.

The two-state theory and its rule of cor-respondence leads to a quite different methodof analysis that exposes an underlying psy-chological process found to be invariantacross the three subject populations. Thisfinding leads to a conclusion opposite to thatobtained through the analysis of variance.When further investigation, through theanalysis of response time distributions andmoments, lends additional support to the fi-nite state assumptions, a better view of theunderlying psychological processes is ob-tained.

The correction for guessing illustrates thefirm theoretical belief that beneath the sur-face level of data lie relations between psy-chological processes that are revealed bestby placing important psychological assump-tions in a suitable mathematical format.Then methods of data analysis are derivedfrom psychological assumptions that aretested across a series of experimenter-in-duced parameter changes. Rather than ap-ply routine statistical techniques appropriateto the fertilized fields of agronomics, thefinite-state methods in the correction forguessing provide a uniquely psychologicalbasis for the analysis of experiments.

Reference Notes

1. Townsend, J. T. Extensive discussions at PurdueUniversity, 1981.

2. Rumelhart, D. E, Discussions on this topic at theUniversity of California, San Diego, 1981.

3. Link, S. W., Ascher, D., & Heath, R. Tests of finitestate theories of response time (Tech. Rep. 36).Hamilton, Ontario, Canada: McMaster University,Department of Psychology, April 1972.

References

Dempster, A. P., Laird, K, & Rubin, D. B. Maximumlikelihood from incomplete data via the EM algo-rithm. Journal of the Royal Statistical Society (Ser-ies B), 1977, 39, 1-38.

Falmagne, J. C. Note on a simple fixed-point propertyof binary mixtures. British Journal of Mathematicaland Statistical Psychology, 1968, 59, 131-132.

Hajek, J., & Sidak, Z. Theory of rank tests. New York:Academic Press, 1967.

Link, S. W. The relative judgment theory of two-choice

486 S. W. LINK

response time. Journal of Mathematical Psychology,1975, 12, 114-135.

Link, S. W., & Tindall, A. D. Speed and accuracy incomparative judgments of line length. Perception andPsychophysics, 1971, 9, 284-288.

Lord, F. M., & Novick, M. R. Statistical theories ofmental test scores. Reading, Mass.: Addison-Wesley,1968.

Lupker, S. J., & Theios, J. Further tests of a two-statemodel for choice reaction times. Journal of Experi-mental Psychology: Human Perception and Perfor-mance, 1977, 3, 496-504.

Murray, G. D., & Titterington, D. M. Estimation prob-lems with data from a mixture. Applied Statistics,1978, 27, 325-334.

Oilman, R. T. Fast guesses in choice reaction time. Psy-chonomic Science, 1966,6", 155-156.

Sperling, G., & Melchner, M. J. Estimating item andorder information. Journal of Mathematical Psy-chology, 1976, 13, 192-213.

Townsend, J. T., & Ashby, F. G. On the stochasticmodeling of elementary psychological processes.New York: Cambridge University Press, in press.

Yellott, J. I. Correction for guessing in choice reactiontime. Psychonomic Science, 1967, 8, 321-322.

Yellott, J. I. Correction for guessing and the speed-accuracy tradeoff in choice reaction time. Journal ofMathematical Psychology, 1971, S, 159-199.

Received June 22, 1981 •

Editorial Consultants for This Issue: Review Articles

William A. BaumFrank BensonSheri BerenbaumBernard L. BloomRussell M. ChurchNorma D. FeshbachFrank A. GeldardWalter R. GoveFrank M. GreshamKenneth HeilmanWilliam C. HirstRobert HoganEarl HuntJulian JaynesRudolf KalinPhyllis A. KatzDavid A. KennyRonald Kessler

John F. KihlstromPeter R. KilleenRalph H. KilmannSuzanne Constance KobasaSigmund KochThomas LandauerJudith Hall LangloisArthur A. LumsdaineBrendan MaherLawrence E. MarksStephen MatthysseJames MazurDonald H. McBurneyWilbert J. McKeachieWilliam P. MilbergMichael MillerPhilip MorsePaul Obrist

Martin T. OrneJ. Bruce OvcrmierCampbell PerryMichael I. PosnerE. C, PoultonS. RachmanMark E. RillingJerry SametBarry SchwartzCharles D. SpielbergerHarold W. StevensonSusan SugarmanPeter VenablesEric WannerEverett WatersMarion A. WengerAllan W. Wicker

Editorial Consultants for This Issue:Quantitative Methods in Psychology

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quantitative methods in psychology correcting response measures for...

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