VOL. OCTOBER 1969 Psychological Bulletin · PDF filecerns primarily the perceptual process,...

VOL. 72, No. 4 OCTOBER 1969

Psychological Bullet in

PERCEPTUAL INDEPENDENCE:

DEFINITIONS, MODELS, AND EXPERIMENTAL PARADIGMS1

W. R. GARNER-

Yale University

JOHN MORTON

Applied Psychology Research Unit, MedicalResearch Council, Cambridge, England

A logical analysis of the problem of the experimental study of perceptualindependence is undertaken. Necessary distinctions are: between the definitionof independence as zero correlation and its definition as performance parity,between normative models of perceptual process and models of perceptualstate, between experimental designs which use orthogonal stimulus inputs andthose which use correlated inputs. In addition, it is often necessary to specifyan organismic decision rule, which can be of many types and can occur withrespect to process or to state. Implications of this analysis are: Perceptualindependence is not a unitary concept; it can be studied most effectively withexperiments using orthogonal inputs; when correlated stimulus inputs are used,so many underlying assumptions have to be made that it is almost impossiblenot to find a model which satisfies both the concept of independence and anygiven experimental result; state independence is intimately related to processindependence; and search for the locus of interaction is probably a moreuseful approach than trying to determine whether independence exists.

A major problem under attack by experi-mental psychologists in recent years is thedetermination of how the organism deals withand makes use of multiple sources of informa-tion. To illustrate some of the many types ofexperimental question being asked: If twotones are presented in succession to one ear,is the resultant probability of detectiongreater than if just one tone is presented?If a tone is presented simultaneously to the

1 Preparation of this paper was supported byGrant MH 14229 from the National Institute ofMental Health to Yale University. Work on thisarticle was first undertaken while the senior authorwas Visiting Scientist at the Applied PsychologyResearch Unit and was continued while the juniorauthor was Research Associate and Lecturer at YaleUniversity. The authors thank Donald W. Taylor,Robert G. Crowder, and Robert A. Rescorla, all atYale, for listening to them and thus helping themclarify their thinking on this paper; and JohnBrelsford, also at Yale, for critically reviewing themanuscript.

2 Requests for reprints should be addressed toW. R. Garner, Department of Psychology, YaleUniversity, 333 Cedar Street, New Haven Connecticut06510.

two ears, is the threshold lower than if thetone is presented to one ear alone? If a set ofstimuli covaries in two attributes, such ascolor and size, are absolute judgments ofthese stimuli more accurate than if theyvary in just one attribute; that is, doesredundant stimulus information improve per-ceptual performance? Does the probabilityof correct identification of a visual form im-prove if it is presented simultaneously to twoparts of the same eye, or to the two eyes to-gether, or perhaps to the same eye in succes-sive time periods? In searching for a targetform among many different forms all on asingle display at the same time, can theorganism simultaneously search for this targetin all locations at once, or must it successivelyscan the different target locations? Still fur-ther, if we present different speech passages tothe two ears, can the organism show adequate

perceptual response to each without inter-ference from the other? Or, as a last example,

suppose the prior verbal context for a word

is given as one source of information, and the

233© 1969 by the American Psychological Association, Inc.

234 W. R. GARNER AND JOHN MORTON

word is then presented under adverse stimulusconditions, how does performance with bothsources of information relate to performancewith either alone?

In all of these examples there is a similarconceptual problem: Does the organism proc-ess the two or more sources or channels ofinformation independently? This question con-cerns primarily the perceptual process, butthe question of independence must necessarilyinvolve the nature of stimulus independence,response independence, and decision independ-ence, since all of these conceptually separableprocesses are involved in any single experi-ment on the problem.

Even though all of the experimental ques-tions mentioned previously are actual casesfrom the experimental literature (most refer-enced in this paper), the purpose of the presentpaper is not to review or criticize this experi-mental literature but to provide a logicalanalysis of the problem itself. For this pur-pose entirely hypothetical data are used in theillustrations, so that special aspects of theproblem can be illustrated more clearly thanis possible with any real experiment, whichnearly always will involve more than oneaspect of the problem of independence. Theuse of such idealized data does not mean thatthe problems do not exist in real experimentsbut only that the several aspects of the prob-lem can be easily separated with idealizeddata.

THE ROLE OF NORMATIVE MODELS

A common and logically sound method ofdealing with this problem is to use a norma-tive model (ordinarily mathematical) whosepurpose is to establish what ideal performanceshould be with the assumption of independ-ence, to provide a base line against which tocompare actual performance. Without such abase line, any given experimental result isso crudely qualitative that we obtain littlegenerality to our understanding of the prob-lem. Thus the normative model providesmeaningful interpretation of any given experi-mental finding.

The need for models in understandingperceptual independence has clearly beenrecognized, since it is a rare experiment in

this area whose results have not been evalu-ated against some model of independence. Therather special role of the normative modelhas perhaps been less well understood, how-ever, since models often are accepted or re-jected on the basis of the fit of the data to aparticular model. But the special role of thenormative model is not to describe a par-ticular perceptual process but to provide anorm against which results are evaluated.Thus we should not ask whether the modelis right or wrong but rather to what extentthe experimental results conform to the cri-terion of independence which the particularmodel implies.

This is not to say that normative modelscannot be wrong; rather, it is to say that thebasis of their being wrong (or right) is notso much in fitting data as in correctly re-flecting the underlying concepts or processeswhich the experimenter intends them to re-flect. Further, normative models may be moreor less useful not insofar as data fit a par-ticular model but insofar as the model helpsclarify the nature of the process when the datado not fit the model. Thus, not only must thenormative model indicate expected values, itmust be expressed in such a form that we canunderstand why the data do not fit the model.Simple rejection of a null hypothesis is muchless valuable than understanding the nature ofthe process which leads to rejection of thehypothesis.

In looking over the many models used inthe study of perceptual independence and theexperimental situations to which they are ap-plied, it seems that several sources of con-fusion have led both to a proliferation ofmodels and to a lack of emergence of a fewbasic explanatory concepts. These particularconfusions do not concern whether variousperceptual processes are in fact independent.Instead, the confusions concern logical andrational aspects of the very question whichis being asked, and these are the things dis-cussed here. There are three basic factorswhich need to be considered: (a) the defini-tion of independence that is used, (b) thetype of model that is used, and (c) the ex-perimental paradigm within which the problemis investigated. The present paper considers

PERCEPTUAL INDEPENDENCE 235

these factors and the interrelations betweenthem in the interest of providing some clarifi-cation of the study of perceptual independ-ence.

INDEPENDENCE: CORRELATION OR PARITY

There are two basic definitions of independ-ence which are used, that of zero correlationand that of performance parity. While thesetwo definitions are related, both concepts canbe operative in a particular experimentalsituation, and the two different definitions ofindependence are themselves capable of beingindependent.

Zero Correlation

The most basic definition of independencein any formal sense is that two variables or setsof events are uncorrelated. We say that out-comes of coin tosses are independent if theoutcome on Trial n is uncorrelated with theoutcome on Trial n— 1. We say that twotests are independent if their scores do notcorrelate across subjects. Even the commonlyused cross-multiplication of probabilities toestablish independence is essentially a zero-correlation criterion, since the cross product isthe value of joint occurrence which obtainswith zero correlation.

Insofar as possible, metric definitions ofzero correlation are desirable, since with suchmeasurable correlations exact limits of maxi-mum and minimum correlation can be speci-fied. Nevertheless, there are many experi-mental problems to which correlation as anexact measure is not applicable, although theidea of independence and even of correlationat a conceptual level is. For example, wespeak of two speech passages as uncorrelatedif they are simply completely different pas-sages. If the same passage in the same voiceis presented to, say, the two ears simultane-ously, we can say that the two ears are re-ceiving correlated messages; but if the pas-sages are simply unrelated, it seems morereasonable to talk about lack of correlationrather than zero correlation, since clear metricspecification of correlation is not possible inthis case.

Performance Parity

An alternative definition of independence isthat of performance parity. Briefly stated,the performance-parity criterion is that twoperceptual processes are considered independ-ent if performance on two perceptual taskscarried out simultaneously is the same as thesum of the performances on each task carriedout separately. The two tasks might requiredealing with information in two modalities, ortwo subaspects of the same modality, or twodimensions of a set of stimuli. With certaintasks, this criterion of independence seemsintuitively reasonable, although in others itdoes not. Certainly it is worth consideringwhy performance parity should be used as acriterion of independence at all, and how ithas come to be accepted as such a criterion ifthe basic definition of independence is zerocorrelation.

Relations between Correlation and Parity

There are definite relations between theperformance-parity and the zero-correlationcriteria of independence that have allowedsubstitution of the derived parity criterionfor the direct correlation criterion. Consider,for example, the relation between the vari-ance of a sum (or difference) and the varianceof the components making up the sum. If wehave just two variables, X and Y, then

SV + y> = S*x + S»y + 2rXYSxSY, [1]

where S2 is a variance. The same relationholds for the difference of two variables ex-cept that the correlation term is negative.This relation says, in words, that the varianceof a sum equals the sum of the variances ofthe individual variables if and only if thecorrelation between the two variables is zero.Thus the criterion of independence as zerocorrelation becomes a criterion of parity ofthe variances; and in many situations, it iseasier to test the parity consequence than tomeasure zero correlation. For example, if wehave two sets of scores with known variancesand also know the variance of their sums, wecan say that the two sets of scores are uncor-related if parity of the variance holds. Thuswe can say that the two sets of scores are


independent without actually calculating thecorrelation between the sets.

Completely similar relations exist with theinformation metric. Using Garner's (1962)notation,

\](X,Y) = U(F) - V(X: Y), [2]

which in words states that the joint un-certainty (or information) of variables X andY equals the sum of the uncertainties of vari-ables X and F separately minus the contin-gent uncertainty between X and Y. Thusparity of information holds if and only ifthere is no contingent uncertainty (a kind ofnonmetric correlation) between X and F.

Further examples of the intimate relationbetween correlation and parity come from themultiple-predictor situation. For example, in abivariate analysis of variance (i.e., two pre-dictor variables plus one criterion variable),the total variance of the criterion equals thesum of the variances due to the two pre-dictor variables only if there is no interaction,which is a kind of correlation between thetwo predictor variables in terms of their ef-fects on the criterion. And a squared multiple-correlation coefficient equals the sum of thesquared correlations of the criterion variablewith each predictor variable if and only if thetwo predictor variables are themselves un-correlated, that is, independent. Once again,completely analogous relations exist with theinformation metric (see Garner & McGill,19S6).

Thus there is a clear formal rationale whichlies behind the use of performance parity asan indirect criterion of independence. Whathas happened, however, is that the perform-ance-parity criterion has become quite inde-pendent of the formal rationale; it has be-come functionally autonomous, so to speak.This functional autonomy is entirely reason-able, since performance parity can be mea-sured in situations where there is no realpossibility of measuring a correlation of anykind, and yet where the need to ask aboutperceptual independence is more than ap-parent. To illustrate, suppose we presentdifferent speech in two ears and ask for totalcomprehension. If both passages can be fullyunderstood (as determined by some per-

formance measure), we say that parity hasbeen satisfied and infer that the processes areindependent. Yet there is no meaningful wayto measure a correlation cofficient of any kind.Or, alternatively, we present a verbal contextplus tachistoscopic exposure of a word, andwe assume independence of the two sources ofinformation if somehow performance withboth sources can be shown to be the sum ofthe performances when each source of in-formation is separate.

Thus there is a real need for a performance-parity criterion of independence, if it can beapplied without serious distortion of thefundamental concept of independence. How-ever, the parity criterion should be usedonly when there is no reasonable possibility ofmeasuring correlation directly. In some cir-cumstances, it is entirely reasonable to askabout perceptual independence both as zerocorrelation and as performance parity in the«ame experimental situation.

Parity and Process Separability

Both the correlation criterion and theparity criterion of independence can usually,although not always, be applied to data insome form; that is, they are operationallydefinable with respect to directly obtained ex-perimental data and derived properties ofthese data. Yet even though these conceptscan be considered external to the organismunder study, the aim of psychologists is ulti-mately to understand the internal processesresponsible for these external concepts.

Related to this entire question of independ-ence is the organismic concept of processseparability, which is simply the assumptionthat two processes act separately in theorganism and thus can go on independently ofeach other. Such an organismic concept, withits connotation of process interaction, can berelated conceptually either to the correlationor the parity criterion of independence; thatis, to say that two processes interact couldmean that they are in some way correlated orthat each process in some way limits perform-ance with respect to the other, perhaps bysharing a common channel. In actual practice,however, the concept of process separabilityis nearly always related to the parity criterion


of independence and is most frequently ap-plied in situations where correlation cannotbe directly measured.

There are many terms used in reference tothe general idea of process separability. Inrecent years, the distinction between paralleland serial processing has received considerableattention (see Neisser, 1967; Posner &Mitchell, 1967, among others). The notionof parallel processing is that two perceptualacts can occur simultaneously with perform-ance parity. Alternatively, the serial process-ing idea is that only one process can occur ata time, so that some form of process timesharing is necessary. Exact interpretation ofthese two concepts in terms of process in-dependence is actually rather tricky, sincemodels which assume parallel processing donot invariably lead to performance parity(see Egeth, 1966). In addition, the assump-tion of a short-term memory which pre-serves information completely makes possiblea serial model with no loss of parity.

Concepts such as serial and parallel process-ing, or the more general idea of process sepa-rability, even though they may be useful intheir own right, often produce only confusionwhen related to the problem of perceptualindependence. Certainly at times there is realcontradiction between the separability ideaand performance parity. Lockhead (1966),for example, introduced the idea of stimulusintegrality (quite the opposite of separability)to explain the circumstances under which per-formance parity will occur in an absolute-judgment task involving multiple stimulus di-mensions. Other authors, for example, Shepard(1964), Hyman and Well (1968), and Handel(1967) introduced the separability idea toexplain greater summative effects of two stim-ulus dimensions in judgments of stimulus dif-ferences. Thus we have the conflicting claimsthat separability is required for greater dif-ferentiation and that it prevents performanceparity. And yet each seems intuitively reason-able in its own context.

As another illustration, consider Broad-bent's (1958) concept of a filter process, aprocess in which one of two input channels iscompletely rejected while the other is passedthrough. Such a concept suggests process

separability, since clearly the organism canprevent interaction between two channels.At the same time, if the organism does rejectone while accepting the other, we have evi-dence of serial processing, a concept whichsuggests nonseparability of process. To someextent, these apparent confusions can be clari-fied by distinguishing between process sepa-rability and channel separability.

However, at the present time it is not clearthat organismic concepts related to eitherprocess or channel separability (and thus ap-pearing to be intimately related to the ques-tion of perceptual independence) are in aposition to clarify the problem of perceptualindependence. And, conversely, it is not clearthat concepts of perceptual independence arein a position to clarify the problem of channelor process separability.

TYPE OF MODEL

With regard to type of model, at least threekinds of distinction need to be made. Further-more, it is not always clear that the distinc-tion should be attributed to the type of modelrather than to the nature of the process as-sumed, since if the model has any hope ofreflecting psychological reality (i.e., processesinside the organism), at least some of theproperties of the model will be reflectingproperties of the organism. Where we chooseto make the distinction is, however, less im-portant than that we make it and recognizethe consequences of the distinction as relatedto our understanding of perceptual independ-ence.

Process versus State Models

The first and most important distinction isbetween models concerned with independenceof the direct process itself and those modelswhich are concerned with independence ofthe states of the possibly separate channels.To illustrate, we can ask whether apparentbrightness is influenced by color, and this isa question which concerns correlations be-tween judgments of brightness and variationsin color. If there is no correlation, we say thatbrightness is independent of color. Thus thisquestion concerns whether the color andbrightness processes are correlated in the


sense that the relevant processes themselvesare influenced.

Alternatively, suppose we ask about theaccuracy of judgments of color and of bright-ness and assume that the organism fluctuatesin its ability to perform these tasks accurately,that is, that the organism has varying stateswhich define its perceptual capability. Nowwe can ask whether there is a correlation be-tween the state of the organism with regardto color judgments and with regard to bright-ness judgments, which is not to ask whetherbrightness is influenced by color but whetherperformance related to each of these at-tributes is correlated over time. Such a cor-relation does not require any process inter-action.

Models of perceptual states are used mostcommonly, but not at all exclusively, in ex-perimental situations in which thresholds arebeing measured. In such tasks, it seemsreasonable that the basic sensitivities of thetwo eyes for example, or of two areas on thesame retina, vary and that they vary in acorrelated fashion. With such models, the cor-relation sought is between right and wrongresponses with respect to the two or morechannels, and in simplest fashion we simplyask whether errors are correlated. Modelsconcerned with the independence of percep-tual states can be considerably more sophisti-cated, however, allowing the assumption ofseveral states, and with explicit statementof decision rules related to these states (seeEriksen, 1966, for an illustration).

The importance of the distinction for pres-ent purposes lies in the fact that assumptionsabout process independence necessarily enterinto models concerned with perceptual states,and vice versa; and these relations can befairly complicated. We will see, for example,that a particular type of process interactioncan lead to performance on a joint perceptualtask which is greater than parity, while atthe same time producing a result with astate model which is less than parity. Per-ceptual processes and states of the organismare really part of the same total problem,and models of either require implict or ex-plicit assumptions about the other.

Logically, the distinction between processand state models can concern either the zero-

correlation or performance-parity criteria ofindependence. However, parity relations be-tween process and state models are generallyquite straightforward, so that there is littleneed to distinguish these two types of modelwith the parity criterion. Where the distinc-tion is potentially very important is in regardto the correlation criterion, since process cor-relation can exist without state correlationand vice versa. Furthermore, the distinctionbetween process and state becomes importantwhen we have to be concerned with real orhypothetical decision rules, since decisionrules can be applied with regard to eitherstate or process.

This distinction between process and stateindependence, while discussed here in thespecific context of perceptual independence, isnot limited to immediate perceptual function-ing at all but is a necessary distinction forany aspect of human performance. For ex-ample, two response systems may be non-independent in the sense that two directprocesses interact, such as right- and left-handsimultaneous responding; or there may becorrelated fluctuations in state produced bytotal organismic alertness, fatigue, etc. Similardistinctions need to be made with respect tomemory processes. To illustrate, if both lettersand numerals are held in memory, there couldbe direct process interaction between theclasses of items, producing either interferenceor facilitation, but there could also be statefluctuations, either correlated for the twoclasses of items or independent, which affectthe overall level of performance on the mem-ory task. Thus the distinction is a very broadone applying to a wide range of psychologicalproblems.

Direct versus Indirect Models

Nearly all models used in tests of per-ceptual independence can be classed as testingfor independence either directly or indirectly.What constitutes directness is not so much aproperty of the model itself but rather is therelation of the model to the kind of independ-ence being tested. Thus if zero correlationis being taken as the criterion of independence,a direct test of this criterion would requireexperimental measurement of correlation insome form. An indirect test would involve


measurement of some consequence of the lackof correlation. For example, suppose geo-metric patterns are presented to four differentretinal locations, and separate identificationof each is required. The number of correctidentifications on a single trial can range fromzero to four, and these should be distributedas the binomial if there is no correlation be-tween performance at any of the retinal posi-tions. This illustration is, of course, one ofstate correlation rather than process correla-tion, so the binominal test in this case is anindirect test of the zero correlation assumptionin a state model (see Eriksen & Lappin, 1967,and Estes & Taylor, 1966, for examples).

The performance-parity criterion of in-dependence can be tested indirectly as well.For example, if performance parity exists,then as we add perceptual tasks to an alreadyexisting one, performance on the first taskshould not decrease. If we take this lack ofinterference on one task as evidence of parityfor both, we are testing the parity conceptindirectly. This paradigm has been usedespecially in studies of memory.

We have already noted that performanceparity can itself be considered as an indirectconsequence of the more fundamental criterionof zero correlation. Yet because it can be andhas been used as a direct criterion so often,it is better to consider models pertaining toperformance parity as direct tests, and onlythose cases where indirect consequences ofparity itself are involved should be consideredas indirect tests.

Measurement Properties

Another distinction concerning models isthat of the type of measurement propertyassumed in the model. Since we are talkingabout mathematical models, the type of modelwhich can be used depends very much on thekind of data which can be collected and theorganismic processes which the data are pre-sumed to reflect.

Models involved in the study of perceptualindependence have involved essentially justthree measurement properties:

1. Probability models, where simple classoccupancy is involved. Models involving per-centage correct responses, or binomial dis-tribution, are of this type.

2. Information models, in which the infor-mation metric is applied to probability dis-tributions. In one sense these models couldbe considered as a subclass of probabilitymodels. However, there are some sets of prob-ability data to which the information metriccannot be meaningfully applied; furthermore,the information model is so intimately as-sociated with the entire problem of perceptualindependence (since the problem is really aquestion of independence of informationprocessing) that it is useful to consider itas a special class of model.

3. Metric models, in which the variablesare measured on some form of metric, usuallyinvolving at least interval scale properties,that is, the metric holds to a linear trans-formation. Models involving product-momentcorrelations, variances, d' (as in signal detec-tion theory), and scales of perceptual distanceare all of the metric type.

The measurement property of the modeldoes not assume an important role as a logicalproblem, because the kind of metric used hasmuch more to do with the specifics of the typeof performance than with any general con-siderations involving type of independenceand experimental paradigm. Any of the typesof measurement can be used with respect toeither criterion of independence, with processand state models, with direct or indirectmodels, and with any experimental paradigm.So the kind of measurement assumed in amodel will enter into this discussion only asspecific illustrations of points being made.

EXPERIMENTAL PARADIGMS

There are many specific experimental ar-rangements which can be and have beenuced in (he study of perceptual independence,depending to a great extent on the par-ticular substantive topic under consideration.Here, however, since the concern is primarilywith the general logic of such studies,four basic experimental paradigms are dis-tinguished, and the distinctions depend en-tirely on the nature of the correlations be-tween stimuli and between responses in theexperiment. If perceptual independence isunder investigation, then the independence in-volved in the stimuli themselves, and the in-dependence in the responses required by the


experimental arrangement, must be crucial inour understanding of any independence whichoccurs between the input and the output ofthe organism. Clearly we cannot attribute in-dependence or the lack of it to internal organ-ismic processes unless we know the kinds ofrelations which exist within the stimuli andwithin the responses themselves.

Complete Orthogonal Experiments

The basic, and in many ways ideal, experi-ment for investigating perceptual independ-ence is one in which the stimuli themselvesare orthogonal or uncorrelated, and in whichthe response arrangement is one which permitsorthogonal response variables corresponding tothe stimulus variables. That is to say, thestimulus variables are independent, and theresponse variables are capable of independ-ence, so that any lack of independence inperformance can reasonably be attributed tointernal organismic limitations rather than tolimitations inherent in the experimental ar-rangement itself.

Process correlation. This experimentalparadigm is ideally suited to the use of modelswhich use zero correlation as the criterion ofindependence and which are concerned withperceptual process. In this paradigm, thereare manifest data in which correlation canexist, and if the correlation in the stimuli isknown to be zero, and the correlation betweenresponses could have been zero, then any ob-tained correlation may be attributed toproperties of the organism.

Nevertheless, the actual measurement ofcorrelations which indicate lack of perceptualindependence is more complicated than atfirst glance it might seem. To illustrate, as-sume that we present a series of visual stimuliwhich vary in both area and brightness andthat all areas occur in combination with allbrightnesses; that is, the stimulus variables arethemselves uncorrelated. Then we require thesubject to make an identifying response, andon each stimulus presentation he must identifyboth area and brightness. The question is,given such data, what kinds of correlationcan be measured which are meaningfully re-lated to the problem of perceptual independ-ence?

This problem is discussed with the informa-tion metric, because relations within such aset of data have been well worked out (seeGarner, 1962). Furthermore, the informationmetric provides a closed analytic system, onein which we can identify the total constraintexisting in a system of variables and then canpartition this constraint into various meaning-ful terms while at the same time being assuredthat no part of the total constraint has beenleft out. Thus it is the logical properties ofthe information metric which are important,rather than the measurement properties perse. Once the various terms have been identi-fied, any of a number of measures of per-formance or correlation can be used withequal effectiveness.

There are four variables to consider: Twostimulus variables, area (A) and brightness( B ) , and two response variables, response toarea (a) and response to brightness ( b ) . Wecan write an expression for the total constraintwithin this system of four variables, \J(a:b:A:B), and this total constraint can then becompletely partitioned into several com-ponents especially meaningful for the ques-tion of perceptual independence:"

V(a:b:A:B) = V(A:Ji}

+ Va(b:A) + V A ( a : VAH(a:b). [3]

The term V(A'.B) is the contingency (or non-metric correlation) between the two stimulusvariables, and it is zero because we use thesevariables orthogonally. The next two termsare the direct contingencies in which each re-sponse variable is correlated with its ap-propriate stimulus variable. These are theterms we would expect to be high, since theyindicate the extent to which the subject can

3 This particular partition of the total constrainthas not been used before. Its derivation can be seenfrom equations given by Garner (1962). FromGarner's Equation 5.8,

U ( a : b : A : B ) = \ ] ( a : l > , A , B ) + V ( b : A ; B ) .From Equation 4.11, the first term on the right be-comes

From Equations 5.3 and 4.4, the second term onthe right becomes

V(b:A:B)=V(A:B)+V(b:B)+Ve(b:A).Combining these terms gives Equation 3 in thepresent paper.


accurately judge area and brightness. And thenormative model for perceptual independenceis that only these two terms are nonzero.

The last three terms all have something todo with the question of perceptual independ-ence. Two of the terms are of the same form,\ J H ( b : A ) and UA(a:B). These are partialcross contingencies (exactly analogous topartial correlation), each of them measuringthe correlation between a response variableand the inappropriate stimulus variable, withthe effect of the appropriate stimulus variablepartialed out. These two terms are directlyrelevant to the question of perceptual in-dependence, since each of them describes asort of "crossing over" from one perceptualsystem to another. To illustrate, supposethat judged brightness is increased with largeareas and that judged area is increased withhigh brightness; the effect would be that theresponse to area is correlated with stimulusbrightness, and the response to brightness iscorrelated with stimulus area. Thus the crosscontingencies are greater than zero.

The last term in Equation 3 is the partialcontingency between the two response vari-ables with both stimulus variables held con-stant, their effects thus being partialed out.At first glance it would appear that such aterm is not concerned with perceptual inde-pendence, since it measures response correla-tion with stimulus effects partialed out. Yetas seen in more detail subsequently, such aterm indicates error correlation, which canclearly be perceptual error as well as responseerror. Furthermore, this term, while being a

TABLE 1

COMPLETE INDEPENDENCE

TABLE 2

DATA JROM TABLE 1 REDUCED TOPERCENTAGE CORRECT

Complete

variable

dibiaibiatbiaibi

E

Orthogonal stimulus variable

AiBi

9664

25

AiBi

6946

25

AiBi

6496

25

AiB,

4669

25

25252525

100

Variable

B+B-E

Variable

A +

362460

A-

241640

2

6040

100

Note.—Data matrices are for the case in which stimulusvariables A and B are each responded to with 60% accuracy,and both perceptual processes and response processes operatecompletely independently. Cell entry is percentage of jointoccurrence of multiply denned stimulus and multiply defined

Note.—"+" is correct and "—" Is incorrect. Cell entry ispercentage of joint occurrence of correctness for stimulusvariables A and B. Tables 1-10 are all of the same form and thecell entries have the same meaning as indicated here. \J(Ao' Be)= 0. Percentage correct for: inclusive disjunction = 84;50-50 on doubtfuls « 60; conjunction = 36.

measure of process correlation, is directly re-lated to state correlation.

In order to clarify the uncertainty analysisand in order to give some additional meaningto the terms which are relevant to the prob-lem of perceptual independence, the authorshave prepared five sets of idealized datawhich illustrate conceptually different typesof process correlation. These data, in the com-plete orthogonal form, are given in Tables1-10; the actual data matrices are given inthe odd-numbered tables, and the reduceddata matrices in the even-numbered tables.Uncertainty calculations from each set ofdata are shown in Table 11. In each case,dichotomized stimulus variables A and B anddichotomized response variables a and b (thesubscripts 1 and 2 indicate the level of eachstimulus or response variable) are used. Alsoin each case, each level of each stimulusvariable always occurs equally often and thetwo variables are orthogonal in keeping withthe immediate concern. In addition, each levelof each response variable occurs equally often,although in three cases there is some cor-relation between responses.

Complete independence is illustrated inTables 1 and 2, and this case is the normativemodel for perceptual independence. Stimulusvariable A is responded to with response vari-able a with 60% accuracy, and there is equiv-alent accuracy for B. These two terms showin Table 11 as 0.029 bit of direct contingentuncertainty each, and the total constraintin the system of four variables is 0.058, thesum of these two terms. There is no crosscorrelation (i.e., b is not correlated with A,


TABLE 3

ERROR CORRELATION

TABLE S

JOINT RESPONSE BIAS

Completeresponsevariable

Q-\D\

(J-iuQ

&2Q\.

azbiL


A\Bi

11446

25

AiBi

41164

25

AiBi

46

114

25

AiB,

644

1125

•s

25252525

100

Note.—Data matrices are for the case in which stimulusvariables A and B are each responded to with 60% accuracy,and there is a tendency for both response variables (a and &) tobe right together or wrong together; that is, errors are correlated.

nor a with B), and there is no correlation be-tween the responses, that is, there is no errorcorrelation.

Error correlation is illustrated in Tables3 and 4, there being perceptual independenceotherwise. Stimulus variables A and B arestill each responded to with 60% accuracy,thus giving 0.029 bit of direct contingentuncertainty in Table 11. However, if a is inerror to A, the probability that b is in errorto B is increased. In Table 11 this relationshows up as the double partial term, UAB( a : b ) , with a value of 0.081 in the actualillustration. There is, however, no cross cor-relation, each response variable respondingonly to its appropriate stimulus variable. Thetotal constraint in this case is 0.139, which isthe constraint for the complete independencecase plus the additional constraint in theresponse system. Note that response or errorcorrelation produced this way does not pro-duce an overall contingency between the tworesponse variables: U(a:6) is zero. Thus it isimportant to differentiate between the simple

TABLE 4

DATA FROM TABLE 3 REDUCED TOPERCENTAGE CORRECT

B +B-£

Variable

A +

441660

A-

162440

6040

100

Complete

variable

flifii<Zl&2

0261#2^2

£


AiBi

11446

25

A\Bi

8728

25

AiBi

8278

25

AiBi

644

1125

2

33171733

100

Note.—Data matrices are for the case in which stimulusvariables A and B are each responded to with 60% accuracy,but there is a joint response bias such that the same a and 6responses are made more frequently than different a and bresponses.

contingency between responses and the con-tingency with the stimulus effects partialedout.

Joint response bias is illustrated in Tables5 and 6 and is a different way in which therecan be constraint in the response system.(Once again, the two direct contingencies eachgive 0.029 bit of contingent uncertainty, in-dicating the same level of 60% accuracy foreach stimulus variable.) In Tables 3 and 4the correlation was between errors in the twosystems, but the proportion of times that eachcombination of responses was made remainedat 0.25. In Tables 5 and 6, however, all re-sponse combinations are not used equallyoften, even though both levels of each re-sponse variable are used equally often. Thisis to say that there is correlation betweenthe responses, measured by U(a:b) in Tables5 and 6. A simple case where this mighthappen is in a threshold test for two visualstimuli at two retinal locations. If the subjectreports seeing both or neither more often thanone and not the other, he is exhibiting a joint

TABLE 6DATA FROM TABLE 5 REDUCED TO

PERCENTAGE CORRECT

B +B-£

Variable

A +

362460

A-

241640

2

6040

100

Note.—UUeiBc) = 0.081. Percentage correct for: inclusivedisjunction = 76; 50-50 on doubtfuls = 60; conjunction «= 44.

Note.—U(Ac:Bc) =0 . Percentage correct for: inclusivedisjunction = 84; 50-50 on doubtfuls = 60; conjunction = 36.


TABLE 7

SINGLE PERCEPTUAL PROCESS

TABLE 9

SINGLE RESPONSE PROCESS

Complete

variable

oibi0162flail02^2£


AiBi

9664

25

AiBs

9664

25

AiBi

4669

25

A,B,

4669

25

S

26242426

100

Note.—Data matrices are for the case in which both responsevariables a and b are in response to stimulus variable A, eachwith 60% accuracy, and the two responses are otherwise madeindependently of each other.

response bias. The consequences of such abias, as seen in Table 11, is also to produce anonzero value for UAs(a:6), and a totalconstraint which is increased from the in-dependence case by this amount. So unlikeTables 3 and 4, in this case there is contin-gency between responses both with and with-out the effects of the stimulus partialed out.

A single perceptual process is illustrated inTables 7 and 8 but with independent responseprocesses. For this case it is assumed thatboth response variables, a and b, are made inresponse to stimulus variable A, each with60% accuracy, and that neither is correlatedwith stimulus variable B. However, beyondthis fact, the two response variables are madeindependently of each other insofar as pos-sible. In uncertainty terms, the direct contin-gency, \J(a:A), remains at 0.029, but insteadof the other direct contingency term havingthis same value, it now appears as the partialcross contingency, UB(b:A), with the samevalue of 0.029. And the total constraint isthe same as it was for the independence case

Complete

aibiaibiaibiciibz£


^4iBi

1500

1025

A,B,

1500

1025

AiBi

1000

1525

AiBi

1000

1525

2

5000

50100

Note.—Data matrices are for the case in which responsevariable a is used with 60% accuracy in response to stimulusvariable A, and response variable b is then copied from a;that is, the two response variables are perfectly correlated andare used in response to A with 60% accuracy.

in Tables 1 and 2. While this arrangementproduces no error correlation, U^B(a:6), thereis a small amount of contingency between thetwo responses when the effect of the stimulusis not partialed out, U(a:6) , and this is dueto the fact that two things each correlatedwith a third must be correlated with eachother, the extent depending on the degree ofcorrelation each has with the third.

A single response process is illustrated inTables 9 and 10. The two response variablesare perfectly correlated, and then this effec-tively unitary response is made only to stim-ulus variable A and with 60% accuracy. Soas with a single perceptual process, there is0.029 bit of direct contingent uncertainty plusan equal amount of partial cross contingentuncertainty. In addition, however, there isnow a large error correlation, MAE(a:b) being0.971 bit. This term also enters into the totalconstraint, which now becomes 1.029 bits.Note that the direct interresponse contin-gency, U(a:b), is 1.000 bit, which is simply a

TABLE 8


B+B-£

Variable

A +

303060

A -

202040

s

5050

100

Note.—U(^4c:Bc) =0. Percentage correct for : inclusivedisjunction = 80; 50-50 on (ioubtfuls = 55 ; conjunction •» 30.

TABLE 10


Variable

B +n

£

Variable

A +

303060

A-

202040

5

5050

100

Note,—U(Acm .B c) = 0. Percentage correct for: inclusivedisjunction = 80; 50-50 on douhtfuh - 55; conjunction = 30.


TABLE 11

UNCERTAINTY ANALYSES TOR THE COMPLETEMATRICES OF TABLES 1-10

U(A:B)+V(a:A)+U(i:5)+Vs(b:A)+ UA(a:B)+tWa:6)= U(a:b:A:B)-TJ(«:6)= V(a,b:A,B)

Data matrix

1

00.0290.029000O.OS800.058

3

00.0290.029000.0810.13900.139

5

00.0290.029000.0820.1400.07S0.065

7

00.02900.029000.0580.0010.057

9

00.02900.02900.9711.0291.0000.029

Note.-—The cell entry is bits of contingent uncertainty. Thesymbols to the left of the notations for the uncertainty termsindicate the mathematical relations between the terms (seeEquations 3 and 7).

description of our statement that the two re-sponse variables are perfectly correlated.

Thus we have illustrated the normativemodel for complete perceptual independence,plus four cases of nonindependence, each ofthese four being based on a different con-ceptualization of how the correlations withinand between the two systems occur. Eachproduces an identifiably different pattern ofuncertainties when a complete analysis is car-ried out. Of the three uncertainty termsidentified with process correlation, it mightseem that the partial response contingency,U^jj(a:6), is possibly not really related toperceptual independence, but that term isintimately related to perceptual correlation instate models.

State correlation. As we noted above, statemodels are not concerned with direct inter-action of the perceptual processes themselvesbut rather with the state of the organism inrespect to the two or more channels of in-formation being processed. The usual experi-mental question is not whether the processesare correlated but whether accuracies arecorrelated.

This problem is handled on a simple levelhere, using the same idealized data presentedin Tables 1, 3, 5, 7, and 9, and asking aboutthe relation between data put in form totest state correlation to data in the originalform. To accomplish this, each of these fivesets of hypothetical data has been reduced

to a 2 X 2 matrix, in which the entry is thejoint percentage of times that A and B, thestimulus variables, were responded to cor-rectly. To illustrate the relation between thereduced and original matrices, consider Tables1 and 2. The entry for A+B+ in the reducedmatrix is the sum of the diagonal entries forAjB^ibi plus A1B2aib2, etc. The entry forA — B — is the sum of the other diagonal, in-dicating that errors are made with respect toboth stimulus variables. The entry for A+B —in the reduced matrix is the sum of the termsin which A was correctly responded to andB was incorrectly responded to, for example,AiBrfibz plus AiB-^ibi, etc. The last entryin Table 2 is computed in like fashion.

These reduced matrices are shown for eachof the five cases. In the note to the reducedmatrix is also shown another contingent un-certainty, indicated as U(AC:BC), which isthe contingent uncertainty or correlation be-tween correctness of response on the two vari-ables. It is the correlation term directly re-lated to state correlation. The other statisticsshown there relate to decision rules and arediscussed later.

First, note the relations between the twomatrices for each set of data. The reducedmatrix has marginal terms which indicatethe average performance level for each of thetwo stimulus variables. We score correct onlyin relation to the appropriate response vari-ables, so these marginal terms indicate thesame thing as the direct contingency termsdo in the complete orthogonal matrix. All ofthe matrices are idealized, so the probabilityof being correct is always independent of thelevel of the stimulus variable. Thus to com-bine these levels into a single measure ofright or wrong does not lose any information.If the probability of being right itself de-pended on the level of the stimulus variable,this information would have been lost in thereduced matrix.

But now consider each of the five reducedmatrices, those which are to tell us aboutstate correlation. In only one case, Table 4, isthe appropriate contingent uncertainty, \J(A0'.Be), nonzero. Thus we would conclude thatthere is perceptual independence in four ofthese five cases, even though the original data


were set up so that four out of the five caseswere clearly indicative of perceptual non-independence or correlation. In Table 4, thecontingent uncertainty between correctness onA and on B is in fact the same as the partialcontingent uncertainty between responses withstimuli held constant in the orthogonal matrix,UAI!(a:b). That case was described as one inwhich there was error correlation, which isafter all what should be happening if thereis state correlation. Note, however, that theexistence of such a nonzero term in theorthogonal matrix does not necessarily meanthat there will be correlation in the matrixreduced to investigate state correlation, sincethere is also such a term in Tables 5 and 9,and neither of them shows state correlationin Tables 6 and 10.

The relationship between the values of theuncertainty terms in the full matrix and thosein the reduced matrix is complex. To some ex-tent this complexity is due to the fact thatthe reduced matrix is obtained by, in effect,arithmetic averaging of terms in the fullmatrix, while the uncertainty terms them-selves are based on logarithmic equivalentsof the various proportions. In addition, how-ever, the relations themselves are complex,so that, for example, it is possible to haveboth correlated error and a joint responsebias, in which case UAB(a:b), U(a:b), andU(AC:BC) will all be nonzero. The usefulnessof the reduced matrix must then lie in thepossibility of distinguishing between the casewhere there is joint response bias alone andthe case where this is accompanied by errorcorrelation. What is certainly clear is thatanalysis of the reduced matrix by itself yieldsvery little information.

Still further, the existence of a correlationin the reduced form does not by itself differ-entiate perceptual process from responseprocess, since such a term indicates only thaterrors are correlated, and they could be cor-related as a consequence of perceptual corre-lation or of response correlation. Neither doesthe analysis from the orthogonal matrix makesuch a differentiation. Nevertheless, a farmore complete and informative picture ofprocess can be obtained from complete analy-

sis of the orthogonal matrix than from thereduced matrix.

Data reduction and decision rules. Aspointed out previously, models may test forperceptual independence indirectly as well asdirectly. In the present case, an easily usedindirect test would be simply to count thetotal number of correct responses rather thanto measure directly correlation in the reducedmatrix. However, before we can carry outsuch a procedure, we (the experimenters, thatis) have to establish some decision rule todetermine what constitutes being correct,since two of the four cells in the reducedmatrix are cases where the subject is cor-rect with respect to one attribute and in-correct with respect to the other.

There are three prototypical rules forhandling these cases of partial correctness,and the percentage correct computed for eachof these rules is shown in the note to each ofthe five reduced matrices:

1. Inclusive disjunction: Credit for beingcorrect is given if the subject is correct oneither or both stimulus variables. This is therule that leads to the familiar normativemodel value of (pA + />« — pApn), wherep,i and pn are the respective probabilities ofbeing correct on the two stimulus variables.

2. 50-50 on doubtfuls: When the subject iscorrect on one variable and incorrect on theother, give him credit for being correct onhalf the total number of such cases, plus thosecases in which he is correct on both.

3. Conjunction: Credit for being correct isgiven only when the subject is correct on both.

It is of interest to compare the percentageof correct responses which would be obtainedfor each of the five idealized matrices, giventhe actual marginal values and correlationsshown. Table 2, representing the case ofcomplete independence, provides the basevalues to be expected with each decision rule.

Two of the matrices, Tables 8 and 10,represent cases where only chance correctnessoccurs with respect to the B variable, andwhere there is no state correlation. Loweredperformance occurs for these two matrices forany of the three decision rules, and thislowered performance simply reflects the lackof accuracy with one stimulus variable,

24-6 W. R. GARNER AND JOHN MORTON

Each of the other two matrices has equalperformance on stimulus variables A and Bseparately. Table 6 also has zero state cor-relation, and thus each decision rule givesthe same performance as expected from thenormative Table 2. Table 4 has state correla-tion, and this correlation produces percentagecorrect values for two of the decision ruleswhich are different from those obtained inTable 2, but these two rules give differencesfrom the independence case in opposite direc-tions. With the inclusive disjunction rule,there is lower performance with correlatedaccuracy than with complete independence,while with the conjunction rule, performanceis better with state correlation than without it.

Just as it is argued that reduced datamatrices should not be used to test directlyfor state correlation if the orthogonal matrixis available, here it must be argued that in-direct tests of state correlation by measure-ment of percentage correct is not reasonable.Not only does it confound performance on theseparate variables with the correlation prob-lem, but it further introduces the complica-tion of having to use a decision rule; andunfortunately the decision rule chosen affectsapparent performance differentially for cor-related and uncorrelated cases. Thus in gen-eral there is little justification for using anyreduced form of data for testing perceptualindependence if it is possible to obtain anduse data in the more complete form.

Orthogonal Input—Unitary Response

Up to this point, the paper has been con-sidering the experimental paradigm whereinthe stimulus variables are orthogonal and thesubject may use his response variablesorthogonally, whether or not he does do so.Now, models of independence when the ex-perimental situation presents orthogonal stim-ulus variables, but the subject must respondwith a single output variable, must be con-sidered. There are two paradigms withinwhich such a requirement would be reason-able:

1. The nature of the stimulus variablesmakes some sort of composite response reason-able. For example, the experimenters might

carry out a threshold experiment with lightspresented or not presented, in all possiblecombinations, to the two eyes separately butsimultaneously. The subject is then requiredonly to state whether he did or did not seea light. Alternatively, stimuli varying in areaand brightness could be used, but with thesubject being required only to give a singleresponse, perhaps indicating a kind of com-posite or total magnitude.

2. The subject is instructed to respondselectively to one stimulus channel or variablealone, even though two orthogonal channelsof information are used as the stimulus. (Ineffect this is what the subject has done forhimself with the idealized data of Tables 9and 10). Ordinarily such a procedure wouldbe used when the experimenters are thinkingin terms of interference effects (i.e., the secondchannel is a distractor, is an irrelevant vari-able, etc.).

Process correlation. When measuring processcorrelation in these two cases, the compositeresponse or selective response are the same. Ineither case this is an undifferentiated responsevariable, symbolized as R, the difference be-tween the two cases being in instructions tothe subject rather than the form of the data.

For a direct test of process correlation,the informational analysis is much simplerthan before, since only three terms, the twostimulus variables (still A and -B) and oneresponse variable (R) are involved. The totalconstraint in this system of three variables,V(R:A:B), can be partitioned into fourterms:

. [4]

The first term on the right is again thecontingent uncertainty between the two stim-ulus variables and is zero because the stimulusvariables are still orthogonal. Each of thenext two terms is the direct contingent un-certainty which indicates the extent to whichthe single response variable is correlated withthe stimulus variable. The last term is aninteraction uncertainty and is the only termdirectly relevant to the question of perceptual


independence. Its relation to this questioncan be seen by writing

V(RAB) = VA(R:B) U(R:B)

V(R:A)

[5]

[6]

Thus the interaction term is a differencebetween the contingent uncertainty relatingthe response and one of the stimulus variablesand that same term with the other stimulusvariable held constant, its effect being thuspartialed out. If the relation between the re-sponse and a stimulus variable is changedby holding the other stimulus variable con-stant, then there is an effect of that secondstimulus variable on the percept pertinent tothe first variable. Note, however, that sinceboth Equations S and 6 are true, the effectcannot be differentiated as being due to A orto B without further experimental controls. Inlike manner, the cross contingencies cannot beidentified, since with an undifferentiated re-sponse it is not possible to tell how much ofthe correlation of the response with a stimulusvariable is appropriate and how much isinappropriate.

Thus the use of a single response in anexperiment, even though at times appropriateto a particular hypothesis, produces a drasticreduction in our ability to determine theamount and nature of perceptual process cor-relation.

State correlation. There is no reasonableway in which a direct test can be made forstate correlation with unitary response data.Since the subject does not separately identifyhis response to A and his response to B, theexperimenter in turn cannot separately identifywhen he is correct on A but not on B, etc.Thus data matrices similar to the reducedmatrices in Tables 2, 4, 6, 8, and 10 cannotbe generated directly from the available data.This fact means that the only possible test ofstate correlation must be indirect.

Given experimental questions for which theuse of a unitary response is appropriate, it israre that a test of state correlation seemsnecessary or desirable. If the subject is di-rected to attend and respond to only onechannel or variable, the experimental interestis normally in parity not in correlation. And if

it is meaningful to make a composite response,the interest is more with the actual responsethan with whether it is right or wrong. Thereis one type of situation in which the conceptof state correlation does make sense, and inwhich a unitary response also makes sense,and that is the threshold case where twostimuli are presented, all combinations ofpresence or absence being used, and the sub-ject simply responds that he did or did notperceive a stimulus. Now there is a criterionby which correctness can be decided; namely,if either stimulus was presented, the subjectshould have responded positively.

An indirect test would then compare theoverall percentage correct responses madeagainst a normative model which assumeszero correlation between the states for the twoseparate stimuli. A hypothetical matrix suchas the reduced matrices of Tables 2, 4, 6, 8,and 10 would be generated, and from them atotal percentage correct figure would be ob-tained. However, with a single experiment theindirect test still could not be made becausethere is no way of determining percentage cor-rect responses for each stimulus separately,and these are required in order to generate thehypothetical matrix with zero correlation.Thus no meaningful test of state correlationcan be made with this experimental paradigmalone.

Parity and orthogonal inputs. In the experi-mental situation in which orthogonal inputsare used, and especially when orthogonal re-sponding is permitted, the data make directcalculation of correlations possible, and thusare ideally suited to direct testing of the cor-relation criterion. Even when data are reducedto percentage correct form, state correlationcan be directly calculated. The parity criterionof independence is simply not applicable withthis experimental paradigm, because anyparity check is simply an indirect test of thecorrelation criterion of independence. Thusperformance parity wilh this paradigm has nomeaning separate from thai of zero correla-tion. It is worth elaborating this problem abit more.

If informational analysis is used again, amultiple contingent uncertainty indicates totalperformance between the two stimulus vari-


ables and the two corresponding responsevariables. It is written as U(a,b:A,B) and isrelated to the total constraint in the systemof four variables as

\J(a,b:A,B) =

-U(a:V)-U(A:B). [7]

Thus total performance in which a multipleresponse is contingent on a multiple stimulusequals the total constraint in the system minusthe constraint within the system of responsevariables themselves and minus the constraintwithin the system of stimulus variables. Sincethe case of orthogonal input is being dis-cussed, then the latter term, U(A:B), is zeroby definition. Thus the only term which pro-duces a difference between the total constraintand the multiple contingent uncertainty is thesimple contingent uncertainty, U(a:b), show-ing the extent to which the response variablesare correlated.

The apparent performance-parity checkwould be to compare the magnitude of themultiple contingent uncertainty, \J(a,b:A,B),with the sum of the two direct contingencies,U(a:A) +U(b:B). A brief comparison foreach of the five data matrices, as shown inTable 11, will clarify the fact that a dis-crepancy between these two measures showsonly that there is correlation somewhere inthe system, although it does not locate thesource of the correlation.

For Table 1, the case of independent proc-essing, the sum equals the multiple term,which indicates that all correlations otherthan the direct contingencies are zero. ForTable 3, the sum is substantially smaller thanthe multiple contingent uncertainty. Thusparity is not satisfied, and it is known thatcorrelation exists but not where. In Table 5,there is also a discrepancy, although com-parison of this result with that of Table 3shows the importance of complete analyses.These two matrices look alike with respectto total constraint but quite different withrespect to total in-out performance. For Table7, the multiple contingent uncertainty isgreater than the sum of the two direct con-tingencies (one of them being zero), so onceagain parity is not satisfied; therefore, thereis correlation somewhere in the total system.

Table 9 is the most interesting, because withit parity is satisfied, the multiple contingentuncertainty being the sum of the two directcontingencies. Thus, one would conclude thatthe processes are independent, when in factthis matrix shows data with the greatest de-gree of perceptual and response interdepend-ence.

Thus the use of parity checks with data inwhich correlation can be computed directlyserves only as an indirect measure of whatcan be measured directly; furthermore, itserves this purpose very inadequately, failingto locate the source of correlation when cor-relation is indicated, and in some cases evenfailing to indicate the existence of correlation.

This limitation in the use of parity checkswith single experiments, because parity isdefined in terms of zero correlation, is in noway restricted to the use of the informationmodel. All metric models have exactly thesame limitation, as shown basically in Equa-tion 1, where parity is defined in terms ofzero correlation. If we use d' measures (seeGreen & Swets, 1966; Tanner, 1956), or someof the dimensional scaling procedures (seeTorgerson, 1958), then parity is defined withthe Pythagorean relation, so that, for example,

d A.B = d A + d 2 B - C8J

This relation, however, holds only if the twoprocesses are uncorrelated. Usually this rela-tion is said to constitute a Euclidean geo-metry, although the primary consideration isthis Pythagorean relation (Euclid's proposi-tion 1:47), and this relation holds (as everyschoolboy knows) only for right triangles, thatis, orthogons, from which the general termorthogonality for zero correlations comes.If we do an experiment and find that paritydoes not hold, then we can assume that therewas a correlation, that is, that the relation-ship was not of a right triangle. Alternatively,of course, one could assume that the model iswrong. On the other hand, if parity appearsto hold, the result could be due to a correla-tion which offsets a true loss of parity.

The limitations of the single orthogonalexperiment in determining whether parity ex-ists are strongest with a linear correlationmodel used with a unitary response system. If


linear correlations are computed, then thesquared multiple correlation coefficient is

7?^ p. A n = r^ j>. A -4~ r^ a u TOT•**- K.AtU ' K.A I ' K.It' L^J

(For the correlations, subscript notation hasbeen used like that used with uncertaintymeasures to emphasize the parallel relations.The subscripted R is the unitary responsevariable,) This equation, again showing thePythagorean relation so common in statisticswhich use variance measures, cannot bewrong. The parity relation holds if the pre-dictor variables are uncorrelated, and in thisexperimental paradigm they are. So if wemeasure each correlation separately, thenmeasure the multiple correlation directly, andfind that the sum of the two squared separatecorrelations equals the square of the multiplecorrelation, we have learned exactly one thing,namely, that we have done our arithmeticproperly. Thus with this particular model andthe experimental paradigm of an orthogonalinput but unitary response, we can ask nomeaningful question about perceptual inde-pendence.

Single Input Experiments as Controls

Since the single experiment with orthogonalinput channels is not satisfactory in providinga check on the performance-parity criterionof independence, some separate measure ofperformance is needed outside of theorthogonal experiment itself. The most nat-ural technique is to run additional controlexperiments in which one channel or variableis used at a time, with responses appropriateto that single channel or variable. There islittle complexity about handling the data inthis case, since they would be handled in theform in which the parity comparison would bemade; for example, either correlations orcontingencies as measures of performance, orpercentage correct responses as in the re-duced matrix, can be used as long as themeasures correspond in the orthogonal andsingle input experiments.

The parity checks themselves are quitestraightforward in most cases and involvecomparison of the performance measure ob-tained in the single input experiment with themost analogous term obtained in the larger

orthogonal experiment. When an orthogonalresponse system has been used, then (usingthe information measures) there will be twodirect contingencies from the orthogonal ex-periment and two simple contingencies fromthe single channel experiments, one for eachvariable or input. If performance measuresare larger with the single input experimentthan with the orthogonal experiment (or pos-sibly smaller), then parity is not satisfied.

If a unitary response has been used withthe orthogonal experiment, then there stillare appropriate direct contingencies availableto compare to the simple contingencies ob-tained in the single input experiments. How-ever, if the unitary response is a selectiveresponse to just one stimulus input, then therewould ordinarily be just the one comparison.This type of comparison, of course, is ideallysuitable when interchannel interference effectsare expected.

There might be some question about whatthe proper comparison is in making theparity check. We have suggested that themeaningful comparison is to compare sepa-rately equivalent terms from the two experi-ments. An alternative that might be con-sidered is to compare total performance fromthe orthogonal experiment, the multiple con-tingent uncertainty \J(a,b:A,B), with thesum of the two simple contingencies obtainedin the two control experiments. Such a com-parison is not valid, however, since the totalperformance measure, as discussed above andas shown in Table 11, contains process cor-relation terms as well as direct performanceterms. To illustrate, Table 3 would give atotal performance greater than the sum ofthe two simple contingencies (assuming themto be the same as the two equivalent directcontingencies in Table 11), but this greaterperformance is due entirely to error correla-tion. Certainly an independent criterion ofperformance parity does not want to be con-cerned with such terms.

While it is possible to make indirect checksof performance parity, by observing some con-sequences of the parity assumption ratherthan the performance itself, it is unusual tofind an example where an indirect check doesnot confound the two kinds of independence.

W. R. GARNER AND JOHN MORTON

The essential dependence of the parity cri-terion on correlation was noted previously. Asone example of a more indirect test, thebinominal distribution of "hits" might bedetermined on the basis of probabilities ob-tained from the single-channel control ex-periments. If the expected distribution doesnot agree with the obtained distribution, thediscrepancy could be due either to a loss ofparity or to the existence of correlation. Ifdirect comparison were then made of theoverall hit rate in the orthogonal experimentwith the average hit rate from the singlechannel experiments, the difference would giveinformation about parity, and the shape ofthe distribution would then give informationabout correlation. But this separation meanswe have gone to a direct test of parity, leavingonly correlation as an indirect test.

The performance-parity criterion of per-ceptual independence, and one which is quiteseparable from the correlation criterion, isespecially necessary when we consider thelarge number of experiments in which itsimply is not feasible to measure correlationin any direct fashion. All of the experimentswhich use time, either reaction time or speed,as the measure of performance fall into thiscategory. Performance as correlation with aninput variable cannot be measured in theseexperiments, because maximum accuracy isexpected of the subject, with differences intime then presumed to indicate differences indifficulty or performance. Also, those experi-ments in which the stimulus inputs do nothave measurable correlation (e.g., separatespeech channels) need to use the performance-parity criterion of independence. Even thoughthe formal logic of the relation between parityand correlation is not applicable in these cases,the conceptual problem of untangling thesetwo aspects still exists.

Other control conditions. There are controlexperiments other than the use of a singleinput channel that are useful in determiningthe existence or nature of perceptual inde-pendence. One ideal set of experiments mightbe worth mentioning because it forms a fairlycomplete and self-contained set of excellentconverging operations (Garner, Hake, & Erik-sen, 1956). There would be, for a two-variable

stimulus such as visual area and brightness,the complete orthogonal experiment. Then twocontrol experiments would be run in whicheach stimulus variable is used alone. Thentwo more control experiments would be runusing the orthogonal stimulus set, but withthe subject being required to respond selec-tively to just one stimulus variable in eachexperiment. Direct calculation of all contin-gent uncertainty terms would be made fromthe orthogonal experiment. Then the two ob-tained direct contingency terms would becompared against the values obtained withthe single input experiments. Suppose thesevalues are lower with the orthogonal experi-ment, so that there is a loss of parity. Nowcompare each of these simple contingenciesfrom the single input experiment to the valuesobtained when orthogonal stimuli with selec-tive responding were used. If the compari-sons show no difference, the conclusion wouldbe that additional perceptual load per se doesnot lead to lower performance, so that thelowered performance obtained with the com-plete orthogonal experiment is due to theextra task requirement of responding to bothvariables. Alternatively, a result showing thatthe loss occurs when the stimulus variablesare orthogonal, regardless of whether responsemust be made to one variable or both, wouldindicate that the parity loss is almost certainlydue to some perceptual incapacity. Still fur-ther, both the complete orthogonal experi-ment and the selective response experimentshave terms which can be identified as crosscontingencies. Comparison of these termswould lead to clarification of the nature ofthe process correlation.

In other words, to gain a reasonably ac-curate picture of whether and what kind ofperceptual independence exists requires notone but several experiments all hopefullyconverging on the same presumed process.(As an illustration using d' measures, seeTaylor, Lindsay, and Forbes, 1967.)

Correlated Input Experiments

The last major experimental paradigm con-sidered is that in which the stimulus variables,inputs, or sources of information are them-selves correlated. For example, if geometric


forms are to be discriminated, the form is al-ways the same regardless of the number ofthem. Or if speech context is one source ofinformation, and tachistoscopic presentation isanother form, the context is appropriate tothe word tachistoscopically presented. Stillfurther, if judgments of visual area andbrightness are required, a given brightness ispaired with a given area, and only a correlatedsubset of the orthogonal combinations is used.

From the viewpoint of understanding per-ceptual independence, the widespread use ofthis experimental paradigm is unfortunate,because learning something about independ-ence of the perceptual process is funda-mentally easier if the stimulus variables arethemselves independent. The use of correlatedstimuli confounds the problem of independ-ence and also provides such a reduced set ofdata that very limited information about per-ceptual independence is obtainable from suchexperiments.

Yet this experimental paradigm must beconsidered, and we must attempt to under-stand it, because the paradigm represents aproblem in and of itself. It is the problem ofthe organism's ability to make use of redun-dant information, and some of the earliestexperiments done on the problem were con-cerned with this very problem (e.g., Eriksen& Hake, 19SS). The question of perceptualindependence enters in because the ability ofthe organism to use redundant informationobviously depends on the extent to whichthis externally correlated information is pro-cessed independently internally, and Garnerand Lee (1962) argued that there should beno gain in perceptual performance with re-dundant stimulus information unless the per-ceptual processes are independent. So theproblem of perceptual independence is acritical aspect of the problem of utilizationof redundant information.

A major problem in studying perceptualindependence with correlated inputs is thatthe data are necessarily so reduced. We haveseen that quite detailed analysis and under-standing of independence are possible if bothstimuli and possible responses are orthogonaland that the analysis becomes quite restrictedif a unitary response system is used. Yet, as

Equation 4 shows, some direct measurementof process correlation is possible with aunitary response if the stimuli are orthogonal,although there is no reasonable way of di-rectly measuring state correlation. If in addi-tion to limiting the response we now alsolimit the input by correlating stimulus values,we completely destroy any possibility of di-rect measurement of either process or statecorrelation because we have only a single,two-dimensional, input-output matrix of dataavailable. Therefore we may use only indirectmodels which state some consequence of theparity and/or correlation criteria of independ-ence, and if correlation, then with respect toprocess and/or state correlation. Further, aselaborated later, decision rules need to beassumed, and these rules likewise can bespecified with regard to perceptual process orperceptual state. Thus an indirect normativemodel which states what kind of performanceshould be obtained with correlated input mustdeal with more conceptual distinctions thanour models can ordinarily handle.

Parity and/or correlation. Performance withcorrelated stimuli will be affected by a correla-tion between the underlying perceptual pro-cesses or states as well as by any parity con-siderations that enter in when more than oneinput channel is used. No direct measure ofeither parity or correlation can be obtainedfrom the single experiment with correlatedstimuli. However, independent estimates ofperformance relevant to the parity criterion ofindependence are usually obtained by runningcontrol experiments with single stimulus vari-ables, just as described above with respect toorthogonal input experiments. These inde-pendent measures of performance level thenprovide a base line against which to compareperformance on the joint task.

Usually, these independent measures of per-formance are used to construct a hypotheticalmatrix of data based on the assumption ofzero correlations; that is, a matrix presumedto reflect process in the organism which can-not be overtly measured, as it is with thecomplete orthogonal experiment. Then if per-formance with correlated stimuli is differentfrom that expected from this normative modelof zero correlation, the discrepancy in effect is


attributed to the existence of correlation.However, performance on the correlated inputtask will be affected by either a loss of parity,by perceptual correlation, or both, and thisexperimental procedure cannot differentiate.A more complete and satisfactory procedurewould be to run the additional control forprocess correlation by obtaining data from theorthogonal stimulus experiment. Given thecorrelations obtained there, a normative modelcould be constructed which assumes perform-ance with correlated stimuli to involve thesame parity and process correlation con-siderations as are found in the orthogonalcase. If performance is different, then we knowthat the use of correlated inputs has somehowchanged the perceptual task for the subject.

Correlation: process and/or state. Whenprocess and state correlation with the com-plete orthogonal experiment was discussed, itwas pointed out that state correlation in re-duced data matrices can be identified witha process correlation term, and in the fivesets of illustrative data matrices used, onlyone showed state correlation, although four ofthem showed process correlation. This dis-cussion now continues within the context ofthe correlated input experiment. What weshall see is that the introduction of correlationin the stimulus variables thoroughly com-plicates the relation between process and statecorrelation and makes essentially impossiblea simple interpretation of a discrepancy ofobtained performance data from a value com-puted from a normative model assuming per-ceptual independence.

Assume that there are the same twodichotomous stimulus variables, A and B, andthat these are now presented in a correlatedfashion. Two responses are allowed, one ap-propriate to each of the two stimuli. Thedata are then reduced to a correct-incorrectform as before (although the term reductionis now less appropriate than transformation,since it is a 2 X 2 matrix to start). Per-centage correct responses should be computedfor the ideal case of complete perceptual in-dependence, and for this purpose a decisionrule based on data in percentage correct formis assumed, although the decision rule is pre-sumed to exist in the organism and to produce

the manifest data, rather than to exist ex-ternal to the organism and to be applied tothe manifest data. Expected performance foreach of the other data matrices is then com-pared in turn:

The reduced matrix in Table 2, for theperceptual independence case, is the samewhether or not correlated stimuli are selected.An assumed internal decision process whichuses an inclusive disjunction rule shows thatthere should be 84% correct when two stim-ulus variables are used in a correlated fashion.That is to say, when another variable isadded, and perceptual independence exists,(hen percentage correct responses increasesfrom 60% to 84%. There has been some gainin perceptual performance.

The reduced data matrix in Table 4 is alsotrue regardless of the subset of correlatedstimuli selected, and this is the matrix inwhich there is state correlation. Thus perform-ance should be less than the normative valueof 84%, and for the inclusive disjunction ruleit does indeed drop to 76%. Thus the statecorrelation has produced lower performancethan expected, given independence; we there-fore reject the assumption of zero state cor-relation, correctly.

For the matrix in Table 5, and the othertwo matrices, new reduced matrices mustbe formed because they are different forcorrelated stimuli and furthermore dependon which subset of correlated stimuli we use.Table 12 shows the two possible matricestaken from Table 5. The reduced matrixin 12A is obtained by considering only thefirst and fourth columns of Table S andtransforming the data to percentage cor-rect as before, except that after summing theappropriate numbers they are changed to per-centages again. Thus the value for A+B+is the sum for A\B^a\b\ and A2B2a2bz fromTable 5, which is 22, but becomes 44 tomaintain the numbers in percentage form. Thereduced matrix in 12B is obtained from thesecond and third columns of Table S, with thecell entries obtained in comparable fashion.Thus the cell entry in Table 12B for A+B+is the sum of A^Bna^b^ plus AnB^a^b^ valuesin Table S.


TABLE 12

DATA MATRIX 5 REDUCED TO PERCENTAGE CORRECTFOR THE Two TYPES or STIMULUS CORRELATION

VariableVariable

A + A-

•£,

A: A and B take same values

B +B-£

441660

162440

6040

100

B:A and B take opposite values

B +B-E

283260

328

40

6040

100

Note.—"+" is correct and " —" is incorrect. Cell entry ispercentage of joint occurrence of correctness for stimulusvariables .A and B. For an actual experiment, these matriceswould be hypothetical, referring to a presumed internal process.Tables 13 and 14 are to be Interpreted In similar way. ForA;\J(A,\Ba) =0.081. Percentage correct for: inclusive dis-junction = 76; 50-50 on doubtfuls = 60; conjunction = 44.For B:U(A,:B,) =0.084. Percentage correct for: inclusivedisjunction = 92; 50-50 on doubtfuls — 60; conjunction — 28,

Note that there was no state correlation inthe reduced matrix from the complete or-thogonal experiment, but that there is sub-stantial correlation (shown as the contingentuncertainty for correctness in the note toTable 12), for each of the two matricesselected for correlated stimuli. Furthermore,the inclusive disjunction rule gives a valuefor correct responses of 76% in one caseand 92% in the other. These two valuesaverage out to the value expected forperceptual independence (as shown for thereduced matrix in Table 6), but we wouldconclude that performance was better thanthat for independence in one case andpoorer in the other. What has happenedhere is that the selection of stimulus subsetshas converted a fairly straightforward jointresponse bias into two state correlation terms,one positive and one negative, depending onwhich subset of correlated stimuli has beenselected. Thus the particular nature of thejoint response bias will produce apparent non-independence in the perceptual state, eventhough none exists in the complete reducedmatrix. The use of correlated stimuli interactsstrongly with the nature of perceptual process

TABLE 13

DATA MATRIX 7 REDUCED TO PERCENTAGE CORRECTFOR THE Two TYPES OF STIMULUS CORRELATION

VariableVariable

A + A-2

:A and B take same values

B+B-E

362460

241640

6040

100

and B take opposite values

B+B-E

243660

162440

4060

100

Note.—For A:U(Ae:Be) = 0. Percentage correct for: in-clusive disjunction — 84; 50-50 on doubtfuls = 60; conjunc-tion - 36. For B:U(A,:Bo) =0. Percentage correct for:Inclusive disjunction •= 76; 50-50 on doubtfuls = 50; con-junction — 24.

correlation to produce apparent state correla-tion where none truly exists.

The data from Table 7 are reduced intothe two separate matrices in Table 13. Inthis case, the stimulus selection does notproduce apparent state correlation, reflectingthe lack of the partial response contingentuncertainty, U^a^), as shown in Table 11.However, note that in this case the inclusivedisjunction rule gives an 84% value (thussuggesting perceptual independence) for thematrix in which stimulus values are the same,while showing the lower value of 76% for thematrix in which the values differ. This latterresult, obtained in an actual experiment,would be interpreted as indicating state cor-relation. Inspection of the two matrices, how-ever, shows that it is due to the paradoxicalresult showing 60% errors to B, offsetting theapparent 60% correct with the other selectedsubset of correlated stimuli. Both of thesevalues are produced only by selection of thesubset of stimuli, since in the original matrixthere is no correlation between B and responseto.B.

The data from Table 9 are shown inthe two separate matrices in Table 14. Thisidealized case is, of course, extreme forpurposes of illustration, since the data were


constructed on the assumption that a is per-fectly correlated with b. And yet it bettermakes the point. Here once again there is astate correlation produced in each of thesematrices by the selection of correlated stimuli.It is negative in one case, positive in theother. Furthermore, we again have the para-doxical result of more than chance errors forB in Table 14B. The inclusive disjunctionrule, for these two cases, gives 60% for onematrix (no improvement over a single vari-able) and 100% in the other, a result whichwould be difficult to interpret, to say the least.

Thus the selection of a correlated stimulussubset introduces changes in apparent averageperformance and introduces state correlationas a result of the existence of process correla-tion not related to state correlation in thecomplete matrix. The use of the correlatedinput experimental paradigm truly makesunderstanding of perceptual independencedifficult.

Decision rule: process and/or state. Wehave seen that the particular decision rulepresumed to exist in the organism greatlyaffects the calculated percentage correct re-sponses which is to represent the value for anormative model of independence, and thatfurthermore the rule used gives differentialexpectations about performance for the fivebasic sets of data used as illustrations. That isto say, which nonindependent case will givegreater performance depends on the decisionrule assumed to be going on in the organism.A still further consideration needs to be dis-cussed and that is whether the decision rulepertains to process or to state.

All of the decision rules discussed so far,and those illustrated in the various tables,are rules pertaining to the organismic state,not the process. Thus, for example, when wespeak of an inclusive disjunction rule, thisrule is being applied to correctness of re-sponses not to the responses themselves or theperceptual processes which they reflect. Theinclusive disjunction rule says that the sub-ject will be correct for the correlated inputcase if he would have been correct on eitherof the two stimuli presented alone. This isthe type of decision rule used most frequentlyin models of perceptual independence, and yet

TABLE 14

DATA MATRIX 9 REDUCED TO PERCENTAGE CORRECTPOR THE Two TYPES OF STIMULUS CORRELATION

VariableVariable

A:A and B take same values

B +B-L

600

60

04040

6040

100

H:A and B take opposite values

R+B-E

06060

400

40

4060

100

Note.—For A:TJ(Ac:Bc) =0.971. Percentage correct f o r :inclusive disjunction =60; 50-50 on doubtfuls =60 ; con-junction = 60. For B:V(AC:BC) =0.971. Percentage correctfor: inclusive disjunction = 100; 50-50 on doubtfuls = 50;conjunction = 0.

it must surely be a questionable procedure,with its implication that somehow the organ-ism makes a joint response based on whetherit was right or wrong with respect to theseparate input channels. If the organismknows whether it is right or wrong, then itcan rectify things. More sophisticated decisionrules can be used that are more reasonable(see Eriksen, 1966), but there is a furtherand more important point to make, namely,that expected outcomes with respect to differ-ent assumed underlying processes are re-arranged if expected correct responses arecalculated with a decision rule based on pro-cess rather than on state.

To illustrate reasonably realistically, assumean experiment on detection of presence of twovisual stimuli, A or B, each of which is presentor absent together on any given trial; that is,there are trials on which no stimulus occursor trials on which two stimuli occur. This isa straightforward redundant stimulus experi-ment. The subject is required to state whetheror not he saw anything on each trial. Thenassume that what happens in the organism isreflected variously in Tables 1-10, so that ifthe subject had been permitted, he would haveat times said he saw both stimuli, one andnot the other, or neither. But since he is al-lowed only a dichotomous response, he must


invoke a decision rule. If his decision rule isbased on whether he would have been correct,then the consequences of different decisionrules for different basic perceptual situationsare what has been shown in the various tables.

Suppose, however, that his decision rule isbased on what his response would have been,not whether that response would have beencorrect; that is, his decision rule is processbased, rather than state based. The use ofthe inclusive disjunction rule assumes he willsay that he saw a stimulus any time he wouldhave said yes to one and/or the other stim-ulus. We look at the columns in the completeorthogonal matrix and find what percentageof the time a correct response would havebeen made. To illustrate, assume the data inTable 1, and that a subscript of 2 means thestimulus has been presented and that theresponse would have been positive. In Column1, with neither stimulus presented, the subjectwould have said he saw a stimulus wheneverhe would have used at least one "2" response,and would be wrong in each of 16 such cases.Alternatively, he would have been right only9 times of the 25 times no stimulus was pre-sented, that being the number of times hewould have said "no" to both stimuli Aand B. In Column 4, with both stimulipresented, he would have said that he saw alight 21 times of the 25 presentations. Thetotal of correct responses for both no stimulusand the redundant positive stimulus is 30, or60%.

This value is furthermore true for each ofthe matrices in Tables 1, 3, 5, 7, and 9. Thusnot only is the calculated value considerablylower than one based on a state rule, but itdoes not differentiate between these variousconditions at all. Still further, suppose the de-cision rule which gives 50-50 on doubtfuls isused, again applied to expected responses, notexpected correctness (i.e., half the responses inthe second and third rows of the completematrices are assumed to lead to a positiveresponse). Every matrix again would pro-duce 60% correct responses. Thus neither therule nor the data matrix gives differentialexpected correct responses for the correlatedstimulus case. And the value expected withcorrelated stimuli is exactly what can be

expected with just one stimulus variableoperating. Perhaps such a result would besurprising if obtained. This result, however,has been obtained in more than one experi-mental situation (see Garner & Lee, 1962).Thus perhaps it needs more considerationthan it has been given.

There are many types of experimentalsituation in which it makes eminently moresense for the decision rule to be based onprocess rather than state, and since we haveseen how great the discrepancy can be be-tween normative values based on process rulesrather than state rules, this distinction re-quires serious consideration. For example,consider experiments in which speech con-text provides one source of information anda brief visual or auditory exposure providesthe other source of information, and let usconsider a realistic illustration. A subject isgiven the sentence context: "The animal thatjust walked into the barn is a " Heconsiders possible alternative responses anddecides that "dog," "cat," "cow," "horse,"and "pig" are possible responses, and over alarge number of such cases he would use all ofthese five possible responses, but they all existfor him as possible responses in somehierarchy of choice on each occasion. Supposethat the correct word (as decided by theexperimenter) is pig and that the subjectuses this word 20% of the time. Then theword pig is presented auditorily, but in noise.The subject considers as alternatives "pig,""big," "tick," and "pick." He chooses pig50% of the time and thus is correct that per-centage of the time. Now given these twovalues for percentage correct for each sourceof information separately, the inclusive dis-junction rule for independence based on per-ceptual state can be used, with the conclusionthat with both sources of information, thesubject, or many different subjects, should becorrect 60% of the time. If then they arecorrect more often than this, we concludethat there is nonindependence, with some sortof facilitating interaction between the informa-tion inputs.

Consider a reasonable rule based on process,however, which is that the subject will givethat response which coincides for the two


possible sets of alternative responses. (Inmilder form, this is the kind of process de-scribed by Tulving and Gold, 1963.) Sinceonly one word coincides, he will use it andwill be correct 100% of the time. Thus if wefind a result less than this, we will concludethat there is nonindependence, with duplica-tion of information in the two sources.

Once again, perhaps such an expected resultseems unreasonable. Yet results close to ithave been obtained (Pollack, 1964; Tulving,Handler, & Baumal, 1964).

Can independence be tested with correlatedstimuli? When we consider the number offactors about which underlying, not directlytestable, assumptions must be made for anormative model of perceptual independence,it is difficult to be sanguine about the valueor possibility of asking whether perceptualprocesses are independent when the experi-mental paradigm using correlated inputs isused. We can assume parity independence,but not correlation independence, in eitherprocess or state form, and we can assumedifferent decision rules in either process orstate form. The number of possible com-binations is so great that surely a normativemodel can be found demonstrating perceptualindependence, at least of some kind, for anygiven experimental result. Equally true is thatwe can disprove the assumption of perceptualindependence for any given experimentalresult. Under such circumstances, the questioncannot be very meaningful.

Furthermore, at best, a normative modelused with correlated inputs can allow us toreject or accept the concept of independencebut would be quite unable to let us knowwhy a particular experimental result is ob-tained if the independence concept is rejected.As pointed out in the beginning, this propertyof a normative model is certainly one of itsstrengths as contrasted to models or theoriesof the psychological process itself, whichare true or false insofar as they correspondto data rather than insofar as they help clarifyan experimental result. Thus it is clear thatthe experimental paradigm using correlatedinputs has limited utility in the study ofperceptual independence.

INTERACTION OR INDEPENDENCE

The question of perceptual independencedoes have some meaning in psychology, par-ticularly when the complete orthogonal experi-ment is feasible. And perhaps it can havesome meaning with redundant or correlatedstimulus inputs if there are logical groundsfor establishing a more sharply delineated setof concepts pertinent to the problem. Thenwe would not test for independence as a gen-eral concept, but as a much more specificconcept.

However, consider briefly the possibilitythat the basic approach is not meaningfuland that other conceptualizations of the per-ceptual problem might further our under-standing more rapidly. In particular, we thinkthat models and experiments aimed at specify-ing the nature of interactive processes in theorganism will probably be more useful thanthose which are especially concerned with theparticular type of interaction called perceptualindependence.

Where Does Interaction Occur?Certainly a major consideration in under-

standing perceptual interaction is the locus ofthe interaction. The assumed locus of decisionrules makes a large difference in assumptionsabout perceptual performance with multiplesources of information. Further considerationof possible loci of interaction may help clarifythe question of perceptual interaction andmay show that some of the apparent con-tradictions in conceptualization of independ-ence in terms of process separability are notnecessarily as confusing as at first glance.

To illustrate, assume two extreme cases oflocus of interaction, and consider how theseassumed loci of interaction will affect thekind of apparent perceptual independence.First, consider two channels which are literallyindependent in the sense that they are com-pletely separate organisms. One organism isgiven stimulus variable A and it respondswith response variable a; the other organismis given stimulus variable B and it respondswith response variable b. Variables A and Bare then presented in correlated fashion, andthe outcomes from a and b are independent.There can actually be four response outcomes

PERC&VTVAL INDEPENDENCE 257

from the two organisms and the distributionof outcomes for stimulus AiBt and for stim-ulus A^Bz can be seen. We will assume thedistributions from Table 1 to hold, and fromthese we (the experimenters, that is) make adichotomized decision. Given on our partknowledge of the outcomes, without knowledgeof the stimuli, there is no decision rule wecan use which will give us greater than the60% accuracy shown to hold with an assumedprocess rule for the organism itself.

Is such an outcome from a single organismever realistically to be expected? If in fact thetwo channels of information are so completelyseparated that a subject can apply a decisionrule only to his responses, the answer is cer-tainly yes. And there are experimental resultswhich show little or no gain due to the useof redundant stimulus information. For ex-ample, for simple multiple visual stimulation,Garner and Lee (1962) found no gain indiscrimination accuracy with increased num-ber of elements; Eriksen and Lappin (196S)found small gain as long as the subject wasuncertain of position for the smaller numberof stimuli. Both of these results have beencorroborated by Garner and Flowers (1969).So there are experimental results which sug-gest that the interaction occurs very late inthe total system between input and output.

But of interest here is the relation of suchresults and the interaction concept to that ofindependence. Usually we assume a gain inperformance with independence, because thetwo or more sources of information, beingprocessed independently, add to improve dis-crimination. If the processes are correlated,thus duplicating information at some stage,little or no gain is expected. Yet this con-ceptualization of interaction only at the re-sponse level carries with it the idea of com-plete independence of the information chan-nels. Thus the use of the independence idealeads to incorrect assumptions about expectedperformance, whereas the use of the inter-action idea at a specified locus does not.

Now consider the opposite end of the con-tinuum. Double stimuli, correlated, but in thesame organism, are presented on the sameretina, on the same location of the retina, andat the same time. If ever a condition where

nonindependence exists was produced, thisis it. Yet what will such a result show? Thatthere is a gain in discriminability, because thetwo stimuli will add in energy. This is whatGarner (1965) called energic redundancy, be-cause there is an actual or potential summa-tion of stimulus energy.

To refer once again to the Garner and Lee(1962) experiments, they found no gain indiscrimination when stimulus elements wereadded at the same time but in different retinallocations. But they found considerable gainwhen the duration of the stimulus was in-creased on the same retinal location. Withinthe present context of discussion, this dif-ferential result would be interpreted as beingdue to the locus of interaction, large gain oc-curring when the locus is at the sensoryperiphery, and no gain occurring when thelocus is at the response periphery.

Again of interest here is the relation ofthe concept of interaction to that of independ-ence. Since literal spatial and temporalduplication of stimuli should produce non-independence, then we should expect littlegain in performance with duplicated stimuluselements. But the opposite effect occurs. Thusit no longer seems clear that independenceis a very useful concept, whereas locus ofinteraction is.

Notice that the assumed locus of interactionwill have great effect on the kind of inter-action which can occur. In our example of adecision process occurring after the response,then the interaction must occur with the dis-crete responses; on the other hand, at thesensory end, the interaction can occur with acontinuum. Thus models with continuousmetrics are appropriate. Models using d' areof this form (see Taylor, Lindsay, & Forbes,1967; Ulehla, Halpern, & Cerf, 1968), andthe underlying process model assumed byGarner and Lee (1962) was a variance modelalso.

Morton (1967) has shown, using the d'metric, that the locus of interaction affects thenormative outcome to be expected with sys-tems otherwise preserving independence. Heshowed that the expected gain in discrimina-tion with correlated inputs is greater when theinformation was combined prior to a decision

258 W. R. GARNER AND JOtiN MORTON

mechanism than it was when a combinatorialrule is applied after decisions are made. Thusthe importance of the locus of interaction inno way depends on the use of the simpleprobabilistic combinatorial rules discussed.

As a further illustration, Morton (1969)has elaborated a Logogen Model for wordrecognition which deals with multiple sourcesof information. In a "logogen," evidencerelevant to a particular response is collectedindifferently with respect to the source ofinformation. The correctness of the responseactually made is a function of the amountof evidence for that response and the amountof evidence for other responses. For the caseof a verbal context followed by the noisypresentation of a stimulus word, the modelpredicts interaction between the two sources as

logitP.. = logitP,+ logitP. + constant [10]

where logit P = log [P/(l - P)], and P., P0,and PS(! are, respectively, the probabilities ofbeing correct with stimulus alone, with con-text alone, and with both sources of informa-tion. This relation, which gives a reasonableaccount of the data, is clearly in the form of aperformance-parity statement. The importantpoint of the above equation, however, is thatthe logit transformation required to make theparity statement is the result of a particularassumed locus of interaction and an assumednature of the decision rules. Other loci ofinteraction and other decision rules wouldrequire different transformations of the datain their parity statements. Thus models whichexplicitly take into account differences inlocus of interaction and in decision rules ap-plied can be used and will hopefully allowgreater clarification of the underlying pro-cesses involved.

This idea of locus of interaction rather thanindependence is useful also in understandingsome of the results found in multidimensionaldistance judgments, where the interstimulusdifference seems to depend on the nature ofthe stimulus dimensions and their relationsto each other (e.g., Handel, 1967; Hyman &Well, 1968; Shepard, 1964). And conceptssuch as Lockhead's (1966) integrality and

Shepard's (1964) separability perhaps aremore meaningfully phrased in terms of locusof interaction of the stimulus dimensions, withlocus of interaction being more toward thereceptor periphery with integral dimensions.

CONCLUSIONS

While the aim in this paper has been lessto draw definitive conclusions about the natureof perceptual independence and how to in-vestigate it than to engage in a discussion ofthe rationale of the problem, certainly somefairly definite implications do emerge:

1. The concept of perceptual independenceis not a unitary concept and cannot be studiedas such. There are too many variations inbasic definition, type of normative model, andexperimental paradigm for it to be so.

2. The only experimental paradigm whichis reasonably effective in dealing with per-ceptual independence is the complete or-thogonal experiment (plus some controls forparity), since several different terms involvedin the independence concept can be separatelyidentified.

3. Even with restricted definitions of per-ceptual independence, experimental paradigmsusing correlated inputs are not effective indealing with the problem. So many assump-tions must be made about type of independ-ence and decision rules for obtaining com-posite responses that some normative modelof independence can be found to satisfy al-most any experimental result.

4. While the concept of state independenceis potentially valuable, it is clearly inter-related with process independence and if pos-sible should not be used without considera-tion of process correlation and parity. Cer-tainly reduction of data to test for state in-dependence destroys much potential informa-tion about perceptual independence and canobscure results clearly indicating noninde-pendence.

5. Our understanding of perceptual pro-cesses will be advanced more effectively withmodels designed to elucidate the nature ofan interaction process, particularly the locusof interaction, than with models which dealonly with that kind of interaction calledindependence.


REFERENCES

BROADBENT, D. E. Perception and communication,New York: Pergamon Press, 1958.

EGETH, H. E. Parallel versus serial processes inmultidimensional stimulus discrimination. Percep-tion and Psychophysics, 1966, 1, 24S-2S2.

ERIKSEN, C. W. Independence of successive inputsand uncorrelated error in visual form perception.Journal of Experimental Psychology, 1966, 72,26-35.

ERIKSEN, C. W., & HAKE, H. W. Multidimensionalstimulus differences and accuracy of discrimina-tion. Journal 0} Experimental Psychology, 1955,SO, 153-160.

ERIKSEN, C. W., & LAPPIN, J. S. Internal perceptualsystem noise and redundancy in simultaneous in-puts in form identification. Psychonomic Science,1965, 2, 351-352.

ERIKSEN, C. W., & LAPPIN, J. S. Independence inthe perception of simultaneously presented formsat brief durations. Journal of Experimental Psy-chology, 1967, 73, 468-472.

ESTES, W. K., & TAYLOR, H. A. Visual detection inrelation to display size and redundancy of criticalelements. Perception and Psychophysics, 1966, 1,9-16.

GARNER, W. R. Uncertainty and structure as psy-chological concepts. New York: Wiley, 1962.

GARNER, W. R. Discussion on data presentation. InF. A. Geldard (Ed.), Communication processes.(Proceedings NATO Symposium on Human Fac-tors, Washington, D. C., 1963) New York: Per-gamon Press, 1965.

GARNER, W. R., & FLOWERS, J. The effect of redun-dant stimulus elements on visual discrimination asa function of element heterogeneity, equal dis-criminability, and position uncertainty. Perceptionand Psychophysics, 1969, 6, 216-220.

GARNER, W. R., HAKE, H. W., & ERIKSEN, C. W.Operationism and the concept of perception. Psy-chological Review, 1956, 63, 149-159.

GARNER, W. R., & LEE, W. An analysis of redundancyin perceptual discrimination. Perceptual and MotorSkills, 1962, IS, 367-388.

GARNER, W. R., & McGnx, W. J. The relation be-tween information and variance analyses. Psycho-metrika, 1956, 21, 219-228.

GREEN, D. M., & SWETS, J. A. Signal detectiontheory and psychophysics. New York: Wiley, 1966.

HANDEL, S. Classification and similarity of multidi-mensional stimuli. Perceptual and Motor Skills,1967, 24, 1191-1203.

HYMAN, R., & WELL, A. Perceptual separability andspatial models. Perception and Psychophysics,1968, 3, 161-165.

LOCKHEAD, G. R. Effects of dimensional redundancyon visual discrimination. Journal of ExperimentalPsychology, 1966, 72, 95-104.

MORTON, J. Comments on "interaction of theauditory and visual sensory modalities." Journalof the Acoustical Society of America, 1967, 41,1342-1343.

MORTON, J. The interaction of information in wordrecognition. Psychological Review, 1969, 76, 165-178.

NEISSER, U. Cognitive psychology. New York: Apple-ton-Century-Crofts, 1967.

POLLACK, I. Interaction of two sources of verbalcontext in word identification. Language andSpeech, 1964, 7, 1-12.

POSNER, M. I., & MITCHELL, R. F. Chronometricanalysis of classification. Psychological Review,1967, 74, 392-409.

SHEPARD, R. N. Attention and the metric structureof the stimulus space. Journal of MathematicalPsychology, 1964, 1, 54-87.

TANNER, W. P., JR. Theory of recognition. Journalof the Acoustical Society of America, 1956, 28,882-888.

TAYLOR, M. M., LINDSAY, P. H., & FORBES, S. M.Quantification of shared capacity processing inauditory and visual discrimination. Ada Psy-chologica, 1967, 27, 223-229.

TORGERSON, W. S. Theory and methods of scaling.New York: Wiley, 1958.

TULVING, E., & GOLD, C. Stimulus information andcontextual information as determinants of tachisto-scopic recognition of words. Journal of Experi-mental Psychology, 1963, 66, 319-327.

TULVING, E., MANDLER, G., & BAUMAL, R. Interactionof two sources of information in tachistoscopicword recognition. Canadian Journal of Psychology,1964, 18, 62-71.

ULEHLA, Z. J., HALPERN, J., & CERF, A. Integrationof information in a visual discrimination task.Perception and Psychophysics, 1968, 4, 1-4.

(Received December 16, 1968)

VOL. OCTOBER 1969 Psychological Bulletin · PDF filecerns primarily the perceptual process,...

Documents

Transcript of VOL. OCTOBER 1969 Psychological Bulletin · PDF filecerns primarily the perceptual process,...