uncoveringmentalarchitectureandrelated ...chapter 11 uncoveringmentalarchitectureandrelated...

CHAPTER 11

Uncovering Mental Architecture and RelatedMechanisms in Elementary Human Perception,Cognition, and Action

JAMES T. TOWNSEND, MICHAEL J. WENGER, AND JOSEPH W. HOUPT

INTRODUCTION: BRIEF HISTORYAND GENERAL CONCEPTION

Psychology, the study of the mind, takesfor itself behavioral as well as surgical andphysiological-recording sets of tools. Thebehavioral and physiological data can beinterpreted, and sometimes predicted, inverbal as well as mathematical languages.

Our approach lies within the confines ofmathematical theory and methodology andconcentrates on behavioral data, althoughsome inroads and prospects for fusion withphysiological techniques will be mentioned.To the extent that psychology and cogni-tive science wish to go beyond an extremebehavioristic catalog of stimulus-responsecorrelations, they are what theoreticallyinclined engineers would call black boxsciences (e.g., see Booth, 1967; Townsend,1984). The very essence of black box sci-ences is to utilize input-output sequence datato uncover the inner subprocesses and theirinteractions that can generate that data.1

1Naturally, the strategies required to identify the mecha-nisms, possibly including germane physics or chemistry,must be drastically different from those involved in, say,electronic circuits or automata in general.

The history of black box psychologymost concretely starts in the 19th-centurylaboratories of pioneering scientists such asF. C. Donders and W. Wundt. These menin particular were perhaps the very first tomanipulate the components of the stimuli andstudy response times (RTs) in order to drawinferences about the underlying mechanismsinvolved in elementary perceptual, cognitive,and action processes. There are many otherroots of embryonic psychological sciencethat complete the background for the currentchapter, including ingenious observationsof that other major dependent (observable)variable of psychology: patterns of accuracy.However, the so-called complication exper-iments carried out by Donders, Wundt, andtheir students, colleagues, and intellectualdescendants constitute the true historicalgenesis of our methodology.

The complication experiment provided atask, such as responding as soon as possibleto the onset of a signal. Then, in separateexperimental trials, the researcher wouldcomplicate the task by adding another sub-task, such as requiring a discriminationbetween two separate signals. A critical pos-tulate was that the extra subtask was placed insequence with the already attendant subtask

Stevens’ Handbook of Experimental Psychology and Cognitive Neuroscience, Fourth Edition, edited by John T. Wixted.Copyright © 2018 John Wiley & Sons, Inc.DOI: 10.1002/9781119170174.epcn511

1

2 Uncovering Mental Architecture and Related Mechanisms

(Woodworth, 1938, Chapter 14). In morecontemporary terminology, this postulate canbe understood in terms of the assumptionsof pure insertion of the new subtask plusserial processing (e.g., Ashby & Townsend,1980; Sternberg, 1966, 1969; for a thoroughdiscussion of many such issues in psychologyand cognitive science see Lachman, Lach-man, & Butterfield, 1979). By subtractingthe mean RT of the simpler task from thatof the more complicated task, an estimate ofthe mean time of the inserted subtask couldbe calculated.

In fact, the pair of opposing hypotheses—serial versus parallel processing, or the ques-tion of mental architecture—though hintedat in research dating back to around 1950,became a primary, essential question in thegrowing cognitive science of the 1960s andbeyond (e.g., Egeth, 1966; Sternberg, 1966).It is important to note that mental architectureis not meant to imply a fixed, unchangeablestructure. For example, there is increasingevidence that some individuals may be ableto alter the processing mode from serial toparallel or vice versa (e.g., Townsend & Fific,2004; Yang, Little, & Hsu, 2014).

Casting the aim of psychological the-ory and methodology in terms of a searchfor mechanisms in a very complex blackbox might bring along with it the worrythat, in the absence of an exact and com-plete dissection of the black box’s innards,different mechanisms and different inter-actions of similar mechanisms might becapable of mimicking one another. Forinstance, in discrete automata theory, giveneven an infinite series of input-output obser-vations, there will still be a set of systems,each of which could generate that obser-vational corpus. These would be distinctfrom the “falsified” class of systems butwould all be capable of producing theobserved data, and thus of mimicking eachother (again, see how this works out in the

well-studied domain of finite state automata:Booth, 1967).

The issue of mimicry has always beenpresent, but only rarely explicitly recog-nized or studied in a rigorous fashion. Inmodern psychology, perhaps its first notableappearance was in the analysis of mimickingof Markov models of memory (Greeno &Steiner, 1964). The analysis of mimicry instochastic accumulator models of percep-tion and decision has come more recently(e.g., Dzhafarov, 1993; Jones & Dzhafarov,2014; Khodadadi & Townsend, 2015; Link &Heath, 1975; Ratcliff, 1978). As will becomeclear, our consideration of theoreticalmimicry raises the question of the exper-imental methodologies needed to potentiallyresolve questions of mimicry. Throughoutmost of the history of psychology, theory andmethodology have been almost entirely seg-regated, with theory being largely verbal andmethodology being concentrated in statistics.Our approach views theory and methodologyas Siamese twins, with the cogent applicationof each requiring consideration of the other.

As already intimated, our approach forover 40 years has been mathematical innature. Furthermore, much of our work hasbeen what we refer to as metatheoretical,in the sense that our major aim has been tostudy simultaneously large classes of mathe-matical models in relation to well-specifiedtypes of experimental designs. The aim isto discover which classes of models mimicone another and which models can be rig-orously tested against one another in theseparadigms. Of course, the paradigms andthe model classes are optimally developed inclose tandem with one another.2 One of thestrengths of the metatheoretical approach isthat it is usually possible to render the classes

2This approach can also be referred to as qualitativemodeling. However, that terminology can be mistaken asreferring to purely verbal inquiries.

Major Characteristics of Elementary Cognitive Systems and Theorems of Mimicry 3

of assayed model in terms of general proba-bility distributions and the assays themselvesin terms of nonparametric statistics and tests.

Allen Newell, a pioneer and luminaryin the development of cognitive science,implicitly brought in the mimicking dilemmain his widely cited article “You Can’t Play 20Questions with Nature and Win . . . ” (1973).Newell called into question the ability ofscientists to effectively test between dichoto-mous properties and issues in cognition.In contrast, the method of strong inference(Platt, 1964), which recommends that scien-tists pose a series of ever more specific alter-native hypotheses to test against one anotherin order to avoid confirmation bias, doesjust this. Interestingly, Greenwald (2012)cogently promotes the indissoluble and mutu-ally supportive nature of method and theory.In “In a Revisit to Newell’s 20 Questions,”(2012), Greenwald finds evidence in ourapproach for a solution to Sternberg’s (1966)issue of parallel versus serial processing.

The metatheoretical approach to date haspreponderantly relied on the two strongestdependent variables, RT and patterns ofresponse probabilities. It is usually acceptedthat RT lies on a ratio scale while probabili-ties lie on an absolute scale (e.g., see Krantz,Luce, Suppes, & Tversky, 1971; Townsend,1992; Townsend & Ashby, 1984). In addi-tion, it turns out that the nonparametric assaysemployed in our metatheoretical approachare invariant up to monotonic transformation,meaning that measurement scales that areonly ordinal can be implemented withoutviolating principles of foundational mea-surement (again, see Krantz et al., 1971, andChapter 12 of Townsend & Ashby, 1983).

The more frequent style of modelingin psychology and cognitive science hasbeen to propose a detailed, parameterizedmodel for certain behaviors and tasks (seealso Chapter 9 in this volume). For example,Ratcliff (1978) proposed a model for memory

search based on the well-known Wienerdiffusion process. That model has sincegained wide recognition and application.Ratcliff’s Wiener process is continuous inboth time and state. Even earlier, Link andcolleagues (e.g., Link & Heath, 1975)invented a mathematically related approach,also based on continuous state but usingdiscrete time: a sequential random walkmodel later referred to as relative judgmenttheory. Interestingly, they emphasized broadassessments of a more qualitative nature thanthe more typical model fits. Both approachesare valuable and complementary. Thus, onesalutary tactic is to use the metatheoreticalstrategy to discern the broad characteristicsof the mental operations and then to bringto bear specific process models that obeythe dictates of the broad elements discoveredthrough the former approach (e.g., see Eidels,Donkin, Brown, & Heathcote, 2010).

This chapter provides the most up-to-dateand general treatment of our theoretical andmethodological philosophy and science. Lim-itations and pertinent references to relatedmaterial are addressed in the conclusion.A somewhat more mathematical treatmentand one that expands on some of the topicsnecessarily abbreviated here is availablein Algom, Eidels, Hawkins, Jefferson, andTownsend (2015).

MAJOR CHARACTERISTICSOF ELEMENTARY COGNITIVESYSTEMS AND THEOREMSOF MIMICRY: THE EVENTSPACE BASIS

According to modern probability theory(dating to Kolmogorov, 1950), a probabilityspace can be informally defined as follows:It is made up of a family of sets—the eventsin question—together with the set operationson them, plus the imposition of a probability


measure. The measure on the entire set ofpoints from which the sets are taken mustbe 1, and the empty set has probability 0.3

The primary elements of the probabilityspaces used for the consideration of mentalarchitecture have been the item-processingtimes combined with the order of completion(e.g., Townsend, 1972; Townsend, 1976a,1976b). Formal representation of this type ofevent space was reported by Vorberg (1977)at a meeting of the Society for MathematicalPsychology, and followed in Townsend andAshby (1983, Chapter 14). The remainder ofthis section will adhere to an examination ofthis type of space, although the discussionwill remain in an informal mode. Subse-quently we will enrich the spaces for serialand parallel models, but again, will keep to avernacular modus operandi.

Because the Sternberg paradigm(Sternberg, 1966) has been one of themost popular and replicated paradigms in thehistory of human experimental psychology,we will utilize it to illustrate the criticalissues in human information processing.In this paradigm, the experimenter presentsa modest number of discrete items; forinstance, letters or numbers. The number istypically varied—say, from one to six—withno repetitions. After a very brief interval, aprobe item is presented and the observer’stask is to indicate whether or not that probewas contained in the original set of items.The observer’s RTs are recorded and the RTmeans graphed as a function of the number ofitems, with yes versus no curves acting as aparameter in the plots and statistical analyses.An analogous procedure can be carried outfor visual search, with the probe presentedfirst, followed by a varied number of itemsin a brief visual display (e.g., Atkinson,

3A succinct, but elegant, introduction to modern proba-bility theory can be found in Chung (2001). Billingsley(1986) is more complete with a nice survey of the topic.

Holmgren, & Juola, 1969). The observeris then required to respond yes versus nodepending on whether the target item wasfound in the visual list or not.

Now, suppose that pure insertion holds foreach of the n items so that memory searchis serial; that is, mean RT is an additivefunction of n, plus a residual time assumedto be occupied by earlier sensory processes,ancillary cognitive terms, and late-termresponse processes, which are postulatedto be independent and invariant across val-ues of n. If in addition the purely insertedtimes are identically distributed, then themean RT functions will be linear. Stochasticindependence of the successive stage times(across-stage independence) is typicallyassumed too, but this constraint is not strictlynecessary. In Sternberg’s (1966) data, themean RT functions were indeed linear, thusbeing consistent with the hypotheses of serialprocessing, pure insertion, and identicallydistributed increments.

Furthermore, Sternberg asked if the meanRT functions for yes and no responsespossessed the same slope. Sternberg rea-soned that no responses necessitated thatall memory items be processed, a so-calledexhaustive stopping rule. However, theobserver could stop on yes trials as soon asthe probe was located in the memory list, aso-called “self-terminating” stopping rule.Under the other assumptions about serialityand identical process-time distributions, theyes curve should thus have one half of theslope of the no curve. Instead, Sternbergfound equal slopes: Mean RT as a functionof n was linear for both yes and no responses.And, the slopes were deemed to be equal,suggesting exhaustive processing for bothtypes of responses. Over the years, the find-ings of the linearity and equal-slopes resulthas often, but not universally, been replicated.Equal slopes are somewhat more rare in thecase of visual search (e.g., see Schneider &


Shiffrin, 1977; Townsend & Roos, 1973;also see Nosofsky, Little, Donkin, &Fific, 2011).

Another indicant of the stopping rule mustbe mentioned, and that is the phenomenonof position effect. If mean RTs are shorterif the probe appears in some positions morethan others, this could be, and usually is,interpreted to mean that self-termination is ineffect and that some paths of serial process-ing are more often taken than others. Thisissue will be discussed further below (alsosee Townsend & Ashby, 1983, Chapter 7).We begin our formal description of the issues,beginning with a set of definitions.

Definition 1: A stochastic model of a systemfor processing positions a and b is standardserial with an exhaustive processing rule ifand only if the density functions of the serialprocessing times satisfy

f [ta1, tb2; (a, b)] = pf (ta1)f (tb2)

and

f [tb1, ta2; (b, a)] = (1 − p)f (tb1)f (ta2).

First, note that Definition 1 includesthe assumption of across-stage indepen-dence. Definition 1 is given for n = 2. If weextend it to arbitrary n, we directly acquirethe increasing straight-line predictions formean RTs from Sternberg (1966, 1975).Of course, if processing is self-terminating

on probe-present trials, then we expect themean RT to possess a slope one half of theprobe-absent trials. Sternberg’s experimentalresults, along with many replications by oth-ers, certainly appear to be very persuasive:The logic is impeccable and the findingsoft replicated with good statistical power.But then how can mimicking occur in theSternberg task? The most antipodal typeof processing that could occur is parallelprocessing. Such processing, in contrast toserial processing, postulates that all itemsare compared with the probe at once, thusseemingly predicting much shorter meanRTs as n is varied. Consequently, we need adefinition for parallel models.

Definition 2: A stochastic model of a systemfor processing positions a and b is parallelwith an exhaustive processing rule if and onlyif the density functions on the parallel pro-cessing times satisfy

ga1,b2[𝜏a1, 𝜏b2; (a, b)] with 𝜏b2 > 𝜏a1

and

ga2,b1[𝜏a2, 𝜏b1; (b, a)] with 𝜏b1 < 𝜏a2.

It can be seen that the parallel processingtimes are, in contrast to serial processing,overlapping, as indicated in Figure 11.1.A structural relationship between the tsand the 𝜏s is that 𝜏r1 = tr1 for r = a, b and

Ta

Ta

Sa

Sa

Sb

Sb

Othersubprocesses Sc

Othersubprocesses

Output

Output

Tb

Tb

Tc

Tc

Input Ii (i = 1, 2, 3, 4)

Input Ii (i = 1, 2, 3, 4) Sc

Figure 11.1 Schematic of serial (upper) and parallel (lower) processing systems. The subprocesses Sa,Sb, and Sc are each selectively influenced by a unique factor (or set of factors), Ta, Tb, and Tc, respectively.


𝜏b2 = ta1 + tb2 and 𝜏a2 = tb1 + ta2. An equiv-alent statement is that tb2 = 𝜏b2 − 𝜏a1 andta2 = 𝜏a2 − 𝜏b1. See Townsend and Ashby(1983, Chapter 1) for details.

A useful special case arises when theparallel processing times are within-stageindependent, which indicates that betweensuccessive completions the parallel chan-nels are operating in a probabilisticallyindependent fashion. It is easiest to pose thiscondition if we use the above interpretation interms of the t’s, which are, after all, the inter-completion times. First, let G(t) = P(T > t)and G∗(t) = 1 − G(t) for any particularchannel. This leads us to our third definition.

Definition 3: A stochastic model of a sys-tem for processing positions a and b isparallel, which satisfies within-stage inde-pendence if and only if the density functionson the parallel processing times during thefirst stage satisfy

ga1,b2[ta1, tb2; (a, b)]= ga1(ta1) G∗

b1(ta1)Gb2(tb2|ta1)

and

ga2,b1[𝜏a2, 𝜏b1; (b, a)]= gb1(tb1)G∗

a1(tb1) ga2(ta2|tb1).

Note that within-stage independence andacross-stage independence are logically inde-pendent notions. Finally, the counterpart tothe standard serial model is the standard par-allel model, which possesses independenceof the actual parallel processing times 𝜏a and𝜏b. This model is described in our fourthdefinition.

Definition 4: A stochastic model of a systemfor processing positions a and b is standardparallel with an exhaustive processing rule ifand only if the density functions on the paral-lel processing times satisfy

ga1,b2[𝜏a1, 𝜏b2; (a, b)] = ga(𝜏a1)gb(𝜏b2)

with 𝜏b2 > 𝜏a1 and

ga2,b1[𝜏a2, 𝜏b1; (a, b)] = ga(𝜏a2)gb(𝜏a2)

with 𝜏b1 < 𝜏a2.

It is important to be aware that inde-pendence of parallel processing times andacross-stage independence are separate andlogically independent concepts. Any model,parallel or serial, could satisfy either or not(Townsend & Ashby, 1983, Chapter 4).

Now, suppose that processing is indeedparallel with stochastic independence (thussatisfying Definition 4), and that every itemis operated on at the same speed (i.e., iden-tically distributed processing times), andthat these distributions are invariant with n,implying unlimited capacity.

Then, Townsend and Ashby (1983,Chapter 4) showed that if processing isexhaustive, mean RTs increase in a curvi-linear manner, with each added item addinga steadily decreasing increment to the totalcompletion time of all the items. If processingis self-terminating, the mean RT function willbe flat. Sternberg’s (1966) results were notwell fit by such curves, therefore falsifyingthe standard type of parallel model with iden-tical processing-time random variables. Evenif the assumption of identically distributedprocessing times for the distinct items isrelaxed, such models will still not mimic thestandard serial predictions.4

However, if the assumption of unlimitedcapacity is dropped so that as n increases,the item-processing times slow down, thenparallel models can easily mimic the linearincreasing-mean RT functions. In fact, sup-pose that as n increases, a fixed amount ofcapacity must be spread across all the mem-ory items (or in visual search, the presenteditems). Then the first item to be processed will

4Note that if distinct positions of items possess dis-tinct processing-time distributions, and if processing isself-terminating, position effects can be readily predictedby these parallel models.


take up the same time as the first serial itemto be processed. Moreover, if the capacitydevoted to the completed item is reallocatedto all the unfinished items, then each succeed-ing item to be processed will add the sameamount of time to the overall completiontime. Given any set of serial distributions,such parallel models can be mathematicallyidentical, at both the level of the means andthe level of the distributions (Townsend,1969, 1971, 1972, 1974). Interestingly, ifonly the mean RTs are considered, ratherthan the entire distributions, the assumptionof reallocation of capacity can be eliminated.Chapter 4 in Townsend and Ashby (1983)shows how architecture, stopping rule, andworkload capacity combine to produce avariety of mean RT curves as functions of n.

There isn’t space to delve into technicalissues of parallel-serial mimicry in detail,much less the complete history. However, inour opinion, mimicry and testability are akinto the yin and yang of Buddhist philosophy:The universe of opposing models cannot befully understood unless both are objects ofdeliberation. Consequently, we need defini-tions that directly address the issue. Let Tij

stand for the random variable for item orposition i = a, b and j = 1, 2.

Definition 5: The fundamental functionalequation of mimicry for n = 2 is

p[ fa1(ta1)fb2(tb2|ta1)]= ga1,b2[𝜏a1, 𝜏b2; (a, b)]= ga1,b2(ta1, ta1 + tb2)= ga1(ta1,Tb1 > ta1)gb2(tb2|ta1)

and

(1 − p)[ fb1(tb1)fa2(ta2|tb1)]= ga2,b1[𝜏a2, 𝜏b1; (b, a)]= ga2,b1(tb1 + ta2, tb1)= gb1(tb1,Ta1 > tb1)ga2(ta2|tb1)

for all tai, tbj > 0, i, j = 1, 2.

If within-stage independence is assumedon the part of the parallel models, then weachieve the next definition.

Definition 6 (Townsend, 1976a): If within-stage independence holds for the paral-lel class of models then the fundamentalequation of mimicry for n = 2 is

Serial Parallel

p[ fa1(ta1)fb2(tb2|ta1)] = ga1(ta1)G∗b1(ta1)

and

(1 − p)[ fb1(tb1)fa2(ta2|tb1)] = gb1(tb1)G∗a1(ta1)

for all tai, tbj > 0, i, j = 1, 2.

For every specification of a parallelmodel (the right-hand side) there exists aserial model; that is, values of p and theserial density functions that can perfectlymimic the parallel model. However, whenthe equation is solved for the parallel-modelfunctions in terms of the serial functions,the result may not result in a distribu-tion whose cumulative function properlygoes to 1. Townsend (1976a) interpretedsuch solutions as violating the definitionof parallel processing. However, Vorberg(1977) pointed out that an alternative isthat the system is still parallel, but that onsome trials, a channel fails to completeprocessing.

The instantiation of within-stage inde-pendence might seem like a stringent con-dition. However, it turns out that any jointdistribution not satisfying within-stageindependence can be transformed intowithin-stage independent distributions (Rao,1992). So far, our emphasis has been on howthese polar-opposite types of architecture canact so much like one another. In the succeed-ing sections, we focus on tactics for testingparallel versus serial processing against oneanother.


STRONG EXPERIMENTAL TESTSON RESPONSE TIMES I: EVENTSPACE EXPANSION

The event spaces employed in the preced-ing section are too coarse for a completetreatment of mental architectures. However,we have seen that even with the former,coarse description, the serial models can bemore general than the parallel if either (a) weremain within the exponential class of mod-els (e.g., Townsend, 1972), or (b) we demandthat the parallel processing-time distributionsare nondefective (Townsend, 1976a).

Here we can expand the form of the proba-bility space to take into explicit considerationwhat an item is made of. For instance, onemight posit that an item is constituted of afinite number of features. The number offeatures completed by a certain time t thenbecomes the nucleus of a state space, whichpermits the assignment of a probabilitymeasure.

Many popular models of human percep-tion, cognition, and action contain a finer suchlevel description of processing, one includ-ing some type of state space that goes be-yond simple uncompleted versus completed.Familiar examples are Poisson countingprocesses (e.g., Smith & Van Zandt, 2000),random walks (e.g., Link & Heath, 1975),and diffusion models (e.g., Busemeyer &Townsend, 1993; Ratcliff, 1978). Each ofthese specifies a state space representingdegree of processing. For instance, a Poissoncounting model might represent an item’sfeatures, with the state of processing refer-ring to the number of completed features.In contrast, the state space for a diffusion orrandom walk process lies on a continuum.For instance, the typical state in such a spaceis a real number.

Finer grained types of spaces were pre-saged in certain earlier papers. Most models,ours included, have typically left out details

of the psychological processes being studied.For instance, though memory and displaysearch patently demand some type of match-ing process, most models leave that out of thepicture. However, Townsend and colleagues(1976b; Townsend & Evans, 1983) constructstochastic processes for search in a waythat the probability event spaces are indeedfounded on such pattern matching. Houpt,Townsend, and Jefferson (2017) present amore detailed and rigorous foundation ofthe underlying and richer measure space forparallel and serial models than is possible todescribe here.

This enrichment unveils ways in whichparallel systems are more general than theserial systems. The enriched event spacesallow for mismatches (also called negativematches) to be associated with distributionsthat differ from matches (positive matches).A fact in search and same–different paradigmsfor decades has been that, modally, positivematches are faster than negative matches.This small and reasonable expansion impliesthat intercompletion times can depend on allof the unfinished items and not only the previ-ous processing time and order of processing,as in the usual models that ignore positiveversus negative matches (e.g., Bamber, 1969;Krueger, 1978). The associated diversity ofparallel and serial models led to the designof strong tests of architecture. This type ofdesign has been dubbed the parallel-serialtester (or PST; see Snodgrass & Townsend,1980; Townsend & Ashby, 1983, Chapter 13,Chapter 15), as shown in Table 11.1.

The basic PST design requires partici-pants to determine whether one or two probestimuli are targets, where the target is eitheran item in memory, a simultaneously dis-played item, or an item revealed after theprobes. In one condition (PST-loc) the targetprobe is always present, and participants areasked to respond affirmatively if the targetprobe is in a specific spatial or temporal

Strong Experimental Tests on Response Times I: Event Space Expansion 9

Table 11.1 The Parallel-Serial Tester

Condition

CI Target = A Comparison Items Response

Trial Type1. AB R22. BA R1

CII Target = A Comparison Items Response

Trial Type

1. AA R12. AB R23. BA R24. BB R2

CIII Target = A Comparison Items Response

Trial Type

1. AA R12. AB R13. BA R14. BB R2

position and negatively otherwise. With +indicating that the time for positive matchesto the target and – indicating the time fornegative matches to the target, the meanRT from a serial self-terminating systemwould be

pE[Ta+] + (1 − p)E[Tb−] target at a

pE[Ta−] + (1 − p)E[Tb+] target at b.

In a parallel, self-terminating system, themean RTs would be

∫∞

0G∗

a+(t)G∗b−(t) dt target at a

∫∞

0G∗

a−(t)G∗b+(t) dt target at b.

In the second condition (PST-and), either,both, or neither of the probes may be targetsand the participant is asked to respond affir-matively only if both probes are targets. Theserial, self-terminating predictions are

p(E[Ta1+] + E[Tb2+])+ (1 − p)(E[Ta2+]+ E[Tb1+])

when both aretargets,

p(E[Ta1+] + E[Tb2−])+ (1 − p)E[Tb1−]

when onlytarget at a,

pE[Ta1−] + (1 − p)(E[Ta2−]+ E[Tb1+])

when onlytarget at b,

pE[Ta1−] + (1 − p)E[Tb1−] when neitherare targets.

With the indicator function given by I(X)equal to 1 if X is true and equal to zero other-wise, the parallel, self-terminating modelpredictions are

∫∞

0G∗

a1+(t)G∗b1+(t) dt

+ E[Tb2+I(Ta1+ < Tb1+)]+ E[Ta2+I(Tb1+ < Ta1+)]

when both aretargets,

∫∞

0G∗

a1+(t)G∗b1−(t) dt

+ E[Tb2−I(Ta1+ < Tb1−)]


∫∞

0G∗

a1−(t)G∗b1+(t) dt

+ E[Tb2−I(Tb1+ < Ta1−)]


∫∞

0G∗

a1−(t)G∗b1−(t) dt when neither

are targets.

In the final condition (PST-or), the sameset of trial types as the second condition isused, but the participant is asked to respondaffirmatively if either of the probes aretargets. The serial, self-terminating modelpredicts

pE[Ta1+] + (1 − p)E[Tb1+] when both aretargets,

pE[Ta1+] + (1 − p)(E[Ta2+]+ E[Tb1−])


p(E[Ta1−] + E[Tb2+])+ (1 − p)E[Tb1+]


p(E[Ta1−] + E[Tb2−])+ (1 − p) × (E[Ta2−]+ E[Tb1−])

when neitherare targets.

The parallel, self-terminating modelpredicts

∫∞

0G∗

a1+(t)G∗b1+(t)dt when both are

targets,

∫∞

0G∗

a1+(t)G∗b1−(t)dt

+ E[Ta2+I(Tb1− < Tb1+)]



∫∞

0G∗

a1−(t)G∗b1+(t)dt

+ E[Tb2+I(Ta1− < Tb1+)]


∫∞

0G∗

a1−(t)G∗b1−(t)dt

+ E[Tb2−I(Ta1− < Tb1−)]+ E[Ta2−I(Tb1− < Ta1−)]

when neitherare targets.

That is, RTs to a present target at a, Ta1+,are modeled with a different random variablethan RTs to a mismatch at a, Ta1−, but both ofthose random variables are used regardless ofwhether a target is present at location b.

Under reasonable assumptions, thisparadigm leads to empirically distinguish-able parallel and serial model mean RTs.5

In particular, if processing is self-terminating,and either the processing time of each probeis equal in distribution across location but notequal in distribution across target/distractorstatus in the parallel model, or if the meanRTs are unequal across target/distractor sta-tus in both positions in the serial model, thenserial models make specific predictions thatcannot be satisfied by parallel models.

Under these assumptions, the serial modelpredicts that the sum of the mean RTs acrossthe two trial types in PST-loc is equal to themean RT when both probes are distractorsin the PST-and added to the mean RT whenboth probes are targets in the PST-or. Underthe same assumptions, the parallel model pre-dicts those two sums to be unequal. The gistof these inferences can be readily seen usingthe equations above; for complete proofs, seeTownsend and Ashby (1983, Chapter 13).

Using the same paradigm, a slightly dif-ferent set of assumptions can lead to the samediscriminable model predictions as well. Onesufficient condition is that the parallel modelshave unequal first stage target and distractorprocessing-time distributions for all possible

5Note that implicit in this notation is the selective influ-ence (defined in the next section) of target presence in agiven location on the processing time of that location.

RTs in both positions; that is, for all t > 0,Ga1+(t) ≠ Ga1−(t) and Gb1+(t) ≠ Gb1−(t).An alternative sufficient condition is that theserial model mean RTs are unequal acrosstargets and distractors in both positions, andE[Tb1+] ≠ E[Tb1−], while the parallel-modeldistributions are either equal for all possibleRTs t > 0, and G∗

b1+(t) = G∗b1−(t), or unequal

for all possible RTs within each position dur-ing the first stage, or and G∗

b1+(t) > G∗b1−(t)

or G∗b1+(t) < G∗

b1−(t).Although the PST has been used rela-

tively much less frequently than the methodsdescribed in the next section, it has beensuccessfully applied in a number of domains,including those outside of cognitive psy-chology. For example, Neufeld and McCarty(1994) used the PST to determine thatincreased stress levels do not generally leadto higher likelihood of serial processing,regardless of stress susceptibility.

STRONG EXPERIMENTAL TESTSON RESPONSE TIMES II:APPLICATIONS OF SELECTIVEINFLUENCE

Perhaps the most experimentally popular andalso the most technically advanced strategiesare those based on what we have termedsystems factorial technology (Townsend &Nozawa, 1995). Of all the strong tests ofhypotheses regarding processing architec-ture, this one has witnessed a degree ofsophisticated investigation and expansionthat is almost unprecedented for a youngscience like psychology. It has also seenwide-ranging applications in a number ofdiverse spheres of psychological research.

Sternberg, who was responsible for therebirth and expansion of the Dondersian andWundt style programs mentioned in the intro-duction, also invented the additive factorsmethod, the direct predecessor to systemsfactorial technology. Sternberg’s striking

Strong Experimental Tests on Response Times II: Applications of Selective Influence 11

insight was to postulate that for twoprocesses, a and b, there could be twoexperimental factors, A and B, each of whichsolely influenced its namesake process. Inparticular, he supposed that the associatedprocessing time means E(Ta|A) and E(Tb|B)could be sped up or slowed down as func-tions of A and B, respectively. The readershould observe that E(Ta|A) is completelyunaffected by changes in B and vice versafor E(Tb|B). This was the first incarnation ofthe notion of selective influence. With thisassumption in place, it was then straight-forward to show that a serial system, witha processed before b, predicts the over-all mean RT to be E(Ta + Tb + T0|A,B] =E(Ta|A) + E(Tb|B) + E(T0). This nice resultshows that the mean RT will be an additivefunction of the experimental factors A, B.The logic was that if an experimenter foundadditive mean RTs, then the inference wasthat processing was serial (see Figure 11.2).

Ashby and Townsend (1980) extended theadditive factors methods to a distribution-level test. They demonstrated that a serialexhaustive system with factors that are selec-tively influencing each stage will lead toequal distributions of the sum of RTs when

both levels are the same and distributionsof the sum of RTs when levels are mixed.That is, if RTab;AB is the RT when the factorinfluencing process a is at level A and thefactor influencing process b is at level B, then

RTab;11 + RTab;22 = RTab;12 + RTab;21

or, equivalently, with * indicating con-volution,

gab;11 ∗ gab;22 = gab;12 ∗ gab;21.

This provides an even stronger test forthe serial exhaustive model as this equal-ity clearly implies equality at the level ofthe mean (by the linearity of the expecta-tion operator). Despite the relatively strongimplication of the extended additive factorsmethod, this test has indicated the serialexhaustive models in a range of applications.For example, Roberts and Sternberg (1993)demonstrated nearly perfect additivity at thedistributional level in a simple detection taskand in an identification task.

The tests from Sternberg (1969) andAshby and Townsend (1980) are powerful.However, if additivity was not supported,then either the architecture was not serial(although what it was, remained unclear),

Serial(with selective influence)

RT1

RT3 RT2

RT1

RT2

RT4

RT3

RT

RT4

Factor Xblevel Factor Xb

level

Factor Xa LevelFactor Xa Level

Xb = xb1

Xb = xb2

Xa = xa1Xa = xa2

Xa = xa1Xa = xa2

RT

Xb = xb1

Xb = xb2

Parallel Exhaustive(with selective influence)

Figure 11.2 Predicted mean response times under selective influence manipulations of each subprocessin a serial system (left) and parallel system (right).


or it might have been serial, but selectiveinfluence was not in force. In addition tothe limitation of the strict part of the logicto serial architectures, other gaps lay in theabsence of a firm technical foundation ofthe selective influence assumption, and ofmeans of testing its satisfaction. We can onlygive a somewhat simplified description of thecurrent state of the art, but it should suffice tocomprehend our discussions herein.

Most significantly, the theory-drivenfactorial methodology has been expandedto include (a) tests based on entire RTdistributions rather than only means (e.g.,Townsend & Nozawa, 1995; Townsend &Wenger, 2004a); (b) serial and parallelarchitectures along with a broad class ofmore complex systems (e.g., Chapter 6 inSchweickert, Fisher, & Sung, 2012);6 and (c)several decisional stopping rules, also iden-tified along with the attendant architecture(e.g., Townsend & Wenger, 2004a).

Definition 7 ensures that (a) effectivespeeding or slowing of the processes occurs,and (b) the effect is of sufficiently strongform to enable theorems to be proven, whichgenerates powerful tests of the architecturesand stopping rules. Even theorems for meanRT interactions currently require selectiveinfluence to act at the distribution-orderinglevel. The reader is referred to Townsend(1984a, 1990) for more detail on this formof selective influence, and Dzhafarov (2003);Townsend, Liu, and Zhang (2017); andSchweickert, Fisher, and Sung (2012) forgeneralizations and discussions of not onlypsychological implications but for connec-tions with other central concepts in sciencein general.

6A basic assumption of the latter is that the graphs ofthe connections of the underlying processes are forwardflow only. No feedback is generally assumed, althoughsome exceptions can be encompassed (see discussion inSchweickert, Fisher, & Sung, 2012).

Definition 7: Distribution-ordering selectiveinfluence holds for processes a and b withregard to experimental factors A and B, whichthemselves form a simple order, if and onlyif Fa(t|A1) = P(Ta ≤ t|A1) < P(Ta ≤ t|A2) =Fa(t|A2) if and only if A1 < A2 and similarlyfor Fb(t|B). And, A has no effect on Fb(t|B)nor does B have any effect on Fa(t|A).

The double factorial paradigm (DFP;Townsend & Nozawa, 1995) offers a furthergeneralization of the additive factors methodusing distribution-ordering selective influ-ence. The main motivation of the DFP is toisolate the effects of workload on RTs fromthe effects of architecture and the stoppingrule but nonetheless test all three constructswithin one paradigm. The first manipula-tion is to factorially combine the presenceor absence of each source of information.Hence, there are trials in which each source ispresented in isolation and trials in which allsources are presented together. The secondmanipulation is meant to factorially speedup and slow down the processing of eachpresent source of information. This manip-ulation is often referred to as the saliencemanipulation, in reference to the fact thatlow-salience perceptual stimuli are slowerto process than high-salience stimuli, butthe manipulations need not necessarily beof salience. Figure 11.3 depicts the possibletrial types for an experiment that requires aparticipant to detect a target either to the rightor left of center, or in both locations. Targetsare made lower salience on the screen bydecreasing the contrast.

Assuming distribution-ordering selectiveinfluence of the salience manipulationsfor each source of information when thatsource is present, serial and parallel mod-els combined with either exhaustive orfirst-terminating stopping rules can be dis-criminated using an interaction contrast ofthe RT distributions. To maintain consistency

Strong Experimental Tests on Response Times II: Applications of Selective Influence 13

HH HL LH LL LA AH ALHA

Figure 11.3 Example of stimuli for a double factorial experiment to test the architecture and stoppingrule of detecting a left and/or right dot. H refers to a high salience level, L refers to a low salience level,and A refers to the absence of the target.

with other papers on systems factorialtechnology, we switch from the G* toS(t) = 1–F(t) for the survivor function—thatis, the complement of the cumulative dis-tribution function. The survivor interactioncontrast is given by

SIC(t) = [SLL(t) − SHL(t)]− [SLH(t) − SHH(t)].

If distribution-ordering selective influ-ence of the salience manipulation holds,then the survivor functions will be orderedsuch that when both processes are sloweddown, the survivor function will be higherthan when either or both of the sourcesare sped up (i.e., the probability of a RTnot occurring yet will always be high-est when both sources are slowed down,(SLL(t) > SLH(t), SHL(t),SHH(t)). Likewise,the survivor function will be at its lowestwhen both sources are sped up (SHH(t) <SLH(t), SHL(t),SLL(t)).

We have already stated that the saliencemanipulation, as long as it selectively

influences the processes, is sufficient fordiscriminating among serial, parallel, exhaus-tive, and first-terminating models, but whyis this the case? Detailed proofs are avail-able in Townsend and Nozawa (1995; forgeneralizations and alternative approachessee Dzhafarov, Schweickert, & Sung, 2004;Zhang & Dzhafarov, 2015). We build basicintuition for the main results here, andsummarize them graphically in Figure 11.4.

First, consider the parallel, first-terminating model. In this model, the fastestprocess to terminate determines the RT.Hence the overall system will be relativelyfast as long as one or both sources are highsalience. In terms of the survivor interactioncontrast (SIC), this means the first difference,[SLL(t) − SHL(t)], will be relatively largerthan the second difference, because onlythe first difference contains a term withouta high salience. Because both terms mustbe positive (due to the effective selectiveinfluence assumption) and the first differenceis a large magnitude, we see that the SIC for

Serial Or

SIC

Time Time Time Time Time

+

SIC

SIC

SIC

SIC

Serial And Parallel Or Parallel And Coactive

Figure 11.4 Schematic representation of the survivor interaction contrast functions (upper panels) pre-dicted by the models associated with the combination of parallel and serial architectures and or and andstopping rules, along with predictions for the coactive model.


the parallel, first-terminating model shouldbe positive.

The logic behind the parallel, exhaustivemodel is similar, but reversed. This model’sRT is determined by the last process to finish.Thus the model will be relatively slow ifeither of the sources are low salience. Hencethe second difference in the SIC will belarger than the first difference, because onlythe second difference contains a term withoutlow salience. Thus, the SIC for the exhaustivemodel should be negative.

Next, we consider the serial, first-terminating model. In these models, theprocess to complete first, and hence the onethat determines the model completion time, isprobabilistically chosen without dependenceon the salience levels. The magnitude of thedifference between and is determined bythe probability that the first position is pro-cessed first (if the second position is processedfirst, then the first position is not processed andthere is no difference in processing times)and the magnitude of the difference betweenthe low and high salience processing timesof the first position. Because both the proba-bility of the first position being processed firstand the magnitude of the salience manipula-tion are the same for both the first and secondparts of the SIC, the overall SIC is thus 0 forall times. The serial, exhaustive model is abit more complicated. The intuition that theoverall area under the SIC curve is 0 followsfrom the fact that the combination of theprocessing times is additive in this model;hence any manipulation of the individualprocessing times will result in additivity (i.e.,0 mean interaction). Indeed, this is essentiallythe result summarized above from Ashby andTownsend (1980). Townsend and Nozawa(1995) go on to show that the SIC is negativefor the earliest RTs, and Yang, Fific, andTownsend (2014) later proved that, as long asat least one of the completion time survivor

functions is log concave, there will only be asingle 0 crossing for the SIC.7

STRONG EXPERIMENTAL TESTSON RESPONSE FREQUENCIES

Back to the State Spaces: Tests Basedon Partial States of Completion

As is the case for the RT literature, there areboth strengths and weaknesses to the uniquefocus on response frequencies; indeed, muchof our more recent work has been driven bythe potential suggested by using the com-plementary strengths of each approach (e.g.,Eidels, Townsend, Hughes, & Perry, 2015).A branch of theory and methodology per-taining to response frequencies also depends,like the methods in the preceding section, onfiner levels of description of the event spaces.

The reader may recall that the PST reliedon the fact that parallel systems allow forintermediate stages of processing for anynumber of uncompleted items, whereas serialsystems only permit one item, at most, tobe in a partial state of completion. In thepresent section, we find that this distinctioncan be utilized in an experimental design onresponse frequencies, rather than RTs, to testparallel versus serial processing.

The experimental design requires low-to-moderate accuracy, and the observer isinstructed to give two responses: his or herfirst, and a second “guess,” just in case his orher first response is incorrect. Parallel modelscan predict that second guesses are correctat a higher level than chance. Serial models,by virtue of allowing at most one item ina state of partial completion, are severely

7For the interested reader, Houpt, Blaha, McIntire, Havig,and Townsend (2014) introduced an R package (avail-able on CRAN: https://cran.r-project.org/web/packages/sft/index.html) and a more thorough tutorial for the DFPand the statistical analyses of the associated measures.

https://cran.r-project.org/web/packages/sft/index.html

https://cran.r-project.org/web/packages/sft/index.html

Strong Experimental Tests on Response Frequencies 15

restricted on second responses. Followingthis logic, Townsend and Evans (1983) calcu-lated nonparametric bounds on the number ofcorrect second guesses that any serial modelcould predict and carried out a preliminarystudy using this paradigm. Although certainconditions and observers were ruled to havebeen parallel processing, for a number ofobservers and conditions, serial processingcould not be ruled out. However, only twoitems were employed, and the serial boundswere not very tight for this small number.This technique should be further exploited,but what has been done so far is in favorof parallel processing (Townsend & Evans,1983; Van Zandt, 1988). We examine thistype of design further next.

Manipulating Process Durationsof Available Information

As noted above, consideration of the constel-lation of issues surrounding the identifiabilityof serial and parallel systems has not beenrestricted to RTs. A much smaller, but stillsubstantial, literature on the use of responsefrequencies (including issues linking processcharacteristics to signal detection measuresof response bias, e.g., Balakrishnan, 1999)has attempted to speak to the issue as well,even above and beyond the above meth-ods based on the differing state spaces ofparallel and serial systems. Nominally, themost frequent issue to be addressed by theuse of response frequencies has been thatof processing capacity and its relation toboth process architecture and processingindependence. With respect to the latterof these questions, Townsend (1981) con-sidered a potential ambiguity presented bywhole-report results of Sperling. Althoughthe serial position data of Sperling seemedto support serial processing with an approxi-mately 10 ms/item scan rate, Townsend noted(as Sperling had acknowledged) that the form

of Sperling’s serial position curves suggestedthe possibility of parallel processing.

Townsend’s analysis of the problempresented by these results hinged on thefact that serial processing models based onPoisson distributions all predicted positiveitem dependencies. As such, a possibilityfor adjudicating the question of architec-ture in this context emerged in the potentialcontrasts between models positing iteminterdependencies and models assumingindependence.

Townsend (1981) developed a set of serialprocessing models for these comparisons.The first assumed strict serial processing,assuming Poisson distributions for item com-pletion. The second allowed for a randomorder of serial processing, again, based onPoisson distributions. The third instantiateda capacity limitation in the form of a fixedsample size for processing. This model isan example of a quite popular model forshort-term memory called a slot model.A fourth assumed independent parallel chan-nels. The first two of these models wereshown to predict positive item interdepen-dencies. The third, a general slot model,predicted negative dependencies. The logicis straightforward: Slot models assume thereexist a finite number of slots available foritem storage. The more items already occu-pying a slot, the less chance another item hasto find a free slot. Even assuming that thenumber of slots can vary from trial to trialdoes not help—the result is simply a mixtureof slot sizes, each of which predicts negativedependence, as does the overall mixture. Themost natural architecture for an independencemodel is parallel, as noted above, althoughother architectures are possible (see, e.g.,Townsend & Ashby, 1983, Chapter 4).

When the data from Townsend’s (1981)whole-report study were plotted against thepredictions of these competing models, itrevealed a high level of agreement between


the data and the predictions of the modelthat assumed independence, consistentwith the idea of parallel processing. Further,Townsend (1981) considered two possibleforms of parallel processing—one assumingindependent rates of information accumu-lation across positions in the display and asecond assuming limitations in the allocationof processing resources—and showed thatthe former offered the superior fit to the data.

Subsequently, possibly the most fre-quently used paradigm to explore questionsregarding processing characteristics usingresponse frequencies has been some varianton what is referred to as the simultaneous-sequential paradigm. This paradigm, intro-duced in its original form by Eriksen andSpencer (1969), has been widely usedto examine questions regarding process-ing capacity at the individual item level(e.g., Duncan, 1980; Fisher, 1984; Harris,Pashler, & Coburn, 2004; Hoffman, 1978;Huang & Pashler, 2005; Kleiss & Lane,1986; Prinzmetal & Banks, 1983; Shiffrin &Gardner, 1972). In its typical form, thesimultaneous-sequential paradigm involvesthe use of varying numbers of stimuli in twoconditions, one in which all of the stimuliare presented at the same time (simultaneouscondition) and one in which they are pre-sented one at a time (sequential). The frameduration, which can be manipulated across arange of values, is held constant across thetwo display conditions.8

In the most felicitous paradigm of thisnature, the experimenter acquires somepreliminary data in order to estimate theaverage duration consumed by the process-ing (e.g., search, identification, etc.) of asingle item. Suppose that time is found to

8In the actual experiment, Eriksen and Spencer (1969)only approximated the simultaneous condition byextremely rapid sequential presentation of the items.However, this manipulation apparently was perceptuallyequivalent to simultaneous presentation.

be approximately t * ms. Thus, we have thebasic setup that one condition presents all nitems simultaneously for t * ms, whereas theother condition presents each item for t * ms,each in succession. Obviously, a standardparallel system should not be seriously hurtin performance in the simultaneous condi-tion as opposed to the sequential condition.In contrast, a standard serial system shouldonly be able to process approximately oneitem in that condition versus about n in thesequential condition.

Eriksen and Spencer’s (1969) introductionof the paradigm was motivated by a verycareful review of the range of issues that hadto be considered in order for an experimentto convincingly address the question of howmuch information can be extracted from astimulus display in a unit of time (capacity),along with how the information is extracted(serially or in parallel; see Estes & Taylor,1964, 1966; Estes & Wessel, 1966; Sperling,1963). The potentially complicating factorsconsidered by Eriksen and Spencer (1969)included capacity limitations at the levels oflow-level stimulus features (duration, noise,boundaries), low-level perceptual character-istics (foveal), and postperceptual, higherlevel aspects of attention and memory.

As Eriksen and Spencer (1969, p. 2)noted, echoing the issues we have discussedabove in the context of RTs, “Definitiveanswers [regarding capacity and processarchitecture] have not been obtained primar-ily because of the numerous problems thatare involved in what at first glance seems tobe a simple methodology. The difficultiesthat are encountered become apparent if theinput-output information flow is consideredin terms of the various distinguishable sub-processes that are involved.” Yet as theynoted, there seemed to be strong and accu-mulating evidence (e.g., Eriksen & Lappin,1965) that there must be some capacitylimitations at some level of processing, with

Strong Experimental Tests on Response Frequencies 17

the general approach at the time suggestinga mapping between capacity limitations andserial processing.

Eriksen and Spencer’s (1969) results, inthis context, were quite simple and com-pelling. Essentially, the only experimentalfactor among the range of those manipulatedto have any regular effect on observer’saccuracy (quantified in terms of signal detec-tion theory) was the number of items inthe display set, with there being a mono-tonic reduction in sensitivity from one item(d′A = 1.64) to nine items (d′A = 0.88), withthis loss of sensitivity being due almostentirely to an increase in the rate of falsepositives. Eriksen and Spencer’s explana-tion for these results was parsimonious andintriguingly similar to Sternberg’s logic.Specifically, they reasoned that if capacitywas constant as a function of display size, thehit and false positive rates for single-item dis-plays could be used, under the assumption ofindependence of item processing, to predictthe level of performance across variations indisplay size. Their predictions were highlyconsistent with their data. Their conclusionillustrates both the potency of this very sim-ple and elegant reasoning on probabilities andthe contemporary conflation of the issues ofcapacity and processing architecture (p. 15;for related research, see Averbach & Coriell,1961; Mayzner, Tresselt, & Helper, 1967;Sperling, 1963): “This would suggest thatthe encoding mechanism proposed in cur-rent theories . . . can scan through or encodenine letters as efficiently in 50 msec. as in25 msec. This interpretation would almostcertainly preclude a serial encoding process.”

Townsend and colleagues (Townsend,1981; Townsend & Ashby, 1983, Chapter 11;Townsend & Fial, 1968) extended the Eriksenand Spencer logic in a visual whole-reportsetting illustrated in Figure 11.5. Visualwhole reports ask the observer to report all

C-I (Both Designs)

C-II (Shiffrin & Gardner, 1972)

C-II (Townsend, 1981)

t*

K

t*

M

t*

2t*

KM

KM

Figure 11.5 Schematic of two related designsfor discriminating parallel and serial processingbased on accuracy. In both designs, participantsare shown a fixed number of items, then asked toreport as many as possible. A baseline of accuracyis estimated by presenting each item in isolation fort* ms. In the Shiffrin and Gardner (1972) design,the baseline accuracy is compared against perfor-mance when all items are presented together for t*ms. In the Townsend (1981) design, all items arepresented together for n times t* ms (where n isthe number of items).

the items she or he is able to from a visualdisplay. Instead of the t * ms simultaneouscondition versus n × t∗ successive item inter-vals, as in the original Eriksen and Spencer(1969), or that of Shiffrin and Gardner’s(1972) design (see below), Townsend andcolleagues added a pair of conditions, withone condition being the same; that is, nitems being presented sequentially for t*


ms each. This was compared with a newcondition where all n items were presentedsimultaneously for n × t∗ ms. In contrast tothe original design, here a standard serialmodel is predicted to do as well, but no bet-ter, in the simultaneous versus the sequentialcondition. On the other hand, the parallelsystem, assuming it’s not working at ceilinglevel, is expected to perform much betterin the simultaneous condition. The results,like those of their independence analy-ses, strongly supported parallel processingchannels.

Shiffrin and Gardner (1972) employed theoriginal Eriksen and Spencer (1969) idea.They found evidence of equality of perfor-mance in the two presentation conditionsacross a set of variations, leading them to aconclusion consistent with that of Eriksenand Spencer: specifically, that processingwas parallel and unlimited in capacity (andby extension, parallel) up to the level ofpostperceptual processing.

More recently, Palmer and colleagues(e.g., Scharff, Palmer, & Moore, 2011)have extended the simultaneous-sequentialparadigm to broaden the class of models ofcapacity that can be addressed in a singleexperiment. Specifically, they added a condi-tion in which all of the test stimuli are in twoframes with equal durations, separated by a1,000-ms blank interval intended to allowsufficient time for attention switching (Ward,Duncan, & Shapiro, 1996). Recall that aswitching interval “recess” was not present inany of the earlier applications of this strategy.They then used a set of probability models toderive contrasting predictions for the variousexperimental conditions in their extendedparadigm.

Most critically, their predictions were evenstronger than those associated with previoususes of the paradigm. The predictions con-trast three classes of models (Scharff et al.,2011, p. 816) across specific subconditions

of their expanded simultaneous-sequentialparadigm. The three classes of models arereferred to as limited, intermediate, andunlimited capacity models. We refer readersto Scharff et al.’s (2011) paper for the spe-cific predictions, noting here that the logicsupporting those predictions does allow,under specific assumptions regarding pro-cessing architecture and independence forboth confirmation and falsification of com-peting accounts, underscoring the recurringtheme that these issues cannot be consideredin isolation.

CONCLUSION

As we asserted earlier, it is our belief thatalmost all of psychological science can,and should be, thought of as a black boxdiscipline. Such a discipline would be anal-ogous to the several engineering, computerscience, and applied physics fields thatattempt, largely through analysis of behav-ioral input-output regularities, to discover,affirm, or disconfirm internal systems thatcan produce the regularities. We can likelyborrow at least some elements directly fromthese fields, such as the theory of the idealdetector from mathematical communicationsscience. However, because of the complexityof the brain and the perversely constrainedmeans we have of studying even input-outputrelationships, the greater part of such strate-gies must be vastly different in detail fromthose of these other sciences.

We hasten to emphasize that this typeof program is hardly behavioristic in thetraditional Skinnerian sense of the term.9

9An intriguing interchange with B. F. Skinner in his laterdays and a wide range of scientists can be found in B. F.Skinner’s “Methods and Theories in the ExperimentalAnalysis of Behavior” (1984). In particular, it containsa small debate between Skinner and Townsend on thepossibilities of mathematical modeling of human psy-chology (Townsend, 1984b).

Conclusion 19

Rather than logging the entries into agargantuan dictionary of stimulus-responsecorrelations, as in Skinner’s early agenda,the overriding goal of our type of programis to move from the input-output regu-larities, garnered from specially designedexperimental paradigms, to sets of internalstructures and processes.10 Most, if not all,of our approaches attempt to thoroughlyadhere to the concept of a psychologicalsystems science, and, in the bargain, to makeourselves aware of when model mimickingis likely to occur and how to avoid it bymetamodeling deeper experiment methods ofassessing important processing mechanisms.The potency of this adherence to a systemsview allows for application to a range ofpsychological phenomena, not just thosein the areas of perception, memory, andperceptual organization in which we haveworked. Space prevents us from giving evena brief description to this range of studies;however, we supply in the appendix a listingof applications of aspects of our approach.

The preponderance of methods intro-duced in this chapter use RTs as the majorobservable variable, although we saw severalbased on response frequencies, and there-fore inclusive of accuracy. A major facetof our armamentarium that we have had toneglect in the present treatment has employedRTs to measure workload capacity, the wayin which efficiency changes as the work-load n is increased (see, e.g., Townsend &

10In the advent of early cognitive science in the 1960s,it was fashionable to vilify behaviorism as a dark side ofscientific enterprise. In truth, it can be forcefully arguedthat a few decades of rather strict behaviorism, along withthe logical positivism of the Vienna Circle and friends,and the allied new emphasis, even in physics, of opera-tionalism, was important in helping psychology to fin-ish pruning itself from its mother philosophical roots.Furthermore, in careful reading of the works of the greattheoretical behaviorist of the day, Clark Hull, therudiments of modern computational and mathematicalpsychology are readily visible.

Nozawa, 1995; Townsend & Wenger, 2004a,2004b). For instance, the change in RTswith increases in memory-set size in ourprototypical Sternberg (1966) paradigm is agood example. Our statistic C(t) provides ameasure where performance as a function ofn is compared against a baseline predicted byunlimited capacity independent parallel pro-cessing. Very recently, we have generalizedthe C(t) statistic to take into account accuracyas well as RTs in measuring performance asa function of n. The new statistic is calledA(t), for assessment function (Townsend &Altieri, 2012).

A “must do” extension of our methodolo-gies is to systems of arbitrary number n ofoperational processes, rather than just n = 2.This target has lately been met for serial ver-sus parallel systems and for exhaustive andminimum time-stopping rules (Yang, Fific, &Townsend, 2014). Another highly valuabletheme of our methodology, one that wasoriginally based on accuracy and confusionfrequencies alone and was co-invented withF. G. Ashby, is general recognition theory(Ashby & Townsend, 1986). Over the yearsAshby and colleagues have developed spe-cial models that bring together concepts fromgeneral recognition theory and RT process-ing (Ashby, 1989, 2000). They also extendedgeneral recognition theory into a powerfultheory of categorization (e.g., Ashby, 1992).

We have recently developed a very generalclass of models entitled response time gen-eral recognition theory (Townsend, Houpt, &Silbert, 2012). Currently, these models canemploy both accurate responses and vari-ous types of confusions in concert with theassociated RTs in order to assess percep-tual and cognitive dependencies, as wellas higher order classification, at the deep-est conceivable level. However, at presentour methodology does not permit tests ofarchitecture or stopping rules, as does sys-tems factorial technology, with RTs as the


dependent variable. Closer to our expositionhere, we are currently extending systemsfactorial technology to incorporate responsefrequencies in identification of architectureand stopping rules.

There is an important omission from ourcurrent set of topics. Virtually all of thesystems under study here obey what has beencalled the Dondersian postulate (Townsend &Ashby, 1983, pp. 5–6, 358, 408); namely, thateach psychological process in a sequence ofsuch processes finishes before the next in thechain begins. A traditional supplement tothe discrete flow assumption is the assump-tion of pure insertion, mentioned earlierin this chapter, wherein the experimenter caninsert or withdraw a stage or process by sometype of empirical manipulation. However,this additional restriction is not mandatory.A more suggestive name for the Dondersianassumption is discrete flow.

However, anyone who has studied nat-ural systems (for example), as approachedor described with differential equations, isaware that many physical, chemical, andbiological systems obey the opposite preceptof continuous flow. The output of a process ina continuous flow system is instantaneouslyinput to the succeeding process. And, theinput space, state space, and output spaceare usually depicted as continuous func-tions (e.g., see Townsend & Ashby, 1983,pp. 401–419). These systems are sometimescalled lumped systems in engineering becausethere is no time lag at all from one stage tothe next.

The simplest such systems, still of extremeimportance in the sciences, are linear innature. Even the output of linear systems,however, can reflect memory of past inputs.A relatively early example of a continuousflow linear system was the cascade modelintroduced by McClelland (1979). Withinsystems factorial technology, even thoughthis type of model does not predict additivity

in mean RTs as do true serial models thatobey selective influence, McClelland showedthat it can approximate such additivity.

The amount of theoretical knowledgeconcerning, for instance, how to identifysuch spaces and when one might predictadditivity of factorial effects in an observablestate space is very modest indeed. Schwe-ickert and colleagues (1989; Schweickert &Mounts, 1998) have derived results for cer-tain continuous flow systems that include, butare not limited to, linear systems. However,their targeted systems do assume a lack ofmemory on past inputs or states. Townsendand Fikes (1995) studied a broad class ofpossibly nonlinear, continuous flow systemsthat do include such memories across time.Miller (1988) has investigated hybrid sys-tems that may only partially process itemsand possess temporal overlap analogous tocontinuous flow systems.

The deeper analysis of continuous flowand hybrid systems and development of toolsfor their identification should be considereda top priority for psychological theorists.

At some level, all of the theory andapplication that we have considered in thischapter must make some form of contactwith measurable aspects of the activity ofthe nervous system. Although quite limitedat the moment, there are examples of howthe foundational ideas considered here mightbe related to neurophysiology. Possibly theearliest consideration of these possibilitiescan be found in Schweickert and colleagues’work on continuous flow systems as men-tioned above (1989; Schweickert & Mounts,1998), where consideration was given to theuse of factorial methods in application toevent-related potentials.

Since that time, a handful of studies haveconsidered the utility of the additive factorsmethod with respect to both event-relatedpotentials (e.g., Miller & Hackley, 1992) andthe blood-oxygen level dependent signal

References 21

in magnetic resonance imaging (e.g.,Cummine, Sarty, & Borokowski, 2010),and the utility of the capacity measures inconjunction with an electroencephalogram(Blaha, Busey, & Townsend, 2009). To ourknowledge, however, there is at present onlyone as-yet unpublished study (initial reportsin Wenger, Ingvalson, & Rhoten, 2017;Wenger & Rhoten, 2015) that has attemptedto apply the complete set of measures asso-ciated with systems factorial technologyto neurophysiological data. In that study,the onset time of the lateralized readinesspotential (an electroencephalogram featurethat reliably precedes choice RT; see, forexample, Mordkoff & Grosjean, 2001) wasused as the dependent variable. Conclusionsdrawn from the analyses of these data wereconsistent, on a subject-by-subject basis,with the conclusions drawn from the RT data.

Although preliminary, these results arequite encouraging with respect to one pos-sible way of connecting extant theory andmethod with cognitive neuroscience.

The ultimate goal of all of these endeavorsis to have in hand a powerful armamentariumwhose tools can utilize both of the strongestobservable dependent variables available:response frequencies and RTs. These toolsare expected to simultaneously assess archi-tecture, stopping rules, attentional and otherkinds of capacity, and finally, several types ofprocess and item dependencies. In addition,it is our hope that we can join with neuro-scientists in employing our methodologiesalong with their diverse strategies, to pinpointunderlying mechanisms and their interac-tions. As we peer into the future, we envisionresearchers from many cognate fields devel-oping rigorous varieties of psychologicalsystems theory, ones that are general (i.e., notglued to specific aspects of processing, suchas particular probability distributions) andconversant with the challenges of modelmimicking.

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APPENDIX: APPLICATIONSOF SYSTEMS FACTORIALTECHNOLOGY

Age-Related Changes in Perceptionand Cognition

Ben-David, B. M., Eidels, A., & Donkin, C.(2014). Effects of aging and distractors on detec-tion of redundant visual targets and capacity:Do older adults integrate visual targets differ-ently than younger adults? PLOS ONE, 9(12),e113551.

Gottlob, L. R. (2007). Aging and capacity in thesame-different judgment. Aging Neuropsychol-ogy and Cognition, 14, 55–69.

Wenger, M. J., Negash, S., Petersen, R. C., &Petersen, L. (2010). Modeling and estimatingrecall processing capacity: Sensitivity and diag-nostic utility in application to mild cognitiveimpairment. Journal of Mathematical Psychol-ogy, 54, 73–89.

Binocular Interaction

Hughes, H. C., & Townsend, J. T. (1998). Vari-eties of binocular interaction in human vision.Psychological Science, 9, 53–60.

Medina, J. M. (2006). Binocular interactions inrandom chromatic changes at isoluminance.Journal of the Optical Society of America A:Optics Image Science and Vision, 23, 239–246.

Medina, J. M., & Diaz, J. A. (2006). Postrecep-toral chromatic-adaptation mechanisms in thered-green and blue-yellow systems using simplereaction times. Journal of the Optical Society ofAmerica A: Optics Image Science and Vision,23, 993–1007.

Categorization

Blunden, A. G., Wang, T., Griffiths, D. W., &Little, D. R. (2015). Logical-rules and theclassification of integral dimensions: Individualdifferences in the processing of arbitrary dimen-sions. Frontiers in Psychology, 5, 1531.

Fific, M., Nosofsky, R. M., & Townsend, J. T.(2008). Information-processing architectures inmultidimensional classification: A validationtest of the systems factorial technology. Journalof Experimental Psychology: Human Perceptionand Performance, 34, 356–375.

Fific, M., Little, D. R., & Nosofsky, R. M. (2010).Logical-Rule models of classification responsetimes: A synthesis of mental-architecture,random-walk, and decision-bound approaches.Psychological Review, 117, 309–348.

Little, D. R., Nosofsky, R. M., & Denton, S. E.(2011). Response-Time tests of logical-rulemodels of categorization. Journal of Experimen-tal Psychology: Learning, Memory, and Cogni-tion, 37, 1–27.

Cognitive Control

Eidels, A., Townsend, J. T., & Algom, D. (2010).Comparing perception of Stroop stimuli infocused versus divided attention paradigms:Evidence for dramatic processing differences.Cognition, 114, 129–150.

Human–Machine Teaming

Yamani, Y., & McCarley, J. S. (2016). Workloadcapacity: A response time-based measure ofautomation dependence. Human Factors 58(3),462–471.

Appendix: Applications of Systems Factorial Technology 27

Identity and Emotion Perception

Fitousi, D., & Wenger, M. J. (2013). Variants ofindependence in the perception of facial identityand expression. Journal of Experimental Psy-chology: Human Perception and Performance,39, 133–155.

Richards, H. J., Hadwin, J. A., Benson, V., Wenger,M. J., & Donnelly, N. (2011). The influenceof anxiety on processing capacity for threatdetection. Psychonomic Bulletin & Review, 18,883–889.

Yankouskaya, A., Humphreys, G. W., & Rotshtein,P. (2014). Differential interactions betweenidentity and emotional expression in own andother-race faces: Effects of familiarity revealedthrough redundancy gains. Journal of Exper-imental Psychology: Learning, Memory, andCognition, 40, 1025–1038.

Yankouskaya, A., Humphreys, G. W., & Rotshtein,P. (2014). The processing of facial identity andexpression is interactive, but dependent on taskand experience. Frontiers in Human Neuro-science, 8, 920.

Individual Differences/ClinicalPopulations

Endres, M. J., Houpt, J. W., Donkin, C., &Finn, P. R. (2015). Working memory capac-ity and redundant information processing effi-ciency. Frontiers in Psychology, 6, 594.

Houpt, J. W., Sussman, B. L., Townsend, J. T., &Newman, S. D. (2015). Dyslexia and configuralperception of character sequences. Frontiers inPsychology, 6, 482.

Johnson, S. A., Blaha, L. M., Houpt, J. W., &Townsend, J. T. (2010). Systems factorial tech-nology provides new insights on global-localinformation processing in autism spectrum dis-orders. Journal of Mathematical Psychology,54, 53–72.

Neufeld, R. W. J., Vollick, D., Carter, J. R., Boks-man, K., & Jette, J. (2002). Application ofstochastic modeling to the assessment of groupand individual differences in cognitive function-ing. Psychological Assessment, 14, 279–298.

Learning and Reward Processing

Sui, J., Yankouskaya, A., & Humphreys, G. W.(2015). Super-Capacity me! Super-Capacityand violations of race independence for self-but not for reward-associated stimuli. Journalof Experimental Psychology: Human Perceptionand Performance, 41, 441–452.

Wenger, M. J. (1999). On the whats and howsof retrieval in the acquisition of a simple skill.Journal of Experimental Psychology: Learning,Memory, and Cognition, 25, 1137–1160.

Memory Search

Donkin, C., & Nosofsky, R. M. (2012). The struc-ture of short-term memory scanning: An investi-gation using response time distribution models.Psychonomic Bulletin & Review, 19, 363–394.

Thomas, R. (2006). Processing time predictions ofcurrent models of perception in the classic addi-tive factors paradigm. Journal of MathematicalPsychology, 50, 441–455.

Townsend, J. T., & Fific, M. (2004). Parallel ver-sus serial processing and individual differencesin high-speed search in human memory. Percep-tion & Psychophysics, 66, 953–962.

Multimodal Interaction

Altieri, N., & Hudock, D. (2014). Hearing impair-ment and audiovisual speech integration ability:A case study report. Frontiers in Psychology, 5,678.

Altieri, N., & Hudock, D. (2014). Assessing vari-ability in audiovisual speech integration skillsusing capacity and accuracy measures. Interna-tional Journal of Audiology, 53, 710–718.

Altieri, N., Pisoni, D. B., & Townsend, J. T.(2011). Some behavioral and neurobiologicalconstraints on theories of audiovisual speechintegration: A review and suggestions for newdirections. Seeing and Perceiving, 24, 513–539.

Altieri, N., Stevenson, R. A., Wallace, M. T., &Wenger, M. J. (2015). Learning to associateauditory and visual stimuli: Behavioral andneural mechanisms. Brain Topography, 28,479–493.


Altieri, N., & Townsend, J. T. (2011). Anassessment of behavioral dynamic informationprocessing measures in audiovisual speech per-ception. Frontiers in Psychology, 2, 238.

Altieri, N., Townsend, J. T., & Wenger, M. J.(2014). A measure for assessing the effectsof audiovisual speech integration. BehaviorResearch Methods, 46, 406–415.

Altieri, N., & Wenger, M. J. (2013). Neural dynam-ics of audiovisual speech integration undervariable listening conditions: An individual par-ticipant analysis. Frontiers in Psychology, 4,615.

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Hugenschmidt, C. E., Hayasaka, S., Peiffer,A. M., & Laurienti, P. J. (2010). Applyingcapacity analyses to psychophysical evalua-tion of multisensory interactions. InformationFusion, 11, 12–20.

Patching, G. R., & Quinlan, P. T. (2004). Cross-modal integration of simple auditory andvisual events. Perception & Psychophysics, 66,131–140.

Perceptual Organization

Donnelly, N., Cornes, K., & Menneer, T. (2012).An examination of the processing capacity offeatures in the Thatcher illusion. Attention, Per-ception, & Psychophysics, 74, 1475–1487.

Eidels, A., Townsend, J. T., & Pomerantz, J. R.(2008). Where similarity beats redundancy: Theimportance of context, higher order similarity,

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Fific, M., & Townsend, J. T. (2010). Information-processing alternatives to holistic perception:Identifying the mechanisms of secondary-levelholism within a categorization paradigm. Jour-nal of Experimental Psychology: Learning,Memory, and Cognition, 36, 1290–1313.

Fitousi, D. (2015). Composite faces are not pro-cessed holistically: Evidence from the Garnerand redundant target paradigms. Attention, Per-ception, & Psychophysics, 77, 2037–2060.

Hawkins, R. X. D., Houpt, J. W., Eidels, A., &Townsend, J. T. (2016). Can two dots forma Gestalt? Measuring emergent features withthe capacity coefficient. Vision Research, 126,19–33.

Houpt, J. W., Townsend, J. T., & Donkin, C.(2014). A new perspective on visual wordprocessing efficiency. Acta Psychologica, 145,118–127.

Little, D. R., Nosofsky, R. M., Donkin, C., & Den-ton, S. (2013). Logical rules and the classifi-cation of integral-dimension stimuli. Journal ofExperimental Psychology: Learning, Memory,and Cognition, 39, 801–820.

Moneer, S., Wang, T., & Little, D. R. (2016).The processing architectures of whole-objectfeatures: A logical-rules approach. Journal ofExperimental Psychology: Human Perceptionand Performance, 42(9), 1443–1465.

Wenger, M. J., & Townsend, J. T. (2006). On thecosts and benefits of faces and words: Processcharacteristics of feature search in highly mean-ingful stimuli. Journal of Experimental Psychol-ogy: Human Perception and Performance, 32,755–779.

Perceptual Detection

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Appendix: Applications of Systems Factorial Technology 29

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Patching, G. R., Englund, M. P., & Hellstrom, A.(2012). Time- and space-order effects in timeddiscrimination of brightness and size of pairedvisual stimuli. Journal of Experimental Psychol-ogy: Human Perception and Performance, 38,915–940.

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Yang, C.-T. (2011). Relative saliency in changesignals affects perceptual comparison and deci-sion processes in change detection. Journal ofExperimental Psychology: Human Perceptionand Performance, 37, 1708–1728.

Yang, C.-T., Chang, T.-Y., & Wu, C. J. (2013).Relative change probability affects the decisionprocess of detecting multiple feature changes.Journal of Experimental Psychology: HumanPerception and Performance, 39, 1365–1385.

Yang, C.-T., Hsu, Y. F., Huang, H. Y., & Yeh, Y. Y.(2011). Relative salience affects the process ofdetecting changes in orientation and luminance.Acta Psychologica, 138, 377–389.

Temporal Order Processing

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order judgments. Perception & Psychophysics,66, 563–573.

Visual Search/Visual Attention

Busey, T. A., & Townsend, J. T. (2001). Inde-pendent sampling vs interitem dependenciesin whole report processing: Contributions ofprocessing architecture and variable atten-tion. Journal of Mathematical Psychology, 45,283–323.

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Sung, K. (2008). Serial and parallel attentive visualsearches: Evidence from cumulative distributionfunctions of response times. Journal of Exper-imental Psychology: Human Perception andPerformance, 34, 1372–1388.

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Yang, C.-T., Little, D. R., & Hsu, C. C. (2014).The influence of cueing on attentional focus inperceptual decision making. Attention, Percep-tion, & Psychophysics, 76, 2256–2275.

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defined in two dimensions is based on saliencesummation, rather than on serial exhaustive orinteractive race architectures. Attention, Percep-tion, & Psychophysics, 71, 1739–1759.

Working Memory/Cognitive Load

Chang, T.-Y., & Yang, C.-T. (2014). Individualdifferences in Zhong-Yong tendency and pro-cessing capacity. Frontiers in Psychology, 5,1316.

Fitousi, D., & Wenger, M. J. (2011). Process-ing capacity under perceptual and cognitiveload: A closer look at load theory. Journal of

Experimental Psychology: Human Perceptionand Performance, 37, 781–798.

Heathcote, A., Coleman, J. R., Eidels, A., Watson,J. M., Houpt, J. W., & Strayer, D. L. (2015).Working memory’s workload capacity. Mem-ory & Cognition, 43, 973–989.

Schweickert, R., Fortin, C., & Sung, K. (2007).Concurrent visual search and time reproductionwith cross-talk. Journal of Mathematical Psy-chology, 51, 99–121.

Yu, J.-C., Chang, T.-Y., & Yang, C.-T. (2014). Indi-vidual differences in working memory capacityand workload capacity. Frontiers in Psychology,5, 1465.

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