The difference engine: A model of diversity in speeded cognition

Copyright 2003 Psychonomic Society Inc 262

Psychonomic Bulletin amp Review2003 10 (2) 262-288

The results of a number of studies suggest that much ofthe variance in individual speeded performance at least onlaboratory tasks may be explained by a single general fac-tor on which diverse tasks load approximately equally(Hale amp Jansen 1994 Hale amp Myerson 1993 Zheng My-erson amp Hale 2000 for further examples see Table 1 inFaust Balota Spieler amp Ferraro 1999) This finding im-plies that if a task is speeded its other characteristics maybe much less important at least for the purpose of pre-dicting the position of an individualrsquos score in the overalldistribution of performances (ie for predicting perfor-mance in standard deviation [SD] units) Even if most of

the variance in individual performance in SD units can beexplained by a single factor however important questionsregarding individual differences still remain For examplewhat task characteristics determine how many seconds ormilliseconds will separate the performance of the fastestand the slowest individuals and what mechanism or mech-anisms underlie the observed variations in the absolutesize of individual differences from task to task

A recent study by Hale and Jansen (1994) provides animportant clue regarding the answers to these questionsThey tested 40 young adults on seven different visuospatialinformation-processing tasks ranging in difficulty fromchoice reaction time (choice RT) to mental paper foldingAll seven tasks loaded heavily on a general speed factor (firstprincipal component) that accounted for nearly two thirds ofthe total standardized variance In order to examine how thedifferences between the performance of fast average andslow individuals changed across tasks individuals wereranked on the basis of their mean z scores and the groupwas divided into quartiles on the basis of these rankings

When the fastest quartilersquos mean response times (RTs)for each of the 21 experimental conditions were regressed

JM and SH were partially supported by Grants AG10197 andAG12996 to SH from the National Institute on Aging LJ was sup-ported by National Institute on Aging training grant AG00030 to MarthaStorandt We thank David Adams Lisa Emery and Shannon Robertsonfor helpful comments on an earlier version For inspiration JM wouldlike to acknowledge Charles Babbage John Crowley William Gibsonand Bruce Sterling Correspondence should be addressed to J MyersonDepartment of Psychology Campus Box 1125 Washington UniversitySt Louis MO 63130 (e-mail jmyersonartsciwustledu)

The difference engineA model of diversity in speeded cognition

JOEL MYERSON and SANDRA HALE Washington University St Louis Missouri

YINGYE ZHENG University of Washington Seattle Washington

LISA JENKINSWyeth Research Collegeville Pennsylvania

and

KEITH F WIDAMANUniversity of California Davis California

A theory of diversity in speeded cognition the difference engine is proposed in which informationprocessing is represented as a series of generic computational steps Some individuals tend to performall of these computations relatively quickly and other individuals tend to perform them all relatively slowlyreflecting the existence of a general cognitive speed factor but the time required for response selec-tion and execution is assumed to be independent of cognitive speed The difference engine correctlypredicts the positively accelerated form of the relation between diversity of performance as measuredby the standard deviation for the group and task difficulty as indexed by the mean response time (RT)for the group In addition the difference engine correctly predicts approximately linear relations be-tween the RTs of any individual and average performance for the group with the regression lines forfast individuals having slopes less than 10 (and positive intercepts) and the regression lines for slowindividuals having slopes greater than 10 (and negative intercepts) Similar predictions are made forcomparisons of slow average and fast subgroups regardless of whether those subgroups are formedon the basis of differences in ability age or health status These predictions are consistent with evidencefrom studies of healthy young and older adults as well as from studies of depressed and age-matchedcontrol groups

THE DIFFERENCE ENGINE 263

on the corresponding mean RTs for the group as a wholethe regression line accounted for more than 98 of thevariance Regressing the slowest quartilersquos mean RTs onthe group mean RTs revealed an equally precise relation-ship (Hale amp Jansen 1994) Similarly precise relationshipswere observed when the mean RTs for the fastest andslowest quartiles were regressed on average RTs as definedby the mean for the two center quartiles again both r2swere greater than 98 As may be seen in Figure 1 the con-sequence of these linear relations is that the difference be-tween the RTs of the subgroups of fast and slow proces-sors increases with the average RT

Because of the strong theoretical implications of thesefindings we followed the old adage ldquomeasure twicemdashcutoncerdquo and assessed their reliability using another largerdata set To this end we combined the data from youngadult controls in previous unpublished and published stud-ies of cognitive aging and development The RTs of 65young adults each of whom had been tested on four tasks(ie choice RT letter classification visual search and ab-stract matching) were reanalyzed using the Hale andJansen (1994) approach Again we found that a principalcomponents analysis of RTs revealed a single generalspeed factor on which all four tasks loaded heavily and

that accounted for a large proportion (75) of the totalstandardized variance (Hale amp Myerson 1993)

Following Hale and Jansen (1994) we selected fast andslow subgroups (bottom and top quartiles) from these 65subjects on the basis of their mean z scores Notably 15 ofthe 16 members of the fast group were faster than averageon all four tasks (the other member was faster than aver-age on three of the tasks) and 15 of the 16 slow groupmembers were slower than average on all four tasks (theother member was slower than average on three tasks)When the mean RTs from the two subgroups were re-gressed on average RTs extremely orderly linear relationswere observed As was the case with the Hale and Jansendata this was true regardless of whether average was de-fined as the mean for the group as a whole or as the meanfor the center quartiles In fact as may be seen in Figure 1the data points representing the Hale and Myerson (1993)data set are collinear with the points representing the Haleand Jansen data set

Further confirmation of this pattern comes from re-analysis of a study by Vernon and Jensen (1984) that usedvery different procedures and samples These authors stud-ied two large samples recruited from students attendingeither a vocational college or a university All subjects weretested on eight different speeded information-processingtasks including choice RT short-term memory scanning

Figure 1 Response times (RTs) of fast and slow subgroupsplotted as a function of the group mean RTs White and graysymbols represent the data from Hale and Jansen (1994) andHale and Myerson (1993) respectively Each point representsperformance by one subgroup in one experimental condition ofone study In order of increasing task difficulty the tasks were asfollows inverted triangles choice RT octagons line-length dis-crimination squares letter classification circles mental rota-tion upright triangles visual search octagons mental paperfolding diamonds abstract matching Each regression line rep-resents the fit to the RTs of the corresponding subgroups (ieboth fast or both slow subgroups) in both studies If a subgroupmean RT for a particular task condition did not differ from thecorresponding group mean RT the data point for that conditionwould fall along the dashed diagonal line

Figure 2 Response times (RTs) of university and vocationalcollege subgroups plotted as a function of the combined studentgroup mean RTs White and gray symbols represent the datafrom Vernon (1983) and Vernon and Jensen (1984) respectivelyEach point represents performance by one subgroup in one ex-perimental condition of one study In order of increasing task dif-ficulty the tasks were as follows circles choice RT (Jensen ap-paratus) octagons Sternberg memory scanning squares samedifferent word judgment diamonds synonymantonym judg-ment If a subgroup mean RT for a particular task condition didnot differ from the corresponding group mean RT the data pointfor that condition would fall along the dashed diagonal line

264 MYERSON HALE ZHENG JENKINS AND WIDAMAN

and samedifferent word and synonymantonym judg-ments For purposes of clarifying the analogy to the twostudies considered previously (Hale amp Jansen 1994 Haleamp Myerson 1993) the two samples may be thought of assubgroups of a larger group of young adult students Asmay be seen in Figure 2 when the RTs for each subgroupare plotted as a function of the mean RT for the wholegroup the Vernon and Jensen results are strikingly simi-lar to those of Hale and her colleagues (Figure 1)

Vernon and Jensen (1984) attributed the differences ininformation-processing speed between the vocational anduniversity students to the fact that the two samples differin academic ability or psychometric g In support of thisclaim they noted that as task difficulty increases the dif-ference between the RTs of the two samples increasesFigure 2 clearly illustrates this finding assuming that themean RT for the whole group provides an index of taskdifficulty It should be noted however that Figure 1 showsthe same pattern of RT differences between fast and slowsubgroups of students all of whom attended the samehighly competitive university Regardless of the strengthor validity of the association between speed and intelli-gence the Vernon and Jensen data taken together with thedata from Hale and her colleagues (Hale amp Jansen 1994Hale amp Myerson 1993) suggest that the relationship wehave observed between the RTs of fast and slow sub-groups is not peculiar to one particular battery of RT tasksnor does it depend on a specific method for dividing agroup into slow and fast subgroups (eg top and bottomquartiles or type of college attended) or a particular methodfor estimating average performance (eg overall mean ormean for center two quartiles)

We hypothesize that the observed orderly increase inthe difference between slow and fast RTs typifies the ef-fect of task difficulty (indexed by the average RT for thegroup) on the absolute size of individual differences inperformance of speeded tasks In subsequent sections wewill propose a model to explain this phenomenon and as-sess its ability to account for both the differences betweenfast and slow groups as well as differences in individualperformance First however it is important to show thateven at the subgroup level the Hale and Jansen (1994) andHale and Myerson (1993) results are not what would beexpected on the basis of the method of constructing sub-groups alone nor do they result simply from a positive re-lationship between group mean RT and SD To demon-strate this and to examine the role of correlations betweenRTs on different tasks we conducted three series of com-puter simulations

With respect to the method by which we have constructedsubgroups it is obvious of course that selecting fast andslow individuals on the basis of their mean z scores wouldtend to produce mean RTs for each subgroup that wouldbe smaller and larger respectively than average RTs How-ever this method would not necessarily lead to a system-atic change in the difference between fast and slow RTs asa function of task difficulty of the kind observed in actualdata (see Figures 1 and 2) To demonstrate that this is true

we conducted 10 simulations in which RTs were randomlyselected for 65 simulated subjects in 10 different task con-ditions so as to produce the same condition means as thosein the Hale and Myerson (1993) study For all task condi-tions the SD was set at 148 msec (the median of the ac-tual SDs for the 10 conditions) and RTs in different taskconditions were uncorrelated Simulated subjects werethen sorted into quartiles on the basis of their mean z scoresand the means for the top and bottom quartiles were re-gressed on overall group means Across the 10 simula-tions in this series the mean of the slopes for the simu-lated fast quartiles was 100 and the mean of the slopesfor the simulated slow quartiles was also 100 with stan-dard errors of 001 in both cases Results for a typical sim-ulation (the one with fast and slow slopes closest in valueto the means) are shown in the upper left panel of Figure 3As may be seen the size of the difference between the RTsof the fast and slow subgroups was very small and re-mained approximately constant across tasks

Notably the method of selecting fast and slow quartilesdid not produce subgroups in which individual simulatedsubjects were consistently fast or slow In repeated simu-lations most simulated subjects from the top and bottomquartiles were faster or slower than the mean in only 6 or7 task conditions (mean for the representative simulationdepicted was 650) just slightly more than the 5 out of 10expected for those of average speed In contrast the actualfast and slow subjects in the Hale and Myerson (1993)study were very consistent Specifically all but 1 subjectfrom the bottom or top quartile was faster or slower thanaverage in at least 9 out of 10 conditions of the four tasks

Further the systematic increase in the difference be-tween the RTs of fast and slow subgroups with task diffi-culty observed in the Hale and Jansen (1994) and Hale andMyerson (1993) data is not simply a consequence of a pos-itive relationship between mean and SD (Hale MyersonSmith amp Poon 1988) To demonstrate this we performeda second set of simulations similar to those just describedexcept that both the mean RTs and the SDs for the 10 taskconditions were set equal to those in the corresponding 10conditions from the Hale and Myerson (1993) study Be-cause of the positive relationship between mean and SDthis set of simulations will be referred to as the dependentSD simulations to distinguish them from the previous setof independent SD simulations As in the independent SDsimulations the correlation between RTs on different taskswas set to zero and the 65 simulated subjects were againsorted into quartiles on the basis of their mean z scores

With the observed relationship between mean RT andSD incorporated into the simulation the difference be-tween the mean RTs for the fast and slow quartiles did in-crease slightly with task difficulty The rate of increasehowever was substantially less than that observed in thereal data To compare the actual and simulated results weconducted 10 complete simulations in order to estimate theslopes for the regression of RTs of the fast and slow quar-tiles on the simulated group mean RTs The mean of theslopes for the simulated slow quartiles was 114 (with a


standard error of 002) which is considerably less than theslope of 145 observed for the slow subgroup in the Haleand Myerson (1993) experiment The mean of the slopesfor the simulated fast quartiles was 085 (standard error of002) which is greater than the slope of 065 for the actualfast subgroup Thus for every increase of 100 msec in taskdifficulty (as measured by group mean RT) the differencebetween the RTs of the actual fast and slow subgroups in-creased by 80 msec This is more than 25 times the rate ofincrease in the difference between the simulated subgroups(29 msec per 100 msec increase in difficulty) The resultsfor a representative dependent SD simulation (the one withfast and slow slopes closest in value to the means for the10 simulations) are shown in the lower left panel of Fig-ure 3 and may be compared with the actual Hale and My-erson results shown in the lower right panel

As in our set of independent SD simulations there wasrelatively little consistency in performance at the individ-ual level in the set of dependent SD simulations despitethe method of selecting fast and slow quartiles Most sim-ulated fast and slow subjects were faster than average orslower than average respectively in only 6 or 7 task con-

ditions just slightly more than the 5 out of 10 expectedfor those of average speed In the typical dependent SDsimulation depicted the mean number of conditions withRTs appropriately faster or slower than average was 675which may be compared with a mean of 966 out of 10 forthe actual fast and slow subjects in the Hale and Myerson(1993) study

We also conducted a third set of 10 simulations in whichRTs in different conditions were correlated in order to as-sess how individual consistency in performance affectsthe pattern of differences between the RTs of fast and slowsubgroups For these correlated RT simulations correla-tions between RTs for different task conditions were pro-duced by adding condition-specific random variables toan individual-specific random variable (for which indi-vidual values were held constant across task conditions)Condition mean RTs were set equal to those observed in theHale and Myerson (1993) study and SDs were set equal tothe median of the observed SDs As was the case for theactual subjects most simulated fast or slow subjects wereconsistently fast or slow (ie 9 or 10 conditions faster orslower than average for the most representative simula-

Figure 3 Simulated and actual fast and slow subgroupsrsquo mean response times (RTs) plot-ted as a function of group mean RTs The two upper panels and the lower left panel presentdata generated by three different computer simulations (see text for description) The lowerright panel presents the data from Hale and Myerson (1993) for comparison purposes In allpanels the white and gray circles represent the slow and fast subgroups respectively


tion M 5 939 out of 10) Unlike the results of the actualstudy however the mean regression slopes for both thesimulated fast and the simulated slow subgroups were100 (standard errors less than 01) The fast and slow sub-group regression results for the most representative cor-related RT simulation (ie the one whose regression slopeswere closest to the mean slopes) are shown in the upperright panel of Figure 3

Comparison of the Hale and Myerson (1993) results(lower right panel of Figure 3) with the results from ourthree sets of simulations (ie the independent SD depen-dent SD and correlated RT simulations) clearly revealsthat none of the alternatives in questionmdashthe method ofcreating fast and slow subgroups based on mean z scoresthe empirical relationship between group means and SDsor the observed correlations between RTsmdashare sufficientin and of themselves to explain the observed pattern of dif-ferences between actual fast and slow subjects The sim-ulation results do suggest however possible roles for bothcorrelations between RTs in different task conditions anda positive relationship between condition mean RT andSD It is possible moreover that consistently fast or slowindividual performance may enhance patterns of differ-ences between fast and slow subgroups (compare the in-dependent SD and correlated RT simulations) and that therelationship between group mean and SD may provide thebasic pattern to be enhanced (compare the independentSD and dependent SD simulations) However the exis-tence of intertask correlations on the one hand and a pos-itive relationship between mean RT and SD on the othermay not necessarily reflect separate empirical constraintsRather as we will propose in subsequent sections a gen-eral cognitive speed factor may play a fundamental role indetermining the relationship between mean and SD aswell as the difference between fast and slow subgroups orindividuals This view is fundamental to our theoretical ac-count of the mechanism that determines the absolute sizeof individual differences on different tasks

Before proceeding to the theoretical development how-ever we wish to point out that the present concern extendsbeyond differences in cognitive speed between healthy in-dividuals of the same age The present effort is also moti-vated by the fact that a growing number of studies involv-ing a variety of special populations have reported findingsrelated to those just discussed suggesting that the mech-anism underlying variation in the absolute size of individ-ual differences in RTs on various cognitive tasks may havea great deal in common with the mechanism that governsthe size of group differences in RTs on such tasks

More specifically when group differences are assessedacross multiple tasks the relations between the RTs of onegroup and those of another (eg control) group are oftenwell described by linear functions similar to those we haveobserved between the RTs of same-age individuals whodiffer in ability For example such relations have been re-ported when children and older adults are compared withyoung adults (for a review see Cerella amp Hale 1994) Re-cently similar results have been observed in studies com-

paring the performance of groups with a variety of condi-tions including mental retardation (Kail 1992) clinicaldepression (White Myerson amp Hale 1997) closed-headinjury (Ferraro 1996) multiple sclerosis (Kail 1997) andAlzheimerrsquos disease (Myerson Lawrence Hale Jenkinsamp Chen 1998) with the performance of appropriate con-trols In all of the cited studies the difference between theRTs of affected groups and appropriate controls increasedsystematically with task difficulty as indexed by the RT ofthe control group

The orderly functional relations seen in these studies ofthe speeded cognitive performance of various groups ap-pear to be strikingly similar to those seen in our studies ofthe diversity of performance within groups (Hale amp Jansen

Figure 4 Nonlexical and lexical response times (RTs) of youngand older adult groups plotted as a function of the correspond-ing RTs of a middle-age group Data are taken from LawrenceMyerson and Hale (1998) In both the upper and lower panelswhite circles represent data from the older adult group (80- to 89-year-olds) and gray circles represent data from the young adultgroup (18- to 21-year-olds) The middle-age group consisted of50- to 59-year-olds


1994 Hale amp Myerson 1993 Zheng et al 2000) In bothcases the RTs of one set of individuals tested on a num-ber of quite different tasks can be predicted directly fromthe RTs of another set of individuals without taking thenature of the tasks into account (other than whether the in-formation that was processed was verbal or spatial) Theparallel may be seen most clearly in Figure 4 in which thegroup mean RTs of young adults and adults in their 80sare each plotted as a function of a group of intermediateage adults in their 50s (data are taken from LawrenceMyerson amp Hale 1998) In both the verbal and spatial do-mains group differences increased systematically with in-creases in task difficulty

Such findings suggest that as with individual differ-ences and differences between fast and slow subgroupsthere may be a general speed factor that underlies the dif-ferences observed between the RTs of various groups atleast within each processing domain (ie verbal or spa-tial) (eg Cerella 1985 1990 Hale Myerson amp Wagstaff1987 Kail 1991) Although orderly relations between themean RTs of different groups can be misleading undercertain circumstances (Fisher amp Glaser 1996) such rela-tions nevertheless raise the possibility that basic modelsdeveloped to explain individual differences in ability mayalso apply to other individual difference variables (eg ageand health status) and perhaps to the interaction of suchvariables as well

In this paper we develop a model of the interaction be-tween individual differences in the ability to process in-formation rapidly on the one hand and the amount of pro-cessing required by different tasks on the other hand Wethen apply this model to the problem of how individualability and task difficulty interact with other individualdifference variables such as age and health status There isa rich database with respect to speeded cognition in youngand older adults that exemplifies the issues involved insuch interactions and such data provide a test case withwhich to evaluate our model We would emphasize how-ever that our ultimate claim is more general and appliesnot just to individual differences and aging but also to de-velopmental changes in processing speed as well as to dif-ferences between healthy individuals and those affectedby various health conditions

In developing our model we will employ simplifyingassumptions that ignore distinctions that may be the focusof other theoretical and experimental efforts We will do thisfor two reasons first because of differences in purposebetween the present effort and other research and secondbecause of differences in scale These two reasons are ob-viously not mutually exclusive After all different pur-poses may require measures and analyses that provide dif-ferent degrees of resolution For example trying to draw(or use) a full-scale map of the world (ie one in which1 foot represents 1 foot) obviously would create problemsThere is usually a tradeoff between resolution and scopeand people tend to select the type of map with the degreeof resolution appropriate for the purpose at hand usingboth more detailed maps of smaller areas (eg cities) andless detailed maps of larger areas (eg countries)

The focus of the present effort is on predicting perfor-mance over a range of task difficulty that spans an orderof magnitude (ie 500ndash5000 msec) at least for somegroups (eg older adults) and differences of an order ofmagnitude or smaller (ie on the order of 50 msec or less)may be neglected in the process Just as makers and usersof maps of whole countries would not claim that maps ofindividual cities are wrong so our use of relatively low de-grees of resolution in order to achieve greater scopeshould not be taken to imply any disagreement with moretightly focused theories and models that offer higher de-grees of resolution

Differences between the purposes of the present modeland those of other efforts also lead us to ignore certain gen-erally accepted distinctions For example the present modeltreats all information-processing steps as if they were equiv-alent even when we know that they involve different cog-nitive operations (eg mental rotation vs visual search)Such distinctions though clearly of fundamental impor-tance for the purpose of understanding how cognitive tasksare accomplished may be less important for purposes ofunderstanding how individuals differ in the efficiencywith which they accomplish these tasks

Some of the simplifying assumptions made to facilitatethe mathematical development of the model will be re-laxed in subsequent computer simulations For exampleinitially we will assume the existence of a ldquogenericrdquo com-putational step and that differences in the RTs for differ-ent tasks or conditions reflect differences in the number of computational steps involved In simulations these as-sumptions will be relaxed so that the different steps involvedin a given task may have different durations and so thatthe correlations between steps from the same task may behigher than those between steps from different tasks re-flecting possible task-specific abilities Through the useof such simulations and more importantly by comparingthe modelrsquos predictions to experimental data we will at-tempt to show that despite its deliberate simplification ofthe problem our model is well suited to the task at handexplaining the linkage between a general speed factor andthe orderly increase in diversity of speeded performancewith task difficulty

THEORETICAL DEVELOPMENTPreliminary Considerations

As Cerella (1990) pointed out orderly relations such asthose shown in Figures 1 and 2 imply correspondence be-tween the processing of the individuals involved That issuch relations suggest that all (or most) individuals per-forming the tasks in question are processing the same in-formation in the same way Thus the major determinant ofthe size of individual differences appears to be quantitativerather than qualitative and involves something that all thetasks have in common albeit something that some tasks (iethose associated with longer RTs) have more of than others

More specifically an orderly relation between perfor-mances implies that these performances are functions ofa third variable (Cerella 1994 Dunn amp Kirsner 1988 Hale


et al 1987) Further understanding of this implication fol-lows from an extension of ideas originally suggested byDunn and Kirsner (1988) and extended to age differencesin RTs by Cerella (1994) Dunn and Kirsner applied thechain rule familiar to calculus students to all cases of or-derly relations between two types of performance Althoughthe chain rule is relevant in all such cases it has special sig-nificance for the specific case of approximately linear re-lations observed here In this case the linear relation im-plies that the performances of fast and slow individualsare isomorphic functions of a third variable such that theeffect of changes in that third variable on the performanceof different individuals differs only proportionally In ad-dition the fact that performance differences appear to berelatively independent of the type of processing involvedbut increase with the average time required for task per-formance suggests that this third variable may be charac-terized as a measure of the amount of processing requiredby a task This insight taken together with evidence for ageneral speed factor (eg Hale amp Jansen 1994 Vernon ampJensen 1984 Vernon Nador amp Kantor 1985 Zhenget al 2000) provides the basis for our model of group andindividual differences

Diversity in performance is typically measured as theSD for the group (rather than as the difference betweenfast and slow subgroups) and statistical theory typicallytreats variability in this form (and as the variance or SDsquared) Therefore we have recast the data represented inFigure 1 in these terms As may be seen in Figure 5 thediversity of individual performances measured as thegroup (or between-subjects) SD increases with the amountof processing required in a task condition

The fact that variability in RT tends to increase with themean may seem unremarkable It should be noted how-

ever that in Figure 5 variability does not merely increasewith the mean but rather increases in a very orderly andhighly specific manner The approximately linear relationbetween mean and SD accounts for more than 95 of thevariance Importantly both the form and the values of theparameters appear to be robust characteristics of RT dataA single regression line describes the data from two dif-ferent studies (Hale amp Jansen 1994 Hale amp Myerson1993) In addition linear relations between SD and meanRT with very similar parameters (slopes of 03 to 04 andnegative intercepts) were reported in a previous meta-analysis comparing the diversity of speed of performancein young and older adults (Hale et al 1988) Even in themeta-analysis where the data came from many differentsamples the linear relation between SD and mean RT ac-counted for more than 85 of the variance Thus the ob-served approximately linear relationship is not peculiar todata from one study one laboratory or even one age group

Moreover the form of the relation between the meanand SD (or variance) can provide important informationabout the nature of the underlying process How this ap-plies to the present problem may be seen by considering afundamental fact from elementary statistical theorySpecifically the variance (VAR) of the sum of n measureswith variances s1

2 s 22 sn

2 is given by the equation

VAR 5 s12 1 s 2

2 1 1 sn2 1 2r12s1s2

1 2r13 s1s3 1 1 2rn n21snsn21 (1)

In the simple case where the means for the different mea-sures are all equal all of the variances are equal and all ofthe correlations between the measures are equal then

VAR 5 ns2 1 rn(n 2 1)s 2 (2)

(We do not suggest that these simple assumptions are cor-rect Rather we wish to see what insights can be gainedfrom considering mathematically tractable cases beforeconsidering cases of sufficient complexity to require com-puter simulations)

Inspection of Equation 2 reveals that if the measures areuncorrelated (ie r 5 0) then VAR 5 ns 2 and SD 5 s n12To apply this to RTs we assume that each measure repre-sents the time required to execute a single computation orone processing step Thus the mean RT for a given taskcondition will be proportional to the number of processingsteps that is RT 5 an where a is the average duration fora single step Substituting for n in the preceding equationyields

SD 5 s (RTa)12 (3)

where s is the SD of step durations across individualsEquation 3 implies that if the durations of the steps are un-correlated then the relation between the mean RT for atask condition and the SD of the RTs for that conditionwill not be linear Rather it will be negatively acceleratedbecause SD is proportional to the square root of the meanRT In contrast if the steps are perfectly correlated (ier 5 10 so that the same individuals who are the fastest at

750

500

250

00 500 1000 1500 25002000

Group Mean RT (msec)

Bet

wee

n-S

ubje

cts

SD

(m

sec)

Hale amp Jansen (1994)Hale amp Myerson (1993)

Figure 5 Between-subjects standard deviation (SD) plotted asa function of group mean response time (RT) As in Figure 1 thedata are taken from Hale and Jansen (1994) and Hale and My-erson (1993) Each circle represents the SD and mean of the RTsfor one experimental condition of one study and the dashed re-gression line represents the fit to all of the data points The solidcurve represents the prediction of the difference engine model(Equation 7)


one step will be the fastest at all other steps) then VAR 5n2s 2 Substituting RTa for n and taking the square rootsof both sides of the equation yields

SD 5 sRTa (4)

Equation 4 reveals that if the durations of the steps areperfectly correlated then the relation between mean RTand SD will be linear In fact the SD will be directly pro-portional to the mean

How correlated do they have to be to yield an approxi-mately linear relation between RT and SD To answer thisquestion we may return to Equation2 and taking the squareroots of both sides and rearranging observe that for any r

SD 5 s [n + n(n 2 1)r]12 (5)

This equation may be used to assess the effect of the cor-relation between step durations on the relation betweenthe number of processing steps and the SD and thus on therelation between the number of steps and the SD of RTsfor a specific task condition

Figure 6 illustrates what happens to the SD of the sumof processing step durations as the number of steps andthe magnitude of the correlation between them are variedAlthough the figure depicts the simple case where thesteps are all equally correlated and also have equal meansand variances the principle represented here undoubtedlyholds as long as these assumptions are at least approxi-mately true That is as may be seen in Figure 6 (which forsimplicity also assumes unit variance) relatively low cor-relations between processing steps may serve to linearizethe relationship between the number of steps and the SD

As the correlation increases so does the slope whereasthe intercept decreases Although at higher correlationsthe SD is approximately proportional to the number ofsteps the general expectation based on Equation 5 is thatintercepts will tend to be positive Thus as a descriptivemodel of RT data Equation 5 suffers from an importantlimitation Although it may explain why the relation be-

tween SD and RT is linear it fails to predict the negativeintercept clearly apparent in the present data (Figure 5) aswell as in previous meta-analyses (eg Hale et al 1988)As will be seen in the following section this limitation iseasily corrected yielding a model that is more in line withintuition as well as providing a better description of the data

THEORETICAL DEVELOPMENTThe Difference Engine

A Two-Compartment Model of Within-Groups Variability

To understand how the observed negative intercept maybe accommodated within the present approach recall thatEquation 5 assumes that any processing step is inter-changeable with any other step of the same task (at least forthe purpose of predicting individual differences) From theusual cognitive perspective steps are obviously not inter-changeable because they involve different computations orprocessing operations In the present usage however stepsare merely arbitrary measures of the amount of processingSuch generic steps serve the mathematically useful func-tion of dividing what may for other purposes be thoughtof as a continuous flow (McClelland 1979) into discreteunits in the same way that a length may be divided into feetor reflecting the arbitrary nature of the division into meters

The negative intercept demonstrates a limit to the useful-ness of the mathematical fiction of generic completely in-terchangeable steps and accordingly the strongest versionof this fiction will be abandoned here in favor of a dis-tinction similar to the familiar one between sensorimotor(or peripheral) and cognitive (or central) processing (egCerella 1985) We assume a two-compartment model inwhich the duration of the response portion of a task is rel-atively uncorrelated with the duration of the central por-tion of that task

Evidence consistent with this assumption is presentedin Table 1 which reports the mean task RTs and intercor-relations from Hale and Myerson (1993) Note that thehighest correlations are between the two easiest tasks(choice RT and letter classification) and between the twohardest tasks (visual search and abstract matching) Thispattern presumably reflects the fact that sensorimotorspeed plays a large role in easier tasks but makes a smallercontribution in more difficult tasks whereas cognitivespeed plays a large role in more difficult tasks but a smallerrole in easier tasks Consistent with previous studies prin-cipal components analysis revealed that all tasks loadedheavily on the first principal component which explained75 of the variance However the second bipolar com-ponent explained an additional 12 of the variance Whenvarimax rotation was used to eliminate negative loadingsand resolve a simple two-factor structure the more diffi-cult tasks (ie visual search and abstract matching)loaded highly on one factor (which may be interpreted ascognitive speed) whereas the easier tasks (ie choice RTand letter classification) loaded highly on the other factor(which may be interpreted as sensorimotor speed)

Figure 6 Standard deviation (SD) of the sum plotted as a func-tion of the number of steps being summed Each curve representsthe relation between SD and number of steps given a particularcorrelation between the different measures Calculations arebased on Equation 5 and the correlation used to calculate eachcurve is given directly to the right of the curve


Similar patterns are seen in the correlations reported byZheng et al (2000) and also in the Hale and Jansen (1994)data In the latter study for example the two easiest tasks(ie line-length discrimination and choice RT) had a cor-relation of 614 and the two most difficult tasks (ie ab-stract matching and mental paper folding) had a correla-tion of 646 whereas the correlation between the easiestand hardest tasks was 397 The same pattern was alsoseen in the correlations reported by Zheng et al In both ofthese studies the second principal component was bipo-lar with loadings that covaried with task difficulty andvarimax rotation yielded one factor that may be inter-preted as cognitive speed for which the loadings werepositively correlated with RT and another factor that maybe interpreted as sensorimotor speed for which the load-ings were negatively correlated with RT Moreover for allthree studies the correlations of both the cognitive andsensorimotor speed factors with task mean RT were 75 orstronger With respect to the Hale and Jansen study forexample the loading of the easiest task (line-length dis-crimination) on the sensorimotor speed factor was 906whereas its loading on the cognitive speed factor was 109In contrast the loadings of the most difficult task (mentalpaper folding) on the sensorimotor and cognitive speedfactors were 265 and 844 respectively

Although the assumption of complete independence be-tween the cognitive and response components of RT is ex-treme it greatly simplifies the mathematical develop-ment We will continue however to assume that withinthe cognitive component the durations of all processingsteps are correlated and equivalently so Thus the modeldeveloped in the previous section continues to apply to allof the cognitive processing that goes into RTs The pres-ent model which we have termed the difference enginemay be summarized as follows

1 Cognitive task architecture It is assumed that tasksand conditions that differ in difficulty also differ in theamount of cognitive processing (ie in the number of el-ementary computations or processing steps) that is re-quired in order for an individual to make an accurate de-cision

2 Cognitive step durations and correlations For any in-dividual performing any task the durations of the stepsthat make up that task are correlated and these correla-

tions are assumed on average to be the same for all indi-viduals and all tasks This assumption provides the basisfor the mathematical modeling that follows Our computersimulations incorporate an additional assumption Morespecifically the correlations between steps of the sametask are assumed to be higher than are those between stepsof different tasks Additionally the within-tasks step du-ration correlations are assumed on average to be equiva-lent for all tasks as are the lower between-tasks step dura-tion correlations

3 Response step durations and correlations The timeit takes to report a decision by selecting and executing aminimal motor response (eg pushing a button) is as-sumed to vary across individuals and to be independent ofthe time they require to execute elementary cognitive pro-cessing steps or to reach a correct decision (ie their totalcognitive processing times)

Before proceeding we should note that we do not assertthat all experimental tasks have a serial architecture likethat assumed here although many may One could obvi-ously find or devise tasks that do not satisfy this constraintand our model would not apply to such tasks Similarly wedo not assert that the time it takes an individual to report adecision is the same on all cognitive tasks Rather we wouldnote that if one wants to study or manipulate processingtime it makes sense to keep the response requirement min-imal as well as to hold it constant as in the present model

The uncorrelated response component may be incorpo-rated into the model as follows According to Equation 1if there are only two uncorrelated items VAR 5 s1

2 1s2

2 In the present application s12 represents the variance

of the duration of the response portion of a speeded taskand s2

2 represents the variance of the duration of the cog-nitive portion As just noted the cognitive portion consistsof a number of correlated processing steps and thereforeEquation 2 may be substituted for s2

2 yielding

VAR 5 sr2 1 nsc

2 1 n(n 2 1)rs c2 (6)

where sr2 is the variance for the response component and

s c2 is the variance for a single cognitive processing stepIn the present two-compartment model the amount of

cognitive processing time (ie the mean RT minus the du-ration of the response component tr) is proportional to thenumber of processing steps (ie RT 2 tr 5 an where a is

Table 1Mean Response Times (RTs) and Standard Deviations (SDs) in Seconds Intertask

Correlations and Loadings on the First and Second Principal Components and First and Second Rotated Factors Extracted From the Hale and Myerson (1993) Data

Intercorrelations Unrotated Rotated

Tasks M RT SD CRT LTR SRC PC1 PC2 RF1 RF2

CRT 0405 0061 833 2470 292 911LTR 0644 0103 701 880 2197 512 743SRC 1079 0233 583 657 863 321 879 321MAT 1711 0511 608 679 740 878 287 840 385

NotemdashCRT choice reaction time LTR letter classification SRC visual search MAT abstract match-ing tasks PC1 and PC2 loadings on the first and second principal components RF1 and RF2 load-ings on the first and second rotated factors


the mean processing step duration) Substituting (RT 2 tr) afor n and rearranging yields

VAR 5 s r2 1 [(1 2 r)sc

2 ](RT 2 tr) a

1 rsc2[(RT 2 tr) a]2 (7)

which is a second-order polynomial (ie VAR 5 b0 1b1RT 1 b2RT 2) Therefore the SD of the RTs is simplythe square root of a second-order polynomial shown as asolid curved line in Figure 5

Although the fit here is only slightly better than that pro-vided by the regression line the theoretical equation (iethe square root of Equation 7) captures what turns out tobe a reliable initial positive acceleration According to thetwo-compartment model this acceleration reflects the factthat as RTs get longer the variance associated with thecorrelated steps of their cognitive component representsan increasingly larger proportion of the total variance Un-fortunately although curve fitting provides estimates ofthe polynomial coefficients these estimates are difficultto interpret because these coefficients represent compos-ites of the parameters of Equation 7 Moreover althoughthe parameters of Equation7 have specific interpretationsthese parameters are highly interdependent and thereforedifficult to estimate Thus although the very good fit andintuitively reasonable form of our theoretical equationprovide support for the logic underlying this equation analternative approach will be needed for some purposes

One such approach capitalizes on the fact that as RTgrows larger the rightmost expression (ie the one con-taining the quadratic term) in Equation 7 comes to domi-nate and the variance is increasingly well approximated by

VAR 5 rs c2[(RT 2 tr) a ]2 (8)

Taking the square root of both sides yields

SD 5 (r 2 sc a)(RT 2 tr) (9)

This equation provides a basis for understanding the pa-rameters of the approximately linear relation between SDand mean RT The slope m 5 r 2 sc a reflects both thecorrelations among processing step durations within a taskand the coefficient of variation for cognitive processing(ie the between-subjects SD of processing step durationsdivided by the mean duration) The y-intercept is simply2mtr and setting SD equal to zero and solving for RT re-veals that the x-intercept is equal to the duration of the re-sponse component tr As may be seen in Figure 5 the ob-served x-intercept of approximately 300 msec is in keepingwith values for simple and choice RTs to highly discrim-inable stimuli reported in the literature (for a review seeTeichner amp Krebs 1974) and is therefore consistent withthe two-compartment assumption of our difference enginemodel

Extending the Model to Performanceof Individuals and Subgroups

To this point the focus has been on the distribution ofspeeded performance across individuals and how this dis-tribution changes with task difficulty (ie with variations

in the number of processing steps) In the present sectionwe focus on the performance of specific individuals or ofsubgroups of individuals of similar ability and follow theirperformances from task to task In subsequent sectionswe will show that the difference engine may be extendedfurther enabling one not only to follow individuals fromtask to task but also in principle to follow their perfor-mance as they change with age and health status through-out the adult life span

We begin by considering the difference enginersquos pre-dictions regarding the interaction of individual differences

Figure 7 Graphical summary of the difference enginersquos pre-dictions regarding individual differences within a group The toppanel illustrates the relation between standard deviation (SD)and group mean response time (RT) the middle panel illustratesthe consequences of this relation for the RTs of subgroups 1 SDfaster and 1 SD slower than average and the bottom panel de-picts the consequences for the relations between the RTs of thesesubgroups and the group mean RT


with task difficulty The logic behind these predictions isstraightforward and is summarized graphically in Figure 7All of the hypothetical data depicted in the figure are cal-culated assuming fast and slow subgroups whose process-ing times are respectively 1 SD shorter and 1 SD longerthan average and also assuming that SD 5 03 RT 2 01(consistent with both Equation 9 and the empirical datapresented previously in Figure 5)

As shown in the upper panel the diversity of individualperformance (indexed by the between-subjects SD) in-

creases linearly with task difficulty (indexed by the meanRT) Because the speed with which individuals executedifferent processing steps is correlated both within andbetween tasks the same individuals tend to have long (orshort) RTs on all cognitive tasks The consequences of thistendency for the RTs of fast and slow subgroups may beseen in the middle panel of Figure 7 Specifically the pat-tern of increasing diversity in the upper panel results in asimilar linear increase in the difference between slow andaverage RTs The difference between fast and average RTs

Figure 8 Response times (RTs) for representative individuals from Hale andMyerson (1993) plotted as a function of group mean RT Each data point rep-resents performance in one experimental condition In order of increasing taskdifficulty the tasks were as follows inverted triangles choice reaction timesquares letter classification upright triangles visual search diamonds ab-stract matching Data from the slowest and the fastest individuals are shown(upper left panel and lower right panel) as well as data from individuals at the10th 33rd 67th and 90th percentiles (middle left lower left upper right andmiddle right panels respectively) If an individualrsquos RT for a particular taskcondition did not differ from the corresponding group mean RT the data pointfor that condition would fall along the dashed diagonal line


also increases linearly with task difficulty although in thiscase the difference is negative

Finally converting the differences shown in the middlepanel into mean RTs for each subgroup has the effect ofrotating the graph so as to produce the graph depicted inthe lower panel Thus as may be seen combining a gen-eral cognitive speed factor with an approximately linearrelation between SD and mean RT leads logically to thecorrect prediction of linear relations between the RTs ofsubgroups of fast and slow individuals and the averageRTs for the group as a whole (Figure 1)

Although plots of difference scores (eg the middle panelof Figure 7) provide an intuitively accessible way to visu-alize individual differences for purposes of parameter es-timation it is preferable to avoid difference scores andconduct regressions using RTs in the manner diagrammedin the bottom panel Interpretation of the parameters ofsuch regressions is straightforward Consider an individ-ual (i) who is consistently slower than the mean by 10 SD(ie RTi 5 RTavg 1 SD) and recall that SD is approxi-mately equal to mRTavg 2 mtr (Equation 9 the linear ap-proximation of the relation between SD and mean RT) Itfollows directly that the RTs of such an individual will beapproximately equal to (1 1 m) RTavg 2 mtr More gen-erally for any individual

RTi 5 (1 1 zi m)RTavg 2 zimtr (10)

where zi is a measure of individual processing speed in SDunits and m 5 r 2 sc a (see Equation 9) This equationprovides a guide to interpretation of the regression para-meters Because fast individuals have negative z scores theywill have slopes less than 10 and positive y-interceptsSlow individuals who have positive z scores will haveslopes greater than 10 and negative y-intercepts More-over individual regression lines will tend to cross the di-agonal (ie RTi 5 RTavg) at a point where RTs are mini-mal (ie when the task is simple and the RT consistslargely of sensorimotor time) although prediction at theindividual level is complicated by both the nonlinearity ofthe relation between SD and RT and the fact that sensori-motor time varies between individuals

The results of analyses of the individual data from the65 subjects in the Hale and Myerson (1993) study are con-sistent with predictions based on Equation 10 Strong lin-ear relations were observed between individualsrsquo condi-tion mean RTs and group condition means as evidencedby very high individual r2s (median r2 5 950) Individ-ual examples (fastest and slowest as well as those at the10th 33rd 67th and 90th percentiles) are presented inFigure 8 As predicted regressions for slow individualstended to have negative intercepts whereas those for fastindividuals tended to have positive intercepts (for furtherexamples of individual data see Hale amp Jansen 1994 andZheng et al 2000) In fact 20 out of the 21 subjects withregression slopes greater than 105 had negative inter-cepts and 34 of the 35 subjects with slopes less than 095had positive intercepts Moreover individual regressionlines tended to cross the diagonal in the region where ex-pected (median crossing point at RT 5 439 msec)

The use of Equations 9 and 10 for predicting the slopesof the linear relation between an individual or subgrouprsquosRTs and the group mean RT may be illustrated using thedata from the fast and slow subgroups in the Hale and My-erson (1993) study For the Hale and Myerson data linearregression of the group SD on RT yielded an estimate ofm 5 0377 (Equation 9) According to Equation 10 theformula for the slope (s) for an individual or subgroup isgiven by s 5 1 1 zm For the fast subgroup the grand meanz (ie the average of the mean z scores for the 16 mem-bers of the subgroup) was 111 Thus substituting for min preceding formula the predicted slope for this sub-group (ie 1 plus 111 times 0377) equals 142 veryclose to the observed slope of 145 For the fast subgroupthe grand mean z was 2101 and thus the predicted slope(1 minus 101 times 0377) equals 062 again very closeto the observed slope of 065

The same method for predicting slopes was also appliedto the data from each of the 65 subjects in Hale and My-erson (1993) With no free parameters the formula si 5 11 0377zi where si and zi represent the slope and mean zfor the ith individual accounted for nearly three fourths ofthe variance in the observed slopes (r2 5 726) Moreoverwhen the observed slopes were regressed on the predictedslopes the slope of the regression did not differ significantlyfrom 10 and the intercept did not differ significantly from00 suggesting that our prediction formula based onEquation 10 was an unbiased predictor of individual re-gression lines

What is important about the present effort is that it of-fers a relatively simple theoretical mechanism to predictand explain a whole set of empirical relationships The ap-proximately linear relation between SD and group meanRT is explained as the consequence of correlations be-tween processing step durationsmdashfaster individuals tak-ing generally less time on all steps of a cognitive task andslower individuals taking generally more time Taken to-gether with the strong general speed factor revealed byprincipal component analysis the relation between SDand mean RT implies that the difference between fast andslow individuals will increase with the task mean RTsomething not predicted by a general speed factor aloneThis increase in individual differences is reflected in thespecific form of the linear relation between individualcondition mean RTs and group condition mean RTs Thusthe difference engine appears to provide an integrated ac-count of various aspects of RT data that are usually treatedas separate phenomena

Computer Simulations of the Difference EngineThe mathematical development presented in the previ-

ous two sections necessitated many simplifying assump-tions In order to assess the consequences of relaxingsome of those assumptions and to set the stage for model-ing performance in different groups we next proceededto develop computer simulations that embodied the prin-ciples of the difference engine As before RTs were as-sumed to consist of two components one sensorimotor andthe other cognitive The cognitive component was assumed


to represent the sum of the duration of individual pro-cessing steps and these step durations were assumed tobe correlated both within and between tasks The sensori-motor component was assumed to be 300 msec on aver-age with a SD of 50 msec across individuals Processingstep durations were assumed to be 50 msec with a SD of25 msec across individuals and tasks and uncorrelatedwith individualsrsquo sensorimotor component

Across tasks the correlation between an individualrsquosprocessing steps was set at 35 on average whereas withintasks the correlation was set at 50 (These assumptionsonly provided target values both because of random vari-ation per se and because such variation occasionally albeitrarely produced negative step durations that were auto-matically replaced with zeros) Correlations were pro-duced by calculating each cognitive step duration as thesum of three random variables More specifically eachsimulated subject was randomly assigned a general cog-nitive speed value as well as an independently chosentask-specific speed for each task Then each step durationwas assumed to be the sum of these first two random vari-ables plus a third independent value This third random

variable may be thought of as a combination of intrinsicvariation and measurement error RTs were calculated asthe sum of the step durations plus a fourth random vari-able that represented subjectsrsquo sensorimotor speed Cor-relations between task steps reflected the shared generaland task-specific variables whereas correlations betweenRTs for different tasks reflected the general cognitive speedvariable and the sensorimotor speed variable

Consider a typical simulation with parameters chosento approximate the tasks used by Hale and Jansen (1994)and Hale and Myerson (1993) For each of 100 simulatedsubjects we generated RTs for seven tasks consisting of 48 12 16 20 24 and 28 processing steps The mean RTsfor these tasks ranged from approximately 500 msec forthe 4-step task (300 msec of sensorimotor time plus200 msec of information processing) to 1700 msec for the28-step task Characteristics of the simulated steps (iemean durations SDs and intercorrelations) were reason-ably close to target values and the first principal compo-nent accounted for 69 of the variance As may be seenin the upper left panel of Figure 9 the relationship of between-subjects SD and task mean RT for this simula-

Figure 9 Between-subjects standard deviation (SD) for computer-simulated data plotted as afunction of group mean response time (RT) The upper left panel presents simulation results for amean step duration of 50 msec and the upper right panel presents results for mean step durationsof 20 100 and 200 msec The lower left panel presents simulation results for tasks consisting of amixture of step durations and the lower right panel presents results for simulations in which themean step duration was varied from 50 to 200 msec but the SD was held constant at 25 msec (seetext for details) For all step durations but those shown in the lower right panel the coefficient ofvariation was 5 Note that to facilitate comparisons we made the ratio of the scale of the x-axis tothe scale of the y-axis the same as in Figure 5


tion closely approximated that seen in the actual data (seeFigure 5)

Additional simulations were conducted varying themean step duration while holding constant the target val-ues for the step intercorrelations and the ratio of mean stepduration to the step duration SD Targets for task meanRTs were also held constant For example with 20-msecsteps (SD = 10 msec) the easiest and hardest tasks consistedof 10 and 70 steps respectively whereas with 200-msecsteps (SD = 100 msec) the corresponding tasks consistedof 1 and 7 steps Consistent with Equation 9 the relation-ship between SD and mean RT generally was not affectedby these manipulations This may be seen in the upper rightpanel of Figure 9

To examine the effect of mixing step durations taskswere simulated using 3 4 5 6 7 or 8 steps with RTsbeing made up of randomly selected mixtures of 20- 50-100- and 200-msec step durations There were 30 tasks inall five for each number of steps Target values for stepduration intercorrelations and the ratio of mean step du-ration to the step duration SD were the same as in previ-ous simulations As may be seen in the lower left panel ofFigure 9 the relationship between SD and mean RT forsimulations in which the steps making up a task had dif-ferent mean durations was similar to the relationship be-tween SD and mean RT for simulations in which the stepsmaking up a task had the same duration

These simulation results suggest that the characteristicsof the difference engine are fairly robust as long as the in-tercorrelations between step durations and the ratio ofmean step duration to the step duration SD are not variedAlthough we did not search the entire space for boundaryconditions we did use simulations to evaluate those con-straints predicted by the model (eg Equation 9) For ex-ample we examined the effect of increasing mean step du-rations without increasing their variability As may beseen in the lower right-hand panel of Figure 9 increasingthe mean step duration from 50 to 100 to 200 msec whileholding the SD of step durations constant at 25 msec hadthe predicted effects of decreasing the slope of the regres-sion of between-subjects SD on mean RT and increasingthe intercept As expected on the basis of Equation 9 sim-ilar effects were observed with different pairs of startingvalues That is increasing the mean step duration whileholding variability constant resulted in regression lineswith parameters that diverged systematically from thoseobserved with real data Also as expected on the basis ofEquation 9 decreasing the correlations between step du-rations (with the target values for mean step duration andSD held constant at 50 and 25 msec respectively) had theeffect of decreasing slopes and increasing intercepts

We conducted one further test of the robustness of thedifference engine model Distributions of RTs across in-dividuals typically are not normal (ie Gaussian) but in-stead are positively skewed That is the difference be-tween the RTs of the slowest individuals and the groupmean is much greater than the difference between the RTsof the fastest individuals and the group mean Because thesimulations just described assumed Gaussian distribu-

tions additional simulations were conducted to verify thatthe present results do not depend on this assumption Ac-cordingly skewed distributions were created by taking thecorrelated step durations generated by our simulation pro-gram and raising them to the 15 power In addition toskewing the distribution this transformation also had theeffect of increasing the mean and therefore all cognitivestep durations were decreased proportionally so that themean was approximately 50 msec

The RTs for seven tasks were generated by adding 4 812 16 20 24 and 28 processing steps as in our initialsimulation and the resulting task mean RTs ranged fromapproximately 500 msec for the 4-step task (300 msec ofsensorimotor time plus 200 msec of information process-ing) to approximately 1700 msec for the 28-step taskWith correlations between cognitive step durations as lowas 30 the distribution of RTs for each task measuredacross 100 simulated individuals was positively skewedand more importantly as predicted by the difference en-gine the relationship of between-subjects SD to task meanRT was linear with a negative intercept With lower cor-relations the relation between SD and mean RT was neg-atively accelerated also as predicted Furthermore forcorrelations that produced linear relations between SDand mean RT the slope was an increasing function of thecorrelation between step durations again as predicted Thesesimulation results obtained with skewed distributions ofstep durations parallel those obtained with our mathemat-ical model and with computer simulations that assumedGaussian distributions

Taken together the results of our simulations with bothnormal and skewed step durations help confirm that the be-havior of the difference engine although robust acrosscertain manipulations does depend on the relationships be-tween certain parameters in the way predicted by the model(see also further simulation results presented in the nextsection) Importantly simple assumptions about serialprocessing other than those incorporated in the differenceengine such as uncorrelated step durations or variabilityin step durations independent of the mean duration didnot result in simulations consistent with the patterns ob-served with real data

THE DIFFERENCE ENGINEAND COGNITIVE SLOWING

Extending the Model to Performance of Different Groups

We are now ready to consider the role of individual char-acteristics other than ability (ie age and health status)Our basic assumptions regarding cognitive task architec-ture step durations and correlations between steps re-main unchanged That is tasks and conditions that differin difficulty are assumed to differ in the number of ele-mentary processing steps required in order to make an ac-curate decision and cognitive step durations are assumedto be correlated with each other but not with the durationof the response step (see Assumptions 1 2 and 3 of thetwo-compartment model of within-groups variability) In


extending the model to different groups for simplicityrsquossake we assume that all individuals affected by the sameage or health status are affected to the same degree on allspeeded processing tasks at least within broad cognitivedomains (eg the verbal domain and the visuospatial do-main) More specifically the difference engine incorpo-rates the following assumptions in addition to the three al-ready described

4 Cognitive step durations and correlations It is as-sumed that aging and many health conditions increase theaverage time that each affected individual requires to com-plete one step of cognitive processing by some proportionsc which is specific to the particular age or condition Asa consequence the between-subjects SD of step durationsis increased by the same proportion so that the mean andSD are sca and scsc respectively Other information-processing parameters of the difference engine (eg themagnitude of the intercorrelations) are unaffected by thisincrease in step durations

5 Response step durations and correlations It is as-sumed that aging and many health conditions may also in-crease the time it takes to report a decision The proportionsr by which this component of performance is increasedis assumed to be specific to a particular age or conditionand may differ from the proportion by which cognitiveprocessing times are increased Although the duration ofcognitive and response steps may covary across groups itis assumed that within a group the time it takes to selectand execute a minimal motor response is independent ofthe duration of an elementary cognitive processing step

It should be noted that the difference engine continuesto assume correspondence that is regardless of age or healthstatus it is assumed that all (or most) individuals per-forming speeded cognitive tasks are processing the sameinformation in the same way (Cerella 1990) Thus themajor determinants of the size of individual and group dif-ferences are assumed to be quantitative It may be possi-ble to find or devise tasks that do not satisfy this con-straint our model would not apply to such tasks Wewould point out however that in order to study group dif-ferences in processing speed as opposed to group differ-ences in strategy it would make sense to focus on tasks inwhich there is correspondence in processing

The preceding assumptions provide the basis for themathematical modeling and computer simulations thatfollow In order to make these efforts more concrete wewill consider age to be the paradigmatic individual char-acteristic and compare the performance of older adultsagainst a young adult standard Assuming that aging af-fects the duration of cognitive processing steps but nottheir intercorrelations we may derive the relation betweenSD and mean RT for an older adult group using the samelogic as that used previously To summarize the differenceengine predicts a relation between SD and mean RT thatis described by the square root of a second-order polyno-mial (Equation 7) Further as the number of processingsteps increases this equation becomes increasingly wellapproximated by a linear function (Equation 9) SD 5(r 2 sc a)(RT 2 tr)

In the present case where there is both cognitive slowing(affecting the mean and SD of cognitive step durationsequally) and response slowing this equation becomes

SD 5 r 2 (scsc sca)(RT 2 sr tr) (11)

Figure 10 Graphical summary of the difference enginersquos pre-dictions for individual differences within different age groupsThe upper panel illustrates the relations between the standarddeviation (SD) and mean response time (RT) for groups of youngand older adults The middle panel illustrates the consequencesof these relations for the RTs of young and older subgroups whoare 1 SD faster and 1 SD slower than average for their age groupThe lower panel depicts the further consequences for the rela-tions between the RTs of these subgroups and the mean RTs ofthe corresponding age groups (ie fast and slow young as a func-tion of average young performance and fast and slow older as afunction of average older performance)


It may be noted immediately that sc the cognitive slowingcoefficient cancels out Thus the slope parameter sim-plifies to m 5 r 2 sc a which is the same as in Equa-tion 9 and Equation 11 may be rewritten as

SD 5 mRT 2 msr tr (12)

The strong and perhaps unexpected implication is thatidentical slopes are predicted for young and older groupsSetting SD equal to zero and solving Equation 11 for RTreveals that the x-intercept for the older group is the youngadult intercept multiplied by sr the coefficient that re-flects the slowing of the response component with ageBecause the degree of sensorimotor slowing in older adultsis typically quite small (Cerella 1985) the age differencein intercepts also will be quite small and as a result maynot be statistically reliable

Moreover because the relationship between SD and meanRT is relatively unaffected by aging the difference enginepredicts that the form of the regression of individual andsubgroup RTs on group mean RT will be the same foryoung and older adults The relationships between these var-ious predictions are summarized graphically in Figure 10which adds a hypothetical older group to the young adultgroup previously depicted in Figure 7 For purposes of cal-culating these hypothetical data we assumed that in olderadults information processing is slowed by a factor of 20typical of nonverbal cognitive tasks (eg Lima Hale ampMyerson 1991 Sliwinski amp Hall 1998) whereas senso-rimotor processing is slowed by a factor of only 115(Cerella 1985)

As may be seen in the upper panel of Figure 10 the di-versity of individual performance (as indexed by the SDfor the group) increases linearly with task difficulty (as in-dexed by the mean RT) for both groups According to thedifference engine the slope of the relation is identical forboth young and older adult groups but the y-intercept forthe hypothetical older adult group is more negative re-flecting age differences in the response component Re-sponse selection and execution are only slightly (ie ap-proximately 15) slower in the older group however andtherefore the difference between the lines for the twogroups is barely discernible

For both age groups the speed with which individualsexecute different processing steps is correlated so that thesame individuals tend to have relatively long (or short) RTsfor their age on all tasks Thus the pattern of increasing di-versity in the top panel leads to linear increases in the dif-ference between average RTs on the one hand and the RTsof fast and slow subgroups on the other hand (see the mid-dle panel of Figure 10) Finally the RTs for each subgroupare equal to the mean for their age group plus the differ-ence from the mean Converting the differences depicted inthe middle panel into the subgroupsrsquo mean RTs producesthe graph depicted in the lower panel in which the differ-ence between the regression lines for the fast older andfast young subgroups which have the same slopes but dif-fer slightly with respect to their intercepts is hard to discernand the same is true for the difference between the regres-sion lines for the slow older and slow young subgroups

Zheng et al (2000) recently reported results consistentwith the present model The young adult and older adultgroups in the Zheng et al study consisted of 40 subjectseach all of whom were tested on seven visuospatial taskschoice RT disjunctive choice RT colored shape discrim-ination two-object samedifferent judgment three-objectsamedifferent judgment and two different types of ab-stract matching tasks Zheng et al reported that a generalspeed factor accounted for 77 and 70 of the variancein standard scores for the young and older groups respec-tively Moreover the composition of the general speedfactor was relatively age invariant as implied by the dif-ference enginersquos assumption that aging does not affect theintercorrelations between step durations Specificallythere was a correlation of 99 between the observed factorscores for individual older adults and the factor scores pre-

Figure 11 Between-subjects standard deviation (SD) plottedas a function of group mean response time (RT) Data are takenfrom Zheng Myerson and Hale (2000) Each circle representsperformance by one age group in one experimental conditionData from older and young adults are presented in the upper andlower panels respectively Note that the scales in the two panelsare different but that the ratios of the scale of the x-axis to thescale of the y-axis are the same as in Figure 5 In each panel thedashed line represents the linear regression equation and thesolid curve represents the prediction of the difference engine(Equation 7)


dicted on the basis of the loadings from the young adultsrsquodata This finding suggests that the composition of thegeneral speed factor in the two age groups was highly sim-ilar (Gorsuch 1983) Furthermore regression of the RTs ofthe fast young and fast older adults on the average RTs fortheir peers yielded virtually identical slopes and interceptsand accounted for more than 98 of the variance in bothcases Regression of the RTs of the slow young and slowolder adults on the average RTs for their peers also yieldednearly equivalent slopes and intercepts and again ac-counted for more than 98 of the variance in both cases

According to the difference engine the necessary the-oretical linkage between the factor analytic findings andthe regression analysis of fast and slow subgroupsrsquo RTs isprovided by the effect of correlated step durations on therelation between SD and mean RT In order to test themodelrsquos predictions regarding the relation between SD andmean RT in young and older adults we reanalyzed theZheng et al (2000) data for each group separately As maybe seen in Figure 11 the data from both the young andolder adult groups are well described by linear functions(Equations 9 and 11 respectively) Notably the x-interceptestimate of the response component for the older groupwas 16 greater than that for the young group very closeto Cerellarsquos (1985) meta-analytic estimate of 15 senso-rimotor slowing The slopes of the regression lines for theyoung and older groups (311 and 337 respectively) areboth very similar to those for the two large young adultsamples (Hale amp Jansen 1994 Hale amp Myerson 1993)shown in Figure 3 as well as to those reported by Hale et al(1988) in their meta-analysis of published studies from anumber of different laboratories further attesting to thereliability of this phenomenon

Because the Zheng et al (2000) study tested subjectson a number of fairly easy tasks that produced short RTsas well as on tasks that produced fairly long RTs the dataafford an opportunity to test for the positive accelerationin the relation between SD and RT predicted by the dif-ference engine Both young and older adults showed sig-nificant positive acceleration (ie in each case polyno-mial regression revealed the presence of a significantquadratic term) as predicted by Equation7 Thus these datasupport the difference enginersquos predictions regarding theform of the relation between SD and mean RT as well asbeing consistent with the modelrsquos assumption that the majoreffect of aging on RTs results from simply increasing thedurations of the steps required to perform information-processing tasks

Extending the Modelto Domain-Specific Slowing

Lima et al (1991) conducted a meta-analysis of studiesof age-related cognitive slowing and they reported thatthe degree of age-related slowing on visuospatial taskswas considerably greater than that on verbal tasks of ap-proximately the same degree of difficulty as indexed bythe RTs of young adults However the studies analyzed byLima et al used either verbal or visuospatial tasks but not

both so that the comparison of verbal and visuospatialslowing necessarily involved different samples and themotor requirements were not always equivalent in differ-ent studies In an experimental validation of the Lima et alfindings Hale and Myerson (1996) tested the same 24young and 24 older adults on both verbal and visuospatialtasks all of which had equivalent response requirements(ie pressing the correct one of two response keys) Therewere four different verbal tasks (ie single lexical deci-sion double lexical decision category membership andsynonymantonym judgment) and four different visuo-spatial tasks (ie line-length discrimination shape classi-fication visual search and abstract matching)

Analyses indicated that the older adult group was morethan 200 slower on average than the young adults atprocessing visuospatial information whereas they wereless than 50 slower than the young adults at processingverbal information Taken together the correspondencebetween these experimental results (Hale amp Myerson1996) and the meta-analytic results reported by Lima et al(1991) attests to the robustness of the phenomenon (ie itis not specific to a particular sample or to one specific lab-oratory and its procedures) More specifically the degreeof age-related slowing (ie the value of sc according tothe difference engine notation) is determined to a consid-erable extent by the domain (ie verbal or visuospatial)rather than by other specific task characteristics Notablythe degree of age-related slowing is much greater in thevisuospatial domain This may be seen by comparing theupper and lower panels of Figure 4 The differences be-tween the RTs of the three age groups on visuospatialtasks (upper panel) are much greater than the differenceson verbal tasks (lower panel)

The finding of greater age-related slowing in the visuo-spatial than in the verbal domain observed at the group levelalso appears to be a reliable characteristic of age-relatedslowing at the individual level When the RTs of individ-ual subjects are regressed on the means for a young adultcontrol group the percentage with greater visuospatialthan verbal slopes increases systematically with age from50 of young adults to 98 of individuals more than60 years of age (Lawrence et al 1998) Because of the dif-ferential effects of age on visuospatial and verbal speed atboth the individual and group levels the two domains pro-vide an important opportunity to test the present differ-ence engine model

The difference engine makes clear predictions for thetwo domains regarding both the regression of SD on meanRT and the regression of the RTs of fast and slow subgroupson the mean RTs for their age group That is just as theslope (m) of the regression of SD on mean RT is predictedto be independent of age so m is predicted to be indepen-dent of domain This is because as already shown the cog-nitive slowing coefficient cancels out of the equation for theregression of SD on mean RT (Equation 12) and thus m isindependent of the degree of age-related cognitive slow-ing The degree of age-related cognitive slowing changesfrom one domain to the other but m does not change


Furthermore because the equation for the regression ofindividual and subgroup RTs on group mean RTs followsdirectly from the regression of SD on mean RT the cogni-tive slowing coefficient does not enter into the relationshipbetween the RTs of an individual or subgroup and the meanRTs for their age group That is it follows from Equa-tion 12 that

RTi 5 (1 1 zim)RTavg 2 zimsr tr (13)

As revealed by Equation13 the difference engine makes thesurprising prediction that the slope of the regression of theRTs of fast and slow individuals and subgroups on the RTsof their peers depends on their z score relative to their agegroup but is otherwise independent of both age and do-main In fact the only difference between Equation13 thegeneral equation for all adult age groups and Equation10the corresponding equation for young adults is in the inter-cept that in the general equation contains an age-specificsensorimotor slowing term (sr)

Computer simulations of the difference engine that re-laxed some of the assumptions of the mathematical modelwere used to test the prediction (Equation 12) that theslope of the regression of SD on mean RT is independentof both age and domain as well as the related predictionregarding the regression of fast and slow RTs on averageRTs (Equation 13) To assess the robustness of the modelwe simulated groups of 100 young and 100 older adults

performing 30 verbal and 30 visuospatial tasks (oneyoung and one older group for each domain) Within eachdomain these tasks consisted of 3 4 5 6 7 or 8 stepswith five tasks at each number of steps For young adultseach task was made up of a randomly selected mixture of20- 50- 100- and 200-msec step durations The results forthe visuospatial simulations were presented previously inFigure 9 (see lower left panel) and thus for present pur-poses we simulated another 30 verbal tasks using thesame target values as those in previous simulations

Both young and older adults were simulated using taskswith the same composition but the degree of slowing wasdifferent for the two domains For example if the easiestvisuospatial task had three steps with mean durations of20 20 and 50 msec for young adults then the durationsof the steps for that task were 40 40 and 100 msec forolder adults Likewise if the easiest verbal task had threesteps of 20 50 and 200 msec for young adults then thedurations of those steps were 30 75 and 300 for older adultsThus older adults were simulated to be 50 slower on ver-bal tasks and 100 slower on visuospatial tasks consis-tent with Hale and Myerson (1996) and Lima et al (1991)

As may be seen in Figure 12 the simulations resultedin very similar relationships between SD and mean RT re-gardless of the age group or cognitive domain being sim-ulated Each simulated age group was then divided intofour quartiles on the basis of individual speed factors and

Figure 12 Between-subjects standard deviation (SD) plotted as a function ofgroup mean response time (RT) Data were generated by computer simulation(see text) The graph presents data from simulated verbal and visuospatialtasks consisting of a mixture of step durations The older adult simulations as-sumed that older verbal processing steps were 50 longer than young verbalprocessing steps whereas older visuospatial processing steps were twice as longas young visuospatial processing steps Note that to facilitate comparisons wemade the ratio of the scale of the x-axis to the scale of the y-axis the same as inFigure 5


the top and bottom quartiles were taken as the slow andfast subgroups respectively Figure 13 shows the regres-sions of subgroup RTs on the corresponding mean RTs foreach age group in each domain In both figures for thesake of clarity and also so as to depict the same number(ie 15) of task conditions as in comparable human datafrom Hale and Myerson (1996) only data from the odd-numbered tasks from each domain are shown

As predicted by the mathematical model the regressionlines for the simulated slow subgroup RTs (fit to the whitecircles) were highly similar regardless of age or domainand the same was true for simulated fast subgroup regres-sion lines (fit to the gray circles) Notably the perfor-mance of simulated individual subjects was fairly consis-tent across task conditions On average the simulatedyoung subjects in the fast and slow subgroups were fasteror slower than average in 136 out of 15 conditions whereasthe simulated fast and slow older subjects were faster orslower than average in 135 conditions

The simulation results demonstrate that the differenceenginersquos predictions regarding the effects (or the lackthereof) of age and domain do not depend on the assump-tion of equal mean step durations that expedited the math-

ematical development but appear to be more robust andhold for tasks that involve different mixtures of step dura-tions as well More specifically regardless of variability instep durations the model predicts that neither the rela-tionship between SD and group mean RT nor the relation-ship between subgroup RT and group mean RT will besubstantially affected by the age group or by the domaininvolved even if different domains involve different de-grees of age-related slowing

In order to test these predictions we conducted a re-analysis of data from Hale and Myerson (1996) Figure 14depicts the SDs of both the 24 young and the 24 olderadults on both the four verbal and the four visuospatial RTtasks (15 task conditions in each domain) and Figure 15depicts the performance of fast and slow subgroups in bothdomains The subgroups were selected to be the fastestand slowest 6 individuals in each age group (ie the firstand fourth quartiles) determined using the same mean z-score procedure as Hale and Jansen (1994) and Hale andMyerson (1993)

As may be seen in Figure 14 there was an approximatelylinear relation between the SD and group mean RT for bothage groups in both the verbal and visuospatial domains The

Figure 13 Response times (RTs) of fast and slow subgroups plotted as a function ofthe group mean RTs Data were taken from the same simulation as in Figure 12 Theleft panels show simulated data from young adults and the right panels show simu-lated data from older adults Fast subgroups are represented by gray circles slow sub-groups are represented by white circles


solid curve is the theoretical function based on Equation 7and the dashed line is the regression line Tests over the samerange of RTs revealed no significant differences either be-tween the verbal and visuospatial regressions or betweenthe regressions for the two age groups These findings pro-vide support for the present model because they are pre-dicted directly by the fact that the cognitive slowing coef-ficient cancels out in the derivation of Equation 11 (whichdescribes the relation between group SD and mean RT)

As may be seen in Figure 15 the RTs of the fast andslow older adult subgroups were linear functions of aver-age older adult RTs Moreover these relations were virtu-ally collinear with those for the corresponding subgroupsof young adults That is there were no significant age dif-ferences either between the regression parameters for theolder and young fast subgroups or between the regressionparameters for the older and young slow subgroups Mostimportantly equivalent results were observed in both theverbal and visuospatial domains despite the much greaterdegree of age-related slowing in the visuospatial domainThat is there were no significant differences either be-tween the regression parameters for the older fast verbaland fast spatial subgroups or between the regression pa-rameters for older slow verbal and slow spatial subgroupsThese findings provide further support for the presentmodel because they are predicted directly by the fact thatthe cognitive slowing coefficient cancels out of the relationbetween older adult individual RTs and average perfor-mance for an older adult group (Equation 13) As a con-sequence the relations between individual (or subgroup)and average RTs are not affected by differences in the de-gree of age-related slowing due to either age or domain

Importantly the subgroup data from Hale and Myerson(1996) presented in Figure 15 accurately reflect the pat-terns of fast and slow performances observed at the indi-vidual level This may be seen by comparing Figure 15with Figure 16 which depicts data from individual fastand slow older adults as well as fast and slow young adultsin each domain In each case subjects were selected so asto be representative of their quartile in terms of the con-sistency of their performance and also in terms of howwell their data were fit by a straight line (r 2) Overall fastand slow young adults were faster or slower than averagefor their age group in 143 out of 15 task conditions andfast and slow older adults were faster or slower than aver-age in 140 task conditions

The results of the present analyses of the Hale and My-erson (1996) data replicate the findings of Hale and hercolleagues (Hale amp Jansen 1994 Hale amp Myerson 1993)with respect to the performance of subgroups of youngadults on visuospatial tasks (Figure 1) as well as Zhenget alrsquos (2000) findings regarding subgroups of both youngand older adults on such tasks The present results also ex-tend these previous findings to both young and older adultsrsquoperformance on verbal tasks Importantly the present re-sults confirm the difference enginersquos prediction that theparameters of certain key relationships between variousaspects of speeded performance are independent of the in-teractive effects of age and processing domain

DISCUSSION

We have presented a model of diversity in speeded in-formation processing This model the difference engine

Figure 14 Between-subjects standard deviations (SDs) of young and olderadult groups plotted as a function of their respective group mean responsetimes (RTs) Data are taken from Hale and Myerson (1996) Each point repre-sents performance by one age group in one experimental condition In the panellabeled ldquoDetailrdquo the data are replotted over a truncated range in order to makeit easier to resolve individual data points Note that to facilitate comparisonswith previous figures we made the ratio of the scale of the x-axis to the scale ofthe y-axis the same as in Figure 5


assumes that for purposes of analyzing individual differ-ences (including differences attributable to age and healthstatus as well as differences in ability) information pro-cessing may be represented by a series of generic compu-tational steps Although an individualrsquos processing stepsmay vary in duration they are correlated in such a waythat all of the processing steps of some individuals tend tobe relatively brief whereas all of the processing steps ofsome other individuals tend to take longer than averageAs a consequence of these intercorrelations the diversity(as measured by the SD for the group) of the cognitivecomponent of RTs increases approximately linearly withtask difficulty (as indexed by the average RT for the group)Because the duration of the cognitive component is un-correlated with the response component however the re-lation between SD and mean RT is actually positively ac-celerated

The difference engine does more however than explainthe form of the relation between SD and mean RT It alsoexplains the approximately linear form of the regressionof the RTs of an individual or subgroup on the RTs of theirpeers and it provides specific interpretations of the para-meters of such regressions Finally the model may be gen-

eralized to differences in processing speed between groupsthat differ in age and health status as well as to the inter-action of such differences with individual differences inability This generalization follows directly from the as-sumption that although groups differ on average in pro-cessing speed other relevant characteristics (ie the num-ber of processing steps required to perform specific tasksas well as the intercorrelations between step durations) areunaffected by age and health status Support for the dif-ference engine appears to be quite broad in that as pre-dicted the same pattern of results was observed in bothyoung and older adults performing tasks from both theverbal and visuospatial domains despite the fact these do-mains are differentially affected by age-related changes inprocessing speed

Comparative Human CognitionAn important theme of the present effort stripped of its

empirical and theoretical specifics is that there exist cer-tain phenomena (all having to do with the magnitude of dif-ferences in speeded cognition) that share common proper-ties even though they manifest themselves in traditionallydistinct research areas (Hale 1997) These phenomena sug-

Figure 15 Response times (RTs) of fast and slow subgroups of young and olderadults plotted as a function of their respective group mean RTs Data are taken fromHale and Myerson (1996) Each point represents performance by a subgroup of oneage group in one experimental condition


gest that there may be fundamental mechanisms underly-ing individual differences in information processing thatoperate in a similar fashion regardless of the cause of theiroperation (eg whether they are enabled by developmen-tal or health-related events) The difference engine repre-sents an attempt to model such fundamental mechanisms

These mechanisms are hypothesized to underlie three setsof findings First when examined across a range of tasksthat differ in difficulty strikingly linear relations havebeen observed between the RTs of selected individuals(eg fast or slow subgroups) and the mean RTs for theirpeers (eg Hale amp Jansen 1994 Zheng et al 2000) Sec-ond orderly linear relations have been observed betweenthe RTs of different age groups (eg children and youngadults or older and younger adults) tested on the sametasks (for a review see Cerella amp Hale 1994) Third lin-ear relations between group mean RTs have also been re-ported in experimental studies and meta-analyses thatcompared the RTs of various special populations with theRTs of control groups Included among these studies arethose that have examined individuals suffering fromAlzheimerrsquos disease (Myerson et al 1998) brain injury(Ferraro 1996) depression (White et al 1997) and mul-tiple sclerosis (Kail 1997)

Similar findings in different populations suggest the pos-sibility of similar explanations and the difference enginerepresents one possible quite general explanation In orderto establish such generality however detailed analyses ofindividual performance will be needed in each case be-cause as is well known relations between group averagesdo not necessarily reflect what is going on at the individ-ual level As a first step examination of the relation betweengroup SD reflecting the diversity of individual perfor-mance and mean RT reflecting task difficulty may provequite useful

For example consider the data from a recent meta-analysis of the effects of depression on speeded cognitiveperformance (White et al 1997) As may be seen in theupper panel of Figure 17 there is a linear relation betweenthe RTs of depressed and control groups which suggeststhat the depressed were approximately 30 slower thancontrols across a variety of information-processing tasks(eg choice RT Stroop color naming and delayed match-ing to sample) As may be seen in the lower panel the re-lation between SD and group mean RT was approximatelylinear for both the depressed and control groups Notablythere were no significant group differences in either theslope or intercept of this relation and for both groups the

Figure 16 Response times (RTs) of representative fast and slow young and older in-dividuals plotted as a function of their respective group mean RTs Data are takenfrom Hale and Myerson (1996) Each point represents the performance of one indi-vidual in one experimental condition


slopes and intercepts were similar to those for young andolder adults (Figures 3 and 10) In all cases the slope ofthe relation between SD and mean was between 30 and35 and the intercept was between 275 and 2100 msecThese results support the difference engine of course butmore broadly they are consistent with the idea that simi-lar mechanisms may underlie the diversity of speeded per-formance in groups of depressed individuals and in groupsof both young and older healthy individuals and the samemay be true for other populations as well

There is a growing awareness among researchers thatthere may be common phenomena research questions andmethodological and theoretical issues involved in what havebeen traditionally distinct research areas each focused onthe cognitive abilities of a distinct population (eg Cerella

amp Hale 1994 Faust et al 1999 Ferraro 1996 Fisher ampGlaser 1996 Kail 1997 Nebes amp Brady 1992 Schatz1998 White et al 1997) This growing awareness suggeststhat a new research area may be emerging one that focuseson what may be termed the study of comparative humancognition (Hale 1997) The present theoretical effort exem-plifies this trend toward the integration of findings regard-ing different aspects of individual and group differences

Comparisons With Other ApproachesThe goals of the present effort are both broad and unique

As a consequence the opportunities to compare the pres-ent approach with previous theoretical models having sim-ilar aims are somewhat limited Recently however three ar-ticles (Faust et al 1999 Ratcliff Spieler amp McKoon2000 Zheng et al 2000) have touched on some of thesame basic theoretical issues considered here We willconsider these recent efforts following a brief discussionof the relevant older literature

The basic approach taken here to the variability of speededperformance (ie that of decomposing task performanceinto a series of generic processing steps) is common inmathematical models (Luce 1986 Townsend amp Ashby1983) and may be traced back to pioneering efforts byMcGill (1963) However the variability of interest in suchmodels is typically the trial-to-trial variability of individ-ual subjects (as reflected in RT distributions) rather thanthe diversity of performance across individuals and tasksAt least under certain circumstances the SDs of such dis-tributions are linear functions of the mean RT (eg HaleFry amp Jessie 1993 Myerson amp Hale 1993) raising thepossibility that the two forms of variability in RTs may bemore than superficially similar

In fact the relation of within-subjects SD (or semi-interquartile range) to individual mean (or median) RT issurprisingly similar to the relation of between-subjects SDto group mean RT in two respects First the relation tendsto hold across tasks second it tends to hold across agegroups (ie children young adults and older adults) withlittle or no change in the value of either slope or intercept(Hale et al 1993 Myerson amp Hale 1993) It is possiblethat trial-to-trial fluctuations in processing efficiency areanalogous to variations in efficiency from individual to in-dividual or from group to group (Smith Poon Hale ampMyerson 1988) in that such trial-to-trial fluctuations mayalso be based on correlated changes in the speed withwhich processing steps are executed If so then this anal-ogy leads directly to application of the present models

However analysis of the trial-to-trial variability of RTsunlike analysis of between-subjects variability must con-sider the problem of sequential dependencies Much of thevariation in RTs across trials is due to adjustments in de-cision criteria following errors and to the effects of stim-ulus andor response repetition (eg Smith amp Brewer1995 for a review see Luce 1986) whereas sequentialdependencies are obviously not an issue with respect tovariation across individuals Because of this limitation onthe analogy between individual differences and trial-to-

Figure 17 Mean response times (RTs) of depressed groupsplotted as a function of the mean RT for healthy control groups(upper panel) and between-subjects standard deviation (SD)plotted as a function of group mean RT for both depressed andcontrol groups (lower panel) Data are taken from the meta-analyses reported by White Myerson and Hale (1997) In theupper panel the solid line represents the regression of the RTs ofdepressed groups on the RTs of control groups In the bottompanel the solid line is the regression of SD on mean RT based onthe data from both depressed and control groups


trial fluctuations we will postpone further considerationof the parallels between them to another occasion

Psychometric approaches The decomposition ofRTs into processing steps has a direct psychometric ana-logue in the decomposition of test scores into performanceon individual items or subtests Despite the analogy be-tween tests and tasks the phenomena of interest from thepsychometric perspective are usually quite different fromthose that are the focus here In particular the psycho-metric approach tends to focus on the relative perfor-mance of individuals (ie their positions in the overalldistribution) and psychometric research on speed hasbeen especially concerned with its relation to intelligenceIn contrast the difference engine is concerned with pre-dicting the absolute size of differences between individualRTs and we remain agnostic in regard to the relation be-tween speed and intelligence

Although the goals of psychometric research have beenquite different from those of the present effort the differ-ence engine incorporates basic psychometric findings Thepresent assumption of a general cognitive speed factor anda separate sensorimotor factor is consistent with the re-sults of psychometric studies (Buckhalt Whang amp Fisch-man 1998 Carroll 1991 Kranzler amp Jensen 1991)Moreover the assumption that age does not change thelevel of individualsrsquo performance relative to their peers isconsistent with the results of longitudinal psychometricresearch (Hertzog amp Schaie 1986)

Information-processing approaches The goals ofthe present model are obviously quite different from thoseof a chronometric approach (eg Posner 1978 Sanders1990) that seeks to decompose processing into qualita-tively different cognitive operations In contrast the pri-mary goal of the difference engine is to predict regulari-ties in performance across tasks that differ substantiallywith respect to the cognitive operations involved For ex-ample what is critically important about visual search andmental rotation from the chronometric perspective is thatthey obviously involve different cognitive processes yet thepresent analyses suggest that individual differences in oneprocess are predictable from individual differences in the other process Moreover aging appears to affect per-formance on both types of tasks (and other visuospatialtasks as well) to approximately the same degree Equiva-lent age-related slowing of visual search and mental rota-tion was first shown in a multitask experiment in whichthe same older and young adults performed both types oftasks (Hale Myerson Faust amp Fristoe 1995) This findinghas since been validated in a meta-analysis using standardregression techniques (Myerson Adams Hale amp Jenkins2003) as well as with more refined meta-analytic methodsbased on hierarchical linear models (Sliwinski amp Hall1998)

This surprisingly robust finding suggests that regard-less of whether individuals are slow because of age abil-ity or both being slow at one process means that one willtend to be slow at the other process Thus making a dis-tinction between visual search and mental rotation al-though essential for some purposes would not contribute

to the present effort Indeed the strength of the present ap-proach is that it can reveal similarities precisely underconditions where differences in the underlying cognitiveprocesses are known to exist We raise these issues here inorder to reemphasize the fact that our simplifying assump-tions which may conflict with some researchersrsquo stronglyheld beliefs regarding the complicated nature of even sim-ple information processing need to be seen as the meansto an end That end is to reveal and understand large-scalequantitative regularities in the patterns of individual andgroup differences To achieve that end the present modelignores processing distinctions that most cognitive psy-chologists accept as valid thereby potentially sacrificingsome degree of detail Yet at the same time the presentmodel strives to provide detail that is missing from otheraccounts in that the difference engine is intended to pre-dict both the magnitude and the distribution of individualperformances

Other general slowing models In the areas of cogni-tive development and cognitive aging the present theo-retical effort is not alone in having sacrificed the usualtask and process distinctions of cognitive psychology(eg Cerella 1985 Kail 1991) Of particular relevance arethe models proposed by Cerella (1985) and Cerella andHale (1994) who divided processing into two compo-nents one cognitive and one sensorimotor for purposes ofanalyzing age-related changes in RTs across the life spanThe difference engine may be viewed as an extension ofthat approach to individual differences and to group dif-ferences attributable to factors other than age This exten-sion is important because such ideas are not a necessarypart of a general slowing hypothesis since general slow-ing could be peculiar to the effects of age or it could de-scribe changes at the group level but not those at the indi-vidual level To the best of our knowledge Zheng et al(2000) were the first to take an explicit position on thisissue and to consider its theoretical implications Theirmagnification hypothesis like the difference engine as-sumes that age-related slowing operates at the individuallevel magnifying everyonersquos processing times and in-creasing the size of individual differences by the sametask-independent factor Their mathematical account washighly simplified however and they called for further the-oretical work along the lines of the present effort

The diffusion model Ratcliff et al (2000) have recentlysuggested applying the diffusion model (Ratcliff 1978) tothe effects of aging on RT The diffusion model describeswithin-subjects RT distributions and speedaccuracytradeoff functions but it has not previously been used tostudy group differences With respect to age-related differ-ences Ratcliff et al suggested that what appear to be dif-ferences in processing speed may be due to differences indecision criteria (ie older adults may be more cautiousrather than slower) This hypothesis however has not gen-erally been supported As Cerella (1990) showed in factolder adults tend to make more rather than fewer errorsthan young adults Moreover as Myerson et al (2003)have pointed out a number of recent studies have shownthat older adults continue to process information more


slowly than young adults even when stimulus-durationthreshold procedures are used to eliminate accuracy dif-ferences (eg Mayr Kliegl amp Krampe 1994 1996 Zacksamp Zacks 1993)

In contrast to the difference engine the diffusion modelmakes no assumptions or predictions regarding the covari-ation between individual RTs on different tasks nor doesit predict or explain the form of the relation between SDand mean RT As Ratcliff et al (2000) have specificallypointed out the diffusion model can accommodate in-creases in between-subjects SD as a function of the meanbut it does not predict them Moreover the diffusionmodel does not predict the approximately linear form ofthe regression of the RTs of an individual or subgroup onthe RTs of their peers (although again the diffusion modelis not inconsistent with such results) and it provides nospecific interpretations of the parameters of such rela-tions Thus although the diffusion model is not inconsis-tent with the results of the analyses reported here its treat-ment of the present findings would be necessarily posthoc For example the simulation of subgroup differencesin Ratcliff et al merely shows that there exist some valuesfor the diffusion modelrsquos parameters that emulate actualdata However Ratcliff et al presented no principled treat-ment of the way in which parameters do (or do not) changeacross tasks and thus they failed to come to grips with thecore of the theoretical question posed by the data

The rate-amount model Faust et al (1999) have pro-posed what they term a rate-amount model that predictslinear relationships between the RTs of individuals andgroups and a linear relation between SD and mean RT Thedifference engine too may be thought of as a rate-amountmodel in which the amount of processing required toreach a decision is represented by the number of process-ing steps and individuals and groups differ in the rate atwhich these steps are completed Despite many similari-ties the two models differ in a number of respects includ-ing some of the fundamental assumptions and the mathe-matical approach taken These differences in part reflectthe different goals of the two efforts The difference en-gine is an explicitly theoretical model of the large-scalestructure of individual and group differences in information-processing speed whereas the rate-amount model is in-tended to provide a basis for statistical testing and isola-tion of small-scale group differences assuming that sucha large-scale structure exists

In terms of generating predictions it is important to notethat the model proposed by Faust et al (1999) is explicitlya one-compartment model although they assume two sep-arate individual difference variables one governing speedand the other governing amount In contrast the differenceengine is a two-compartment model that distinguishes re-sponse selection and execution from cognitive processingThe difference engine then further subdivides the cogni-tive compartment into generic processing steps withequivalent correlations between step durations Althoughthe two models make a number of predictions in commonas a consequence of these differences only the differenceengine correctly predicts the initial positive acceleration

of the approximately linear relation between SD and meanRT The difference engine also provides an interpretationof the x-intercept of the linear approximation of this rela-tion in terms of the time required for response selectionand execution Additionally this interpretationmdashwhichthe rate-amount model specifically eschews (Faust et al1999)mdashis consistent with observed estimates and cor-rectly predicts the approximate point at which the regres-sion of individual or subgroup RTs on group mean RT willintersect the equality diagonal

Faust et al (1999) have suggested that their rate-amountmodel provides a statistical basis for asking important ques-tions such as whether all younger adults become slowed tothe same degree as they grow older Their model howeverdoes not predict what the answer to this question will beIn contrast the difference engine does propose an answerto this question as well as other related questions Our modelspecifically proposes that at least within broad cognitivedomains (eg verbal and visuospatial tasks) age-relatedchanges effectively multiply the speed coefficients of allindividuals fast or slow by an equivalent factor withoutaffecting other aspects of processing On the basis of thistheoretical assumption the difference engine makes theunique prediction that differences in slowing factors be-tween domains or age groups will not affect other key re-lationships That is such slowing factors will affect nei-ther the slope of the relation of the grouprsquos SD to mean RTnor the relationships of individual and subgroup RTs tothe mean RT for their age group The results of the pres-ent analyses of the Zheng et al (2000) and Hale and My-erson (1996) data strongly support these unique predictions

It may be noted that the precise details of the theoreti-cal machinery that are the focus of the difference enginemay be largely irrelevant for the statistical purposes thatFaust et al have in mind The major goal of their effortwas to justify transforms that could facilitate the isolationof small-scale group differences The justification of thesetransforms depends on linear or approximately linear re-lations between group SDs and individual RTs and linearrelations between group mean RTs The two models are inagreement as to the approximate form of these relationsand thus we see the efforts as complementary one moredirected to solving theoretical problems in large-scalestructure and the other more directed to solving the statis-tical problems raised by the existence of such a structureWhereas the transformations justified by the rate-amountmodel may facilitate the identification of small-scalegroup differences the parameters of the equations for in-dividual performance (Equations 10 and 13) easily inter-pretable in terms of individual z scores may facilitate theidentification of clinically significant individual differ-ences (Schatz Hale amp Myerson 1998)

ConclusionThe most obvious potential limitation of the difference

engine model is that it relies on strong simplifying as-sumptions although computer simulations suggest thatthe modelrsquos predictions are quite robust when some butnot all of these assumptions are relaxed The important


question with regard to the difference engine of course isnot whether the modelrsquos assumptions are precisely trueThey are not They are merely simplifying assumptionsintended to facilitate the mathematical development sothat we can examine the consequences of certain theoret-ical ideas Rather the important question with respect tothe difference engine is whether it leads to an increase inthe ability to predict performance and to new insights intowhy performance conforms to these predictions

With respect to the accuracy of the difference enginersquospredictions the results of a number of new analyses ofpreviously published data provide considerable supportfor the present model These analyses examined the inter-correlations and factor structure of RTs as well as theirmeans and SDs the relationship between these variablesand the relations between the RTs of individuals andgroups that differ in age or ability With respect to the in-sight gained from the model the difference enginersquos mostimportant contribution may be to provide theoretical link-age between these seemingly disparate phenomena Takentogether the findings to date suggest that the theoreticalmodel presented here provides a uniquely integrated ac-count of individual and group differences on speeded cog-nitive tasks Future research will be needed to establish thegenerality of the difference enginersquos predictions and theirapplicability to additional populations including childrenand individuals with various health conditions

REFERENCES

Buckhalt J A Whang P S amp Fischman M G (1998) Reactiontime and movement time relationships with intelligence in three dif-ferent simple tasks Personality amp Individual Differences 24 493-497

Carroll J B (1991) No demonstration that g is not unitary but therersquosmore to the story Intelligence 15 423-436

Cerella J (1985) Information processing rates in the elderly Psy-chological Bulletin 98 67-83

Cerella J (1990) Aging and information-processing rate In J E Bir-ren amp K W Schaie (Eds) Handbook of the psychology of aging(3rd ed pp 201-221) San Diego Academic Press

Cerella J (1994) Generalized slowing in Brinley plots Journals ofGerontology 49 P65-P71

Cerella J amp Hale S (1994) The rise and fall in information-processing rates over the life span Acta Psychologica 86 109-197

Dunn J C amp Kirsner K (1988) Discovering functionally indepen-dent mental processes The principle of reversed association Psycho-logical Review 95 91-101

Faust M E Balota D A Spieler D H amp Ferraro F R (1999)Individual differences in information processing rate and amount Im-plications for group differences in response latency PsychologicalBulletin 125 777-799

Ferraro F R (1996) Cognitive slowing in closed-head injury Brainamp Cognition 32 429-440

Fisher D L amp Glaser R A (1996) Molar and latent models of cog-nitive slowing Implications for aging dementia depression develop-ment and intelligence Psychonomic Bulletin amp Review 3 458-480

Gorsuch R L (1983) Factor analysis (2nd ed) Hillsdale NJ Erlbaum Hale S (1997) Introduction [Special issue Comparative adult cogni-

tion] Aging Neuropsychology amp Cognition 4 155-156Hale S Fry A F amp Jessie K A (1993) Effects of practice on speed

of information processing in children and adults Age-sensitivity andage-invariance Developmental Psychology 29 880-892

Hale S amp Jansen J (1994) Global processing-time coeff icientscharacterize individual and group differences in cognitive speed Psy-chological Science 5 384-389

Hale S amp Myerson J (1993 November) Ability-related differences in

cognitive speed Evidence for global processing-time coefficientsPoster presented at the annual meeting of the Psychonomic SocietyWashington DC

Hale S amp Myerson J (1996) Experimental evidence for differentialslowing in the lexical and nonlexical domains Aging Neuropsychol-ogy amp Cognition 3 154-165

Hale S Myerson J Faust M amp Fristoe N (1995) Convergingevidence for domain-specific slowing from multiple nonlexical tasksand multiple analytic methods Journals of Gerontology 50B P202-P211

Hale S Myerson J Smith G A amp Poon L W (1988) Age vari-ability and speed Between-subjects diversity Psychology amp Aging3 407-410

Hale S Myerson J amp Wagstaff D (1987) General slowing ofnonverbal information processing Evidence for a power law Journalof Gerontology 42 131-136

Hertzog C amp Schaie K W (1986) Stability and change in adult in-telligence 1 Analysis of longitudinal covariance structures Psychol-ogy amp Aging 1 159-171

Kail R (1991) Developmental change in speed of processing duringchildhood and adolescence Psychological Bulletin 109 490-501

Kail R (1992) General slowing of information-processing by personswith mental retardation American Journal on Mental Retardation 97333-341

Kail R (1997) The neural noise hypothesis Evidence from process-ing speed in adults with multiple sclerosis Aging Neuropsychologyamp Cognition 4 157-165

Kranzler J H amp Jensen A R (1991) The nature of psychomet-ric g Unitary process or a number of independent processes Intelli-gence 15 397-422

Lawrence B Myerson J amp Hale S (1998) Differential decline ofverbal and visuospatial processing speed across the adult life spanAging Neuropsychology amp Cognition 5 129-146

Lima S D Hale S amp Myerson J (1991) How general is generalslowing Evidence from the lexical domain Psychology amp Aging 6416-425

Luce R D (1986) Response latencies Their role in inferring elemen-tary mental organization New York Oxford University Press

Mayr U Kliegl R amp Krampe R (1994) Timendashaccuracy functionsfor determining process and person differences An application tocognitive aging Cognitive Psychology 26 134-164

Mayr U Kliegl R amp Krampe R (1996) Sequential and coordina-tive processing dynamics in f igural transformations across the lifespan Cognition 59 61-90

McClelland J L (1979) On the time relations of mental processesAn examination of systems of processes in cascade Psychological Re-view 86 287-330

McGill W J (1963) Stochastic latency mechanisms In R D LuceR R Bush amp E Galanter (Eds) Handbook of mathematical psy-chology (Vol 1 pp 309-360) New York Wiley

Myerson J Adams D R Hale S amp Jenkins L (2003) Analysisof group differences in processing speed Brinley plots QndashQ plotsand other conspiracies Psychonomic Bulletin amp Review 10 224-237

Myerson J amp Hale S (1993) General slowing and age-invariance incognitive processing The other side of coin In J Cerella W HoyerJ Rybash amp M Commons (Eds) Adult information processingLimits on loss (pp 115-141) San Diego Academic Press

Myerson J Lawrence B Hale S Jenkins L amp Chen J (1998)General slowing of lexical and nonlexical information processing indementia of the Alzheimer type Aging Neuropsychology amp Cogni-tion 5 129-146

Nebes R D amp Brady C B (1992) Generalized cognitive slowingand severity of dementia in Alzheimerrsquos disease Implications for theinterpretation of response-time data Journal of Clinical amp Experi-mental Neuropsychology 14 317-326

Posner M I (1978) Chronometric explorations of mind The thirdPaul M Fitts lectures Hillsdale NJ Erlbaum

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

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


Sanders A F (1990) Issues and trends in the debate on discrete vs con-tinuous processing of information Acta Psychologica 74 123-167

Schatz J (1998) Cognitive processing eff iciency in schizophreniaGeneralized versus domain-specific deficits Schizophrenia Research30 41-49

Schatz J Hale S amp Myerson J (1998) Cerebellar contributionsto linguistic processing efficiency revealed by focal damage Journalof the International Neuropsychological Society 4 491-501

Sliwinski M J amp Hall C B (1998) Task-dependent cognitive slow-ing in the elderly A meta-analysis using hierarchical linear modelswith random coefficients Psychology amp Aging 13 164-175

Smith G A amp Brewer N (1995) Slowness and age Speedndashaccuracymechanisms Psychology amp Aging 10 238-247

Smith G A Poon L W Hale S amp Myerson J (1988) A regularrelationship between old and young adultsrsquo latencies on their best av-erage and worst trials Australian Journal of Psychology 40 195-210

Teichner W H amp Krebs M J (1974) Laws of visual choice reactiontime Psychological Review 81 75-98

Townsend J T amp Ashby F G (1983) The stochastic modeling of el-ementary psychological processes New York Cambridge UniversityPress

Vernon P A (1983) Speed of information processing and general in-telligence Intelligence 7 53-70

Vernon P A amp Jensen A R (1984) Individual and group differ-ences in intelligence and speed of information processing Personal-ity amp Individual Differences 5 411-423

Vernon P A Nador S amp Kantor L (1985) Reaction times andspeed of information-processing Their relationship to timed and un-timed measures of intelligence Intelligence 9 357-374

White D A Myerson J amp Hale S (1997) How cognitive is psy-chomotor slowing in depression Aging Neuropsychology amp Cogni-tion 4 166-174

Zacks J L amp Zacks R T (1993) Visual search times assessed with-out reaction times A new method and an application to aging Jour-nal of Experimental Psychology Human Perception amp Performance19 798-813

Zheng Y Myerson J amp Hale S (2000) Age and individual differ-ences in visuospatial processing speed Testing the magnification hy-pothesis Psychonomic Bulletin amp Review 7 113-120

(Manuscript received June 1 1999 revision accepted for publication April 30 2002)


on the corresponding mean RTs for the group as a wholethe regression line accounted for more than 98 of thevariance Regressing the slowest quartilersquos mean RTs onthe group mean RTs revealed an equally precise relation-ship (Hale amp Jansen 1994) Similarly precise relationshipswere observed when the mean RTs for the fastest andslowest quartiles were regressed on average RTs as definedby the mean for the two center quartiles again both r2swere greater than 98 As may be seen in Figure 1 the con-sequence of these linear relations is that the difference be-tween the RTs of the subgroups of fast and slow proces-sors increases with the average RT

Because of the strong theoretical implications of thesefindings we followed the old adage ldquomeasure twicemdashcutoncerdquo and assessed their reliability using another largerdata set To this end we combined the data from youngadult controls in previous unpublished and published stud-ies of cognitive aging and development The RTs of 65young adults each of whom had been tested on four tasks(ie choice RT letter classification visual search and ab-stract matching) were reanalyzed using the Hale andJansen (1994) approach Again we found that a principalcomponents analysis of RTs revealed a single generalspeed factor on which all four tasks loaded heavily and

that accounted for a large proportion (75) of the totalstandardized variance (Hale amp Myerson 1993)

Following Hale and Jansen (1994) we selected fast andslow subgroups (bottom and top quartiles) from these 65subjects on the basis of their mean z scores Notably 15 ofthe 16 members of the fast group were faster than averageon all four tasks (the other member was faster than aver-age on three of the tasks) and 15 of the 16 slow groupmembers were slower than average on all four tasks (theother member was slower than average on three tasks)When the mean RTs from the two subgroups were re-gressed on average RTs extremely orderly linear relationswere observed As was the case with the Hale and Jansendata this was true regardless of whether average was de-fined as the mean for the group as a whole or as the meanfor the center quartiles In fact as may be seen in Figure 1the data points representing the Hale and Myerson (1993)data set are collinear with the points representing the Haleand Jansen data set

Further confirmation of this pattern comes from re-analysis of a study by Vernon and Jensen (1984) that usedvery different procedures and samples These authors stud-ied two large samples recruited from students attendingeither a vocational college or a university All subjects weretested on eight different speeded information-processingtasks including choice RT short-term memory scanning

Figure 1 Response times (RTs) of fast and slow subgroupsplotted as a function of the group mean RTs White and graysymbols represent the data from Hale and Jansen (1994) andHale and Myerson (1993) respectively Each point representsperformance by one subgroup in one experimental condition ofone study In order of increasing task difficulty the tasks were asfollows inverted triangles choice RT octagons line-length dis-crimination squares letter classification circles mental rota-tion upright triangles visual search octagons mental paperfolding diamonds abstract matching Each regression line rep-resents the fit to the RTs of the corresponding subgroups (ieboth fast or both slow subgroups) in both studies If a subgroupmean RT for a particular task condition did not differ from thecorresponding group mean RT the data point for that conditionwould fall along the dashed diagonal line

Figure 2 Response times (RTs) of university and vocationalcollege subgroups plotted as a function of the combined studentgroup mean RTs White and gray symbols represent the datafrom Vernon (1983) and Vernon and Jensen (1984) respectivelyEach point represents performance by one subgroup in one ex-perimental condition of one study In order of increasing task dif-ficulty the tasks were as follows circles choice RT (Jensen ap-paratus) octagons Sternberg memory scanning squares samedifferent word judgment diamonds synonymantonym judg-ment If a subgroup mean RT for a particular task condition didnot differ from the corresponding group mean RT the data pointfor that condition would fall along the dashed diagonal line









































2 s 22 sn


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SD 5 s (RTa)12 (3)


750

500

250

00 500 1000 1500 25002000


Bet

wee

n-S

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cts

SD

(m

sec)





SD 5 sRTa (4)



SD 5 s [n + n(n 2 1)r]12 (5)




















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VAR 5 sr2 1 nsc

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DISCUSSION
















































REFERENCES
































































































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SD 5 s (RTa)12 (3)


750

500

250

00 500 1000 1500 25002000


Bet

wee

n-S

ubje

cts

SD

(m

sec)





SD 5 sRTa (4)



SD 5 s [n + n(n 2 1)r]12 (5)




















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2 yielding

VAR 5 sr2 1 nsc

2 1 n(n 2 1)rs c2 (6)








CRT 0405 0061 833 2470 292 911LTR 0644 0103 701 880 2197 512 743SRC 1079 0233 583 657 863 321 879 321MAT 1711 0511 608 679 740 878 287 840 385




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2 ](RT 2 tr) a

1 rsc2[(RT 2 tr) a]2 (7)




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SD 5 (r 2 sc a)(RT 2 tr) (9)




























































































DISCUSSION
















































REFERENCES


























































SD 5 sRTa (4)



SD 5 s [n + n(n 2 1)r]12 (5)




















2 1s2




2 yielding

VAR 5 sr2 1 nsc

2 1 n(n 2 1)rs c2 (6)








CRT 0405 0061 833 2470 292 911LTR 0644 0103 701 880 2197 512 743SRC 1079 0233 583 657 863 321 879 321MAT 1711 0511 608 679 740 878 287 840 385




VAR 5 s r2 1 [(1 2 r)sc

2 ](RT 2 tr) a

1 rsc2[(RT 2 tr) a]2 (7)




VAR 5 rs c2[(RT 2 tr) a ]2 (8)


SD 5 (r 2 sc a)(RT 2 tr) (9)




























































































DISCUSSION
















































REFERENCES

































































2 1s2




2 yielding

VAR 5 sr2 1 nsc

2 1 n(n 2 1)rs c2 (6)








CRT 0405 0061 833 2470 292 911LTR 0644 0103 701 880 2197 512 743SRC 1079 0233 583 657 863 321 879 321MAT 1711 0511 608 679 740 878 287 840 385




VAR 5 s r2 1 [(1 2 r)sc

2 ](RT 2 tr) a

1 rsc2[(RT 2 tr) a]2 (7)




VAR 5 rs c2[(RT 2 tr) a ]2 (8)


SD 5 (r 2 sc a)(RT 2 tr) (9)




























































































DISCUSSION
















































REFERENCES


























































VAR 5 s r2 1 [(1 2 r)sc

2 ](RT 2 tr) a

1 rsc2[(RT 2 tr) a]2 (7)




VAR 5 rs c2[(RT 2 tr) a ]2 (8)


SD 5 (r 2 sc a)(RT 2 tr) (9)




























































































DISCUSSION
















































REFERENCES













































































































































DISCUSSION
















































REFERENCES





































































































REFERENCES
























































The difference engine: A model of diversity in speeded cognition

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Transcript of The difference engine: A model of diversity in speeded cognition