eric - education resources information centerphilosopher of science, terry l. smith, discussed this...

MATCHING IN AN UNDISTURBED NATURAL HUMAN ENVIRONMENT

J. J MCDOWELL AND MARCIA L. CARON

EMORY UNIVERSITY

Data from the Oregon Youth Study, consisting of the verbal behavior of 210 adolescent boys determinedto be at risk for delinquency (targets) and 210 of their friends (peers), were analyzed for theirconformance to the complete family of matching theory equations in light of recent findings from thebasic science, and using recently developed analytic techniques. Equations of the classic and moderntheories of matching were fitted as ensembles to rates and time allocations of the boys’ rule-break andnormative talk obtained from conversations between pairs of boys. The verbal behavior of each boy in aconversation was presumed to be reinforced by positive social responses from the other boy. Consistentwith recent findings from the basic science, the boys’ verbal behavior was accurately described by themodern but not the classic theory of matching. These findings also add support to the assertion thatbasic principles and processes that are known to govern behavior in laboratory experiments also governhuman social behavior in undisturbed natural environments.

Key words: generalized matching law, Herrnstein’s hyperbola, modern matching theory, human socialbehavior, verbal behavior, juvenile delinquency

_______________________________________________________________________________

The application of basic behavior-analyticprinciples to socially relevant human behaviorhas most often taken the form of therapeuticintervention, where basic principles are usedto engineer human social environments toproduce desired outcomes (e.g., O’Leary &O’Leary, 1977). Far less common are purelyobservational studies in natural environments,where the question under investigation iswhether basic principles discovered in thelaboratory also govern behavior in unengi-neered natural environments. Applied re-search has shown that many basic behavior-analytic principles can be used to changeclinically significant human behavior in socialenvironments. Although such findings areobviously of practical value, it does notnecessarily follow from them that the basicprinciples are also operating in the absence ofspecific interventions. From a basic scienceperspective this is an important issue. Specif-ically, do the variables and processes that

govern behavior in laboratory environmentsalso govern naturally occurring behavior inundisturbed environments? It might seem thatthe second of Newton’s well known rules ofscientific reasoning would be sufficient toanswer this question in the affirmative:

To the same natural effects we must, as far aspossible, assign the same causes. As to respirationin a man, and in a beast; the descent of stones inEurope and in America; the light of our culinaryfire and of the sun… (Newton, 1999/1726).

This rule is usually interpreted to mean that,in the absence of contrary evidence, likephenomena must have like causes. Hence, justas the light of our culinary fire and of the sunmust have the same causes and must operatein the same way, so too must adaptive behaviorin the laboratory and in the natural environ-ment. This rule will be sufficient for manystudents of human behavior, as it certainly wasfor Skinner, judging from books like Scienceand Human Behavior. But the caveat in the ruleis the condition, ‘‘in the absence of contraryevidence.’’ Many philosophers and psycholo-gists would no doubt argue that behavior innatural environments, especially human be-havior in natural social environments, may bejust the sort of phenomenon where contraryevidence, such as intentional choice, could befound. This makes it necessary to demonstratespecifically that the basic principles governbehavior in both engineered environmentsand undisturbed natural environments.

We are grateful to the Oregon Social Learning Centerfor access to the high quality data set from the OregonYouth Study that was analyzed in this study. We benefittedfrom discussions with John Berg, Saule Kulubekova, andAndrei Popa, and we thank Andrei Popa for his commentson an earlier version of the manuscript. We also thankTom Critchfield, Tony Nevin, and two anonymousreviewers for their careful consideration of our submittedmanuscript, and for their helpful guidance in revising it.

Address correspondence to either author at Dept. ofPsychology, Emory University, Atlanta, GA 30322 (e-mail:[email protected]; or [email protected]).

doi: 10.1901/jeab.2010.93-415

JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR 2010, 93, 415–433 NUMBER 3 (MAY)

415

Philosopher of science, Terry L. Smith,discussed this issue in the context of theanalogy between operant conditioning andevolution by natural selection (McDowell,1991, 2004: Skinner, 1981; Staddon & Simmel-hag, 1971). Smith (1983) pointed out thatselective animal breeding, that is, artificial orengineered selection, was known to be usefulin producing desired animal phenotypes longbefore it was suggested that the same processmight occur naturally and be responsible forphenotypic differences among species. Giventhe known results of selective animal breeding,one might think again that Newton’s secondrule of scientific reasoning would occasionwidespread support for the idea of naturalselection, but this did not happen. Evidently,the caveat in the rule also was salient in thecase of natural selection. The diversity ofspecies may be just the sort of phenomenonwhere contrary evidence, such as independentacts of creation, could be found. It was notuntil Darwin’s extensive documentation in Onthe Origin of Species that the idea of naturalselection began to be taken seriously. Smith(1983) noted that

Darwin discussed animal breeding in ChapterOne of a fifteen chapter book [On the Origin ofSpecies]—as a way of motivating the plausibilityof his claim that natural selection plays acentral role in the origin of species. The nextfourteen chapters, however, contained themain argument, which was designed to showthat natural contingencies play the same role asthe artificial contingencies practiced by breed-ers (p. 151, italics in the original).

Behavior analysts are still mostly at work ontheir Chapter One. There is no doubt that it isan extensive and valuable Chapter One,however, and that it is an important sourcefor the remaining chapters, the main argu-ment.

A large number and wide variety of basicbehavior-analytic phenomena, principles, andtheories can be studied in unengineerednatural environments. Among the most chal-lenging of these are the mathematical ac-counts that began appearing in the basicscience literature around 1960, and that havebecome increasingly prevalent. Probably themost successful mathematical account in basicbehavior analysis is matching theory, althoughthere are many others, such as contingencydiscriminability theory (Davison & Jenkins,

1985), behavioral momentum theory (Nevin,1992), stay-switch theory (MacDonall, 2000),and hyperbolic discounting theory (Mazur,1997), to name a few. Matching theory consistsof a family of equations, some with absoluteresponse rates as dependent variables, andsome with relative response rates, that is,proportions or ratios of response rates, asdependent variables. A few studies haveevaluated absolute response rate forms ofmatching theory in natural unengineeredenvironments (e.g., Fernandez & McDowell,1995; Martens & Houk, 1989; McDowell, 1981,1982), and a few have evaluated relativeresponse rate forms of matching theory insuch environments (e.g., Borrero & Vollmer,2002; Houston, 1986; Oliver, Hall, & Nixon,1999; Snyder & Patterson, 1995; Symons,Hoch, Dahl, & McComas, 2003). Behaviorconsistent with the matching theory equationswas reported in all of these studies. Reed,Critchfield and Martens (2006) and Vollmerand Bourret (2000) found that the behavior ofathletes in sporting events was also consistentwith a relative response rate equation ofmatching theory. Although athletic competi-tions take place in unengineered environ-ments, they entail the additional constraintsof game rules, which complicate the interpre-tation of findings from these competitions.

In the research presented in this article, theverbal behavior of 210 adolescent boys whowere determined to be at risk for delinquentbehavior (targets) and 210 of their friends(peers) was studied. Pairs of boys engaged inconversations and the verbal behavior of eachtarget boy and each peer boy was recorded andcategorized as either ‘‘rule-break’’ or ‘‘norma-tive’’. Positive social responses from each boyto the other were also recorded. These datawere then analyzed for their conformance tothe complete family of matching theoryequations in light of recent findings from thebasic science, and using recently developedanalytic techniques. Previous researchers havereported that human verbal behavior support-ed by social consequences in engineeredenvironments was well described by matchingtheory (Borrero et al., 2007; Conger & Killeen,1974; Pierce, Epling, & Greer, 1981). Theprincipal purpose of the present article is toadd to and extend the literature on thegovernance by matching theory of naturallyoccurring human behavior, which in this case

416 J. J MCDOWELL and MARCIA L. CARON

happens to be verbal behavior. A secondpurpose is to introduce applied researchersto modern theoretical developments, and tonew analytic procedures from the basic sciencethat are relevant to applications of matchingtheory. And a final purpose is to add to thebasic-science literature that compares theclassic version of matching theory with amodern version of the theory.

Classic and Modern Matching Theory

The classic theory of matching consists ofthree equations, namely,

B1

B2~

r1

r2, ð1Þ

B1~kr1

r1zr2zre, ð2Þ

and

B2~kr2

r1zr2zre, ð3Þ

where the Bs in all equations representresponse rates, the rs represent reinforcementrates, and the numerical subscripts refer toeach of two response alternatives. Equation 1 isthe original matching equation (Herrnstein,1961) expressed in ratio form, and states thatorganisms allocate their behavior to each oftwo response alternatives in the same propor-tion or, equivalently, in the same ratio, asreinforcement is distributed across thosealternatives. Equations 2 and 3 express theabsolute response rate on each of the twoconcurrently available alternatives as a func-tion of the rates of reinforcement obtainedfrom both alternatives. These single-alternativeforms were derived from Equation 1 byHerrnstein (1970), who assumed that there isan ever present rate of background behavior,Be , supported by a rate of background rein-forcement, re , in a two-alternative environ-ment, and that the sum of the rates ofresponding on the two alternatives and onthe background alternative, B1 + B2 + Be, isconstant with respect to changes in reinforce-ment properties. This sum is represented bythe parameter, k, in Equations 2 and 3(McDowell, 1986). This constant-k assumptionis necessary in order to obtain Equations 2and 3 from Equation 1. Note that when thereis no second source of arranged reinforcement

(r2 5 0), Equation 2 reduces to the familiarHerrnstein hyperbola (Herrnstein, 1970) forsingle-alternative responding. Similarly, whenthere is no first source of arranged reinforce-ment (r1 5 0), then Equation 3 also reduces tothe familiar Herrnstein hyperbola. Equations1–3 and the derivation of Equations 2 and 3 arediscussed in detail by McDowell (1986). Equa-tions 1–3 can also be expressed in timeallocation forms, where the dependent vari-ables are T1 and T2, the amount of time spentengaging in each alternative. Note finally thatEquations 1–3 contain two parameters, k and re .

All translational applications of matchingtheory either have entailed Equation 1, some-times expressed in an equivalent proportionalform, and sometimes expressed in a powerfunction form (as in Equation 4 below), orthey have entailed a single-alternative equationlike Equations 2 and 3, with one source ofreinforcement contingent on the behavior ofinterest, as just described. No translationalapplications of matching theory have exam-ined Equations 2 and 3 in the context of aconcurrent schedule and, interestingly, veryfew basic-science applications have done so.Yet the strongest test of matching theory ondata from concurrent schedules entails con-sidering all three equations, one that describesthe ratio of response rates, and two thatdescribe the absolute rates of responding inthe two components (McDowell, 1986). This isa strong test of the theory because it engagesthe theory’s constant-k assumption, and alsobecause it constrains the data more severelythan does considering Equation 1 alone. Notethat no additional experimentation or datacollection is necessary to consider all threeequations, as compared to considering justEquation 1. Whatever data are collected to testEquation 1 are simply further constrained byapplying Equations 2 and 3 as well.

A key vulnerability of classic matchingtheory is its constant-k assumption (Herrn-stein, 1974; McDowell, 1986). As noted earlier,this assumption is required in order to deriveEquations 2 and 3 from Equation 1. But asearly as 1984 experimental evidence began toappear that suggested the assumption did nothold (McDowell & Wood, 1984). In the past20 years or so, a considerable amount ofevidence has accumulated indicating that theassumption is in fact false (McDowell, 2005,but cf. McLean, 2006). But if the constant-k

MATCHING IN AN UNDISTURBED NATURAL HUMAN ENVIRONMENT 417

assumption is false, then Equations 2 and 3cannot be derived legitimately from Equation1, and this means that the so-called quantita-tive law of effect (Herrnstein, 1970), which is aform of Equations 2 and 3, as explainedearlier, does not follow from Equation 1. Ithas been known for some time that Equation 1fails to describe much of the data from two-alternative experiments (Baum, 1974, 1979;Wearden & Burgess, 1982). Based in part onthe extensive literature showing the limitedapplicability of Equation 1, and on theexperimental evidence against the constant-kassumption, McDowell (2005) concluded thatall three equations of classic matching theorywere false, or at best applied only in limitedcircumstances.

The sole survivor of the disconfirmatoryevidence is the power-function version ofEquation 1, which was discussed by Staddon(1968) and Baum and Rachlin (1969), and wasformalized and studied extensively by Baum(1974, 1979), and later by Wearden andBurgess (1982). McDowell (1986) noted thatthe logic Herrnstein (1970) used to obtainsingle-alternative forms from Equation 1 alsocould be used to obtain single-alternativeforms from the power function version ofEquation 1. In other words, matching theorycould be rebuilt on a new foundationalequation that was known to accurately describea large body of data from many vertebratespecies, including humans (Baum, 1974, 1979;Wearden & Burgess, 1982).

Like the classic version of matching theory,the rebuilt, or modern, version also consists ofthree equations, which are

B1

B2~

c2e

c1e

r1

r2

� �a

, ð4Þ

B1~k c1e1

r1

� �a

zc1e

c2e

r2

r1

� �a

z1

� �{1

, ð5Þ

and

B2~k c2e1

r2

� �a

zc2e

c1e

r1

r2

� �a

z1

� �{1

, ð6Þ

where the Bs, rs, and subscripts, including thee subscript, which refers to the backgroundalternative, have the same meanings as inEquations 1–3. Equation 4 is the familiarpower function matching equation, also re-

ferred to as the generalized matching law. It isthe foundation of the modern theory ofmatching, and introduces to the classic Equa-tion 1 an exponent, a, on the reinforcementrate ratio, and a bias parameter, here repre-sented by c2e/c1e , as a factor of the exponen-tiated reinforcement rate ratio. The single-alternative Equations 5 and 6 were obtainedfrom Equation 4 by McDowell (1986, 2005)using the Herrnstein (1970) logic, as notedearlier. The parameter, re , is incorporated inthe two c parameters in Equations 5 and 6because it is not independently estimable inthese equations (McDowell, 2005). Equations5 and 6 also include a simplification recom-mended by McDowell (2005), namely, that thethree exponents appearing in their originalderivation be made equal.

Although Equations 5 and 6 are notationallycomplicated, they are conceptually straightfor-ward. Just as for Equations 2 and 3, responserates in Equations 5 and 6 are functions ofreinforcement rates obtained from both alter-natives of the concurrent schedule, but now theexponent, a, that appears in Equation 4 iscarried over by the derivation to the single-alternative forms. In addition, there are threeparameters that include bias, one for eachcomparison of a target alternative with thebackground, c1e and c2e , and one for thecomparison of the two target alternatives witheach other. McDowell (2005) showed that thisthird bias parameter necessarily equals c2e/c1e

when the recommended exponent simplifica-tion is used. Notice that Equations 4, 5, and 6reduce to Equations 1, 2, and 3 when theexponent and all bias parameters in theequations equal unity. As noted for Equations1–3, Equations 4–6 can also be expressed withtime allocation as the dependent variable. Notefinally that Equations 4–6 contain four indepen-dently estimable parameters, k, a, c1e , and c2e .

McDowell (2005) showed that the moderntheory of matching predicts the violation ofthe constant-k assumption in the classic theory,and accurately describes at least some data thatviolate that theory’s constant-k assumption. Inaddition, Dallery, Soto, and McDowell (2005)directly tested the classic and modern theoriesof matching on data from human subjectsworking on concurrent schedules of monetaryreinforcement in the laboratory. They used anensemble method (McDowell, 2005) to f it allthree equations of the classic and the modern


theories simultaneously to sets of data fromindividual subjects using shared parameters.Dallery et al. found that the modern, but notthe classic, theory of matching accuratelydescribed the behavior of their subjects.

Just as for the classic theory, the strongesttest of the modern theory of matching entailsconsidering all three of its equations. In thepresent article, Equations 1–3 and Equations4–6 were fitted as ensembles to the boys’verbal-behavior data. To be consistent withfindings from the basic science, the latter, butnot the former set of equations must describethe data well.

METHOD

All procedures described in this section,including the coding of videotapes, wereconducted by researchers at the Oregon SocialLearning Center (OSLC).

Subjects

The subjects were 210 13- to 14-year-old boysparticipating in the fifth phase of the longitu-dinal Oregon Youth Study (OYS; Capaldi &Patterson, 1987), plus 210 similarly-aged malefriends who served as their partners. Each pairconsisted of one target child from the OYS anda friend, the peer child, who participated inthis portion of the study only. The target boyshad been recruited for the OYS at ages 9 and10 in two waves during 1983 and 1984 from 10elementary schools in high-crime neighbor-hoods of a medium-sized city in the northwest-ern United States. These boys were consideredto be at risk for juvenile delinquency becausethey came from neighborhoods with higherthan average rates of juvenile delinquency.The resulting samples were 90% EuropeanAmerican, of lower socioeconomic status, andwith a relatively high percentage of unem-ployed parents. The peer boys were selected byasking each target boy and his parents to namethe male friend with whom the target childspent the most time. Boys from the first of twowaves of the OYS comprised sample e5 (n 5129); boys from the second wave comprisedsample n4 (n 5 81).

Procedure

During the first four phases of the OYS, thetarget boys and their parents completed

various surveys and experimental tasks onmultiple occasions at the OSLC. During bothwaves of the fifth phase, each pair of boysparticipated in a videotaped 25-min session. Atthe start of each 25-min session, a researcherseated the 2 boys in front of a video camera setup in a room at the OSLC. In accordance withthe directions of the Peer Interaction Task(Forgatch, Fetrow, & Lathrop, 1985; Panella &Henggeler, 1986), the researcher told the boysthat they were to engage in conversation for25 minutes about a variety of topics to beannounced by the researcher. Each sessionstarted with a 5-min warm-up discussion aboutplanning an activity together, followed by four5-min segments with four randomly-ordered,assigned problem-solving discussions relatedto self-selected problems with the target child’sand peer child’s parents and peers. Theresearcher was present in the room only tointroduce each topic; during the discussionsthe boys were alone.

Videotapes of the boys’ conversations weretranscribed and coded by trained observers.The boys’ verbal behavior was coded into twomutually exclusive categories: rule-break talkand normative talk. As defined for this study,rule-break talk contained some element thatindicated a violation of legal and/or conven-tional norms of conduct. Examples of rule-break talk included climbing out of bedroomwindows, lying to parents, cheating at school,engaging in criminal behavior, and behavingdefiantly. Examples of normative talk includedgossiping about friends, and talking aboutschool, family, or what to do for fun. Thesecategories constituted two mutually exclusive,concurrently available response alternatives.The frequencies and durations of verbal boutswere recorded for each boy. Each bout beganwhen a boy started speaking and ended whenthe boy stopped speaking. This permitted thecalculation of both response rates and timeallocations.

In addition to coding for content, socialconsequences from the ‘‘other’’ boy (namely,the peer child when the target child’s behaviorwas under consideration, and the target childwhen the peer child’s behavior was underconsideration) for bouts of both rule-breakand normative talk were also coded into twomutually exclusive categories: positive socialresponses and nonpositive social responses. Apositive social response was assumed to rein-


force a bout of talk if it immediately followedthat bout in the behavior stream. Examples ofpositive social responses included makingapproving statements, smiling, nodding,laughing, and giving a thumbs-up. Nonpositivesocial responses included anything other thanpositive social responses, including remainingsilent. Frequencies of positive social responsesfor each category of verbal behavior wererecorded for each boy. This permitted thecalculation of reinforcement rates for eachcategory of verbal behavior.

Samples e5 and n4, which corresponded tothe two waves of the OYS, were codedseparately by two sets of coders following theinstructions from the Topic Code (Poe, Dish-ion, Griesler, & Andrews, 1990). Coders wereblind to all other data related to the partici-pants, and they were involved in only thecoding of the data. Reliability scores werecalculated for a randomly-selected 12% ofpairs in sample e5 and 15% of pairs in samplen4. Mean percent agreements of 90.0 and 94.2,and kappa coefficients of 0.735 and 0.674 wereobtained for samples e5 and n4, respectively(Dishion, Spracklen, Andrews, & Patterson,1996).

RESULTS

All analyses were conducted using datasupplied by the OSLC in the form of SPSSfiles. Data were missing from 15 target boys inthe e5 sample and from 1 peer boy in the n4sample. The entire 20 minutes of conversationfor each boy was used to generate a single datarecord for that boy, consisting of two rein-forcement rates, r1 and r2, two response rates,B1 and B2, and two time allocations, T1 and T2.Equations 1–3 and Equations 4–6 were fitted asensembles to the response rates and timeallocations of the e5 target boys (n 5 114), thee5 peer boys (n 5 129), the n4 target boys(n 5 81), and the n4 peer boys (n 5 80).Hence, the data were separated into foursubsets, a target-boy and a peer-boy subsetfrom each sample. During their conversations,the target boys’ verbal behavior was presumedto be reinforced by the peer boys’ positivesocial responses, and the peer boys’ verbalbehavior was presumed to be reinforced by thetarget boys’ positive social responses, as de-scribed in the Methods section. In all analyses,rule-break talk was taken as response alterna-

tive 1 and normative talk was taken as responsealternative 2.

Ensemble fits of the classic theory to eachsubset of the data entailed finding a singleshared k and re for Equations 1–3 thatminimized a quantity that might be referredto as the normalized total residual sum ofsquares, RSSNT, given by

RSSNT ~RSS1

SS1z

RSS2

SS2z

RSS3

SS3, ð7Þ

where the RSSs represent residual sums ofsquares, the SSs represent total sums ofsquares, and the numerical subscripts refer toEquations 1 through 3. For example, RSS2 isthe sum of the squared residuals for the fit ofEquation 2 to a dependent variable (B2 or T2),and SS2 is the sum of the squared deviations ofthe dependent variable from its own mean.The logarithmic transformation of Equation 1was used in all fits, as is customary in the basicscience. This method of obtaining a single setof shared parameter estimates by simulta-neously fitting a set of equations to multiplesources of variance was developed and dis-cussed in detail by McDowell (2005). Percent-ages of variance accounted for (%VAF) byeach equation were calculated from theindividual RSSs left by the ensemble fit. Inaddition, an overall %VAF for the ensemblewas calculated from

%VAF~ 1{RSS1zRSS2zRSS3

SS1zSS2zSS3

� �|100: ð8Þ

This %VAF evaluates the sum of the indi-vidual residual sums of squares against the sumof the individual total sums of squares, andthus provides an overall assessment of thequality of the ensemble fit (McDowell, 2005).The same ensemble method was used tosimultaneously fit the three equations of themodern theory, yielding a single sharedestimate of k, a, c1e , and c2e , for each subsetof the data. The logarithmic transformation ofEquation 4,

logB1

B2

� �~a log

r1

r2

� �z log

c2e

c1e

� �, ð9Þ

was used in all fits, as is customary in the basicscience. Notice that the slope of the best-fittinglog transform is the exponent, a, of Equation4, and that the antilog of its y-intercept is the


bias parameter, in this case, c2e/c1e . A detailedexample of using this ensemble method ofcurve fitting is given in the Appendix.

Goodness of Fit

Listed in Table 1 are the percentages ofvariance accounted for by each set of equa-tions, and its ensemble, for the response ratesand time allocations from each of the fourdata subsets, e5-Target, e5-Peer, n4-Target,and n4-Peer. The %VAFs in the e5-Averageand n4-Average rows of the table will bediscussed later. The majority of %VAFs forthe equations of classic matching theory andtheir ensembles were negative for the fourdata subsets (11 of 16 for both response ratesand time allocations). A negative %VAF meansthe data deviate more from the fitted functionthan from their own mean, in which case ofcourse the mean is a better predictor of thedata than the fitted function. The average%VAFs across the four subsets of data for eachequation and for the ensemble are plotted onthe left sides of the top and bottom panels inFigure 1. These are not the quantities thatappear in the e5-Average and n-4 Average rowsof Table 1, which refer to a different kind ofaveraging that will be described later. As shownin the figure, on average, the equations ofclassic matching theory either accounted foressentially no variance in the response ratesand time allocations (Equations 1 and 2), oraccounted for a negative amount of variance.

Hence, in none of these cases did the fittedfunction on average predict the data betterthan the mean.

The %VAFs for the equations of modernmatching theory and their ensembles are alsolisted in Table 1. Over the four data subsetsthey ranged from 25% to 61% with a medianof 48% for the response rates, and from 4% to56% with a median of 43% for the timeallocations. The average %VAFs across thefour data subsets for each equation and theirensemble are plotted on the right sides of thepanels in Figure 1. Again, these are not thequantities that appear in the e5-Average andn4-Average rows of Table 1. As shown in thefigure, the equations of modern matchingtheory and their ensemble on average ac-counted for about half the variance in theboys’ verbal response rates and time alloca-tions.

Properties of the Boys’ Verbal Behavior

Parameter estimates from the ensemble fitsof the classic and modern matching theoryequations are listed in Table 2 for fits toresponse rates and time allocations. Again,the parameter estimates in the e5-Average andn4-Average rows of the table will be discussedlater. Notice that estimates of the sharedexponent, a, in Equations 4–6 for the fourdata subsets consistently fell between about 0.7and about 0.9, and averaged about 0.8 for bothresponse rates and time allocations. This

Table 1

Percentages of variance accounted for by Equations 1, 2, 3, and their ensemble, (the classictheory), and by Equations 4, 5, 6, and their ensemble (the modern theory) for response and timeallocation data.

Sample-Child

Classic Modern

Eq. 1 Eq. 2 Eq. 3 Ensemble Eq. 4 Eq. 5 Eq. 6 Ensemble

Responses

e5-Target 218 216 248 237 49 59 41 46e5-Peer 229 212 276 258 55 56 37 42n4-Target 28 15 243 225 49 59 25 36n4-Peer 39 36 10 17 61 61 27 37e5-Average 220 234 2121 293 97 95 82 86n4-Average 46 25 22 0 92 96 85 90

Time

e5-Target 23 28 228 221 47 50 40 42e5-Peer 220 223 258 248 51 49 40 42n4-Target 24 19 285 233 44 56 4 30n4-Peer 30 9 21 17 55 46 41 43e5-Average 6 43 282 268 94 92 92 92n4-Average 46 4 250 221 87 95 72 84


means that the boys’ verbal behavior was notperfectly controlled by the reinforcement rateratios (in which case the exponent wouldequal unity), a finding that is referred to asundermatching, and is almost always obtainedin laboratory experiments with human andanimal subjects (Baum, 1974, 1979; Davison &McCarthy, 1988; Wearden & Burgess, 1982).Also noteworthy is the ratio of the two cparameters, c2e/c1e, which consistently fell

between about 0.3 and about 0.5, indicatinga strong bias in favor of normative talk (recallthat rule-break talk is in the numerator ofEquation 4) for all boys in both samples andfor both response rates and time allocations.This means that all boys had a constantproportional preference for normative talkover rule-break talk.

Figure 2 is a plot of the response rate (toppanels) and time allocation (bottom panels)data for the e5 target boys, along with plots ofEquations 4, 5 and 6 from the ensemble fit.Because the latter two equations are functionsof two variables they describe surfaces, asshown in the center and right panels of thefigure. The %VAFs and parameters for all fitsin Figure 2 are given in Tables 1 and 2.

The straight lines in the left panels ofFigure 2 are plots of the log transforms ofEquation 4 from the ensemble fit. Notice thatfor both response rates and time allocationsthe line intersects the y-axis at a negative value,reflecting the strong bias in favor of normativetalk (which appears in the denominator of theratios). The slopes of these lines are alsoslightly less than one, reflecting exponents ofabout 0.8. The scatter around the fitted linesreflects the %VAFs of about 50%.

The coordinate origins of the surface plotsin the center and right panels of Figure 2 arethe bottom front corners where the two zerotick labels are adjacent. This is the front cornerof the reinforcement rate floor. Reinforce-ment rate for rule-break talk, r1, increasesalong the right edges of the floor, andreinforcement rate for normative talk, r2,increases along the left edges. The vertical z-axes represent response rates (top panels) ortime allocations (bottom panels). Notice thatthe response rates and time allocations ofthese boys’ rule-break talk (center panels)increased with the rate of reinforcement forrule-break talk provided by their peers, r1, anddecreased with the rate of reinforcement fornormative talk, r2. This decrease in respondingon one alternative of a concurrent schedule asthe reinforcement rate increases on the otheralternative is the well known contrast effectthat is routinely observed on concurrentschedules in laboratory experiments.

The plots in the right panels of Figure 2show that the rate and time allocation of theboys’ normative talk increased with the rate ofreinforcement for normative talk (r2) provided

Fig. 1. Percentages of variance accounted for (%VAF)by classic (left portions) and modern (right portions)matching theory equations and their ensembles, averagedover the four subsets of data for response rates (top panel)and time allocations (bottom panel). Error bars represent6 1 standard error.


Table 2

Shared parameter estimates from ensemble fits of Equations 1, 2, and 3 (first two columns) andfrom ensemble fits of Equations 4, 5, and 6 (columns 3 through 6). The ratios of the two cparameters are listed in the last column.

Sample-Child

Classic Modern

c2e/c1ek re k a c1e c2e

Responses

e5-Target 8.00 2.26 9.14 0.76 3.03 1.01 0.33e5-Peer 8.52 6.11 9.14 0.72 3.57 1.11 0.31n4-Target 7.58 3.41 8.41 0.76 3.32 1.13 0.34n4-Peer 7.60 1.91 8.19 0.88 3.30 1.53 0.46e5-Average 7.88 4.75 9.63 0.64 4.29 1.42 0.33n4-Average 7.84 1.98 8.64 0.74 3.68 1.48 0.40

Time

e5-Target 25.60 0.34 26.78 0.83 0.00 0.00 0.35e5-Peer 27.41 3.56 28.88 0.78 1.91 0.59 0.31n4-Target 26.12 0.40 25.60 0.74 0.00 0.00 0.33n4-Peer 28.21 2.65 30.04 0.87 5.78 2.48 0.43e5-Average 26.08 2.17 29.58 0.69 2.21 0.76 0.34

n4-Average 27.19 0.69 27.55 0.71 1.19 0.47 0.40

Fig. 2. Response rates (top panels) and time allocations (bottom panels) for the e5 target boys, along with plots ofthe best fitting Equations 4, 5, and 6. The percentages of variance accounted for, parameter values, and residual analysesfor these fits are given in Tables 1, 2, and 3. Bias in favor of normative talk is indicated by the negative y-intercept of thebest fitting log transform of Equation 4 in the left panels, and by the generally higher levels of normative than rule-breaktalk (right panels vs. center panels).


by their peers, and decreased with the rate ofreinforcement their peers provided for rule-break talk (r1), again showing the expectedcontrast effect. Notice the generally higherlevels of normative talk, shown in the rightpanels, than rule-break talk, again reflectingthe strong bias in favor of normative talk. Plotsof the other three data subsets showed similarfeatures, including the contrast effect that iscommonly observed in laboratory experi-ments, and a strong bias for normative talk.

Residuals

While the equations of modern matchingtheory appeared to provide a reasonabledescription of these boys’ verbal behavior, itis important to examine the residuals forsystematic trends that would indicate incorrectfunction forms. Standardized residuals werecalculated for each fit and were plotted againstthe dependent variable predicted by the fittedequation. These plots were then examined forlinear trends by calculating the correlationbetween the standardized residual and thedependent variable predicted by the equation,which is a common method of testing forlinear trends. Correlation coefficients for theclassic and modern fits are listed in Table 3.For the classic theory, 23 of the 24 correlationsfor the four data subsets were negative. Themean correlations across the four data subsetsare plotted in the left portions of Figure 3 forresponse rates (top panel) and time alloca-tions (bottom panel). The mean correlation

over all four data subsets and fits was 20.34 forthe response rates and 20.32 for the timeallocations. Given that the fits of the classictheory equations failed to describe these datawell, it is not surprising that they also showedstrong trends in their residuals. The correla-tions for the modern theory fits varied aroundzero, as is evident from the entries in Table 3.The mean correlations across the four datasubsets are plotted in the right portions ofFigure 3 for each equation. The mean corre-lation over all data subsets and fits was zero totwo decimal places for both response rates andtime allocations. Hence, there was no evidencefor systematic trends in the residuals left by fitsof the modern theory equations.

Averages Over a Quartile Grid

The modern theory of matching provided areasonable description of these boys’ verbalbehavior. In addition, fits of the equationsevidently left random residuals, which indicat-ed that the boys’ verbal behavior conformed tothe equations’ specific function forms. Adisappointing feature of these analyses, how-ever, was that the equations of modernmatching theory accounted for only abouthalf the variance in the response rate and timeallocation data. In laboratory experiments,including experiments with human subjects,at least 80% of the variance is typicallyaccounted for by matching theory equations,and it is common to find 90% or more of thevariance accounted for (Baum, 1974, 1979;

Table 3

Correlations of standardized residuals with values predicted by ensemble fits of Equations 1–3(the classic theory) and ensemble fits of Equations 4–6 (the modern theory).

Sample-Child

Classic Modern

Eq. 1 Eq. 2 Eq. 3 Eq. 4 Eq. 5 Eq. 6

Responses

e5-Target 20.31 20.59 20.24 0.01 20.06 0.04e5-Peer 20.32 20.58 20.30 0.07 20.06 20.02n4-Target 20.26 20.53 20.28 0.04 0.00 20.10n4-Peer 20.03 20.46 20.13 0.13 20.08 20.01e5-Average 20.97 20.52 20.94 20.21 0.10 0.13n4-Average 20.94 20.72 20.67 20.17 20.01 0.22

Time

e5-Target 20.18 20.49 20.30 0.00 20.08 0.03e5-Peer 20.20 20.54 20.26 0.07 20.13 0.06n4-Target 20.19 20.42 20.61 0.09 0.22 20.29n4-Peer 0.03 20.51 20.21 0.18 20.13 20.02e5-Average 20.94 20.75 20.85 20.29 0.28 0.14n4-Average 20.91 20.83 20.55 0.31 20.26 0.32


Davison & McCarthy, 1988; Wearden & Bur-gess, 1982). An important difference betweenthese verbal behavior data and typical labora-tory data is that the former are between-subjects data whereas the latter are almostalways within-subject data. It may be that theunaccounted-for variance in the boys’ data wasdue to individual differences in the between-subjects data sets. Indeed, fits to within-subjectdata from laboratory experiments typicallyproduce at least slightly different parameterestimates for different individuals, whichwould add error variance to a between-subjects

data set that might be constructed from within-subject data from multiple subjects.

One way to remove presumably randomindividual differences from a data set is toaverage across subjects. This is problematic inthe case of the present data because it entailsdependent variables that are functions of twoindependent variables. This means that it isnot possible to average over simple indepen-dent–dependent variable pairs. One approachto averaging across subjects in this type of dataset is to divide the range of each of the twoindependent variables into quartiles. Thisproduces a grid of 16 rectangles of varyingsizes on the reinforcement rate floors (of, e.g.,plots of Equations 2, 3, 5, and 6). Each pointfalling in a rectangle has associated with itvalues of r1, r2, B1, B2, T1, and T2 for oneparticipant. Averaging each of these valuesover all points that fall in a given rectangleproduces a single data record that is anaverage over participants whose independentvariable values fall in that rectangle of the grid.This method is preferable to using a grid withequal-sized square cells because, for thepresent data at least, the equal-grid methodproduces cells with very different ns and leavesmany cells empty.

The quartile-grid method of averaging thedata was used on the total e5 sample and thetotal n4 sample. In other words, for eachsample, data from the target boys and the peerboys were pooled. The equations of classic andmodern matching theory were then fitted tothe averaged data using the ensemble methoddescribed earlier. The %VAFs for these fits arelisted in Table 1 in the rows labeled e5-Average and n4-Average. Evidently, averagingthe data did not rescue the classic theory. Itdid, however, considerably improve the good-ness of fit of the modern theory. For Equations4–6, over both samples and both independentvariables, the %VAFs ranged from 72% to97%, averaging 90%. These %VAFs are wellwithin the range typically obtained in labora-tory experiments with animal and humansubjects (Baum, 1974, 1979; Davison &McCarthy, 1988; Wearden & Burgess, 1982).Parameter estimates from these fits are listedin Table 2 in the rows labeled e5-Average andn4-Average. The exponents, a, are slightlylower than those obtained for the four subsetsof the data, but are nevertheless well within therange typically found in laboratory experi-

Fig. 3. Correlations of standardized residuals withresponse rates (top panel) and time allocations (bottompanel) predicted by the equations of classic (left portions)and modern (right portions) matching theory. Correla-tions not equal to zero indicate linear trends in theresiduals. Error bars represent 6 1 standard error.


ments. The bias parameters, c2e/c1e , for theaveraged data fell within the range of biasparameters obtained for the four data subsets,indicating, as for the data subsets, a strong biasin favor of normative talk.

Figure 4 is a plot of the response rate andtime allocation data for the pooled, averagede5 data, along with the best ensemble fits ofEquations 4, 5, and 6. The improved %VAFsare reflected in the greatly reduced scatter inall plots in the figure. Notice, however, thatthe same properties that characterized theplots and fits for the separate data subsets alsocan be seen in the plots of the averaged data.These include y-intercepts ,0 in the leftpanels, reflecting the strong bias in favor ofnormative talk, generally higher levels ofnormative talk than rule-break talk, shown inthe center and right panels of the figure, and acontrast effect for both rule-break talk andnormative talk. Plots of the data and theensemble fits for the pooled, averaged n4

sample are similar to those shown in Figure 4,and reflect all the response and time alloca-tion properties described for the averaged e5sample. Evidently, quartile-grid averagingacross subjects produces results that arerepresentative of the pooled individual-subjectdata subsets, and that are as good as thosetypically obtained from within-subject datagenerated in the basic science laboratory.

Residuals from fits to the averaged data werecalculated and examined as described for thedata subsets. Correlations of the standardizedresiduals with values predicted from the fittedequations are listed in Table 3 in the rowslabeled e5-Average and n4-Average. Not sur-prisingly, fits of the classic theory showedstrong linear trends in the residuals. All ofthe large negative correlation coefficients forthese fits were statistically different from zero(a 5 0.05, two tails). For fits of the moderntheory, the correlation coefficients acrosssamples, response rates, and time allocations

Fig. 4. Response rates (top panels) and time allocations (bottom panels) for the e5 target and peer boys, pooled andthen averaged over a quartile grid. The percentages of variance accounted for, parameter values, and residual analyses forthese fits are given in Tables 1, 2, and 3. Averaging considerably reduced the scatter about the fitted functions. Bias infavor of and higher levels of normative talk are still evident in the averaged data, as are the lips on the rule-break surfacesin the center panels.


ranged from 20.29 to 0.32, averaging 0.05.None of these correlation coefficients wasstatistically different from zero (a 5 0.05, twotails). These results indicate that the averagedresponse rates and time allocations conformedto the specific function forms required by theequations of modern matching theory.

DISCUSSION

Dishion et al. (1996) analyzed these data ina different way. They considered each pair ofboys to be a dyad that responded as a unit andthat reinforced its own behavior. Hence, forexample, a conversational rule-break exchangethat shifted back and forth between boysconstituted one instance of rule-break talk.Similarly, a positive social response that mightshift back and forth between boys counted asone reinforcer. Dishion et al. plotted theproportion of dyadic rule-break responsesagainst the proportion of positive dyadic socialresponses for rule-break talk, and found whatthey described as a linear relationship betweenthe two proportions. The relationship, shownin their Figure 1, was actually more interest-ing—it was a sigmoidal relationship immedi-ately recognizable as indicating a degree ofundermatching in the dyad’s responding. Theauthors did not attempt to fit Equations 1 or 4to their data. Dishion et al.’s analyses areinteresting, but they differ from the typicalbehavior-analytic focus on the behavior ofindividual organisms.

One issue with the analyses reported here isthat the positive social responses from thetarget and peer boys for their partners’ verbalbehavior were only presumed to be reinforc-ers. Although there is overwhelming evidencethat these kinds of social responses in fact actas reinforcers (e.g., O’Leary & O’Leary, 1977),it would nevertheless be ideal to demonstratethat this was the case for each study ofnaturally occurring behavior in an undis-turbed environment. This could be accom-plished after the naturalistic data were collect-ed, by making the putative reinforcerscontingent on behavior in experiments witha subset of subjects using, perhaps, a reversaldesign to show control of behavior by theputative reinforcer (Snyder & Patterson, 1995;St. Peter et al., 2005).

A second concern that may be raised aboutthese data and analyses bears on the ratio of

responses to reinforcers. Considering thereciprocal nature of conversation, it seemspossible that the ratio of responses to rein-forcers could be close to 1:1 for these data, inwhich case there would be too few ‘‘extra’’behaviors to be affected by reinforcementallocation, and hence behavior could not strayfar from the matching relation. For example, ifevery response was reinforced, a reinforce-ment allocation of, say, 2:1 would be associatedwith a response allocation of exactly 2:1. Butresponse allocation is a true dependentvariable only when it can vary from thereinforcer allocation, and of course this ispossible only when there are more responsesthan reinforcers. The additional responsesmight or might not be allocated in a way thatis consistent with matching. To examine thisissue in the present data, the normative andrule-break talk of the target and peer boys inthe e5 and n4 samples were combined. Onaverage, the boys in the combined sample,across response categories, emitted 20 respons-es for each reinforcer. This response/reinforc-er ratio is comparable to ratios found inlaboratory experiments. For example, Dallery,McDowell, and Lancaster (2000) studied thelever pressing of 7 rats working on variable-interval schedules of sweetened water. The ratsin their 0.32M condition, to take a relativelyhigh response-rate example, emitted an aver-age of 12 responses for each sweet-waterreinforcer. Evidently, the verbal behavior ofthe boys in the present study had ample leewayto deviate from matching.

A third issue that might be raised aboutthese analyses is that the equations of classicmatching theory entail two free parameterswhereas the equations of modern matchingtheory entail four free parameters. It mightseem that this gives the latter theory anadvantage and that the tradeoff between themodern theory’s additional %VAF and theadditional parameters in its equations shouldbe assessed using an F-test or the AkaikeInformation Criterion (Motulsky & Christopo-lous, 2004). But as McDowell (2005) pointedout, this sort of comparison is necessary onlywhen the competing theories both providereasonable descriptions of the data, that is,when both account for a respectable amountof data variance and leave random residuals.When this is the case, the only issue thatremains is whether the additional variance


accounted for by one of the theories is worththe additional parameters it entails. But theclassic theory of matching accounted for novariance and left systematic residuals. Henceits account of these data cannot be correctunder any circumstance.

A fourth issue with these data is theconsiderable amount of scatter around thefunctions fitted to the four data subsets. Thisproduced %VAFs that were not as large asthose typically observed in laboratory experi-ments. Averaging over subjects producedmuch larger %VAFs, which suggested thatthe scatter in the data subsets was likely dueto variance caused by individual differences inthe between-subjects data. But there is still achance that each boy’s verbal behavior wasonly loosely governed by the equations ofmodern matching theory, in which case within-subject data would also exhibit considerablescatter. The 20-minute conversations fromwhich the present data were obtained weretoo short to permit calculating reliable within-subject response rates and time allocations forthese boys, although obviously it would bedesirable to conduct naturalistic studies thatpermitted the collection of within-subject data.A different way to investigate whether thescatter is due to individual differences is to askif within-subject data that showed minimalscatter would show greater scatter whenpooled with other individual-subject data in aconstructed between-subjects data set.

Such an investigation was conducted bycompiling data from all laboratory studies ofconcurrent schedules using human subjectspublished through 1990 that reported at leastfive response and time allocations for eachsubject. This yielded 24 individual-subject datasets from five experiments (Bradshaw, Szabadi,& Bevan, 1979; Bradshaw, Szabadi, Bevan, &Ruddle, 1979; Cliffe & Parry, 1980; Ruddle,Bradshaw, & Szabadi, 1981; Ruddle, Bradshaw,Szabadi, & Foster, 1982). The behavior of theindividual subjects in these experiments waswell governed by Equation 4, which accountedfor between 72% and 100% of the individualsubjects’ data variance, with a median of 98%of the variance accounted for by the equation.In other words, the data from these subjectsshowed very little scatter around the best fit ofEquation 4. One data point was drawn atrandom from each of the individual-subjectdata sets, and these 24 data points were then

pooled to form a between-subjects data set towhich Equation 4 was fitted. This randomsampling from the individual within-subjectdata sets was repeated a total of 20,000 times.The resulting %VAFs for the 20,000 fits ofEquation 4 ranged from 15% to 89%, with amedian of 66%. By comparison, the %VAFsshown in Table 1 for fits of Equation 4 to theboys’ response rate and time allocation dataranged from 44% to 61%, with a median of50%, which are roughly comparable figures.Plots of the constructed between-subjects datasets that yielded the median %VAFs are shownin Figure 5 for response and time allocationdata, along with the best fits of the logtransform of Equation 4. Clearly, these plotsare similar in appearance to the plots in theleft panels of Figure 2. It is not unreasonableto suppose, then, that individual boys’ verbalbehavior in the present study was governedclosely by the equations of modern matchingtheory.

A fifth issue that may be raised about theseanalyses is that they apparently ignore theinterlocking nature of conversation betweenindividuals. The verbal behavior of one boy isaffected by the verbal behavior of the otherboy, which in turn affects the verbal behaviorof the first boy, and so on. Basic behavioranalysis deals with the interlocking nature ofbehavior and environment by means of feed-back functions (Baum, 1992; McDowell &Wixted, 1986, 1988; Soto, McDowell, & Dal-lery, 2006). In the laboratory, the organism’senvironment is arranged and presented bymachinery. In the case of naturally-occurringconversation, however, each boy’s environ-ment is ‘‘arranged’’ and presented by theother boy’s social behavior. But these twosituations are not so different, for even in thelaboratory behavior and environment changeand are changed by each other, and hence areengaged in an interlocking conversation ofsorts. Behavior–environment feedback in lab-oratory experiments is far from completelyunderstood (e.g., Baum, 1992), and often it issimply ignored. For example, feedback iscompletely ignored in all the equations ofboth classic and modern matching theory. Theindependent variables in these equations arerates of obtained reinforcement, which are notknown in advance, and which can be, andoften are, very different from their scheduledrates. The reinforcement rates that are ob-


tained depend on the organism’s behavior,and yet that behavior is considered to dependon the very reinforcement rates the behavioritself generates. The ultimate solution to thisproblem is to incorporate feedback functionsinto the equations of matching theory thatspecify how behavior produces obtained rein-forcement rates from scheduled reinforce-ment rates, thus restoring true independentvariables to the equations. The same must betrue in applications of the equations to socialbehavior. There is no doubt that additionalresearch on how feedback works in both basicand applied environments is needed.

The findings reported here add to theliterature showing that human behavior innatural, unengineered environments is gov-erned by matching theory, and hence lendsfurther support to the assertion that theprinciples and processes that govern behaviorin laboratory environments also govern behav-ior in undisturbed natural human environ-ments. These findings are especially interestingbecause they entail verbal behavior and socialreinforcement in reciprocal social interactionswhere each person’s behavior is regulated byconsequences naturally provided by the otherperson. This type of social exchange is thehallmark of human social interaction.

These analyses also make use of contempo-rary theoretical and analytic developments in

the basic science that may be of interest toapplied workers. If modern matching theorysupplants its classic version, as McDowell(2005) contends it must, then the classicHerrnstein (1970) hyperbola, the quantitativelaw of effect, must be replaced by the morecomplicated form derived from Equation 4:

B~kr a

r azr ae

b

:

This is modern matching theory’s version ofthe quantitative law of effect. McDowell (1986,2005) explained how this single-alternativeform is derived from the power-functionmatching equation (Equation 4), and how itis related to Equations 5 and 6. McDowell andCaron (2007) discussed some of the difficultiesthat may be encountered when attempting tofit this equation to data.

Another feature of the present analyses thatis novel in translational research is the simulta-neous fitting of all three equations of matchingtheory. As noted earlier, this is a much strongertest of the theory than is usually undertakenwhen only the relative response rate equation(viz., Equation 1 or 4) is fitted to data. Thesimultaneous fitting entails a new, ensemble,method that minimizes a normalized totalresidual sum of squares by adjusting parametersthat are shared by all the equations.

Fig. 5. Human between-subjects data sets constructed by random sampling from 24 within-subject data sets thatshowed little scatter around the best fits of Equation 4. The response rate and time allocation data shown are for the twoconstructed between-subjects data sets out of 20,000 that yielded the median %VAFs for fits of the log transform ofEquation 4 (straight lines drawn through the plots). Evidently, the scatter in these plots is similar to that for plots of theboys’ verbal response rates and time allocations (as shown, for example, in the left panels of Figure 2).


And finally, these analyses are interestingfrom a basic science perspective because theyadd support to the assertion that the classictheory of matching is false (McDowell, 2005).It is safe to say that no basic scientist will besurprised to learn that Equation 1 did notdescribe the boys’ verbal behavior well. Therestricted applicability of this equation hasbeen known for at least 35 years (Baum, 1974,1979; Davison & McCarthy, 1988; Wearden &Burgess, 1982). But, upon reflection, it shouldbe surprising to basic scientists that Equations2 and 3 did not describe the boys’ behaviorwell or, in fact, at all. This is because, as notedearlier, Equations 2 and 3 are forms of thesingle-alternative hyperbola that has beenstudied extensively in many experiments, andthat appears to describe the behavior of manyvertebrate species, including humans, quitewell. But McDowell (2005) showed that thisappearance can be deceiving; that is, thehyperbola can appear to accurately describebehavior when in fact it does not.

Further research on behavior in natural,undisturbed environments may improve ourunderstanding of how important classes ofnaturally-occurring human behavior are regu-lated, and this in turn may lead to a wider andfuller acceptance of the idea that basic princi-ples and processes discovered in the laboratoryalso govern human behavior in everyday life.

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Received: July 1, 2009Final Acceptance: January 20, 2010


APPENDIX

The ensemble method of simultaneouslyfitting Equations 4–6 to a set of data usingshared parameters is illustrated in Figure A1.Estimated values of the shared parameters, k,a, c1e , and c2e are given in the top left of thefigure. The parameters are shared across thethree equations, also shown in the figure,which means that each parameter is requiredto have the same value in every equation inwhich it appears. The ratio of the c parameters,shown to the right of the four estimatedparameters, is not itself estimated by the fittingprocedure; instead, it is calculated from theestimated values of the two c parameters.

The block beneath each equation containsinformation pertaining to the least-squares fitof that equation. The vertical ellipses indicatedata that were omitted to save space in thefigure. The total sum of squares, SS, iscalculated for the three dependent variables,B1, B2, and B1/B2. The SS for B1 is indicated inthe figure by a label and an arrow. The thirdcolumn in each block contains the value of thedependent variable predicted by the applica-ble equation, using the parameter values listedin the top left of the figure. In an actualspreadsheet, this column contains the formulafor the applicable equation, where the inde-pendent variables and parameter values of theequation are represented by appropriate cell

references. The fourth column in each blockcontains the residuals, which are the differ-ences between observed and predicted valuesof the dependent variable. The residuals aresquared and then summed to obtain theresidual sum of squares, RSS, which is identi-fied in the figure for B1 by a label and anarrow. The percentage of variance accountedfor (%VAF) by each equation, also identifiedin the figure for B1, is given by

%VAF~ 1{RSS

SS

� �100:

The %VAF values shown in the figure can beverified using the RSS and SS values given ineach block.

The quantity in the bold box in the lowerright of Figure A1 is the normalized totalresidual sum of squares, calculated fromEquation 7. Its value in the figure can beverified by substituting the RSS and SS givenfor each block into Equation 7. The normal-ized total residual sum of squares is thequantity that a fitting algorithm must mini-mize by iteratively adjusting the four parame-ters entailed by Equations 4–6. Every iteration(i.e., change) of the parameter values causesthe predicted values of the dependent vari-ables in the three blocks to change, and hencealso causes the residuals and RSSs in the threeblocks to change. As a consequence, all three

Fig. A1. Example of fitting Equations 5, 6, and 4 simultaneously as an ensemble with parameters, k, a, c1e and c2e,shared across all equations. The total sum of squares (SS), residual sum of squares (RSS), and percentage of varianceaccounted for (%VAF) are identified for the first data block. The value in the bold box in the lower right is the sum of thethree RSS/SS ratios and is the quantity that is minimized by adjusting the four parameters. The quantity underneath thebold box is the ensemble %VAF.


equations are fitted simultaneously by mini-mizing the single quantity in the bold box. Avariety of optimization algorithms can be usedto carry out this ensemble fit. The mostconvenient is the generalized reduced gradi-ent algorithm developed by Lasdon andWaren (1978) and implemented by MicrosoftExcel’s Solver tool, for which extensive help is

available. The quantity beneath the bold boxin Figure A1 is the %VAF for the entireensemble, and is calculated from Equation 8.Again, this quantity can be verified using theRSS and SS values given in the figure.

The three equations of classic matchingtheory are fitted as an ensemble in the sameway, using the shared parameters, k and re.


eric - education resources information centerphilosopher of science, terry l. smith, discussed this...

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