A DISCRIMINATION ANALYSIS OF TRAINING-STRUCTURE …

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JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR 1999, 72, 117–137 NUMBER 1 (JULY)

A DISCRIMINATION ANALYSIS OF TRAINING-STRUCTUREEFFECTS ON STIMULUS EQUIVALENCE OUTCOMES

RICHARD R. SAUNDERS AND GINA GREEN

SCHIEFELBUSCH INSTITUTE FOR LIFE SPAN STUDIES,UNIVERSITY OF KANSAS AND

NEW ENGLAND CENTER FOR CHILDREN,E. K. SHRIVER CENTER FOR MENTAL RETARDATION, AND

NORTHEASTERN UNIVERSITY

Experiments designed to establish stimulus equivalence classes frequently produce differential out-comes that may be attributable to training structure, defined as the order and arrangement ofbaseline conditional discrimination training trials. Several possible explanations for these differenceshave been suggested. Here we develop a hypothesis based on an analysis of the simple simultaneousand successive discriminations embedded in conditional discrimination training and testing withineach of the training structures that are typically used in stimulus equivalence experiments. Ouranalysis shows that only the comparison-as-node (many-to-one) structure presents all the simplediscriminations in training that are subsequently required for consistently positive outcomes on alltests for the properties of equivalence. The sample-as-node (one-to-many) training structure doesnot present all the simple discriminations required for positive outcomes on either the symmetry orcombined transitivity and symmetry (equivalence) tests. The linear-series training structure presentsall the simple discriminations required for consistently positive outcomes on tests for symmetry, butnot for symmetry and transitivity combined (equivalence) or transitivity alone. Further, the differencein the number of simple discriminations presented in comparison-as-node training versus the othertraining structures is larger when the intended class size is greater than three or the number ofclasses is larger than two. We discuss the relevance of this analysis to interpretations of stimulusequivalence research, as well as some methodological and theoretical implications.

Key words: stimulus equivalence, stimulus classes, simple discrimination, conditional discrimination,discrimination learning, stimulus relations

Experimental analyses of stimulus equiva-lence based on the Sidman model (1971,1986, 1994) expose subjects to match-to-sam-ple (MTS) training designed to establish con-ditional discriminations among stimuli thatare not physically similar to one another. Typ-ically, two or more conditional discrimina-tions with some stimuli in common aretrained through differential reinforcement.For example, subjects may learn to respondto a comparison stimulus designated B1 ifand only if Sample Stimulus A1 is present,

Development of this paper was supported in part byNational Institute of Child Health and Human Develop-ment Grants HD25995-06 to the E. K. Shriver Center andHD18955-13 and HD02528 to the Schiefelbusch Institutefor Life Span Studies, University of Kansas, and by theNew England Center for Children. We acknowledge JoeSpradlin and Julie McEntee for their helpful commentson preliminary drafts. Portions of this paper were pre-sented at the third European meeting of the Experimen-tal Analysis of Behaviour Group, July, 1997, in Dublin,Ireland and at the meeting of the Association for Behav-ior Analysis, May, 1997, Chicago.

Requests for reprints should be addressed to RichardR. Saunders at the Parsons Research Center, 2601 Gabri-el, Parsons, Kansas 67357 (E-mail: [email protected]).

Comparison Stimulus B2 if and only if Sam-ple Stimulus A2 is present, and ComparisonStimulus B3 if and only if Sample StimulusA3 is present. In addition, subjects may learnto respond to Comparison Stimuli C1, C2,and C3 only in the presence of Sample Stim-uli B1, B2, and B3, respectively. If successful,such training establishes conditional relationsbetween each sample (conditional stimulus)and its corresponding correct comparison(discriminative stimulus, or S1), in this case,A1B1, A2B2, A3B3, B1C1, B2C2, and B3C3.

Performances indicating the developmentof conditional relations are prerequisites fortesting the possibility that the trained (orbaseline) relations have the properties ofequivalence, as defined in mathematics: re-flexivity, symmetry, and transitivity. The testsconsist of MTS trials on which subjects havethe opportunity to demonstrate all possibleuntrained conditional relations among thestimuli involved in training. Symmetry wouldbe evaluated by reversing the sample andcomparison stimuli relative to training (i.e.,B1A1, B2A2, B3A3, C1B1, C2B2, C3B3 in ourexample). Transitivity would be evaluated by

118 RICHARD R. SAUNDERS and GINA GREEN

presenting sample and comparison stimulithat were related indirectly in trainingthrough a common trained relation with oth-er stimuli (i.e., A1C1, A2C2, and A3C3). Testsfor the untrained relations C1A1, C2A2, andC3A3 would constitute simultaneous tests forthe properties of symmetry and transitivity(also called combined tests, or simply equiv-alence tests). Reflexivity would be evaluatedvia conditional identity matching tests withthe stimuli involved in training (e.g., A1A1,A2A2, A3A3, B1B1, etc.). If outcomes of alltests are positive, the inference is made thatthe trained relations are equivalence rela-tions and that the stimuli so related constituteequivalence classes (e.g., Green & Saunders,1998; R. R. Saunders & Green, 1992; Sidman,1986, 1994; Sidman et al., 1982; Sidman &Tailby, 1982).

TRAINING STRUCTURE

The foregoing example represents theminimal conditions required to determinewhether MTS training produces stimulusequivalence: training on two arbitrary condi-tional discriminations with one set of stimuliin common (AB and BC in the example), fol-lowed by testing for all possible conditionaldiscriminations that were not trained directly.The minimum number of stimuli in an equiv-alence class is three. In our example, the ABconditional discrimination was trained first,followed by the BC conditional discrimina-tion; the B stimuli were common to both dis-criminations. Of course, equivalence classescan and often do include more than threestimuli each, and the baseline conditional dis-criminations can and often are trained in dif-ferent sequences and with different commonstimuli than in our example. The term train-ing structure has been used to refer to the se-quence of conditional discriminations andthe arrangements of common or ‘‘linking’’stimuli presented to subjects in baseline train-ing. Various terms have been coined to de-scribe specific training structures. For exam-ple, Sidman, Kirk, and Willson-Morris (1985)described the situation in which two stimuliare mutually related to a third (e.g., AB, BC)as a ‘‘three-stage’’ training arrangement;training another linked conditional discrimi-nation such as CD created a ‘‘four-stage’’ ar-rangement (underscored letters designate

common, or linking, sets of stimuli). Fieldsand Verhave (1987) referred to a stimulusthat is related to only one other stimulus intraining as a ‘‘single,’’ and called a stimulusthat is related to more than one other stim-ulus a ‘‘node.’’ To Fields and Verhave, thefour-stage arrangement of Sidman et al.(1985) would be a two-node arrangement,with the B and C stimuli serving as nodes.These authors also suggested that the struc-ture of equivalence classes can be describedin terms of four parameters: the number ofstimuli in each class, the number of nodes,the pattern of singles relative to nodes, andthe pattern formed by assignment of stimulito the roles of samples and comparisons dur-ing training (i.e., directionality of training).

The original Sidman analysis of stimulusequivalence did not suggest that equivalencetest outcomes should vary as a function oftraining structure, order, or direction (Sid-man & Tailby, 1982). On the contrary, it im-plied that if training established the intendedconditional relations and concurrently pre-vented the development of extraneous stim-ulus control, then responding on all tests foruntrained relations should be consistent withequivalence, regardless of the order and ar-rangement of the trained conditional dis-criminations (Carrigan & Sidman, 1992;Green & Saunders, 1998; Sidman, 1994).Some investigators, however, have reporteddifferential outcomes on equivalence teststhat appear to be due to training structure.For example, one recent study with preschoolchildren found that five-member equivalenceclasses were more likely following one train-ing sequence than another (R. R. Saunders,Drake, & Spradlin, 1999). All of the childrenwere exposed to two-choice MTS training de-signed to establish four conditional discrimi-nations among 10 arbitrary visual stimuli. For6 children, two stimuli served as the samplesin all four conditional discriminations; thetrained relations were designated AB, AC,AD, and AE. This kind of training structurehas been referred to as a ‘‘sample-as-node’’or ‘‘one-to-many’’ structure. Five other chil-dren received training with one pair of stim-uli serving as comparisons with each of fourdifferent pairs of sample stimuli. The trainedconditional relations were designated BA,CA, DA, and EA. This training structure hasbeen dubbed a ‘‘comparison-as-node’’ or

119DISCRIMINATION LEARNING AND STIMULUS EQUIVALENCE

‘‘many-to-one’’ structure (cf. K. J. Saunders,Saunders, Williams, & Spradlin, 1993; Urcuio-li & Zentall, 1993; Urcuioli, Zentall, Jackson-Smith, & Steirn, 1989). Positive outcomes onequivalence tests were found for all 5 chil-dren who had comparison-as-node training,but positive outcomes were found for only 2of 6 children who had sample-as-node train-ing.

The results obtained by R. R. Saunders etal. (1999) replicated those of previous exper-iments with adolescents and adults with men-tal retardation (Drake & Saunders, 1987, cit-ed in K. J. Saunders et al., 1993; R. R.Saunders, Wachter, & Spradlin, 1988; Sprad-lin & Saunders, 1986). Across these studies,five-member equivalence classes were estab-lished in only 1 of 7 subjects trained with thesample-as-node procedure, but they were es-tablished in 6 of 6 subjects trained with thecomparison-as-node procedure. Apparentstructure-related differences have also beenreported with normally capable adult subjects(Barnes, 1992, as cited in Barnes, 1994;Fields, Hobbie, Adams, & Reeve, in press). Incontrast, a recent study with normally capableadults suggested that sample-as-node trainingwas more likely to produce three-memberequivalence classes than was comparison-as-node training (Arntzen & Holth, 1997). Fi-nally, some investigators have reported nega-tive outcomes on tests for some properties ofequivalence following baseline training inwhich several conditional discriminationswere trained in sequence, with multiple nod-al or linking stimuli (e.g., AB, BC, CD, DE;see Arntzen & Holth, 1997; Fields, Landon-Jimenez, Buffington, & Adams, 1995; Holth& Arntzen, 1998). This has been described asa linear-series training structure (e.g., Green& Saunders, 1998).

SOME HY POTHESES ABOUTTRAINING-STRUCTURE EFFECTS

Why might different training structuresyield different outcomes on equivalencetests? Several possibilities have been suggest-ed. With respect to linear-series trainingstructures, Fields and colleagues offered anaccount based on nodal or ‘‘associative’’ dis-tance. They suggested that the larger thenumber of nodes potentially linking stimuliindirectly in training, the less robust the per-

formances on tests for the untrained relationsamong those stimuli were likely to be (e.g.,Fields, Verhave, & Fath, 1984). Although theresults of some experiments seem to supportthis hypothesis (e.g., Fields, Adams, Verhave,& Newman, 1990; Kennedy, 1991; Kennedy,Itkonen, & Lindquist, 1994), procedural var-iations and stimulus characteristics in thoseexperiments make them difficult to interpret.(These will be discussed later.) This accounthas also been criticized because it invokes la-bels (nodal distance, associative distance) fora structural property (the number of nodes)that may imply that other hypothetical struc-tural properties are at work (Sidman, 1994,p. 539). Further, the account falls short of ex-plaining different outcomes on tests forequivalence in basic behavioral terms.

With respect to sample-as-node versus com-parison-as-node differences, Sidman (1994,pp. 527–528) raised the possibility that theformer might establish differential contextualcontrol of trained conditional relations bynegative stimuli (incorrect comparisons),which could lead to negative outcomes ontests for the properties of equivalence. An al-ternative account was postulated by Spradlinand his colleagues (K. J. Saunders et al., 1993;Spradlin & Saunders, 1986; also see Sidman,1994, pp. 526-527). They speculated that sam-ple-as-node training did not consistently pro-duce equivalence classes because the trainingcontingencies did not require all the simplediscriminations subsequently called for ontests for equivalence. Spradlin and colleaguesnoted that when AB and AC are trained, forexample, subjects need only discriminate theA samples from one another, the B compari-sons from one another, and the C compari-sons from one another; the training contin-gencies do not require discrimination of eachB stimulus from each C stimulus. The B ver-sus C discriminations are called for, however,on the BC and CB trials that constitute testsfor the properties of equivalence. In contrast,comparison-as-node training (e.g., BA andCA) requires successive discrimination of allB and C stimuli across trials and the simul-taneous discrimination of all A comparisonswithin trials to fulfill the training contingencyrequirements. That is, this training structurepotentially establishes all of the simple dis-criminations required for consistently positiveoutcomes on tests for the properties of equiv-


alence (K. J. Saunders et al., 1993; R. R. Saun-ders et al., 1999; Spradlin & Saunders, 1986;also see Barnes, 1994; Sidman, 1994).

THE DISCRIMINATIONACCOUNT: AN ELABORATION

AND EXPANSION

We propose that, with some additional de-velopment and elaboration, the discrimina-tion analysis suggested by Spradlin and col-leagues provides a parsimonious account ofthe differential effects of training structureson equivalence test outcomes, one that is con-sistent with basic principles of stimulus con-trol and does not invoke constructs like as-sociative distance or mediation (cf. McIlvane& Dube, 1992; Sidman, 1986, 1994). Here weundertake such an elaboration by examining(a) how simple simultaneous and successivediscriminations are necessarily embedded inconditional discriminations; (b) which andhow many of those component simple dis-criminations are presented to subjects in eachof the training structures commonly used instimulus equivalence experiments; and (c)the component simple discriminations re-quired for consistently positive outcomes ontests for equivalence following training witheach structure. Finally, we reanalyze the re-sults of several published studies of stimulusequivalence from this perspective.

Assumptions and Definitions

For our analysis, we adopt a critical as-sumption that has been stated or implied byseveral other authors (e.g., McIlvane & Dube,1996; K. J. Saunders et al., 1993; Spradlin &Saunders, 1986; Sidman, 1994): For perfor-mances to meet criteria for acquisition of the trainedbaseline relations as well as criteria for positive out-comes on all tests for stimulus equivalence, eachstimulus must be discriminated from every otherstimulus in the experiment. On its face, this as-sumption might appear counterintuitive. Itmight seem that in order to ‘‘pass’’ all testsfor stimulus equivalence, subjects need onlydiscriminate all stimuli in each class from allstimuli in the other classes (between-class dis-criminations, such as A1 vs. A2 vs. A3, B1 vs.B2 vs. B3), and need not discriminate stimuliwithin classes from one another (A1 vs. B1,B2 vs. C2, C3 vs. D3, etc.). The reasoningmight go like this: If training contingencies

establish that two sample stimuli (e.g., B1 andC1) both control a response to the same com-parison (e.g., A1), might not the subject thentreat B1 and C1 as if they are the same—thatis, fail to discriminate them? And would thisnot suffice to produce apparently positiveoutcomes on all tests for equivalence (e.g.,trials testing B1C1 and C1B1), as long as thesubject discriminates B1 and C1 from thestimuli in the other prospective equivalenceclasses? We maintain that the answer to thesequestions is no, for the following reasons: Atthe beginning of a well-designed stimulusequivalence experiment, the subject is ex-posed to a group of unsorted stimuli that donot bear any consistent physical resemblanceto one another. The initial training contin-gencies are designed to establish a relationbetween each comparison stimulus and a par-ticular sample stimulus. Therefore, the con-tingencies require discrimination of everysample from every other sample presentedsuccessively across trials, all samples from allcomparisons, and all comparison stimuli pre-sented simultaneously within trials (Sidman,1986). Training procedures that enhance theprobability that every stimulus will be discrim-inated from every other stimulus—such ascontingencies that specify a different re-sponse to each conditional (sample) stimulusand to each discriminative (comparison)stimulus—should foster the development ofconditional discriminations and, therefore,the development of equivalence classes (Sid-man, 1994, pp. 413–414).

Like training trials, tests for stimulus equiv-alence also present conditional discrimina-tions that are composed of simple successiveand simultaneous discriminations among ex-perimental stimuli. Therefore, for consistent-ly positive outcomes on all tests for the prop-erties of equivalence, every stimulus must bediscriminated from every other stimulus.More important, however, test trials may pre-sent some simple successive and simultaneousdiscriminations that were not presented at allin training, or that are presented differentlyon tests than in training (e.g., successively vs.simultaneously). If the relevant simple dis-criminations that compose the tested condi-tional discriminations are not made, negativeoutcomes on some or all tests for the prop-erties of equivalence will result (see McIlvane& Dube, 1996; K. J. Saunders et al., 1993; R.


R. Saunders et al., 1999; Sidman, 1986, 1994;Spradlin & Saunders, 1986). We argue herethat different training structures make suchoutcomes more or less probable because theyare more or less likely to establish the nec-essary simple discriminations.

To bolster our contention that consistentlypositive outcomes on tests for stimulus equiv-alence require discrimination of every stim-ulus from every other stimulus in the exper-iment, it is important to note that althoughsuch outcomes imply that the stimuli withinclasses are substitutable for one another, itdoes not necessarily follow that they are in-discriminable from one another. On the con-trary, from a practical standpoint it is vital fororganisms to discriminate that stimuli thatcan substitute for one another in certain con-texts nonetheless have distinctive features.For example, we may observe that a childmatches an apple, a picture of an apple, andthe printed word APPLE to one another inevery possible sample-comparison arrange-ment. That observation might lead us to con-clude that those stimuli are substitutable forone another, and constitute an equivalenceclass. But that observation should not lead tothe conclusion that those stimuli are not dis-criminated. Indeed, for the class to be func-tional for the child, it is necessary for discrim-inations among the members to bemaintained—so that the child does not try toeat the picture or the printed word, for ex-ample—at the same time as the stimuli aretreated as equivalent (substitutable) in cer-tain contexts. (Whether discriminationsamong stimuli are in fact maintained duringor after demonstrations that they are equiva-lent could be evaluated empirically via con-ditional identity matching tests; cf. Dube,McIlvane, & Green, 1992.) Of course, it isalso necessary for all members of the ‘‘apple’’class to be discriminated from all members ofother stimulus classes. In other words, per-formances that consistently demonstrate stim-ulus equivalence entail discriminationsamong stimuli within as well as between clas-ses.

For purposes of our analysis, we refer tothe grand total of all possible simple simul-taneous and successive discriminationsamong the stimuli in a stimulus equivalenceexperiment as required for consistently posi-tive outcomes on all training and test trials.

We describe the types and proportions ofthose discriminations that are presented andnot presented in each of the training structurestypically used in stimulus equivalence exper-iments, to reflect the understanding thatwhether the training contingencies actuallyestablish those discriminations is always anempirical question.

Our analysis also assumes (a) the use of si-multaneous MTS procedures; (b) that base-line conditional discriminations are mixed atsome point before testing for equivalence be-gins (e.g., if AB and AC relations are trained,all of those trial types are presented togetherwithin one or more training session beforetesting); (c) that all trial types testing for aparticular property of equivalence (e.g., BCand CB) are presented together within thesame session; and (d) that training and test-ing are conducted with balanced MTS con-ditional discrimination procedures (Green &Saunders, 1998). For example, ComparisonC1 is always presented with Comparisons C2and C3; C1 is never presented as a compari-son with any stimuli from other sets (e.g., B2,D3). A final assumption is that no differentialconsequences are arranged for responses ontest trials.

Simple Discriminations Embedded inConditional Discriminations

We turn now to a review of simple and con-ditional discrimination training contingen-cies, to reiterate precisely why acquisition ofconditional discriminations necessitates ac-quisition of the component simple discrimi-nations.

Simple discriminations. In the presence of aparticular antecedent stimulus, if a particularresponse is followed by a particular conse-quence, the response may come to occurmore often in the presence of the stimulus(S1) than in its absence or in the presenceof another stimulus (S2). When repeated ap-plication of these contingencies leads to re-sponding in the presence of the S1 but notin the presence of the S2, one can infer thata simple discrimination has been established.When the S1 and the S2 are presented con-currently on each of a series of trials, the pro-cedure is termed a simultaneous simple discrim-ination. When the stimuli are presented oneat a time, on unsystematically alternating tri-


Fig. 1. Representations of three basic training struc-tures used in stimulus equivalence experiments: compar-ison as node (upper panel) with three potential classesof five stimuli each; sample as node (middle panel) withtwo potential classes of four stimuli each; and linear se-ries (lower panel) with four potential classes of threestimuli each. Arrows point from stimuli used as samplesto comparison stimuli.

als, the procedure is termed a successive simplediscrimination.

Conditional discriminations. In conditionaldiscrimination training, simple discrimina-tions are brought under the control of addi-tional antecedents, or conditional stimuli(Sidman, 1986). A common and reliablepreparation for establishing conditional dis-criminations is MTS training. The MTS pro-cedures used in most stimulus equivalenceexperiments involve a minimum of two dif-ferent conditional stimuli (samples) and twodifferent discriminative stimuli (compari-sons) per conditional discrimination. Thesame comparison stimuli are presented on ev-ery trial while the sample stimulus varies un-systematically from trial to trial. In the leastcomplicated procedure, contingencies are ar-ranged so that each comparison stimulus isdiscriminative for reinforcement (S1) in thepresence of one and only one conditional(sample) stimulus, and is not discriminativefor reinforcement (S2) in the presence of asecond sample stimulus. That is, the func-tions of the discriminative stimuli changefrom trial to trial, depending on which sam-ple stimulus is present. Although three ormore samples and comparisons are generallydesirable (Carrigan & Sidman, 1992; Green& Saunders, 1998; Johnson & Sidman, 1993;Kelly, Green, & Sidman, 1998; Sidman, 1987),we use just two samples and comparisonshere and elsewhere for ease of illustration.

In a typical stimulus equivalence experi-ment, multiple training trials are presented ina session or block. A different sample is pre-sented on each trial, and the same compari-sons are presented on every trial. Each sam-ple is usually presented equally often withina session or block of trials, in unsystematicorder. The positions of the comparisons, es-pecially the S1, vary unsystematically fromtrial to trial. Sample stimuli are presentedone at a time across trials, with some timeelapsing between presentations; thus, dis-criminating among them involves successivesimple discriminations. Comparison stimuliare presented together on every trial, as arethe samples and comparisons that constituteeach trial type; thus, discriminating amongthem involves simultaneous simple discrimi-nations. In short, performing conditional dis-criminations necessarily involves a number ofsimple discriminations, some simultaneous,

some successive (cf. Green & Saunders, 1998;K. J. Saunders & Spradlin, 1989, 1993; Sid-man, 1986).

Common Stimulus EquivalenceTraining Structures

Many variations of the training structuresdescribed by Fields and Verhave (1987) ap-pear in the stimulus equivalence literature,but there are three basic prototypes. The toppanel of Figure 1 shows a comparison-as-node(many-to-one) structure for potentially devel-oping three equivalence classes with five stim-uli per class. This training structure usuallyinvolves the use of three-choice MTS proce-


Fig. 2. Comparison-as-node training to produce twothree-member equivalence classes. Solid arrows indicatetrained relations, dashed-line arrows indicate tests for theproperty of symmetry, and the double-pointed dotted-line arrows indicate combined tests for the properties ofsymmetry and transitivity (or equivalence tests). Trialtypes for training trials and tests for symmetry and equiv-alence are shown below the schematic. Asterisks indicateexperimenter-designated correct comparisons (for posi-tive test outcomes).

dures. The middle panel of Figure 1 showsthe sample-as-node (one-to-many) structurefor potentially developing two equivalenceclasses with four stimuli per class. The train-ing represented is usually arranged using two-choice MTS procedures. The linear-seriesstructure is shown in the lower panel of Fig-ure 1. The schematic depicts four potentialclasses of three stimuli each, which usually in-volves four-choice MTS procedures. Moststimulus equivalence experiments in the lit-erature have used comparison-as-node, sam-ple-as-node, or linear-series training struc-tures, or variations thereof, to produce two ormore equivalence classes (Green & Saunders,1998; for some examples of variations, seeFields, Adams, & Verhave, 1993; Kennedy,1991; Pilgrim & Galizio, 1995; Spradlin, Cot-ter, & Baxley, 1973; Wetherby, Karlan, &Spradlin, 1983).

Simple Discriminations Presented inVarious Training Structures

Comparison as node. Figure 2 representscomparison-as-node training (BA and CA)leading potentially to the development of twoequivalence classes of three stimuli each. Tri-al types for this training, and for tests of sym-metry and equivalence, are shown beneaththe schematic. In this structure, 15 simple dis-criminations are presented in training, as fol-lows: Comparison Stimuli A1 and A2 are pre-sented simultaneously on training trials (onesimple discrimination). Each comparisonstimulus is presented simultaneously witheach sample stimulus; that is, A1 is presentedwith B1, B2, C1, and C2 (four discrimina-tions), and A2 is presented with B1, B2, C1,and C2 (four discriminations). The two Bstimuli and the two C stimuli are presentedsuccessively across training trials as sampleswhen BA and CA trials are mixed (six dis-criminations). For the training contingencyrequirements to be fulfilled consistently, all15 of these simple discriminations must bemade.

Tests for stimulus equivalence—BC and CBtrial types—also present certain simple dis-criminations as components of the testedconditional discriminations: the successivediscrimination of all B and C sample stimuli,two simultaneous discriminations betweenthe comparison stimuli (B1 vs. B2 and C1 vs.C2), and four simultaneous discriminations

between samples and comparisons (i.e., eachB stimulus from each C stimulus). In thiscase, all of those simple discriminations werepresented during baseline training. Althoughthe B and C stimuli were not presented assimultaneous discriminations, they were pre-sented as successive (sample) discriminationswhen the BA and CA training trials weremixed prior to testing. Because simultaneousdiscriminations are generally easier than suc-cessive discriminations (Brady & Saunders,1991; Carter & Eckerman, 1975; Urcuioli et


Fig. 3. Sample-as-node training to produce two three-member equivalence classes. Solid arrows indicatetrained relations, dashed-line arrows indicate tests for theproperty of symmetry, and the double-pointed dotted-line arrows indicate combined tests for the properties ofsymmetry and transitivity (or equivalence tests). Trialtypes for training trials and tests for symmetry and equiv-alence are shown below the schematic. Asterisks indicateexperimenter-designated correct comparisons (for posi-tive test outcomes).

al., 1989), we would expect most subjects tohave little difficulty with the BC simultaneousdiscriminations called for on the equivalencetests in this example. Similarly, all of the sim-ple discriminations involved in tests for sym-metry (AB and AC) were presented duringtraining. Only one (A1 vs. A2) might pose aproblem for some subjects, because the dis-crimination between those stimuli was pre-sented simultaneously in training, but it ispresented successively on the symmetry tests.Most of the other simple discriminationscalled for on the symmetry tests are exactlythe same as in training; two (B1 vs. B2 andC1 vs. C2) that were presented successively intraining appear as simultaneous discrimina-tions on symmetry tests. The discriminationsbetween the A stimuli and the B and C stim-uli remain the same as in training, exceptthat sample and comparison roles are re-versed. In short, training with this compari-son-as-node structure presents all of the sim-ple discriminations involved in everysymmetry and equivalence test trial type.(Tests for transitivity alone are not possiblefollowing training with this structure.)

Sample as node. Quite a different pictureemerges from an analysis of the simple dis-criminations presented during sample-as-node training. Figure 3 represents sample-as-node training (AB and AC) leadingpotentially to the development of two equiv-alence classes of three stimuli each. In train-ing, Comparison Stimuli B1 and B2 are pre-sented simultaneously, as are Comparisons C1and C2. Samples A1 and A2 are presentedsuccessively, and are presented simultaneous-ly with the B and C comparisons. Althoughthe B and C stimuli are presented in pairssuccessively across trials when AB and ACtraining trials are mixed, the B and C stimuliare never pitted against one another within atrial (see Barnes, 1994), nor are they pre-sented successively as samples. Thus, success-ful training with this sample-as-node structurepotentially results in every stimulus being dis-criminated from every other stimulus exceptthe B stimuli from the C stimuli. Discrimi-nations among the B and C stimuli are firstcalled for either on tests for equivalence (BC,CB) or on tests for symmetry (BA, CA),whichever is presented first. As the lower por-tion of Figure 3 shows, all four B and C stim-uli must be discriminated from one another

as samples across trials on the symmetry tests,and as samples across trials as well as samplesand comparisons within trials on the tests forequivalence. On symmetry tests, the simulta-neous discrimination of A1 from A2 as com-parisons is not likely to be problematic, be-cause those stimuli were presented insuccessive discrimination format in training.In summary, negative results on equivalenceand symmetry tests are expected to be morelikely following sample-as-node training thancomparison-as-node training, because sam-


Fig. 4. Linear-series training to produce two three-member equivalence classes. Solid arrows indicatetrained relations, dashed-line arrows indicate tests for theproperty of symmetry, and the double-pointed dotted-line arrow indicates the test for the properties of transi-tivity (AC) and symmetry and transitivity combined, orequivalence (CA). Trial types for training trials and testsfor symmetry, transitivity, and equivalence are shown be-low the schematic. Asterisks indicate experimenter-des-ignated correct comparisons (for positive test outcomes).

ple-as-node training does not present some ofthe simple discriminations that make up theconditional discriminations called for on thetests.

Linear series. Examination of the simple dis-criminations involved in linear-series training,shown in Figure 4, reveals yet another pat-tern. In AB and BC training to establish twoclasses of three stimuli each, the B stimuli arepresented simultaneously as comparisons, asare the C stimuli. The A and B stimuli arepresented simultaneously as samples andcomparisons on the AB training trials, andthe B and C stimuli are presented simulta-neously on the BC training trials. The A andB stimuli are presented successively as sam-ples when the AB and BC training trials aremixed. On the symmetry tests (BA and CB),

the B and C stimuli are presented successivelyas samples; they were presented simulta-neously in training. As noted previously, thismight pose a problem for some subjects. Inaddition, the CB symmetry test trials call forthe successive discrimination of C1 from C2,which was never presented in training, where-as the BA symmetry test trials involve succes-sive (B1 vs. B2) as well as simultaneous (A1vs. A2, B stimuli vs. A stimuli) discriminationsthat were presented in training. Thus, differ-ent outcomes may occur on these two typesof symmetry test trials. Note that with linear-series training designed to establish largerequivalence classes (e.g., AB, BC, CD, DE),only the test for symmetry of the final con-ditional discrimination (e.g., ED) will sufferfrom the potential problem just described forCB test trials.

Referring to the lower portion of Figure 4,it is evident that the tests for transitivity andequivalence following linear series trainingpresent both simple and successive discrimi-nations among stimuli (A and C) that are nev-er pitted against one another in training. Theonly context in which those stimuli are pre-sented together in training is when the ABand BC training trials are mixed; there the Aand C stimuli are presented successivelyacross trials within a session, but as samplesand comparisons, respectively, on differenttrial types. Thus, as with sample-as-node train-ing structures, linear-series structures are ex-pected to result in a higher probability of fail-ure on equivalence tests than training withcomparison-as-node structures.

Simple Discriminations in LargerTraining Structures

Table 1 summarizes the preceding analysis.To reiterate, in any training structure de-signed to produce two equivalence classes ofthree stimuli each, the grand total of possiblesimple simultaneous and successive discrimi-nations among all the stimuli is 15. The thirdcolumn of the table contrasts the proportionof those discriminations not presented in eachof the training structures. It shows that all ofthe potential simple discriminations are pre-sented in comparison-as-node training, mean-ing that the subject encounters no novel dis-criminations on tests for equivalence andsymmetry. Training with the other structures,however, omits some of the simple discrimi-


Table 1

Simple discriminations in each of three training structures designed to produce two three-member equivalence classes.

Structure Grand total

Number notpresented intraining/total

Number notpresented intraining, butpresented on

equivalence tests


symmetry tests

Comparisonas node

Sample as nodeLinear series

151515

0/154/154/15

044

040

Table 2

Simple discriminations in each of three training structures designed to produce two four-member equivalence classes.




equivalence tests


symmetry tests

Comparisonas node


282828

0/2812/2812/28

01212

0120

nations that are presented on tests for equiv-alence, as indicated by the entries in thefourth column. The entries in the fifth col-umn show that the discriminations not pre-sented in sample-as-node training are pre-sented in tests for symmetry. This is not thecase for linear-series training.

Table 2 compares the numbers of simplediscriminations presented in the three train-ing structures when training is designed toproduce two equivalence classes of four stim-uli each. Comparing the entries in Table 3with those in Table 2 reveals that increasingclass size from three to four stimuli increasesthe total number of simple discriminationsfrom 15 to 28. It also changes the proportionof those discriminations that are presented intraining, as well as the proportion of simplediscriminations not presented in training butcalled for on tests for the properties of equiv-alence. In comparison-as-node training, allthe simple discriminations among the stimuliare presented in training. In sample-as-nodeand linear-series training structures, 12 sim-ple discriminations are not presented intraining. Following sample-as-node training,

all of those discriminations are presented onboth equivalence and symmetry tests. Follow-ing linear-series training, all of the simple dis-criminations not presented in training arepresented on equivalence tests, but none ofthem are presented on symmetry tests.

Table 3 shows the effects of holding pro-spective class size at four stimuli per classwhile increasing the number of prospectiveclasses to three and employing three-choiceMTS training procedures (to insure balancedtrial types per the assumptions underpinningour analysis). The total number of possiblediscriminations increases to 66. As in the pre-ceding examples, comparison-as-node train-ing presents all of them. In contrast, increas-ing the number of prospective equivalenceclasses from two to three alters the propor-tion of the total simple discriminations thatare presented in sample-as-node and linear-series training structures. As Table 3 indi-cates, 27 of the possible simple discrimina-tions among the experimental stimuli are notpresented in sample-as-node and linear-seriestraining; all of those discriminations appearon the equivalence tests that follow training


Table 3

Simple discriminations in each of three training structures designed to produce three four-member equivalence classes.




equivalence tests


symmetry tests

Comparisonas node


66

6666

0/66

27/6627/66

0

2727

0

270

Table 4

Simple discriminations in sample-as-node training with 12 stimuli and three conditional dis-criminations (AB, AC, AD) to produce three four-member equivalence classes.

Discrimination Stimuli involved Simple discriminations

Presented in trainingSuccessive Samples A1 vs. A2 A1 vs. A3 A2 vs. A3Simultaneous Samples and

comparisonsA1 vs. B1A1 vs. C1A1 vs. D1A2 vs. B1A2 vs. C1A2 vs. D1A3 vs. B1A3 vs. C1A3 vs. D1

A1 vs. B2A1 vs. C2A1 vs. D2A2 vs. B2A2 vs. C2A2 vs. D2A3 vs. B2A3 vs. C2A3 vs. D2

A1 vs. B3A1 vs. C3A1 vs. D3A2 vs. B3A2 vs. C3A2 vs. D3A3 vs. B3A3 vs. C3A3 vs. D3

Comparisons B1 vs. B2C1 vs. C2D1 vs. D2

B1 vs. B3C1 vs. C3D1 vs. D3

B2 vs. B3C2 vs. C3D2 vs. D3

Not presented in training Comparisons B1 vs. C1B1 vs. D1B2 vs. C1B2 vs. D1B3 vs. C1B3 vs. D1C1 vs. D1C2 vs. D1C3 vs. D1

B1 vs. C2B1 vs. D2B2 vs. C2B2 vs. D2B3 vs. C2B3 vs. D2C1 vs. D2C2 vs. D2C3 vs. D2

B1 vs. C3B1 vs. D3B2 vs. C3B2 vs. D3B3 vs. C3B3 vs. D3C1 vs. D3C2 vs. D3C3 vs. D3

with both of those structures. Those 27 dis-criminations are also presented on symmetrytests following sample-as-node, but not linear-series, training.

Table 4 shows the complete map of the sim-ple discriminations summarized in Table 3for sample-as-node training. This table distin-guishes the simple discriminations that areand are not presented in training, and re-flects the roles of the stimuli (i.e., samples,comparisons) involved in each simple dis-crimination. A similar map for linear-seriestraining would be organized differently, but

the category totals would be the same. Forexample, the discrimination of A1 from C3would not be presented in linear-series train-ing, but the discrimination of B1 from C3would be.

As the series of preceding tables suggests,the differences between the proportion ofsimple discriminations presented in compar-ison-as-node training and those presented inthe other structures increase as prospectiveclass size and number of classes increase. Ta-ble 5 shows that for sample-as-node and lin-ear-series training, the number of simple dis-


Table 5

Simple discriminations by class size and number ofclasses for linear series and sample-as-node training struc-tures.

Stimuliper class

Number ofclasses

Grand totaldiscriminations

Number ofdiscriminationsnot presented

in training

3 2345

15a

3666

105

4a

91625

4 2345

28b

66c

120190

12b

27c

4875

5 2345

45105190300

245496

150

6 2345

66153276435

4090

160250

a From Table 1.b From Table 2.c From Table 3.

criminations not presented in trainingincreases as a direct function of class size andclass number. Moreover, given a particularnumber of classes, the proportion of discrim-inations not presented to the total number ofdiscriminations also increases as class size isincreased. The proportion is relatively unaf-fected by increases in number of classes, how-ever, when class size is held constant. Whetherincreases in the proportion of discriminationsnot presented will have different effects thanincreases in number alone is a question forfuture research.

In sum, sample-as-node or linear-seriestraining structures for large equivalence clas-ses present only a fraction of the simple dis-criminations that make up the conditionaldiscriminations encountered on tests for theproperties of equivalence. It is possible thatsome new discriminations will be acquiredover the course of repeated testing, which isoften conducted when positive outcomes arenot seen on initial tests (see R. R. Saunders& Green, 1992). Because these additional dis-criminations must be acquired in the absenceof trial-by-trial differential consequences, theyare not likely to be acquired rapidly, if they

are acquired at all. This may account for thegradual emergence of equivalence that hasbeen reported by a number of investigators,a point to which we will return later.

APPLICATION OF THEDISCRIMINATION ANALYSIS TO

EQUIVALENCE RESEARCH

The analysis presented here suggests sometestable hypotheses about the results of stim-ulus equivalence experiments conducted withvarious training structures. The most generalone is that, all other things being equal, sam-ple-as-node and linear-series training are lesslikely to yield positive outcomes on tests forequivalence than is comparison-as-node train-ing. In the following sections we examine rel-evant published studies for evidence bearingon this and related hypotheses.

Studies Involving a Small Number ofSmall Classes

Investigators have reported mixed out-comes on tests for equivalence in young chil-dren following either sample-as-node training(Barnes, Browne, Smeets, & Roche, 1995;Barnes, McCullagh, & Keenan, 1990; Devany,Hayes, & Nelson, 1986; Pilgrim, Chambers, &Galizio, 1995; Sidman et al., 1985; Sidman,Willson-Morris, & Kirk, 1986) or linear-seriestraining (Lazar, Davis-Lang, & Sanchez, 1984;Michael & Bernstein, 1991) to establish twothree-member classes. In an experiment with3 children aged 6 to 7 years, a sample-as-nodestructure was used to establish three three-member classes of tactile stimuli. Equivalenceclasses were eventually established, but onlyafter extensive retesting (O’Leary & Bush,1996). Another study, using comparison-as-node training to establish three three-mem-ber classes of arbitrary visual stimuli with chil-dren aged 7 to 12 years, produced positivetest outcomes immediately following training(Williams, Saunders, Saunders, & Spradlin,1995). In contrast, Eikeseth and Smith (1992)reported that two three-member classes wereestablished in none of 4 young children withautism following sample-as-node training. It isimportant to note that the objective of theinitial testing and training in all of these ex-periments was to establish just two or threethree-member equivalence classes, so that the


total number of simple discriminations in-volved was quite small.

Most experiments using sample-as-nodeand comparison-as-node training to establishsmall equivalence classes with adults haveproduced positive outcomes. Green (1990)reported development of two auditory-visualas well as two all-visual equivalence classes ofthree members each in adults with mild men-tal retardation following sample-as-nodetraining. In another study using a sample-as-node structure, three three-member classeswere established in 15 of 16 normally capableadults (Innis, Lane, Miller, & Critchfield,1998). Pilgrim and Galizio (1990) also re-ported positive outcomes in normally capableadults following sample-as-node training.More recently, the same investigators report-ed establishing two four-member equivalenceclasses in normally capable adults using amixed comparison-as-node/sample-as-nodetraining structure (Pilgrim & Galizio, 1995).

Investigators using linear-series trainingstructures designed to establish small equiv-alence classes have reported mixed results.Fields, Adams, Newman, and Verhave (1992)reported that immediately after linear-seriestraining, positive outcomes on tests for allproperties of equivalence were evident foronly 3 of 14 adults; repeated or specially de-signed testing was necessary to yield positiveoutcomes in the other subjects. Similar re-sults were obtained in the same laboratorywith other adult subjects, and successful lin-ear-series training and testing with one set ofstimuli was not always followed by positiveoutcomes following linear-series training witha second set of stimuli (Adams, Fields, & Ver-have, 1993; Buffington, Fields, & Adams,1997; Fields et al., 1997). The studies byFields and colleagues employed consonant-vowel-consonant trigrams as stimuli. In con-trast, three-choice linear-series training witholfactory stimuli, trigrams, and arbitraryforms (Annett & Leslie, 1995) and a mixtureof trigrams and haptic stimuli (Tierney, deLargy, & Bracken, 1995) produced positiveresults in most adult subjects.

In summary, the observation that sample-as-node and linear-series training establishedequivalence classes in some subjects suggeststhat the successive discriminations that werecalled for on the tests for equivalence wereestablished, even though they were not ex-

plicitly presented in baseline training. Wenoted previously that although sample-as-node training (e.g., AB, AC) does not explic-itly require discrimination of comparisonstimuli from different trial types (e.g., B stim-uli from C stimuli) because those stimuli arenever pitted directly against one another, theyare presented successively across trials as pairsof comparisons when training trials aremixed. With small potential classes, this ex-posure may suffice for acquisition of the Bversus C discriminations by some subjects.The same possibility applies to linear-seriestraining for a small number of small classes.For example, the A stimuli may be discrimi-nated from the C stimuli when AB and BCtrials are mixed during training. If so, positivetest outcomes would seem equally likely fol-lowing training with either structure whenthe total number of simple discriminationsinvolved is relatively small. Alternatively, pos-itive outcomes might result from interspersingtest trials with training trials in test sessions.Previously untrained simple discriminationscould develop over the course of testing dueto (a) the additional exposure to training tri-als, (b) the juxtaposition of test trials that in-clude those discriminations with training tri-als, or (c) both. This might explain thegradual emergence of equivalence-consistenttest performances that has been documentedin a number of studies (e.g., Lazar et al., 1984;Sidman et al., 1986; Spradlin et al., 1973).

Studies Involving Larger Classes

With sample-as-node or linear-series train-ing designed to produce more than two po-tential classes or classes with more than threemembers, the number of new discriminationspresented to subjects on tests for equivalenceis greater than with smaller classes (refer toTable 5). Therefore, we would expect struc-ture-related differences in test performancesto be more likely or more marked when train-ing is designed to produce larger classes. Re-sults of a series of studies by R. R. Saundersand colleagues, discussed previously, are con-sistent with this prediction: In 23 of 28 youngchildren and individuals with mental retar-dation, four- and five-member equivalenceclasses were established following compari-son-as-node training; in only 3 of 13 subjectswere such classes established following sam-ple-as-node training (Drake & Saunders,


1987, cited in K. J. Saunders et al., 1993; K.J. Saunders et al., 1993; R. R. Saunders et al.,1999; R. R. Saunders, Saunders, Kirby, &Spradlin, 1988; R. R. Saunders, Wachter, &Spradlin, 1988; Spradlin & Saunders, 1986).A study with adult subjects, in which five-member equivalence classes were establishedin only 2 of 12 cases following linear-seriestraining, also corroborates this prediction(Fields et al., 1995).

Results of other studies, however, appear tobe inconsistent with our prediction. For ex-ample, Spradlin, Saunders, and Saunders(1992) reported successful development oftwo five-member classes in normally capablechildren immediately following linear-seriestraining. Kennedy (1991) employed a‘‘branching’’ linear-series structure, or whatmight be considered a combination of linear-series and sample-as-node training, to estab-lish three seven-member classes with normal-ly capable adults. Gradual emergence ofequivalence was reported for most subjects,with multinode relations emerging last. Sim-ilar results were reported for typically devel-oping children following sample-as-nodetraining (K. J. Saunders et al., 1993). As wenoted above and elsewhere (R. R. Saunders& Green, 1992), simple discriminations thatwere not presented in training may be ac-quired over the course of testing, even in theabsence of differential trial-by-trial conse-quences. When there are a large number ofthese, acquisition may be more likely to occurin normally capable adults and older childrenthan in individuals with severe learning dif-ficulties or young children. This could ac-count for the results reported by Kennedy(1991), K. J. Saunders et al. (1993), andSpradlin et al. (1992).

Gradual Emergence of Equivalence

Next, we pick up the thread of our earlierdiscussion about gradual emergence of equiv-alence, this time in the context of linear-se-ries training structures. As Table 5 shows, lin-ear-series training (e.g., AB, BC, CD, DE) andtesting for two five-member equivalence clas-ses yield 24 simple discriminations that arenot presented in training but that are calledfor on test trials (e.g., A1 vs. C1, B2 vs. D2,C1 vs. E2, etc.). In the terminology of thenodal distance literature, the A versus C, Bversus D, and C versus E discriminations are

presented in the one-node tests, the A versusD and B versus E discriminations in the two-node tests, and the A versus E discriminationsin the three-node tests. A typical test sessionintersperses test trials (AC, BD, CA, EC, etc.)among baseline trials (AB, BC, CD, DE) witheach trial type appearing equally often. Thismeans that the B, C, and D stimuli will appearthree times more often than either A stimulusand 50% more often than either E stimuluson baseline trials during each test session. Ifpreviously untrained discriminations developover the course of testing, as we speculatedpreviously, then we hypothesize that the or-der in which those untrained discriminationsare acquired will correspond to the frequencyof reexposure to particular stimuli on base-line trials during testing. That is, one-nodetests (BD and DB) should produce positiveresults first. Other tests involving B, C, and Dstimuli (i.e., one-node tests for AC, CA, CE,and EC and two-node tests for AD, DA, BE,and EB) should produce positive results next,and tests involving the A and E stimuli (i.e.,the three-node AE and EA tests) should bethe last to produce positive results. In otherwords, patterns of gradual emergence thathave been attributed to nodal or associativedistance (e.g., Fields et al., 1990; Kennedy,1991; Kennedy et al., 1994) may instead re-flect gradual acquisition of simple discrimi-nations as a function of frequency of stimuluspresentation during testing.

To evaluate this possibility, we looked forstudies reporting nodal distance effects thatconformed to the procedural assumptionsunderlying our analysis (simultaneous MTSprocedures, all baseline trials mixed beforetesting, test-trial types for each property ofequivalence presented together within a ses-sion, balanced MTS trials in training and test-ing, and no differential consequences on testtrials). Unfortunately, we found none. Theseminal experiment on nodal distance (Fieldset al., 1990), for example, employed unbal-anced test trials on which comparisons fromdifferent stimulus sets were mixed (e.g., CAtest trials had A and C stimuli as compari-sons). Kennedy (1991) also employed unbal-anced trial types. Other studies purporting toshow nodal distance effects had other meth-odological features that made them unsuit-able for our discrimination analysis. For in-stance, extensive pretesting and the use of


stimuli (words) with which subjects hadpreexperimental histories likely confoundedthe results (Kennedy et al., 1994). Thus, newexperiments will be necessary to evaluate ourhypothesis about gradual emergence.

Differential Responding

Exposure to test trials that involve noveldiscriminations may give rise to behavior thatwas not the direct product of training contin-gencies, but that may nonetheless foster ac-quisition of the new discriminations. Forsome subjects, preexperimental repertoiresmay emerge in the presence of novel trialtypes or arrangements, generating differen-tial responding. Among verbally sophisticatedhumans, stimulus naming is a common formof such behavior. Although perhaps not nec-essary for untrained simple discriminations toemerge, differential responding such as nam-ing could foster the acquisition of those dis-criminations and, therefore, the emergenceof equivalence-consistent performances overthe course of testing (see Dugdale & Lowe,1990; Eikeseth & Smith, 1992; Lowe & Beasty,1987; McIlvane & Dube, 1996; Sidman,1994).

McIlvane and Dube (1996) suggested thatnaming or other differential responding mayoccur even when experimental proceduresdo not explicitly require it. Several proce-dures may promote naming. For instance, K.J. Saunders et al. (1993) noted that in priorstudies comparing sample-as-node with com-parison-as-node training structures (R. R.Saunders, Wachter, & Spradlin, 1988; Sprad-lin & Saunders, 1986), subjects with mentalretardation were given unique names foreach stimulus in four ‘‘instructed’’ trials atthe very beginning of conditional discrimi-nation training. On those trials, the experi-menter named each of the stimuli twice, butdid not do so thereafter, and the subjectswere never required to repeat the names. Pos-itive outcomes on equivalence tests were seenfor all subjects given comparison-as-nodetraining but for only 1 subject given sample-as-node training. In a partial replication, K. J.Saunders et al. (1993) exposed 11 subjectswith mental retardation to comparison-as-node training to establish two four-memberequivalence classes. Six subjects received fourinitial instructed trials with stimulus namesprovided by the experimenter, and 5 did not.

Positive outcomes on equivalence tests wereseen for 5 of the 6 instructed subjects but foronly 1 of 5 uninstructed subjects. A follow-upexperiment showed that when previously un-successful uninstructed subjects were giveninstructed training with new stimuli, they toopassed equivalence tests. These results sug-gest that differential naming of stimuli by theexperimenter fostered the establishment ofequivalence classes in subjects with mental re-tardation, including those who received com-parison-as-node training (K. J. Saunders et al.,1993). Demonstrations that equivalence clas-ses emerged more readily following sample-as-node training with auditory samples thanwith visual samples may reflect similar pro-cesses (Green, 1990; Sidman et al., 1986).That is, presenting auditory samples (names)in training may set the occasion for subjectsto produce names for the other experimentalstimuli, thereby fostering simple discrimina-tions among them. These possibilities warrantdirect empirical testing.

In a related recent study, R. R. Saunders etal. (1999) provided preschool children with‘‘instructed’’ training like that provided inprevious experiments in the same laboratory(i.e., differential experimenter-provided oralnames for each stimulus on the first fourtraining trials). Some subjects had compari-son-as-node training, and others had sample-as-node training. The former required moretrials for performance to reach the trainingcriterion than did the latter. Similar differ-ences were reported by R. R. Saunders, Wach-ter, and Spradlin (1988) and Fields et al. (inpress). Thus, across comparable experiments,preschool children, normal adults, and ado-lescents with mental retardation acquiredbaseline conditional discriminations morerapidly in sample-as-node training than incomparison-as-node training, but were lesslikely to produce positive outcomes on equiv-alence tests immediately after training. Onthe other hand, as noted above, equivalenceclasses were not established in all the subjectswith mental retardation who had comparison-as-node training without instructions that in-cluded differential names for the stimuli inthe study by K. J. Saunders et al. (1993). Tak-en together, these results suggested to R. R.Saunders et al. (1999) that instructions mayenhance training-structure differences. Be-cause comparison-as-node training presents a


larger number of successive (sample) discrim-inations than does sample-as-node training, itmay be more likely to set the occasion fordifferential sample naming (at least by ver-bally skilled subjects). This might foster thedevelopment of simple discriminations, notjust among the samples but also among sam-ples and comparisons presented simulta-neously, which is then likely to produce pos-itive outcomes on tests for the properties ofequivalence (cf. Sidman, 1994, pp. 413–414).In contrast, sample-as-node conditional dis-criminations require fewer successive discrim-inations than does comparison-as-node train-ing, which might decrease the likelihood thatsubjects who are capable of naming stimuliwill do so during training. This may in turnmake them ill-prepared to perform the newdiscriminations called for on tests. Either orboth of these possibilities may account fortraining-structure differences reported insome stimulus equivalence studies (R. R.Saunders et al., 1999).

Response Speed and Verbal Self-Reports

A recent study used linear-series training(AB, BC, CD, DE, EF, FG) in an attempt toestablish three seven-member equivalenceclasses with 12 college students (Spencer &Chase, 1996). The investigators reported thataccuracy of responding on tests for the prop-erties of equivalence decreased with increas-ing nodal distance for most subjects, but theeffect was small and transient. In contrast,speed of correct responding on tests of tran-sitivity and combined tests of symmetry andtransitivity was inversely related to nodal dis-tance for nearly all subjects, a relation thatwas maintained with repeated testing. Sub-jects responded considerably faster on base-line and symmetry trials than on transitivityand combined trials during test sessions, andsomewhat faster on baseline trials than onsymmetry test trials. There was no overall dif-ference in speed of responding to transitivityand combined trials.

It is interesting to speculate whether thedifferential response speeds reported bySpencer and Chase (1996) might have beena function of differential acquisition duringtraining of the simple discriminations re-quired on the tests. The GF symmetry tests,for example, required successive discrimina-tions among G stimuli, which had only been

presented simultaneously (as comparisons)during training. The A, B, C, D, E, and F stim-uli, on the other hand, were all presentedsuccessively (as samples) during training, sosymmetry test trials with those stimuli likelyposed little difficulty for the subjects. Becausethe GF symmetry tests constituted one sixthof all symmetry test trials in the study, de-creases in speed on those trial types alonecould account for the slight overall differenc-es in speed of responding between baselineand symmetry trials. Indeed, it seems plausi-ble that the decrease in response speed onequivalence test trials that Spencer and Chaseattributed to nodal distance was instead afunction of the numbers and types of new,untrained discriminations presented on thevarious test trials. Further, it is possible thatsome of the simple discriminations that werenot explicitly presented in the conditionaldiscriminations trained early in the linear se-ries were nonetheless acquired over thecourse of subsequent training, whereas thosetrained late in the series were not. The orderin which conditional discriminations weretrained was AB, BC, CD, DE, EF, FG. As wesuggested earlier, simple discriminationsamong some stimuli that were never present-ed together as samples within a session or ascomparisons within trials (e.g., the B and Dstimuli) might develop when those stimuliare presented successively across trials (e.g.,in sessions mixing AB, BC, and CD trainingtrials, as in the Spencer and Chase study).This possibility could not arise, however, forstimuli introduced late in the linear series(e.g., F and G stimuli). Our analysis wouldpredict, therefore, that subjects might re-spond faster on two-node tests involving stim-uli introduced early in the series (e.g., BD,DB) than on two-node tests involving stimuliintroduced later (e.g., DG, GD), because theformer involve simple discriminations thatwere more likely to have been acquired thanthe latter. Response speed on the trials withstimuli from the middle of the training series(BE, EB, CF, FC) should fall in between. Allthree-, four-, and five-node tests would involvesome of the stimuli introduced late in thetraining series, so they should be excludedfrom such an analysis.

The differential response speeds reportedby Spencer and Chase (1996) seem to paral-lel verbal self-reports of accuracy on symme-


try and equivalence tests demonstrated in an-other study. College students who were givensample-as-node training and testing to poten-tially establish two four-member equivalenceclasses were asked to report verbally whetherthey believed their test-trial responses werecorrect or incorrect (Lane & Critchfield,1996). Symmetry and equivalence test results(i.e., MTS performances) were highly accu-rate. Self-reported evaluations of accuracyranged from 96% to 100% on symmetry testsand 79% to 100% on equivalence tests. Thatis, despite nonverbal responses on equiva-lence test trials that were highly consistentwith equivalence, verbal reports seemed to re-flect some uncertainty about the accuracy ofthose responses, more so than on symmetrytests. Lane and Critchfield’s results are diffi-cult to interpret, however, because of thecomposition of the training trials. Althoughtraining was designed to establish only twoclasses, three-choice MTS procedures wereemployed instead of balanced two-choice pro-cedures. This meant that stimuli from differ-ent stimulus sets were mixed as comparisons:AB trials had the C stimuli as the third com-parisons, AC trials included D comparisons,and AD trials had B comparisons. Thus, un-like sample-as-node training with standardprocedures in which the B, C, and D stimuliare never juxtaposed within trials, Lane andCritchfield’s training procedures might haveestablished simple discriminations among allthe experimental stimuli prior to testing. Fur-ther research will be necessary to determinehow such unbalanced trial configurationsmight influence the numbers and types ofsimple discriminations established in train-ing.

SUMMARY ANDCONCLUSION

We have suggested, based on an analysis ofthe simple discriminations that make up thetrained and tested conditional discrimina-tions in typical stimulus equivalence experi-ments, that sample-as-node and linear-seriestraining are less likely to produce positiveoutcomes on all tests for the properties ofequivalence than is comparison-as-node train-ing. These differential outcomes should bemore likely or more marked when training isdesigned to establish relatively large classes or

a large number of classes. The discriminationanalysis also suggests an alternative interpre-tation of some findings in the stimulus equiv-alence literature that have been attributed tosuch variables as nodal distance. For exam-ple, we have suggested that differential re-sponse speeds or latencies on various types oftest trials may reflect differential acquisitionof the component simple discriminations thatare due to training structure. Unlike nodaldistance, this is a characteristic of experimen-tal procedures that relates to familiar behav-ioral principles.

Further, some specific, testable predictionsfollow from our main hypothesis:

1. Sample-as-node training should yield lessaccurate performances on tests for symmetrythan does linear-series training, providedclass size and number of classes are equated.

2. Following sample-as-node training, sym-metry tests present successive discriminationsamong former comparison stimuli serving assamples, and simultaneous discriminationsamong former samples serving as compari-sons. Positive outcomes on these symmetrytests entail demonstration of the simple suc-cessive and simultaneous discriminations nec-essary for positive results on equivalence tests.Thus, following sample-as-node training, ifsubjects are given equivalence tests only afterproducing positive outcomes on the symme-try tests, all test outcomes are likely to be pos-itive. If equivalence tests are given first, how-ever, positive outcomes are less likely.

3. Any procedure that leads to differentialresponding to each of the stimuli in the ex-periment, whether explicitly arranged by theexperimenter or arising from subjects’ preex-perimental histories, should establish simplediscriminations among all the stimuli, there-by mitigating the differential effects of train-ing structures.

Although our review suggests that pub-lished research on stimulus equivalence pro-vides some support for the discriminationanalysis of training-structure effects, the evi-dence is neither overwhelmingly confirma-tory nor discomfirmatory. This may reflectthe complexities of equivalence research atleast as much as the validity of our analysis.In this now sizable body of research, proce-dural and subject variability is so great that itis very difficult to make comparisons acrossstudies or to draw general conclusions. For


example, we suggested that experiments de-signed to produce small numbers of smallequivalence classes should produce positiveoutcomes regardless of training structure, be-cause of their relatively low discrimination de-mands in terms of both the total number andthe types of component simple discrimina-tions required. Yet Fields and his colleagueshave consistently reported negative outcomesfrom such experiments, even though theirsubjects were normally capable adults whoshould not have had difficulty learning therequisite discriminations (e.g., Fields et al.,1992). One striking difference between thoseexperiments and many others in the basicstimulus equivalence literature is that thestimuli were printed trigrams rather than ar-bitrary forms or sounds. This suggests thatspecific characteristics of the stimuli might ac-count for the high rate of equivalence testfailures in the Fields laboratory. In particular,the presence of some identical or physicallysimilar letters among trigrams might makefor especially difficult simple discriminationsamong experimental stimuli that we contendare necessary for positive equivalence out-comes. We have not reanalyzed the experi-ments from the Fields laboratory for this pos-sibility, but our analysis suggests it may be aplausible explanation for the reported equiv-alence failures (and see K. J. Saunders et al.,1993).

Another study that appears to contradictour analysis was reported recently by Arntzenand Holth (1997). Their data are consistentwith our contention that linear-series trainingis not very likely to produce positive equiva-lence outcomes, but are inconsistent with ourpredictions about the outcomes of sample-as-node and comparison-as-node training struc-tures. Following training to establish twothree-member classes, Arntzen and Holth re-ported positive test outcomes in fewer sub-jects following comparison-as-node trainingthan sample-as-node training, and in evenfewer following linear-series training. As R. R.Saunders et al. (1999) noted, however, theseinvestigators conducted their tests in isola-tion, that is, in blocks of test trials rather thanwith test trials interspersed among trainingtrials. This raises a question as to how well thetrained conditional relations were main-tained during testing. Given the observationthat comparison-as-node training arranges

more difficult discriminations (i.e., more suc-cessive discriminations among samples) thansample-as-node training, it seems plausiblethat the stability of the baseline performancesengendered by the two training structures dif-fered at the point at which the isolated testtrials were presented. This might account forthe smaller number of subjects in whomequivalence classes were established followingcomparison-as-node training than followingsample-as-node training.

A more recent study by the same investi-gators poses some interesting challenges toour discrimination analysis. Holth and Arntz-en (1998) reported that different mixtures ofarbitrary stimuli and familiar stimuli (e.g.,pictures of common objects) in linear-seriestraining structures (AB, BC) produced differ-ent results with normally capable adult sub-jects. Recall that this structure yields test trialsfor transitivity (AC) and equivalence (CA)that require both simultaneous and succes-sive discriminations among the A and C stim-uli not presented in training (see Table 1 andFigure 4). When familiar pictures constitutedthe A and C stimulus sets, the B set, or allthree sets, positive equivalence test outcomeswere produced more often than when onlythe A or the C stimuli were familiar pictures,or when all the stimuli were arbitrary. En-hanced outcomes with familiar pictures inboth the A and C positions or in all threepositions in the linear series are entirely con-sistent with our analysis: Subjects presumablycould discriminate among all those stimulibefore the experiment began, so they shouldhave had no difficulty making the simple dis-criminations called for on the AC and CAtests. When just the A stimuli or the C stimuliwere familiar to subjects prior to the experi-ment, there may not have been enough pre-existing simple discriminations to carry themthrough the AC and CA tests. It is not readilyapparent to us, however, why equivalence testoutcomes were also enhanced when just theB stimuli, but not the A and C stimuli, werefamiliar to the subjects (i.e., discriminated)when they entered the experiment, becausethe B stimuli did not appear at all on the testsfor transitivity and equivalence. Whatever theexplanation, the Holth and Arntzen experi-ment suggests an interesting approach to ex-amining simple discrimination effects withinvarious training structures.


We hope the analysis presented here willprompt research to resolve some of the ap-parent inconsistencies in the stimulus equiv-alence literature, particularly inconsistenciesthat appear to be the bases for ongoing de-bates about such issues as naming, gradualemergence, nodal or associative distance, andof course, training-structure effects. We donot mean to suggest that all of the reporteddifferences in stimulus equivalence outcomescan be accounted for by differences in thesimple discriminations presented in the vari-ous training structures; other potential sourc-es of extraneous stimulus control should beexamined as well. We do mean to suggest thatthe role of simple discriminations as inherentcomponents of conditional discriminationshas been underappreciated, and that carefulattention to this factor can lead to improvedexperimental designs for stimulus equiva-lence work.

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Received January 23, 1998Final acceptance March 29, 1999

138

ERRATUM

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