A four-member equivalence class (A----B----C----D) can be formed by training AB, BC, and CD. The nodal stimuli, B and C, mediate all of the derivative (transitive and equivalence) relations in the class. The derivative relations AC, CA, BD, and DB are separated by one node, whereas AD and DA are separated by two nodes. How do the number of nodes that separate the stimuli in a derivative relation influence the induction of stimulus control exerted by that relation? Seven college students learned two four-member classes made up of nonsense syllables. After training, all derivative relations were presented repeatedly without informative feedback. Stimulus control exerted by each derivative relation was assessed concurrently. For the 7 subjects, control exerted by the derivative relations increased gradually with repeated presentations. With 6 of the 7 subjects, the one-node relations exerted more control than the two-node relations during the process. However, the disparity between the one- and two-node relations decreased with repeated presentations. Eventually, all derivative relations exerted complete control. The control exerted by derivative relations during induction was inversely related to the number of nodes separating the terms in the derivative relations. These results demonstrate that nodal distance is a determinant of the relatedness of stimuli in equivalence classes. The findings are discussed in terms of remote association, semantic memory networks, and the study of transitive inference.

Thirty college students attempted to form three 3-node 5-member equivalence classes under the simultaneous protocol. After concurrent training of AB, BC, CD, and DE relations, all probes used to assess the emergence of symmetrical, transitive, and equivalence relations were presented for two test blocks. When the A-E stimuli were all abstract shapes, none of 10 participants formed classes. When the A, B, D, and E stimuli were abstract shapes and the C stimuli were meaningful pictures, 8 of 10 participants formed classes. This high yield may reflect the expansion of existing classes that consist of the associates of the meaningful stimuli, rather than the formation of the ABCDE classes, per se. When the A-E stimuli were abstract shapes and the C stimuli became S(D)s prior to class formation, 5 out of 10 participants formed classes. Thus, the discriminative functions served by the meaningful stimuli can account for some of the enhancement of class formation produced by the inclusion of a meaningful stimulus as a class member. A sorting task, which provided a secondary measure of class formation, indicated the formation of all three classes when the emergent relations probes indicated the same outcome. In contrast, the sorting test indicated "partial" class formation when the emergent relations test indicated no class formation. Finally, the effects of nodal distance on the relatedness of stimuli in the equivalence classes were not influenced by the functions served by the C stimuli in the equivalence classes.

The structure of equivalence classes can be completely described by four parameters: class size, number of nodes, the distribution of "singles" among nodes, and directionality of training. Class size refers to the number of stimuli in a class. Nodes are stimuli linked by training to at least two other stimuli. Singles are stimuli linked by training to only one other stimulus. The distribution of singles refers to the number of singles linked by training to each node. Directionality of training refers to the use of stimuli as samples and as comparison stimuli in training. These four parameters define the different ways in which the stimuli in a class can be organized, and thus provide a basis for systematically characterizing the properties of stimuli in a given equivalence class. The four parameters can also be used to account for the development of individual differences that are commonly characterized in terms of "understanding" and connotative meaning.Methods are described for generating all possible combinations of parameter values, and a formula is introduced which specifies all of the parameter values for an equivalence class. Its utility for interrelating experimental procedures is demonstrated by analyzing a number of representative experiments that have addressed equivalence-class formation.Key words: equivalence class, distribution of singles, node, single, directionality of training, individual differences, derived relations, equivalence relation, training cluster (TC) At first, a picture of a peach, the spoken word "peach," the written word PEACH, and the smell of a peach are stimuli that are unrelated in terms of physical properties and meaning. With exposure to appropriate contingencies, these stimuli become interrelated and form an equivalence class. Because the stimuli in the class do not necessarily share any physical properties, their relatedness is very likely to be the result of training, either in the laboratory or in a natural setting. In this article we consider how groups of physically disparate stimuli come to be functionally linked to form classes of equivalent stimuli, how different forms of linkage can be quantitatively described and summarized, and how each variation in linkage may influence the degree of relatedness exhibited by the stimuli in the class.

Twelve subjects were trained to select one of two stimuli from a pair (the B pair) when presented with one of two stimuli from another pair (the A pair), thus establishing two AB relations, Al-BI and A2-B2. In a similar fashion, additional stimuli were used to establish BC, CD, and DE relations. Trials used to train all relations occurred in each session. Once performances were established, probe trials were introduced that tested for the emergence of untrained relations (e.g., BI-DI or Al-El).These emergent relations were categorized according to nodal distance (i.e., the number of stimuli across which transitivity would have to hold in order for the relation to emerge). For example, a test for A2-C2 crosses one node (B2), whereas a test for Al-El crosses three nodes (Bl, Cl, and Dl).Only 2 of the subjects formed equivalence classes. The evocation of class-appropriate responding by each emergent-relation probe was an inverse function of nodal distance for all 12 subjects. In addition, performance on the originally trained relations was disrupted by the introduction of probes. The 2 subjects who exhibited equivalence classes were then trained to make different numbers of key presses in the presence of each of the four A and E stimuli. In a response-transfer test, the B, C, and D stimuli evoked the responses trained to the A and E stimuli in the same equivalence class. Likelihood of class-appropriate responses was an inverse function of nodal distance, and this pattern persisted across testing. Reaction times in the transfer test were an inverted U-shaped function of nodal distance. Because training of the baseline relations occurred concurrently and the B, C, and D stimuli were presented an equal number of times before the transfer test, the test performances illustrate effects of nodal distance that were not confounded by order or amount of experience with the stimuli. The results imply that ordered, sequential exposure to individual stimulus relations may facilitate the development of equivalence classes and that the relatedness of stimuli within an equivalence class is a relatively permanent inverse function of nodal distance. Willson-Morris, 1985). It has also been supported by more recent studies that systematically measured the choice performances (Dube, Green, & Serna, 1993;Fields, Adams, Newman, & Verhave, 1992;Fields, Adams, Verhave, & Newman, 1990;Kennedy, 1991;Kennedy, Itkonen, & Lindquist, 1994) and the reaction times (Bentall, Dickins, & Fox, 1993;Dickins, Bentall, & Smith, 1993;Wulfert & Hayes, 1988) occasioned by tests of emergent relations. The most recent support was provided by Fields, Adams, Verhave, and Newman (1993), who showed that nodal distance influenced transfer of responding among members of an equivalence class.In all of these studies, however, the baseline relations were trained in a serial manner. As a result, nodal distance was confounded with the order of stimulus introduction during training and most likely with the number of times each stimulus was presented in

When a number of two-stimulus relations are established through training within a set of stimuli, other two-stimulus relations often emerge in the same set without direct training. These, termed "transitive stimulus relations," have been demonstrated with a variety of visual and auditory stimuli. The phenomenon has served as a behavioral model for explaining the emergence of rudimentary comprehension and reading skills, and the development of generative syntactic repertoires. This article considers the range of relations that can arise between a given number of stimuli in a class, the number of directly established two-stimulus relations necessary for the emergence of transitive relations, the forms that training sets of stimuli can take, and the number of transitive two-stimulus relations that can be induced without direct training. The procedures needed to establish and assess transitive stimulus control, the possible interactions between the training and testing procedures, and the constraints these interactions place upon the analysis of transitive stimulus control are also examined. The present analysis indicates that in a transitivity test, choice among such stimuli may be controlled by (1) the relation between the sample and the positive comparison stimulus (transitive stimulus control), (2) the relation between the sample and the negative comparison stimulus (S-rule control), and (3) possible discriminative properties that may inadvertently be established in the positive and negative comparison stimuli (valence control). Methods are described for distinguishing these three forms of stimulus control.

The emergence of equivalence classes in college students is unlikely when all baseline relations are trained concurrently and all probes for emergent relations are then introduced concurrently (the simultaneous protocol). This experiment showed how the number of nodes and the size of previously established equivalence classes enhanced the emergence of new equivalence classes under the simultaneous protocol. First, one-node three-, five-, or seven-member classes or three-node five-or seven-member classes were established with college students. A sixth group received no pretraining. Then, the simultaneous protocol was used to establish new three-node five-member equivalence classes with all students. The speed and variability with which the baseline relations were established in the simultaneous protocol were inverse functions of number of nodes in the previously established classes, but not of their size. The percentage of subjects who showed the emergence of new equivalence classes under the simultaneous protocol was a direct function of number of nodes and size of pretrained classes. The additional time spent for pretraining greatly reduced the total training time needed to produce individuals who showed the emergence of classes under the simultaneous protocol. The total time saved was a direct function of number of nodes and number of stimuli in the pretrained classes.

Many students struggle with statistical concepts such as interaction. In an experimental group, participants took a paper-and-pencil test and then were given training to establish equivalent classes containing four different statistical interactions. All participants formed the equivalence classes and showed maintenance when probes contained novel negative exemplars. Thereafter, participants took a second paper-and-pencil test. Participants in the control group received two versions of the paper-and-pencil test without equivalence-based instruction. All participants in the experimental group showed increased paper-and-pencil test scores after forming the interaction-indicative equivalence classes. Class-indicative responding also generalized to novel exemplars and the novel question format used in the paper-and-pencil test. Test scores did not change with repetition for control group participants. Implications for behavioral diagnostics and teaching technology are discussed.

This study investigated how the learning of one set of equivalence classes enhances the learning of new equivalence classes. Fifty-two undergraduate students were divided into four groups. Subjects in Group 1 received no pretraining. Using the simple-to-complex procedure followed by incremental expansion of class size, subjects in Groups 2, 3, and 4 learned 3-, 4-, and 5member equivalence classes, respectively. After pretraining, two new 3-member equivalence classes were established by the concurrent training of all baseline relations and the concurrent presentation of all emergent relations probes to assess class formation (the simultaneous protocol). With no pretraining, 58% of subjects formed the new classes under the simultaneous protocol. After pretraining of the 3-, 4-, and 5-member classes, the new classes were formed by 62, 85, and 100% of the subjects, respectively. Pretraining of 4-and 5-member classes produced a small increment in the percentage of subjects who showed the immediate emergence of the new classes. Pretraining of the 5member classes produced a large increment in percentage of subjects who formed classes with repeated testing. Thus , pretraining influenced immediate and delayed emergence of equivalence classes. With no pretraining, during the tests used to assess the formation of the new classes, 12% of subjects showed disruption of baseline performances, relational responding produced by symmetry probes was lower than that produced by baseline relations, and very low levels of relational responding were evoked by 1-node probes. These data demonstrated the effects of nodal distance. Pretraining did not ameliorate the disruption of baseline performances. Pretraining of 4-and 5-member classes produced moderate increments in the relational responding evoked by symmetry probes. Pretraining of 5-member classes produced large increments in the relational responding evoked by 1-node probes.