An equivalence class consists of a set of N stimuli that bear no resemblance to each other and are initially unrelated to each other. They come to function as members of a class when the training of N-1 relations results in the emergence of all possible relations among the stimuli, without additional training. Many studies of equivalence classes have identified procedural variables that enhance the formation of these classes (Arntzen, 2012), primary among them being the protocol used for training and testing (Adams, Fields, & Verhave, 1993; Fienup, Wright, & Fields, 2015) and the nodal structure of the to-be-formed class (Arntzen, Grondahl, & Eilifsen, 2010; Arntzen & Hansen, 2011; Arntzen & Holth, 1997, 2000; Fields, Hobbie-Reeve, Adams, & Reeve, 1999). Class formation is also enhanced by the prior establishment of other classes that contained larger sizes and nodal numbers (Buffington, Fields, & Adams, 1997; Fields et al., 2000). The likelihood of class formation can be maximized if training and testing are conducted using the simple to complex protocol (Fields et al., 1997). In contrast, the likelihood of class formation is much lower when training and testing are conducted using the simultaneous protocol, which involves the training of all baseline relations followed by testing for all of the derived relation probes in the same test blocks. The low yields produced by the simultaneous protocol have proven to be valuable because they have permitted the discovery of many variables that enhance equivalence class formation. Because the simultaneous protocol simulates the “train-all test-all” procedures that characterize many traditional educational settings, identifying these variables could be used to enhance the success of education conducted in a traditional context.

Members of equivalence classes can be meaningless and/or meaningful. Meaningless stimuli are those that do not have any specific discriminative functions, while meaningful stimuli bear some relation to other classes of stimuli. Other experiments have shown that the likelihood of class formation is influenced by the inclusion of at least one meaningful stimulus in a set of otherwise meaningless stimuli. If they have positive or neutral valences, their inclusion enhances the likelihood of class formation (Bentall, Dickins, & Fox, 1993; Bortoloti & de Rose, 2009), and if they have negative or conflicting valences (attractiveness and aversiveness in combination), they interfere with class formation (Grehan, 1998; Leslie, Tierney, Robinson, Keenan, & Watt, 1993; McGlinchey & Keenan, 1997; Peoples, Tierney, Bracken, & McKay, 1998; Plaud, 1995; Watt, Keenan, Barnes, & Cairns, 1991).

Some experiments have shown that equivalence class formation is enhanced by testing for emergent relations using trials administered in a delayed matching-to-sample format (Arntzen, 2006; Arntzen, Galaen, & Halvorsen, 2007; Saunders, Chaney, & Marquis, 2005; Vaidya & Smith, 2006). For example, Arntzen (2006) showed that equivalence class formation increased as function of increasing delays. Another line of research studied how the likelihood of equivalence class formation was influenced by the delays used during preliminary training that involved the establishment of identity (C ➔ C) or arbitrary (C ➔ X) relations, after which the training of necessary conditional discriminations and the testing for emergent relations were presented using trials having a simultaneous matching-to-sample format (Arntzen, Nartey, & Fields, 2014, 2015). In Arntzen et al. (2014b), 40 participants were taught the baseline relations for three 5-member classes with a linear series training structure (A → B→ C → D → E). The participants were assigned to four different groups. The experimental design included two reference groups, all abstract stimuli (ABS) and C stimuli as pictures (PIC), and two groups with preliminary training of C stimuli, identity simultaneous matching-to-sample (Identity-S) and identity 6-s delayed matching-to-sample (Identity-D). The main findings were that 0 and 80% formed equivalence classes in the reference groups, ABS and PIC, respectively. Furthermore, 0 and 60% formed classes in the preliminary groups, Identity-S and Identity-D, respectively. Arntzen et al. (2015) replicated and extended the results from Arntzen et al. (2014) by including two more preliminary groups; Arbitrary conditional discriminations were formed between C and X stimuli using simultaneous (Arbitrary-S) and 6-s delayed matching-to-sample procedures (Arbitrary-D). The main findings were that preliminary training with the delayed groups (Identity-D and Arbitrary-D) increased equivalence class formation more than the two simultaneous groups (Identity-S and Arbitrary-S).

Although this two-point comparison of simultaneous matching vs. delay showed a difference in likelihood of equivalence class formation, it did not clarify how delay, in general, would enhance equivalence class formation. The potential importance of this issue is clarified by considering the following possibilities. We know that a 6-s delay in the preliminary training substantially enhances class formation while a 0-s delay does not. If delay is reduced to a very short duration, at what point would the class enhancement effect cease to exist? Also, what level of delay less than 6 s would have the same enhancement effect as the 6-s delay? Furthermore, if yield increases with delay between 0 s and 6 s, will that increment be stepwise or gradual? Also, would class enhancement peak at some delay level less than 6 s? If delay duration were to be increased beyond 6 s, would class enhancement continue to increase with longer delays (e.g., 9 s)? Would some delay duration maximize class enhancement and then become asymptotic with further increases in delay duration, or would yield decrease beyond some maximal delay value?

Thus, we are conducting an experiment to address the following: (1) It will determine how the yield obtained in the ABS group is related to the yield obtained during the PIC group. In general, this comparison will designate how the inclusion of a meaningful stimulus enhances class formation. (2) It will determine how yields obtained after preliminary training are related to the yield obtained during the ABS group. Overall, this comparison will designate how preliminary training influences the enhancement of class formation. (3) It will determine how yields obtained after preliminary training are related to the yield obtained during the PIC group. In general, this comparison will indicate whether preliminary training approximates the enhancement of class formation by the inclusion of a meaningful stimulus. (4) It will determine how the likelihood of equivalence class formation is influenced by the duration of the delays used when establishing the identity relations during preliminary training. (5) It will determine how the likelihood of equivalence class formation is influenced by the duration of the delays used when establishing the arbitrary relations during preliminary training. (6) For each delay, it will determine how the establishment of identity and arbitrary relations during preliminary training influence the likelihood of equivalence class formation. We expect somewhat greater yields to follow the establishment of the arbitrary relations than the identity relations. (7) Finally, it will determine how preliminary training influences the intactness of the baseline relations while testing for the emergence of the equivalence classes. Our expectation is that resistance to disruption of the baseline relations will be enhanced by preliminary training.

Answers to these questions are important for three reasons: First, they would characterize the systematic effects of delay on the enhancement of equivalence class formation. Second, they would be a necessary first step for development of a theoretical understanding of the effect of delay on the enhancement of class formation. Finally, answers to these questions could be of applied value since they would inform the development of optimal procedures for the establishment of content-relevant equivalence classes in educational settings.

Method

Participants

A net of 120 university students (50 males and 70 females) from the University of Ghana participated in this study. Their average age was 20.6 years (SD = 1.76), and none of them had any prior experience with stimulus equivalence research. Each participant was assigned to one of 12 experimental groups in the following manner. First, all participants were assigned to one of 12 squads of 10 participants each, after which each participant in a squad was assigned to one of the 12 groups on a random basis without replacement. Finally, a debriefing was conducted with each participant upon completion of the experimental session to inform them about the study. A participant was excused for further participation in the experiment if he or she did not acquire the baseline relations for the equivalence classes.

Apparatus

Setting, Hardware, and Software

An office space measuring about 5 m × 5 m was used as the laboratory for conducting the experiments. The experiment was run with two Windows®-based laptop computers with a 16.8 in diagonal length screen with a 16 × 9 horizontal-to-vertical ratio. An external mouse was used to control the position of the cursor and to make comparison stimuli choices throughout the experiment. A custom-made matching-to-sample (MTS) software controlled the presentation of trials and recorded selection responses during the training and testing of the conditional discriminations.

Stimuli

Figure 1 displays the stimuli used in the experiment. Fifteen stimuli were used for establishing the necessary conditional discriminations for testing of three 3-node, 5-member equivalence classes. Some were abstract glyphs while others were Cyrillic, Greek, or Arabic letters. Collectively, all of which will be referred to as abstract shapes. In the PIC group, the A, B, D, and E members of the classes were abstract shapes while the C members were familiar pictures. In the remaining 11 groups, all of the stimuli were abstract shapes. In addition to being used as members of the to-be-formed equivalence classes, the abstract C stimuli were also used in the 10 preliminary training groups. Finally, Fig. 1 displays the X stimuli, which were used in preliminary training to establish the C→X relations, but were not also used as members of the to-be-formed 5-member equivalence classes.

Fig. 1
figure 1

The stimuli used in baseline conditional discrimination training and testing for equivalence class formation. The top section shows the 15 abstract stimuli while the two lines in the bottom section show the meaningful stimuli that replaced the abstract C-stimuli in the PIC group and the X stimuli used in the preliminary training

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The abstract stimuli were displayed in black on white backgrounds, while the pictorial stimuli were presented in color on white backgrounds. The size of the touch sensitive areas on the screen was 9.4 cm × 3.4 cm. Finally, 3.8 cm × 3.8 cm plastic-laminated pictures depicting each of the 15 abstract stimuli and the three pictures were used for participants to sort prior to exposure to the experimental protocol.

Design

Table 2 outlines the organization of 12 groups with 10 participants per group in the experiment. Participants in the ABS and PIC groups were taught the baseline relations for three 3-node 5-member equivalence classes with no preliminary training. Thus, both groups served as references against which to measure the effects of preliminary training on class formation. Participants in the 10 other groups received preliminary training which was conducted with the abstract C stimuli which were also used in the to-be-formed equivalence classes. In the five Identity-X groups, preliminary training involved the establishment of identity conditional discriminations (C → C) using 0-,1-, 3-, 6-, or 9-s delayed matching-to-sample procedures in the Identity-0 s, Identity-1 s, Identity-3 s, Identity-6 s, and Identity-9 s groups, respectively. In the five Arbitrary-X groups, preliminary training involved the establishment of arbitrary conditional discriminations (C → X) using 0-, 1-, 3-, 6-, and 9-s delays in the Arbitrary-0 s, Arbitrary-1 s, Arbitrary-3 s, Arbitrary-6 s, and Arbitrary-9 s groups, respectively.

The detailed arrangement of stimuli in the delay paradigm is described below in the section titled, “Preliminary Training of Identity and Arbitrary Conditional Relations.” The likelihood of class formation obtained in each group was measured in terms of the percentage of participants in a group who formed the classes. Referred to as “yield,” it was the primary dependent variable in the experiment.

Procedure

Informed Consent

Each participant was first welcomed, asked to take a seat, and was given an informed consent form, which indicated that the experiment would last approximately 2–3 h and that all that would be required of them was to read the information presented on the computer screen and to make computer mouse clicks. The consent form also indicated that there were no known harmful effects caused by participating in the study, and that anyone was free to withdraw from the experiment at any time without any negative consequences. Participants who agreed to the terms enumerated in the consent document signed it and began the experiment.

Sorting Task

To begin, a participant was given 15 plastic-laminated cards that corresponded to the stimuli to be used in their assigned group and told to “put them into stacks that correspond to categories.” Thus, a sorting task was administered to determine whether participants demonstrated the pre-experimental existence of any of the experimenter-defined classes (e.g., Arntzen, Norbom, & Fields, 2015). Once completed, the contents of each sorted stack were recorded by the experimenter.

Instructions Prior to all Groups

Upon completion of the sorting test, the participants were shown the following instructions presented on the computer screen:

“In a moment, a stimulus will appear in the middle of the screen. Click on this by using the computer mouse. Three stimuli will then appear in three corners of the screen. Choose one of them by clicking on it with the mouse. If you choose the stimulus we have defined as correct, words like “very good,” “excellent,” and so on will appear on the screen. If you press a wrong stimulus, the word “wrong” will appear on the screen. At the bottom of the screen, the number of correct responses you have made will be counted. During some stages of the experiment, the computer will NOT tell you if your choices are correct or wrong. However, based on what you have learned so far, you can get all of the tasks correct. Please do your best to get everything right. Thank you and good luck!”

Trial Structure and Contingencies

The following procedures were used to establish conditional relations with varying delays during preliminary training for all of the identity and arbitrary groups, and for the baseline relations of the equivalence classes in all 12 groups. All training and testing trials used a matching-to-sample format. In each trial, the sample stimulus was presented first in the center of the screen. When participants made a mouse click on the sample stimulus (an observing response), the three comparison stimuli were presented in three of the four corners of the screen, with their locations randomized across trials. The participant then selected one of the comparison stimuli with a mouse click on one of them. When an experimenter-designated correct choice was made, the words correct, very good, super, or excellent were presented on the screen while the selection of the other comparisons produced the word wrong on the screen. Programmed textual consequences were displayed in the center of the screen for 1 s. Thereafter, a 0.5-s inter-trial interval occurred before the presentation of the next sample stimulus, which was always accompanied by the resetting of the cursor to the center of the screen at the start of a trial.

Simultaneous and Delayed Matching

All trials used to establish the necessary baseline relations and test for the emergence of the equivalence classes were presented in the simultaneous matching-to-sample (SMTS) format. On each trial, the observing response to the sample stimuli produced the comparison stimuli but the sample remained on along with the comparisons. All stimuli were terminated with the selection of a comparison stimulus.

The trials used during all of the preliminary training were presented in the delayed matching-to-sample (DMTS) format and differed from the SMTS trials in terms of the effect of the observing response on the presence of the sample stimulus. Specifically, the observing response to the sample resulted in its termination for X-s, after which the set of comparison stimuli was presented. The value of X, however, differed depending on group and took on values of from 0-, 1-, 3-, 6, or 9-s.

Preliminary Training of Identity and Arbitrary Conditional Relations

After the sorting task, participants in 10 of the 12 groups (Identity-X and Arbitrary-X) received preliminary training with DMTS procedures that used the abstract C-stimuli listed in Fig. 1. For the Identity-0 s, Identity-1 s, Identity-3 s, Identity-6 s, and Identity-9 s groups, preliminary training was designed to teach C-based identity conditional discriminations (C → C). The preliminary training trials were presented in the formats C1/C1-C2-C3, C2/C1-C2-C3, and C3/C1-C2-C3, with the stimulus that precedes the / as the sample while the three stimuli that follow the / were the comparisons. In each trial format, the bold letter-number pair represents the sample stimuli and the underlined letter-number pair represented the correct comparison.

In the Arbitrary-0 s, Arbitrary-1 s, Arbitrary-3 s, Arbitrary-6 s, and Arbitrary-9 s groups, preliminary training was designed to teach arbitrary conditional discriminations using the C stimuli from the conditional discrimination training and another abstract shape not used in the equivalence classes (i.e., C → X). The three C stimuli were presented as sample stimuli during training in the following format: C1/X1-X2-X3, C2/X1-X2-X3, and C3/X1-X2-X3. Each of the five identity and five arbitrary groups received preliminary training with only one delay value which depended on group (see Table 2).

In each group, training was conducted in blocks of nine trials (three presentations of each of the trials described above). Within each block, the trial order was randomized without replacement. The training block was repeated until a participant responded with 100% accuracy for all trials in one block (the mastery criterion). During training, comparison selections during all trials produced programmed textual consequences (i.e., 100% programmed consequences). Thereafter, maintenance of these trained relations was assessed with sets of blocks that programmed a systematic reduction in the percentage of trials per block that produced programmed consequences, starting with a probability of 75%, then 25% and finally 0%. Each block was repeated with the specified level of consequences until the mastery criterion was attained or for three blocks. If mastery was not achieved in three consecutive blocks, the percentage of programmed consequences was increased to the previously used value in the next block.

Training of Baseline Relations for Equivalence Classes

After acquiring the C ➔ C (identity groups) or C ➔ X relations (arbitrary groups), all participants in these 10 groups, along with those from the ABS and PIC control groups, advanced to training of the baseline relations of the ABCDE equivalence classes (see Table 1). A linear series training structure was used (i.e., A ➔ B ➔ C ➔ D ➔ E). Baseline relations were trained using trials presented in a simultaneous matching format.

Table 1 An overview of the phases in the simultaneous protocol, listing the conditional discriminations presented during baseline training and derived relations testing
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The AB, BC, CD, and DE baseline relations contained only abstract stimuli for the participants in the 10 groups who received preliminary training (identity and arbitrary groups) and the participants in the ABS group. In contrast, AB, BC, CD, and DE baseline relations for the PIC group contained A, B, D, and E abstract stimuli but the C stimuli were meaningful pictures.

All training blocks contained 36 trials with three presentations each of the 12 trial types (see Table 1). All trials in a block were presented in a randomized order without replacement. Thus, the training of the baseline relations was conducted on a concurrent basis. The training block was repeated until correct comparisons were selected on at least 33/36 trials or 92% of the trials in a single block (the mastery criterion). During acquisition, all comparison selections were followed by the presentation of the previously described programmed textual consequences.

Maintenance of Baseline Relations

Following acquisition of the baseline relations, their maintenance was assessed and established during a systematic reduction of informative consequences across a minimum of four blocks (with 36 trials in each block) that reduced the probability of informative consequences from 75% to 50%, 25% and finally to 0% of the trials in a block (see details in Table 1). Each block was repeated with the specified programmed consequence level until the participant achieved the mastery criterion (at least 33/36 trials or 92% in a block).

Emergent Relations Test Blocks

Following the successful acquisition of the training of baseline relations, participants were given two 180-trial derived relations test blocks. Each block contained 36 baseline trials, 36 symmetry trials, 54 1-node trials, 36 2-node trials, and 18 3-node trials. These trials were presented in a random order without replacement in each test block, and no informative consequences was provided for selection responses. Testing was conducted using the simultaneous protocol.

Equivalence class formation in a given test block was documented if class-indicative selections occurred in at least 92% of the baseline relations trials, the symmetrical relations trials, the transitive relations trials, and the equivalence relations trials (mastery). Mastery on the first test block denoted immediate emergence of the equivalence classes. If this was followed by mastery in the second block it documented the maintenance of the classes. Delayed emergence of the classes was denoted if performance was below mastery in the first test block and was then was followed by mastery in the second test block. Finally, failure of class formation was defined by sub-mastery performances in both test blocks. In this experiment, equivalence class formation was taken as being formed by a participant if their test performances showed immediate or delayed emergence of the classes, a criterion also used by Nartey, Arntzen, and Fields (2014).

Results

Pre-Experimental Performances

Sorting

The participants sorted the stimuli into clusters that contained from two to seven stimuli from the 15 available cards. Because none of the clusters contained the five stimuli in an experimenter-defined class, no participant demonstrated the presence of any of the three experimenter-defined equivalence classes prior to their subsequent training and testing.

Attrition

Of the 120 participants, four (all from the 9-s group) did not acquire the baseline relations after 120 min of training, and were excused from further participation. They were replaced with four additional participants.

Preliminary Training

Figure 2 presents the median number of trials needed to acquire the identity (C → C) and arbitrary (C → X) conditional relations plotted as a function of delay during preliminary training. For all delays but the longest (i.e., 9 s), more trials were needed to acquire the arbitrary relations than the identity relations. In addition, the trials to acquisition across the five identity groups was statistically different from those needed to acquire them in the five arbitrary groups, t(4) = 3.959, p = .016, with relational type accounting for 80% of the variance, r2 = .7967.

Fig. 2
figure 2

Median number of trials needed to learn arbitrary or identity relations during preliminary training plotted as a function of the delay duration that characterized the relations being learned. Identity relations; Arbitrary relations

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At the 9-s delay, the identity relations were acquired as quickly as the identity relations that were established at the shorter delays. In contrast, the arbitrary relations with the 9-s delays were acquired faster than the arbitrary relations with shorter delays, and also matched that required for the acquisition of the identity relations. Thus, the convergence of trials to acquisition at the 9-s delay was produced by a reduction in number of trials needed to acquire the arbitrary relations.

Across the identity groups, acquisition of identity relations was a gradual increasing function of delay as value increased from 0 to 3 s, which then became asymptotic at all of the longer delays. Across the arbitrary groups, delay duration from 0 to 6 s had minimal effect on the number of trials needed for their acquisition. Surprisingly, the longest delay (i.e., 9 s) resulted in faster acquisition of the C ➔ X relations than did all of the four shorter delays.

Class Formation Performances

Duration of Training and Testing

Figure 3 is a box and whisker representation of the number of minutes needed to complete the simultaneous protocol for the participants in each group. The lower and upper edges of each box indicate the number of minutes needed to acquire the baseline relations by participants in the first and third quartile in a group, and the lower and upper whisker represent the number of minutes needed to acquire the baseline relations by 10 and 90% of the participants in each group. The horizontal line in the box indicates the median number of trials needed to establish the baseline relations for the participants in a given group.

Fig. 3
figure 3

Box and whisker renditions of the number of minutes required to complete the simultaneous protocol, plotted as a function of the delay duration that characterized the identity and arbitrary relations established during preliminary training. Identity relations; Arbitrary relations; ABS, all abstract shapes; PIC, C stimuli as pictures.

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Across all groups, a median of 81 min were needed to complete training and testing, with group medians that varied from 70 to 92 min. The minimum and maximum durations ranged from 120 to 200 min. A visual analysis of these data shows no systematic effects of preliminary training on the time needed to complete the training and testing protocol.

Acquisition and Maintenance of Baseline Relations

Figure 4 shows how the two reference groups, as well as the delays and relational types established during preliminary training influenced the number of trials needed to acquire the baseline relations for the equivalence classes. The baseline relations were acquired slowest in the ABS group and fastest in the PIC group. For participants in all preliminary training groups (i.e., Identity and Arbitrary) baseline relations were acquired faster than they were for participants in the ABS group. Thus, it appears that the learning of any conditional relations during preliminary training enhanced the learning of the new baseline relations. Also, the number of trials needed to acquire the baseline relations did not vary systematically with type of preliminary training. Finally, all forms of preliminary training produced baseline acquisition speeds that were similar to that observed during the acquisition of the baseline relations in the PIC group. Thus, the learning of the baseline relations was enhanced to the same degree by the inclusion of a meaningful stimulus or by the prior establishment of identity or arbitrary conditional relations regardless of the delays that characterized these relations.

Fig. 4
figure 4

Box and whisker renditions of the number of trials required to learn the baseline relations plotted as a function of the delay duration that characterized the identity and arbitrary relations established during preliminary training. Identity relations; Arbitrary relations; ABS, all abstract shapes; PIC, C stimuli as pictures.

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Once the baseline relations were acquired, level of programmed consequences was systematically reduced from 100 to 0% of the trials in a block. Mastery was maintained throughout this phase of the experiment. At the end of programmed consequence reduction, baseline relations were maintained in the absence of any programmed consequence.

Baseline Performances During Emergent Relations Testing

Figure 5 illustrates the effects of preliminary training on the maintenance of the baseline relations for each delay used during preliminary training and for the reference ABS and PIC groups. Since there were no differences across relational types used in preliminary training, the data were averaged across these groups for each delay duration. The data presented in Fig. 6 were derived from the information contained in Table 2a–c. An intact set of baseline relations was defined by responding correctly on at least 90% of the baseline trials in a test block.

Fig. 5
figure 5

Percentage of blocks that contained intact baseline relations (BL+), and percentage of blocks that show class formation for blocks that contained intact baseline relations (ECF+/BL+)

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Fig. 6
figure 6

The percentage of participants who formed classes in the reference (Refs) ABS (all abstract shapes) and PIC (C stimuli as pictures) conditions and for each delay and relational type established during preliminary training. Identity relations; Arbitrary relations; Avg, average

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Table 2 Outline of the experimental design. ABS, all abstract shapes; PIC, C stimuli as pictures
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In the ABS and PIC groups, none and 75% of the test blocks showed intact baseline relations, respectively. If the delay values used during preliminary training are considered, the percentage of test blocks with intact baseline relations was an inverted U-shaped function of the duration of the delay used to establish the C → C or C→ X relations. When compared with the outcome of the PIC group, the percentage of intact baseline relations following preliminary training with 0-, 1-, 3-, and 9-s delays was lower than the percentage that occurred when a meaningful stimulus (i.e., C stimulus picture) was included as a class member with one exception; In the 6-s delay groups, the percentage of blocks that produced accurate baseline relations approached that seen in the PIC group.

Effect of Relational Type and Delay on Equivalence Class Formation

The information in Table 3a–c was used to produce Fig. 6, which displays the percentage of participants who formed classes after learning identity or arbitrary relations with the C stimuli as a function of delay duration, and in the two control groups. The upper and lower functions of Fig. 6 show the results for preliminary training of Arbitrary and Identity relations, respectively. In addition, the middle function displays the yields averaged across relational types for each delay value.

Table 3 Group, participant number, number of baseline trials, indices for Tests 1 and 2, and overall scores of immediate emergence, delayed emergence or failure to show equivalence class formation
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In the ABS group, none of the participants formed the classes. In the PIC group, 70% of the participants formed the classes. This significant increase in yield, χ2 (1, N = 20) = 10.77, p < 0.001 showed that the inclusion of one meaningful stimulus in a set of otherwise meaningless stimuli enhanced equivalence class formation.

When the main effect of delay duration was considered, the likelihood of equivalence class formation was indexed by averaging across both relational types, for each delay value. The 0-s delay produced a low likelihood of equivalence class formation, but was 10% greater than that seen in the absence of preliminary training with the ABS group.

Increasing the duration of the delay in the C-based conditional relations from 0 to 6 s produced a nearly linear increase in the likelihood of equivalence class formation, a statistically significant effect, χ2 (3, N = 80) = 14.48, p < 0.01. A further increase in delay duration from 6 to 9 s produced a significant decline in likelihood of class formation, χ2 (1, N = 40) = 6.4, p = 0.05. Thus, the enhancement of equivalence class formation was an inverted U-shaped function of the delay between the sample and comparison stimuli that characterized the pre-class formation conditional discriminations.

Preliminary training involved the formation of either Arbitrary or identity conditional relations, as indicated by the functions in Fig. 6. When type of preliminary training relation is considered at the 0-s delay, the same low yields were obtained regardless of relation type (i.e., identity or Arbitrary). When the 1-, 3-, and 6-s delays were used, class formation was 10% more likely to occur with Arbitrary relation preliminary training was used instead of identity relation preliminary training. When the 9-s delay was used, class formation was 20% more likely to occur following the establishment of Arbitrary instead of identity relations.

Thus, a higher proportion of all participants from the 10 preliminary training groups formed classes after exposure to the learning of arbitrary relations than identity relations. The binomial probability of obtaining this same directional effect of relational type on class formation was equal to 0.032 or 0.55. Thus, the type of relation established in preliminary training influenced the subsequent likelihood of forming new equivalence classes.

Figure 6 also illustrated how the yields in the preliminary training groups compared to the yields obtained in the two control groups. No participants formed classes in the ABS group. With the exception of the groups with 0- and 1-s delays, all of the remaining delays produced higher yields of class formation than the ABS group. Thus, preliminary training with delays that were at least 3 s in duration enhanced the formation of equivalence classes relative to class formation in the absence of preliminary training. The greatest yield occurred when each class contained one meaningful stimulus (the PIC group). A similar yield was produced by the preliminary training group that experienced the establishment of C-based identity relations with 6-s delays.

Baseline Relations as Predictors of Class Formation

As seen in Figs. 5 and 6, the delays used in preliminary training had similar effects on the retention of the baseline relations during testing and class formation yields. On average, however, the number of testing blocks that showed retention of the baseline relations was 7% greater than those that showed class formation.

When these data were viewed in terms of individual test blocks, intact baseline performances were observed in 77 of the 100 test blocks in the 10 groups that involved preliminary training. In contrast, class formation was documented in 64 of the same 100 test blocks. A one-tailed Fisher’s exact test (FET) confirmed that these two yields were significantly different, FET, p = 0.0312. Thus, the fact that a test block included maintained baseline relations did not necessarily result in class formation.

Discussion

In this experiment, participants in 12 groups were taught the baseline relations for three 3-node 5-member equivalence classes, with training and testing conducted using the simultaneous protocol. Participants from 10 groups received preliminary training in which identity relations (CC) were established in five identity groups, and arbitrary relations (CX) were established in five arbitrary groups. Five different delays (i.e., 0, 1, 3, 6, and 9 s) were used with each of the five arbitrary and identity groups. In addition, participants from two reference groups received no preliminary training. The participants in the ABS group were taught the baseline relations for the equivalence classes. The participants in the PIC group were taught the baseline relations for the equivalence classes in which one of the five stimuli (C) was a meaningful picture.

Because the classes did not form in the ABS group, and classes were formed by 70% of participants in the PIC group, the likelihood of class formation was enhanced by the inclusion of a meaningful picture as a member of a potential equivalence class. When preliminary training was conducted, the likelihood of class formation was an inverted U-shaped function of the delay durations used in preliminary training: starting at a very low level that was similar to that produced without preliminary training, peaking at 6 s with a yield similar to that produced when classes included one meaningful picture, and finally declining with longer delays. In addition, the entire delay-yield function was somewhat higher when preliminary training involved the formation of arbitrary relations instead of identity relations.

Preliminary training also influenced the performances produced by the baseline relations that were included in the derived relations tests. When preliminary training was conducted, however, the maintenance of the baseline relations during the tests was also an inverse U-shaped function of the delay used in preliminary training. Thus, preliminary training had the same general effect on the resistance to disruption of the baseline relations that were presented in the derived relations test blocks. However, when participants showed intact baseline relations in the tests, only some of them showed the formation of the equivalence classes. Theoretical analyses of many of these findings are discussed below in sections labeled “Basic Processes,” “Meaningful Stimuli and Enhanced Equivalence Class Formation”, and “General Implications.”

Basic Processes

Differential Effects of Identity and Arbitrary Relations

For each delay value, the likelihood of subsequent equivalence class formation was somewhat higher after the training of arbitrary relations than identity relations. A similar finding was also reported by (Arntzen et al., 2015) when studying the effect of 0-s delay vs a 6-s delay. The fact that two experiments produced similar outcomes provides additional evidence that equivalence classes are more likely to emerge following the formation of arbitrary conditional relations than for identity relations.

This differential advantage can be understood in terms of generalization between the relational types used in preliminary training and those used during class formation. Specifically, arbitrary conditional discriminations were established in the arbitrary groups during preliminary training. In addition, arbitrary conditional discriminations also characterized the baseline relations for the classes, as well as the symmetry, transitivity, and equivalence probes used during the derived relations tests. Thus, the same trial format was used during preliminary training and during establishing conditional discriminations and equivalence class testing. In contrast, identity conditional discriminations were established in the identity groups, while arbitrary relations were used during equivalence class training and testing. Thus, trials of different formats were used during preliminary training and class formation. It follows from this analysis that more generalization should occur among relations of the same type (arbitrary in preliminary training and in the class formation protocol), than between relations of different types (identity relations in preliminary training and arbitrary relations in the class formation protocol). Because this prediction matches the outcome noted in the present experiment, generalization across trial formats appears to be a plausible account of the superiority of yields obtained after arbitrary rather than identity training.

In contrast, the acquisition of the baseline relations did not influence the acquisition of the baseline relations for the equivalence classes. Thus, generalization of relational type would appear to have differential effects on class formation and the acquisition of the baseline relations for the classes. The reason for such an outcome, however, is not clear.

Preliminary Training and Baseline Relations During Testing

Baseline Relations at the End of Training and During Testing

The baseline relations for the equivalence classes were acquired by all participants and then maintained as reinforcement was thinned and eventually eliminated. Preliminary training, then, did not influence these outcomes.

In contrast, preliminary training had a pronounced effect on the maintenance of the baseline relations in the derived relations test blocks. In the ABS group, all of the participants showed the disruption of the baseline relations in the test blocks. This disruption can be attributed to the inclusion of the derived relations probes in the test blocks, as noted by others (e.g., Arntzen & Hansen, 2011; Fields, Landon-Jimenez, Buffington, & Adams, 1995; Fields & Watanabe-Rose, 2008). The mechanisms of this effect are yet to be discovered.

In addition, some of the participants in the preliminary training groups showed the maintenance of the baseline relations during the derived relations test blocks. Thus, preliminary training appeared to inoculate the baseline relations from disruption by the concurrent presence of derived relations probes in a test block. The mechanism of this effect will have to be determined with additional research.

Baseline Relations During the ABS and PIC Groups

The baseline relations were maintained in many of the test blocks in the PIC group and by none of the test blocks in the ABS groups. This finding suggests that the inclusion of a meaningful stimulus in a class increased the resistance to disruption of the baseline relations during testing.

Baseline Relations in the ABS Group and After Preliminary Training

The percentage of test blocks with intact baseline relations was an inverted U-shaped function of the duration of the delay used during preliminary training. All of these outcomes exceeded that observed in the ABS group. Thus, preliminary training increased resistance to disruption of the baseline relations during the test blocks. The mechanisms responsible for these effects will have to be uncovered in new research.

Baseline Relations in the PIC Group and After Preliminary Training

The percentage of intact baseline relations following preliminary training with 0-, 1-, 3-, and 9-s delays was lower than the percentage that occurred in the PIC group: when a meaningful stimulus (i.e., C stimulus picture) was included as a class member. In contrast, the 6-s delay group showed as much maintenance of the baseline relations as the PIC group. Thus, delay during preliminary training systematically reduced the disruption of the baseline relations during the derived relations test blocks.

Maintained Baseline Relations and Equivalence Class Formation

Given the similarities in the effects of delay on the maintenance of baseline relations during the test blocks and the emergence of the classes, it could be argued that delay had its primary effect on retention of the baseline relations and a minimal effect on class formation. If that assumption is true, 100% of the test blocks that contained intact baseline relations would also have to give rise to class formation. In the 10 preliminary training groups, then, the 79 test blocks that included intact baseline relations would also have to show class formation: i.e., 79 hits and 0 misses. In actuality, class formation occurred in only 63 of those blocks: i.e., 63 hits and 16 misses. Since these two data sets (79–0 vs 63–16) were significantly different, Fisher’s exact test, p < 0.0001, the effect of delay on the formation of the equivalence classes could not be attributed solely to its impact on the maintenance of the baseline relations in the test blocks. Rather, delay value had to have influenced both the maintenance of the baseline relations during testing, and the emergence of the derived relations that documented the formation of the equivalence classes. Thus, while intact baseline relations were necessary for class formation, they were not sufficient for class formation.

When baseline relations were intact during the derived relations tests blocks, 7% of them did not show class formation. This suggests that preliminary training had differential effects on baseline maintenance and a constant effect on the induction of the derived relations that documented the formation of the equivalence classes.

As noted above, all participants responded with at least 94% accuracy during baseline relations trials in the absence of programmed consequences prior to the tests for the retention of baseline relations and for the emergence of the derived relations. In those tests, some of the participants did not respond in accordance with stimulus equivalence, but responded correctly to the baseline relations probes. Further experiments should study whether this lack of retention of the baseline relations is influenced by the number of consecutive correct training trials before testing or by the introduction of the novel tests trials.

Mechanisms of Delay on Class Enhancement: An Opponent Process Account

In the present experiment, the enhancement of equivalence class formation was an inverted U-shaped function of the delay that separated the stimuli in the C-based relations established during preliminary training. As delay value increased from 0 to 6 s, and yield increased accordingly. A further increase to 9 s, however, resulted in a substantial decrement in yield. When a U-shaped or inverted U-shaped functions are obtained, they can be accounted for in terms of the summative effects of two processes or functions (A and B) each working in opposite directions. That is, two monotonic functions are operating in opposition to each other and sum together to produce an inverted U-shaped function. This is illustrated in Fig. 6. With increases in the value of some independent variable, Function A has a direct effect on some dependent variable. In addition, with increases in the value of the same independent variable, Function B has a negative or suppressive effect on the same dependent variable. For each value of the independent variable, the net effect on the dependent variable is equal to the sum of the two opposing effects of Functions A and B. Thus, as the value of the independent variable increases from 0 to 5, the net effect of the two processes produces an increase in the value of the dependent variable. Beyond an independent variable value of 6, however, the negative effect of Function B eliminates the further increase in the value of the dependent variable. With further increases in the value of the independent variable, the negative effect of Function B produces an increasing decrement in the value of the dependent variable.

In the present experiment, we are assuming that the independent variable of note is the duration of the delay used to establish the CC or CX relations during preliminary training. As it is increased, it increases attention to the relations between stimuli that are members trained relations (to be called within relation stimulus control topographies), and also to relations among stimuli that are not directly trained but are members of the same set of nodally interrelated cues (to be called across relation stimulus control topographies). This process corresponds to Function A. We are also assuming that these stimulus control topographies then generalize to the baseline and derived relations that are trained and emergent during the simultaneous protocol. Thus, Function A should increase the likelihood of inducing equivalence classes during the simultaneous protocol.

At the same time, the increase in delay used during preliminary training induces a participant to pay an increasing amount of attention to the relation being established between the stimuli in the trained CC and CX relations and less and less attention to relations among stimuli that are not directly trained. This corresponds to Function B. This process also generalizes to the stimuli and relations that are trained or induced during the subsequently administered simultaneous protocol. This process then suppresses attention to untrained relations among the stimuli that are members of the baseline relations established during the simultaneous protocol.

The combined effects of these historically induced stimulus control topographies are to produce the inverted U-shaped function that relates the delays used during preliminary training and the subsequent likelihood of equivalence class formation. The validity of this speculative analysis would depend on additional research that could track the isolated processed mentioned above. Outcomes that would predict the outcomes observed in the present experiment would then support the analysis presented above.

Meaningful Stimuli and Enhanced Equivalence Class Formation

Class-Enhancement by Meaningful Stimuli

We and others have shown that a meaningful stimulus enhances the likelihood of equivalence class formation when it is included as a member of a prospective class (e.g., Arntzen et al., 2015; Bortoloti, Rodrigues, Cortez, Pimentel, & de Rose, 2013). In addition, meaningful stimuli most likely exert a variety of previously acquired stimulus control functions. Thus, some of those stimulus control functions served by meaningful stimuli could be responsible for the class enhancing effects of meaningful stimuli. One of these variables is the delay that characterizes the identity or arbitrary relations learned during pre-class formation training.

The results of the present experiment support that line of argument and refine it in an unexpected way. Specifically, the class enhancing property of meaningful stimuli should be maximized when they have served as members of conditional relations that were acquired with intermediate delays of about 6 s. Thus, if a meaningful stimulus produces minimal class enhancement, perhaps it had acquired its conditional relational function under delays that were shorter or longer than 6 s.

Class Expansion, Meaningful Stimuli, and Enhanced Class Formation

Equivalence classes were unlikely to form when the class members were all initially meaningless stimuli (i.e., ABS group). In contrast, classes were likely to form when they contained one meaningful stimulus along with a remainder of initially meaningless stimuli (i.e., PIC group). Presumably, the meaningful stimulus could be a member of an already established category before it was imbedded in the prospective equivalence class. Thus, what appears to be the de-novo formation of a new equivalence class may actually represent the expansion of an already established category. While plausible, we are not aware of direct evaluations of this notion. Positive outcomes could identify other mechanisms by which meaningful stimuli enhance the formation of equivalence classes.

Meaningful Stimuli and Stimulus Complexity

The inclusion of a picture as the C stimulus increased the likelihood of equivalence class formation. The picture differed from the abstract C stimuli in terms of its presumed meaningfulness, color, and compositional complexity, all of which were confounded with each other. While we have attributed the enhancement effect to the meaningfulness of the picture, the other stimulus characteristics might also have produced the enhancement effect. Indeed, this circumstance is also relevant with virtually all of the experiments that have shown class enhancement by the inclusion of meaningful stimuli.

The individual and combined effects of these stimulus characteristics on the enhancement of equivalence class formation will have to be determined with additional research designed to isolate the effects of these factors (Arntzen, 2012). Some experimental strategies that have achieved such a goal have been described by Fields and Spear (2012), McIlvane and Dube (2003), and Spear and Fields (2015).

General Implications

Effects of “Constructed Histories” on Equivalence Class Formation

Multiple studies have shown that if equivalence classes with a linear structure are trained with the simultaneous protocol, then equivalence classes are unlikely to emerge (e.g., Arntzen, 2012; Fields et al., 1997). Prior studies, however, have shown that a variety of pre-class formation procedures substantially enhanced the formation of equivalence classes under the simultaneous protocol. These include the acquisition of (a) simple successive and simultaneous discriminative functions alone and in combination (Arntzen et al., 2015; Fields, Arntzen, Nartey, & Eilifsen, 2012; Nartey et al., 2014; Travis, Fields, & Arntzen, 2014), (b) the overtraining of the successive discriminative function (Travis et al., 2014), (c) the acquisition of identity and arbitrary conditional discriminative functions (Nedelcu, Fields, & Arntzen, 2015), (d) the number of arbitrary conditional discriminative functions (Nedelcu et al., 2015) with a single delay value used in the conditional relations established prior to equivalence class formation (Arntzen et al., 2014b, 2015a), and (e) the prior establishment of other equivalence classes that varied in size and nodal number (Buffington et al., 1997; Fields et al., 2000).

The present experiment showed how likelihood of equivalence class formation established under the simultaneous protocol was substantially enhanced by the prior establishment of identity or arbitrary relations. All previous studies mentioned above have shown monotonic effects on class enhancement. This present experiment is the first to show class enhancement effects with a non-monotonic function of a parameter of preliminary training: namely, delay duration used while establishing identity or arbitrary conditional relations. These results, then, substantially expand our knowledge of the historical variables that influence the likelihood of equivalence class formation. In addition, as noted in the section titled, “Mechanisms of Delay on Class Enhancement: An Opponent Process Account,” we discussed two opposing processes that could account for the effects of delay in preliminary training on the subsequent formation of equivalence classes. Experimental confirmation of these mechanisms would also expand our understanding of the manner in which historically established stimulus control topographies could influence the formation of new equivalence classes.

Potential Applications

Equivalence-based instruction (EBI) is being used with increasing frequency to establish conceptual categories in an increasing wide range of academic subject matters, such as descriptive and inferential statistics (Albright, Reeve, Reeve, & Kisamore, 2015; Critchfield & Fienup, 2010; Fields et al., 2009; Fienup & Critchfield, 2010), neuroanatomical structures, functions, brain–behavior relations, and pharmaceutical agents (Fienup, Covey, & Critchfield, 2010; Fienup, Mylan, Brodsky, & Pytte, 2015; Fienup, Wright, & Fields, 2015; Zinn, Newland, & Ritchie, 2015), along with experimental design and etiologies of developmental disabilities (Arntzen, Halstadtro, Bjerke, & Halstadtro, 2010; Arntzen, Halstadtro, Bjerke, Wittner, & Kristiansen, 2014; Lovett, Rehfeldt, Garcia, & Dunning, 2011; Walker & Rehfeldt, 2012; Walker, Rehfeldt, & Ninness, 2010).

For obvious reasons, EBI should be designed to optimize the formation of these classes. In some cases, however, it might not be practical to use training and testing protocols that maximize the likelihood of forming equivalence classes. Rather, class formation may have to be established using less than optimal training and testing procedures that are like the simultaneous protocol. The present experiment shows that the likelihood of class formation obtained using the simultaneous protocol was enhanced substantially by the preliminary training of particular conditional discriminations where those relations are established with particular delays. Thus, if EBI has to be conducted using the simultaneous protocol, conceptual learning might be optimized by the preliminary establishment of arbitrary conditional relations that are established using 6-s delays: the groups that optimized class enhancement in the present experiment. It is also possible that this might not occur if all of the stimuli in the potential class were meaningful, as is often the case during EBI with academic content.

Summary

Likelihood of equivalence class formation was an inverted U-shaped function of the delay that characterized the relations trained in preliminary training. Also, somewhat higher yields were produced when preliminary training involved forming Arbitrary instead of identity relations. The yields peaked at 6 s and equaled those produced when classes contained one meaningful picture. The delays used during preliminary training did not influence the likelihood of acquiring the baseline relations. In contrast, the resistance to disruption when presented in the test blocks was an inverse U-shaped function of delay used in preliminary training. Delay, then influenced the likelihood of class formation by increasing resistance to disruption of baseline relations and by fostering the emergence of the derived relations that documented class formation.