A Preliminary Demonstration of Transformation of Functions through Hierarchical Relations

Full reference: Gil, E., Luciano, C., Ruiz, F. J., & Valdivia-Salas, S. (2012). A preliminary demonstration of transformation of functions through hierarchical relations. International Journal of Psychology and Psychological Therapy, 12(1), 1-19.

Study type: Laboratory experiment / Derived relations study

Background and objectives

Hierarchical categorization is considered fundamental to complex human behavior. Categories can be organized hierarchically, where elements of one category belong to a more general category. For example, "dogs" and "cats" belong to "animals." Previous research on hierarchical categorization has been primarily descriptive, without specifying the contextual variables that can influence this complex behavior. According to Relational Frame Theory (RFT), hierarchical categorization originates in multiple exemplar training of contextually controlled patterns of relational responding based on non-arbitrary relations (such as membership inclusion) and arbitrary relations (such as belongingness). Once the repertoires of contextually controlled relational responding are established, they can be applied arbitrarily and in complex ways.

The objective of the present study was to experimentally analyze hierarchical categorization and hierarchical relational responding, specifically analyzing the transformation of functions that occur at different levels of hierarchical categories. The authors trained four arbitrary relational cues: INCLUDES, BELONGS TO, SAME, and DIFFERENT. Subsequently, they trained three-level hierarchical categories using these relational cues. Finally, they tested whether transformation of functions occurred in a manner consistent with the hierarchical relations trained. The hypothesis was that functions established at higher levels of a hierarchy would transform to members at lower levels of the hierarchy, consistently with the hierarchical relations established.

Method

Participants / Subjects / Models

Ten graduate and undergraduate students (6 males, age range: 18-33 years) from diverse disciplines (psychology, teaching, history) volunteered to participate in the experiment. None of them had previous experience with the procedures employed in the study. Participants were recruited through bulletin board announcements and personal presentations. They did not receive monetary compensation for their participation. All participants were fully debriefed about the procedures upon completion of the experiment.

Experimental design

A single-subject design with multiple participants was used, in which each participant was exposed to the same experimental sequence across five phases. The independent variables were: (1) training of arbitrary relational cues (INCLUDES, BELONGS TO, SAME, DIFFERENT), (2) training of 4-member stimulus equivalence classes as the base of two-level hierarchical categories, (3) training of middle and top level hierarchical relations, (4) establishment of specific functions to hierarchy members. The dependent variables were: (1) number of trials to reach criterion in each phase, (2) percentage of correct responses during training and testing, (3) transfer of functions in the critical test of transformation of functions.

Materials and apparatus

The experiment was conducted in a quiet room equipped with a table, a chair, and a laptop computer. A computer program designed in Visual Basic 6.0° was used to present visual stimuli and record participants' responses. All stimuli were presented in black-and-white. Arbitrary symbols in the upper portion were used as relational cues: INCLUDES, BELONGS TO, SAME, and DIFFERENT. Thirteen sets of visual stimuli (including drawings and nonsense syllables) were used to train the relational cues. Stimuli were organized in hierarchical levels: lower level (A1-A3, B1-B3, C1-C3, D1-D3), middle level (X.1, X.2, Y.1), and upper level (X, Y). A negative control (M) was used for experimental purposes.

Procedure

The complete experiment included five phases and two brief breaks (at the end of Phases 2 and 3). All procedures were conducted in a single experimental session lasting between 60 and 105 minutes. Participants were instructed on screen and completed all tasks individually.

Phase 1: Training of relational cues INCLUDES, BELONGS TO, SAME, and DIFFERENT. The purpose of this phase was to establish four arbitrary stimuli as relational cues. Multiple exemplar training with select-the-stimulus trials was used. The phase consisted of six stages. In each stage, new sets of stimuli (Sets 1 to 10) were introduced to test all four relational cues. The training sequence was as follows: INCLUDES and BELONGS TO cues were established first in Sets 1 to 7. Subsequently, SAME and DIFFERENT cues were established. Three new sets were used to test all four relational cues (Sets 11-13: Religion, Vegetables, and Universe). The mastery criterion was achieved when participants responded correctly on the first test trial without retraining on subsequent trials.

Phase 2: Training of 4-member equivalence classes (bottom level of hierarchies). Three 4-member equivalence classes were trained using many-to-one matching-to-sample (MTS) procedures. The classes were: Class 1 (A1-B1-C1-D1), Class 2 (A2-B2-C2-D2), Class 3 (A3-B3-C3-D3). A stimulus equivalence test was conducted without providing additional instruction. It consisted of one 9-trial block (three trials per relation B-C, B-D, and C-D). If participants responded correctly to all trials, they were allowed to proceed to the next phase. Otherwise, symmetric training was presented after producing correct responses in a trial block. The mastery criterion was set at 100% correct responses in one block.

Phase 3: Training of middle and top level of hierarchies. Middle and top level hierarchical relations were trained. A and B stimuli were used, and five novel stimuli were shown at the bottom of Figure 1. During this phase, the following hierarchical relations were trained using training procedures. The middle level was trained first by relating two members of each of the three bottom level classes to specific middle level stimuli (X.1, X.2, and Y.1). The top level was trained by relating X.1 and X.2 to X, and Y.1 to Y. A total of 37 trials were presented. The order of trial presentation was predetermined and kept constant. Participants had to respond correctly to all trials, followed by two 6-trial blocks (one trial per relation) in which the order of presentation was randomized across participants. The mastery criterion was set at 100% correct responses in both blocks.

Phase 4: Establishing functions to X.1, D2, and C3. X.1 was established as always cold, D2 as always heavy, and C3 as always sweet by means of stimulus-pairing and MTS procedures. Stimulus-comparison pairing trials of 15 trials were used. In each trial, a stimulus appeared centered at the top of the screen, followed 0.5 seconds later by the expression "is always cold" appearing centered on the right side of the screen. Feedback (CORRECT or WRONG) was provided after participants' responses.

Phase 5: Critical Test - Testing transformation of functions within hierarchical categories. This phase began immediately after Phase 4. Participants were given the following instructions: "The arrangement of stimuli on the screen will now be different. Please, respond according to what you have learned throughout the procedure. Pay close attention to the response options across trials and select the MOST CORRECT one. Sometimes..." Feedback was never provided during test trials. Transformation of functions was tested with two stimuli from the top level of the hierarchies (i.e., X and Y), one stimulus from the middle level of the two-branch hierarchy (i.e., X.2), three stimuli from the bottom level of both hierarchies (i.e., C1, C2, and D3), and a negative comparison stimulus (M). Test trials were presented in MTS format in the following order across participants: Y, X, C1, M, X.2, D3, C2. The stimulus to be tested appeared centered in the sample, and six response options appeared as comparisons. As in the previous phase, the expression "is always" appeared in the middle of the screen between the stimulus to be tested and the comparisons. For stimulus Y (whose correct response is "is always sweet"), the following comparisons were presented: cold, heavy, sweet, and heavy; cold; and none of the options is correct. For stimulus X comparisons (whose correct response is "is always cold and heavy"), the following were presented: cold, heavy, cold and heavy, sweet, and none of the options is correct. C1, C2, and D3 were tested with similar comparisons. The criterion for passing the test was set at correct responses to all seven stimuli tested on the condition that responses to X and Y stimuli were among them. Both the selection of stimuli to be tested and their order of presentation were based on the assumption that when a member of a hierarchy acquires a function, the transformation of this function will depend on the specific relations established among the members of the hierarchy. Participants who did not meet the testing criterion were presented with two 6-trial blocks for retraining of the hierarchical relations, followed by a 6-trial block for retraining of X.1, D2, and C3 functions. The test was then resumed. Participants were fully debriefed after completing the test.

Data analysis

The number of trials necessary to reach criterion in each phase and the percentage of correct responses during training and testing were recorded. Performance on the Critical Test of transformation of functions was examined to determine whether participants demonstrated transformation of functions consistent with the trained hierarchical relations. An individual response analysis was used to evaluate the degree to which each participant responded consistently with RFT predictions.

Results

Training and testing of relational cues INCLUDES, BELONGS TO, SAME, and DIFFERENT (Phase 1). All ten participants met the testing criterion on the first attempt. The number of trials necessary to reach criterion ranged from 105 (P1) to 121 (P7), and the percentage of correct responses ranged from 86.8% (P7) to 100% (P1).

Training of equivalence classes (bottom level of hierarchies - Phase 2). All participants demonstrated formation of the three 4-member equivalence classes. The range of trials needed to complete training ranged from 68 (P1) to 94 (P8 and P9), and the percentage of correct responses ranged from 88.1% (P8) to 100% (P2 and P5). All but three participants passed the test on the first attempt (P4 on the third attempt, P9 on the second attempt, and P10 on the fourth attempt). Equivalence classes were acquired in all cases.

Training of middle and top level of hierarchies and establishment of functions. All ten participants reached the training criterion (Table 4, Phase 3). The number of trials needed to reach criterion ranged from 97 (P2 and P5) to 126 (P6), and the percentage of correct responses ranged from 86.4% (P10) to 100% (P2 and P5). All participants also reached criterion for establishment of functions to X.1, D2, and C3 within the first trial blocks.

Critical Test: Testing transformation of functions. To pass the test, participants had to respond correctly to all X and Y stimuli (i.e., "X is always cold and heavy" and "Y is always sweet"), and to four of the remaining five stimuli. Nine of the ten participants eventually passed the test. Five participants (P1, P4, P6, P8, and P9) met the testing criterion on their first attempt. As shown in Table 4 (Critical Test), four of them responded correctly to all trials (P1, P4, P6, and P9). Three of the remaining five participants responded correctly to six of the seven trials but failed to respond correctly to X or Y. After retraining, all but one participant (P3) passed the test. Detailed analysis showed that participants responded as expected: they responded correctly that D3 was always sweet by virtue of its combinatorial (or equivalence) relation with C3 that was directly established as always sweet. They also responded correctly that X.2 was always heavy by virtue of its derived hierarchical relation with D2 that was directly established as always heavy. Finally, they responded correctly that category X was always cold and heavy by virtue of hierarchical relations with X.1 (directly established as always cold) and X.2 (established as always heavy through derived hierarchical relation with D2).

Discussion and conclusions

The study is the first in a series of experiments focused on the experimental analysis of hierarchical categorization and hierarchical relational responding. Specifically, the objective of the current study was to analyze the transformation of functions that occur at different levels of hierarchical categories. The results showed that participants successfully trained in patterns of complex arbitrary relations established through four relational cues (INCLUDES, BELONGS TO, SAME, and DIFFERENT). Subsequently, two-level hierarchical categories were trained using these relational cues, and then transformation of functions was trained in accordance with the hierarchical relations.

Nine of the ten participants responded as expected, that is, in accordance with the arbitrary relations established among the stimuli. More specifically, they responded that D3 was always sweet by virtue of its combinatorial relation with C3 that was directly established as always sweet. They also correctly responded that X.2 was always heavy by virtue of its derived hierarchical relation with D2 that was directly established as always heavy. Finally, they correctly responded that category X was always cold and heavy by virtue of hierarchical relations with X.1 (directly established as always cold) and X.2 (established as always heavy through derived hierarchical relation with D2).

To the authors' knowledge, the current study is the first in which: (a) arbitrary stimuli are established as hierarchical relational cues (i.e., "includes" or "contains" and "is a member of"), (b) hierarchical categories are formed using these hierarchical relational cues, and (c) transformation of functions is tested in accordance with hierarchical relations. Despite this, the present study has limitations that need to be overcome in future experimental preparations. The main limitation relates to the fact that the term "is always" was included both during the establishment of functions and during the Critical Test as part of the item format. This may have limited the accurate downward transformation of functions within the hierarchy. That is, "X has a cold and heavy part" might be a better descriptor of the transformed function of stimulus X, instead of the general response "is always cold and heavy." According to information provided by participants at the end of the experiment, both forms ("X is always cold and heavy" and "X has a cold and heavy part") are technically more precise, although the option "X is always cold and heavy" is more colloquial. This is because, in some contexts, the expression "is" can be functionally equivalent to "has a"; it is used to express hierarchical relations. However, future studies might implement procedures to allow participants to discriminate between both types of relations (equivalence vs. hierarchical) more precisely.

Another limitation of the present experiment is that hierarchical relations were trained in both directions (e.g., X includes X.1, and X.1 belongs to X). Thus, it could be argued that the transfer of functions from X.1 to X is not based on derived relations. Future research should consider training hierarchical relations in only one direction. In our case, X could include X.1, or X.1 belongs to X, but not both.

Finally, an important feature of the present experiment is that hierarchical relations were trained in both directions (e.g., X includes X.1, and X.1 belongs to X). Thus, it could be argued that the transfer of functions from X.1 to X is not based on derived relations. Future research should consider training hierarchical relations in only one direction. In our case, X could include X.1, or X.1 belongs to X, but not both.

Significance and contribution

This study significantly contributes to the functional analysis of human hierarchical categorization by providing a systematic experimental demonstration of how stimulus functions transform within hierarchical networks. The study validates mechanisms proposed by Relational Frame Theory for explaining how contextually controlled relational response repertoires can be applied arbitrarily in complex structures. The theoretical implications include the potential to understand how complex cognitive processes such as categorization, problem-solving, and abstract reasoning develop. By demonstrating that functions can transform through specific hierarchical relations, the study provides an experimental basis for understanding how the functional meaning of stimuli can change through multiple and structured relations.