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ABA Concepts

Stimulus Equivalence and Derived Relational Responding

Stimulus equivalence and derived relational responding explain how learners can show new skills they were never directly taught. Stimulus equivalence, described by Murray Sidman, occurs when relations among stimuli demonstrate reflexivity, symmetry, and transitivity. This guide explains each property with practical examples, how to teach with matching-to-sample, and how equivalence classes allow efficient instruction, such as teaching a few relations and getting many more for free. We also introduce relational frame theory (RFT) and its focus on arbitrarily applicable relational responding, including frames of coordination, opposition, comparison, and hierarchy. You will learn how derived relations support language, reading comprehension, and generative learning, and how these concepts appear in clinical programs for children with autism and in acceptance and commitment therapy. Exam tips, examples, and a quick decision process will help you identify reflexivity, symmetry, transitivity, and equivalence in BCBA scenarios and apply them to instruction.

May 20, 2024

Why Derived Relations Matter

Typical teaching requires direct instruction for every skill, which is slow and costly. Yet humans routinely show behavior that was never taught directly. If you are told that a "kiwi" is a green fruit with fuzzy brown skin, you can find one in a market without ever having been trained to do so. Stimulus equivalence and derived relational responding describe how this generative learning arises, and they are a significant area in advanced behavior analysis.

Stimulus Equivalence

Murray Sidman's research in the 1970s showed that after being taught certain conditional discriminations, learners could demonstrate other relations that were not directly taught. A set of stimuli is said to form an equivalence class when the relations among them demonstrate three properties: reflexivity, symmetry, and transitivity.

Reflexivity (A = A)

Reflexivity is generalized identity matching. Given a sample, the learner selects the identical comparison stimulus without being taught that specific relation. If shown a picture of a cat and asked to match it to one of three pictures, the learner picks the same cat picture.

Symmetry (if A = B, then B = A)

If a learner is taught to point to a picture of a dog when hearing the spoken word "dog" (A to B), and is then able, without further teaching, to select the spoken word "dog" when shown the picture (B to A), symmetry has emerged. The relation works in both directions even though only one direction was trained.

Transitivity (if A = B and B = C, then A = C)

Suppose a learner is taught to match spoken "dog" (A) to a picture of a dog (B), and the picture of a dog (B) to the printed word "dog" (C). If the learner then matches spoken "dog" (A) to printed "dog" (C) without direct teaching, transitivity has emerged.

Equivalence

The combination of symmetry and transitivity is sometimes tested as equivalence: for example, given the printed word, the learner can select the spoken word (C to A). Showing all three properties demonstrates that A, B, and C are members of an equivalence class. Importantly, stimuli in an equivalence class are not physically similar. The relation is based on training and not on form.

Teaching with Matching-to-Sample

Equivalence is typically established with matching-to-sample (MTS) procedures. A sample stimulus is presented, and the learner selects the correct comparison from an array. The training structures include:

  • Linear series: Train A-B, then B-C, then C-D. Test for transitive relations like A-C.
  • One-to-many: Train A-B, A-C, and A-D (the sample is the same in each). Test for B-C and C-D, which relies on symmetry and transitivity.
  • Many-to-one: Train B-A, C-A, and D-A. Test for B-C and similar relations.

Research suggests that one-to-many and many-to-one structures often yield more reliable equivalence than linear series, although outcomes vary across learners.

Practical Value

Equivalence-based instruction offers efficiency. By teaching a limited number of relations, you can produce many untaught relations.

Example

You teach a child to match spoken names of coins to pictures of coins (A-B), and pictures of coins to their values (B-C). The child may then match spoken names to values (A-C), say values when shown coins (B-C reverse), and match values to coin pictures (C-B), without more training. That is six relations from two trained.

Equivalence has been used to teach reading, mathematics, music, and professional content such as pharmacology and statistics, and often yields quicker acquisition than teaching each relation separately.

Relational Frame Theory

Relational frame theory (RFT), developed by Steven Hayes and colleagues, expands on equivalence. RFT proposes that humans learn to respond to stimuli in terms of their relation to other stimuli, not just their form. This arbitrarily applicable relational responding is controlled by contextual cues rather than physical properties. Equivalence is one type of relation (the frame of coordination). Others include:

  • Opposition: Hot is the opposite of cold.
  • Distinction: This is not the same as that.
  • Comparison: A dime is worth more than a nickel, even though a nickel is bigger.
  • Hierarchy: A dog is a kind of animal.
  • Temporal and spatial: Before and after, above and below.
  • Deictic: I and you, here and there.

RFT proponents argue that derived relations underlie language, reasoning, and many clinical phenomena. Acceptance and commitment therapy (ACT) is built on RFT principles, targeting how verbal relations can create psychological inflexibility.

Key RFT terms

  • Mutual entailment: If A is related to B, then B is related to A, which is similar to symmetry.
  • Combinatorial entailment: If A relates to B and B relates to C, then A relates to C, similar to transitivity.
  • Transformation of stimulus functions: Functions of one stimulus transfer to others in a relational network. If a stimulus is paired with a reinforcer, other stimuli in its equivalence class may acquire similar functions.

Transfer and Transformation of Function

Stimulus functions can transfer across members of an equivalence class. If one member of a class acquires a discriminative function, other members may serve a similar function without direct training. Clinically, this explains why a feared word can evoke anxiety similar to the feared object, and why teaching new uses for stimuli can transfer broadly.

Clinical Guidance

  • Check prerequisites such as the ability to perform matching-to-sample and identity matching.
  • Use mastery criteria for baseline relations before testing derived relations.
  • Test for emergent relations without reinforcement or with intermittent reinforcement so you can see whether the relation was truly derived.
  • If the relations do not emerge, provide direct training for them, then retest.
  • Be mindful that many learners, including some with developmental disabilities, may require extra training in multiple exemplars before derived responding emerges.

Exam Strategies

  1. If the item says a skill was never directly taught but appears, think of derived relations.
  2. Use the AB, BC, AC logic. Mapping each trained relation as an arrow helps you identify symmetry and transitivity.
  3. Remember that stimuli in a class are not necessarily similar; that is what makes equivalence powerful.
  4. Associate Sidman with equivalence, Hayes with RFT.

Key Takeaways

Stimulus equivalence is shown through reflexivity, symmetry, and transitivity, and allows learners to demonstrate untaught relations. RFT extends the idea to many relational frames and supports generative language. Use equivalence-based instruction to make teaching efficient, and be ready to identify each property in exam scenarios.

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