Matching law

Matching compares relative shares

Suppose a person can request help through two available routes. If 60% of requests use route A and 40% use route B, response allocation is 60/40. If 60% of obtained helpful responses follow route A and 40% follow route B, the observed shares match.

The comparison uses proportions, not raw counts alone. Thirty responses and three responses can have the same percentage while providing very different precision.

Obtained reinforcement is part of the relation

Matching analyses often use consequences actually obtained rather than consequences merely programmed. A helper may be scheduled to respond equally through two channels, yet delays, missed messages, or unavailable staff change what the person contacts.

Reed and Kaplan’s tutorial explains basic and generalized matching equations for practitioners. It also shows why logarithms, bias, sensitivity, and data requirements matter before interpreting a fitted line.

Many variables can shift allocation

Two options may differ in response effort, outcome delay, amount, quality, predictability, accessibility, social history, or instructions. A person may consistently prefer one route for reasons outside the measured reinforcement rate.

Generalized matching models can describe bias, a preference unexplained by the measured reinforcement ratio, and sensitivity, how strongly allocation changes with that ratio. These parameters summarize data under a model. They do not identify every controlling variable.

Concurrent availability must be real

The alternatives need to be available during the same relevant periods. Comparing weekday clinic behavior with weekend home behavior introduces setting, people, tasks, and time along with the schedule difference.

Record when each option was usable, not simply when it existed in policy. Device failure, inaccessible communication, staff absence, or a closed door changes the choice set.

A fictional allocation record

Lina uses two accessible help messages during 30 eligible opportunities. She chooses route A in 18 of 30 and route B in 12 of 30, a 60/40 response split.

Partners provide the requested help on 9 of 15 obtained outcomes through A and 6 of 15 through B, also 60/40. The proportions match in this small record. They do not establish the matching law, because response allocation and obtained outcomes influence each other and because effort, delay, message form, partner, and task may differ.

The team reports the raw counts, outcome delays, unfulfilled requests, and time each route was available. Lina’s report about ease and preference remains a separate and essential source.

Clinical use is a systems question

A matching analysis can reveal that a support system reliably reinforces one response while rarely responding to an accessible alternative. The first action may be to improve partner availability, reduce effort, or honor communication more consistently.

Do not manipulate basic needs, pain, safety, communication, social connection, or essential care to create a cleaner ratio. Choice is meaningful only when options are safe, available, understandable, and acceptable.

Borrero and colleagues provide an applied example of concurrent-schedule analysis. Small applied studies illustrate methods and do not make matching a universal clinical law for every response.

Measurement needs stable units

Define:

• each response and eligible opportunity
• each consequence and delivery window
• how simultaneous or repeated responses count
• the period when both alternatives were available
• prompts, delays, effort, and unmatched outcomes
• the analysis window and minimum observation volume

Use the same unit across alternatives. Comparing time allocated to one option with response counts for another makes the ratio uninterpretable.

The operant-conditioning review by Staddon and Cerutti covers concurrent choice and schedule relations in a broad experimental context. The BACB BCBA Test Content Outline, 6th edition lists schedules and related concepts as examination content, not a required clinical model.

A trend is stronger than one matched point

A matching analysis typically needs several stable observations across different reinforcement ratios. With only one 60/40 point, many fitted relations remain possible. Predeclare each analysis window, preserve raw counts, and show points that depart from the pattern.

When system changes alter the obtained outcome ratio, examine whether allocation changes afterward. Keep other differences visible, including new rules, staff, delays, effort, and access. A fitted line can summarize association across conditions; it cannot prove that changing one consequence will produce a precise clinical shift.

Use model fit as one source beside direct client input, social validity, safety, treatment integrity, and meaningful outcomes. A mathematically close match can still describe an inaccessible or unwanted arrangement.