Why Pair Subjects?
In Randomized Block Design, you learned that researchers can group similar experimental units and randomly assign treatments within those groups. A matched pairs design uses that idea in a particularly focused way: experimental units are arranged in pairs, and the treatments are compared within each pair. A pair may consist of two similar subjects, or it may consist of two measurements from the same subject under different treatments.
Pairing is useful when subjects differ in ways that may affect the response. If two similar subjects are placed together, or the same subject experiences both treatments, the comparison can account for some of those differences. The treatment comparison is made within the pair rather than relying only on a comparison between two large, unrelated groups.
There are two common ways to create pairs. In the first, researchers pair two different subjects who are similar in a relevant characteristic, such as starting weight, age, or initial plant height. Within each pair, one subject is randomly assigned to Treatment A and the other to Treatment B. In the second, each subject receives both treatments, usually in separate sessions or under separate conditions. The order of the treatments for each subject is randomly determined.
Pairing does not replace random assignment. With two different subjects, chance must decide which member of each pair receives which treatment. With the same subject receiving both treatments, chance should decide which treatment comes first. As discussed in Why Random Assignment Is the Key to Causation, random assignment helps make treatment comparisons fair; it is different from randomly selecting subjects from a population.
Two Ways to Make Matched Pairs
| Approach | How the pair is formed | Where chance is used | What is compared |
|---|---|---|---|
| Pair similar subjects | Two different subjects with similar values of a relevant characteristic are placed together. | Within each pair, chance assigns one subject to each treatment. | The responses of the two subjects in each pair. |
| Use each subject twice | One subject receives both treatments, typically at different times or under different conditions. | Chance determines which treatment comes first for each subject. | The subject’s response under the first treatment versus the second. |
For the first approach, the matching characteristic should be related to the response, or otherwise help make the two subjects in a pair comparable. For example, if the response is how much a plant grows, initial height may be useful when forming pairs. The characteristic used to match subjects is not the treatment; it is a way to make the comparison more informative.
For the second approach, each subject serves as their own comparison. This controls for stable differences between subjects because the same person, animal, or object is measured under both treatments. However, the first treatment might affect the response to the second, or subjects might improve simply through practice. Randomizing treatment order helps prevent one treatment from always being linked to a particular position in the sequence, but it does not guarantee that every possible order effect disappears.
Planning a Matched Pairs Experiment
A clear description should identify the experimental units, the treatments, the response, how pairs are formed, and exactly where chance is used. The plan should also explain why the matching characteristic is relevant or why each subject can reasonably receive both treatments.
State the treatments and the response variable, including how the response will be measured.
Pair similar subjects using a relevant characteristic, or plan for each subject to receive both treatments.
For different subjects, randomly assign one member to each treatment. For repeated measurements, randomly determine the order of treatments for each subject.
Apply treatments and measure responses consistently. Consider whether timing, practice, or lingering treatment effects could influence the comparison.
A useful way to describe what the design produces is a within-pair difference. If the response is measured in the same units for both treatments, a researcher could define \(d_i\) as the response under Treatment A minus the response under Treatment B for pair \(i\). The sign convention must be stated and used consistently. This description does not tell us which treatment works better; it clarifies how the observations are linked. The purpose here is to plan the experiment, not to carry out an inference procedure.
Worked Example: Pairing Similar Seedlings
Worked Example: Two Fertilizers and Matched Seedlings
A fictional gardening team wants to compare two fertilizers, A and B, for their effect on the height gained by tomato seedlings over four weeks. The experimental units are 20 seedlings. Before treatment, the team measures each seedling’s height and pairs seedlings with similar starting heights. The team forms 10 pairs.
Identify the design: This is a matched pairs design using different subjects. The seedlings are paired by initial height, a characteristic that could be related to later growth. Fertilizer is the treatment factor, with levels A and B. The response is height gained in centimeters over four weeks.
Randomize within each pair: For each pair, the team flips a coin. Heads assigns the first seedling in the pair to Fertilizer A and the second to Fertilizer B; tails reverses those assignments. The team repeats this separately for all 10 pairs. Each pair therefore has one seedling assigned to each fertilizer, giving 10 seedlings per treatment overall.
Keep the comparison fair: The team should use the same growing conditions, measurement method, and four-week treatment period for all seedlings. It should follow the assigned fertilizer plan consistently. These steps help avoid making fertilizer assignment systematically different from another condition.
Conclude: This is a matched pairs design because similar seedlings are paired and chance assigns the two fertilizers within every pair. The treatment comparison can focus on the difference in growth between the two seedlings in each pair, rather than treating all 20 seedlings as if they were unrelated to one another.
Worked Example: Each Subject Receives Both Treatments
Worked Example: Comparing Two Study Environments
A fictional learning researcher wants to compare a quiet room with a room playing soft instrumental music. Sixteen volunteers each complete two separate study sessions, one in each environment. In each session, the volunteer studies a different but similarly designed set of material and then takes a 20-point quiz. The response is the quiz score.
Identify the design: This is a matched pairs design in which each subject receives both treatments. The subject is the experimental unit, the two study environments are the treatments, and quiz score is the response. Each volunteer contributes one score under each environment.
Randomize treatment order: For each volunteer, use a chance process to decide which environment comes first. For example, flip a coin: heads means quiet first and music second; tails means music first and quiet second. Apply this process separately to all 16 volunteers. The assignment gives each volunteer both environments, but randomizes the order.
Consider order effects: A volunteer might perform better in the second session because of practice, or worse because of fatigue. If all volunteers had the quiet session first and the music session second, treatment and order would be tangled together. Randomizing the order prevents the researchers from deliberately or systematically assigning one environment to one position. Using comparable study materials and similar session conditions also helps, though it cannot guarantee that there is no order effect.
Conclude: This is a matched pairs design because each volunteer experiences both study environments, and chance determines the order. The direct comparison is each volunteer’s quiz score in the quiet room versus that same volunteer’s score with music. The design controls for stable differences among volunteers, such as their general level of academic preparation.
Worked Example: When the Pairing Plan Does Not Fit
Worked Example: One Treatment Cannot Be Repeated
A fictional recreation center wants to compare two one-time training programs for 24 new climbers. Each program consists of a single, intensive session, and the center wants to compare the climbers’ safety-procedure scores afterward. A planner suggests pairing each climber with a similar climber and having both members of every pair complete Program A and then Program B.
Spot the problem: The proposed plan has each climber receive both programs, but Program A is an intensive one-time session. Learning from the first program could affect performance after the second. If the first session teaches skills that carry over, the second score would not reflect only the second program. Randomizing which program comes first would help balance order across climbers, but it would not undo learning that persists from one session to the next.
Revise the plan: Pair the 24 climbers into 12 pairs using a relevant pre-study measure, such as a short baseline safety assessment. Within each pair, randomly assign one climber to Program A and the other to Program B. Each climber receives only one program. The response is the climber’s score on the same type of post-session safety assessment.
Conclude: The revised plan is a matched pairs design using similar subjects. It avoids having every climber receive both intensive programs, while still comparing treatment outcomes within pairs. The matching measure should be chosen before treatment and should help identify climbers likely to have similar safety scores.
This example illustrates an important planning question: can the same experimental unit reasonably receive both treatments without the first treatment changing the response to the second? If not, pair different but similar subjects instead. For some treatments, such as a lasting training program or a treatment with a lingering effect, a repeated-measures plan may be inappropriate even when treatment order is randomized.
Common Mistakes and AP Exam Tips
- Calling any experiment with groups matched pairs. The design must create pairs and make the treatment comparison within each pair. Merely dividing subjects into several broad groups is not enough.
- Forgetting random assignment. When there are two different subjects in a pair, say how chance determines which subject receives each treatment. When each subject receives both, say how chance determines treatment order.
- Pairing on an irrelevant characteristic. Explain why the characteristic used to match subjects could be related to the response or otherwise help make them comparable. Do not describe the treatment itself as the matching variable.
- Assuming the same-subject design has no limitations. The same subject may experience practice, fatigue, or a lingering treatment effect. Name a plausible concern in context and explain how the plan addresses it, if possible.
- Confusing order randomization with random selection. Randomizing treatment order is part of assigning treatments. It does not, by itself, make volunteers a random sample or support generalizing to a population. As in Scope of Inference: Four Combinations, consider selection and assignment separately.
- Describing only the treatments and not the response. A full description names what is measured and, when useful, its units—for example, height gained in centimeters or quiz score out of 20 points.
- Claiming that pairing guarantees a treatment advantage. Pairing helps organize a comparison and can account for differences between subjects. It does not determine which treatment will produce the larger response.
For a strong AP response, identify the pair structure, state the matching basis or explain that each subject receives both treatments, and describe the chance process within the pairs. Then name the response and mention a relevant concern, such as order effects, when repeated treatments are used.
Check Your Understanding
For each situation, identify the pair structure and describe where chance should be used.
- A gardener pairs 18 plants by initial height and compares two watering schedules. How many pairs are formed, and how should the schedules be assigned?
- Eight volunteers each test two versions of a phone interface in separate sessions. What should be randomized, and what is one possible order effect?
- A researcher matches people by age, then assigns everyone in the first half of the pairs to Treatment A and everyone in the second half to Treatment B. Is this a sound matched pairs assignment? Explain what should change.
- Why might a repeated-measures matched pairs design be a poor choice when the first treatment has a lasting effect?
- In a sentence or two, distinguish a matched pairs design using similar subjects from one in which each subject receives both treatments.