Why Randomize Within Pairs?
Suppose researchers want to compare two treatments, but they expect some units to differ substantially before treatment begins. In an experiment with people, age or baseline health might matter; with plants, starting size might matter. If those differences are not accounted for in the design, one treatment group could happen to contain more units with a characteristic related to the outcome.
One approach is to form pairs of units that are similar on relevant characteristics before treatment begins. The researcher then uses chance to assign one unit in each pair to Treatment A and the other to Treatment B. This is a pair-randomized experiment. Each pair serves as a block: the randomization is carried out separately within it.
Pairing and randomization do different jobs. Pairing deliberately groups units that are alike on chosen baseline features. Randomization then decides which member of each pair receives which treatment. Similarity by itself does not make an experiment randomized, and random assignment does not guarantee that the two treatment groups will match on every characteristic.
How the Design Is Organized
A pair-randomized design has several distinct parts. The experimental units are the entities to which treatments are assigned. The treatments are the conditions being compared. The pairs are formed before assignment, and each pair is treated as a block. The random assignment is made within each block, with exactly one unit assigned to each treatment.
For a simple design with two treatments, the researcher can use a fair random mechanism for every pair: for example, flip a coin to decide which member receives Treatment A, and give the other member Treatment B. The coin flip is repeated for each pair. The pair—not a preference, a schedule, or the order in which units arrive—determines which units are randomized together.
Specify what receives a treatment and state the two conditions being compared clearly enough that they can be applied consistently.
Use relevant baseline information to group units into pairs. Pairing decisions must not be based on outcomes observed after treatment.
Use a chance mechanism to assign one member to each treatment. Keep a record of the assignment and pair identifier.
Implement the assigned conditions consistently and record outcomes in a way that preserves each unit’s pair membership.
This design guarantees that each pair contributes one unit to each treatment group. Across all pairs, the treatment groups therefore have the same size. It does not guarantee identical groups on characteristics not used to form the pairs, nor does it remove the need for sound measurement and consistent treatment implementation.
Worked Example: Pairing Similar Seedlings
A hypothetical greenhouse experiment compares two nutrient solutions for their effect on seedling growth. Researchers select 12 pairs of seedlings with similar starting height and age. They will apply one solution to each seedling in a pair. Describe the design and check its defining features.
Units and treatments. The experimental units are the 24 seedlings. The treatments are nutrient solution A and nutrient solution B. The outcome could be a specified measure of growth collected after a fixed period.
Blocks. Each of the 12 pairs is a block. The pairs are made using starting height and age, both measured before the solutions are applied. Those features motivate pairing; they do not determine which seedling gets which solution.
Random assignment. For each pair, a fair coin can assign one seedling to A and the other to B. If the coin has one outcome for “first seedling gets A” and the other for “second seedling gets A,” each member has probability \(1/2\) of receiving A. The other member necessarily receives B.
Design check. There are 24 units, 12 blocks, and one unit assigned to each treatment in every block. The design is pair-randomized only if the allocation within each pair is determined by chance, not by which seedling appears healthier or is easier to reach. To keep the experiment interpretable, the solutions should be applied as specified, and the seedlings should not affect one another’s growth through the treatment process.
What Randomization Does and Does Not Guarantee
Random assignment makes the treatment allocation independent of researchers’ preferences and of the units’ observed or unobserved features, in the sense that each eligible allocation is selected by the stated chance mechanism. In a pair-randomized design, it also ensures that each block is represented in both treatment groups. The design controls the allocation within each pair; it does not make the paired units identical.
If there are \(B\) pairs and, independently for each pair, a fair coin determines which of its two members receives Treatment A, there are two possible assignments per pair. The total number of possible full assignments follows by multiplying the choices across pairs:
For example, with 12 pairs there are \(2^{12}=4096\) possible assignment patterns under this mechanism. Each pattern has probability \((1/2)^{12}=1/4096\), because it requires one specified coin outcome in each of 12 independent flips. This count describes the randomization scheme; it is not a statement about the outcomes the experiment will produce.
The calculation relies on the stated design: each pair has two possible orientations, each orientation is equally likely, and the random choices for different pairs are independent. A researcher could instead use a restricted randomization scheme, but then the set of possible assignments and their probabilities would need to be described according to that scheme.
Worked Example: Randomizing Within Five Pairs
In a hypothetical study of a new training protocol, researchers form five pairs of participants using a prespecified baseline fitness measure. Within each pair, one participant will receive the new protocol and the other the standard protocol. Suppose the assignment uses independent fair coin flips.
Number of units and blocks. There are \(2 \times 5=10\) participants and five blocks. Each block contains two participants, with one assigned to each protocol.
Possible assignments. Each of the five pairs has two possible orientations: either member can receive the new protocol. Thus the number of complete assignments is
Probability of one specified pattern. A particular pattern requires five specified coin outcomes, one for each pair. Therefore,
Every pair contributes one participant to each protocol, so there are five participants per treatment. The assignment remains random within each pair: matching on baseline fitness did not give the researchers permission to choose the treatment based on which participant seemed more likely to improve.
Example: Blocking in a Laboratory Study
Blocking is a general design idea: first group units into relatively similar sets, then randomize treatments within those sets. Pair randomization is the special case in which every block contains two units and there are two treatments, with one unit assigned to each treatment. The block can be based on people, animals, plants, machines, or another kind of experimental unit.
Worked Example: Matching Laboratory Machines
A hypothetical laboratory compares two measurement procedures on 8 machines. Before assignment, the researcher forms four pairs of machines with similar calibration readings. Within each pair, a random number generator assigns one machine to Procedure A and the other to Procedure B.
Identify the blocks and units. The machines are the experimental units. Each matched pair of machines is a block, so there are four blocks. Procedure A and Procedure B are the treatments.
Check the allocation. In every pair, one machine receives A and one receives B. The assignment within each pair is random, so the researcher cannot systematically give A to the machine with the more favorable calibration reading. The pairing feature is therefore used as a block in the randomization, rather than only as a way to describe the machines.
Preserve the design in the records. Each machine’s procedure and measured outcome should be stored together with its pair identifier. Without that identifier, the record would no longer show which two machines were randomized against one another. The design is defined by the assignments made, not merely by a later decision to place similar machines side by side in a table.
Design Checks and Common Mistakes
Before an experiment begins, write down the pair-forming rule and the randomization procedure. The rule should use information available before treatment, and the randomization should specify how one member of each pair is selected for one treatment. These details make it possible to tell whether the experiment actually followed the intended design.
- Pairing without randomization. Similar units are placed together, but a researcher chooses which one receives each treatment. This is a matched allocation, not a pair-randomized experiment. Use a chance mechanism within each pair.
- Randomizing first and pairing later. Units are assigned to treatments, then similar units are linked in pairs afterward. Such pairs did not govern treatment assignment and are not blocks in the randomization.
- Assigning both members of a pair to the same treatment. This fails the defining allocation rule for this two-treatment pair design. Each pair must contribute one unit to each treatment.
- Pairing on a post-treatment feature. A characteristic changed by treatment cannot serve as a baseline pairing variable. Define pairs before treatment using eligible baseline information.
- Treating pairing as a guarantee of balance on everything. Units in a pair can still differ, and pairing on selected features does not ensure balance on every other characteristic. Randomization is still essential.
A further design concern is whether one unit’s treatment can affect another unit’s outcome. If, for example, two paired animals share an environment in which one animal’s assigned treatment changes the other’s exposure, the intended treatment comparison may be difficult to interpret. Researchers should consider how units are housed, treated, and measured, and prevent such cross-unit effects when feasible.
Check Your Understanding
Answer each question by referring to the design, not to a later analysis of the outcomes.
- A researcher matches 20 participants into 10 pairs using baseline age, then lets participants choose their treatment. Is this a pair-randomized experiment? Explain.
- In a pair-randomized experiment with 7 pairs, how many units are assigned to each of two treatments? If independent fair coin flips are used, how many complete assignment patterns are possible?
- Why must pair formation occur before the treatments are assigned?
- A study pairs similar machines after treatment assignments have already been made. Do these pairs function as blocks in the randomization? Why or why not?
- Name two design details that should be recorded so the pair-randomized structure is preserved.