Why Treatment Order Matters
In earlier tutorials on paired data, you learned that paired observations are linked within a subject and that a paired analysis uses one difference for each pair. A crossover experiment is one way to create paired data: each subject receives both treatments, but the treatments are given at different times, or periods. The order can matter because subjects may change over time, become more practiced, get tired, or respond differently after an earlier treatment.
Suppose treatments are called A and B. A subject assigned to sequence AB receives A in the first period and B in the second. A subject assigned to sequence BA receives B first and A second. Randomizing the treatment order means using a chance process to assign subjects to these sequences, rather than giving everyone the treatments in the same order.
Random order matters because the period itself can affect the response. For instance, participants might perform better on a task the second time because they have practiced. If every participant receives A first and B second, treatment B is always paired with the second period. An apparent difference between A and B could then reflect the treatment, a practice effect, or both. The treatment and the period are confounded in that design.
With subjects in both sequences, each treatment is used in both periods. In a balanced design, A is used by half the subjects in period 1 and by the other half in period 2; B is represented in both periods as well. This makes the comparison less sensitive to a general period effect than giving all subjects the same order. It does not guarantee that period effects disappear or that the treatments have identical effects for every subject.
How to Assign the Sequences
The researcher first decides which subjects will take part in the crossover and identifies the two treatment sequences. A chance process then assigns subjects to AB or BA. For example, a random number generator could assign each subject to one of the two sequences. If the sample size is even, a restricted random assignment can randomly choose which subjects go into AB while assigning the rest to BA, so the sequence groups have equal sizes.
Equal numbers in each sequence are useful, but balance is not automatic whenever order is randomized. With a small sample, assigning each subject independently by a coin toss could, by chance, put many more subjects in one sequence than the other. A researcher can use a random assignment method that ensures equal sequence sizes while still using chance to decide which subjects receive which sequence.
The table shows why balanced sequences address a common order concern: in each period, both treatments are represented. The comparison is not based on “first treatment versus second treatment.” Each subject receives A and B, and the treatment comparison must keep track of which measurement belongs to which treatment, regardless of its period.
For example, if the paired difference is defined as \(d_i=\text{response under B}-\text{response under A}\), use that order for every subject. Do not define the difference as “second period minus first period,” because the second period is B for AB subjects but A for BA subjects. As in “Defining the Difference Variable,” state the subtraction order before interpreting the sign of a difference.
What Randomized Order Can—and Cannot—Do
Random assignment of sequence helps prevent subject characteristics from being systematically tied to one order. It also reduces the risk that a period effect will consistently favor one treatment. In a balanced design, each treatment occurs equally often in each period, so a general effect of being in period 1 or period 2 is not automatically credited to just one treatment.
However, randomizing the sequence does not make the treatment periods independent in every important sense. The first treatment might have an effect that continues into the second period. This is a carryover effect. If treatment A continues to affect a subject after the subject switches to B, the second-period response may reflect both B and the lingering effect of A.
Researchers may schedule a period without treatment, called a washout period, between treatments when it is reasonable to expect the first treatment’s effects to fade. A washout period is not a guarantee: its adequacy depends on the treatments and the outcome being measured. If carryover is plausible and cannot be addressed by the design, a crossover experiment may not give a clear comparison of the treatments. Randomizing order does not repair that limitation.
Likewise, balanced sequences help with a general period effect but cannot ensure that every subject responds to the treatments in the same way. An interaction between treatment and period—for example, a treatment working differently depending on whether it is given first or second—may still complicate interpretation. The design should be planned with the setting in mind, not treated as a mechanical fix.
Worked Examples
Worked Example: Comparing Two Keyboard Layouts
A fictional researcher wants to compare the time, in seconds, that students need to complete a typing task using keyboard layouts A and B. Twelve students will each use both layouts in two separate periods. The researcher expects students may get faster during the second period because they have practiced. Explain an appropriate treatment-order assignment and why it helps.
Solution. The researcher can randomly assign six students to sequence AB and six to sequence BA. One chance method is to randomly select six of the twelve students for AB and assign the remaining six to BA. The assignment of particular students to each sequence is determined randomly.
In period 1, six students use A and six use B. In period 2, the same counts are reversed: six use B and six use A. Thus, each layout is used equally often in each period. If students generally improve in period 2 through practice, that general improvement is not attached only to layout B. The randomized, balanced order makes the comparison less vulnerable to confusing a practice effect with a layout effect.
For a paired comparison, the researcher could define \(d_i=\text{time under B}-\text{time under A}\), in seconds, for each student. That definition stays the same whether a student used B first or second. A negative difference means that student took less time under B than under A. Randomizing the order does not establish that there is no carryover; the researcher should consider whether using one layout could affect performance with the other.
Worked Example: Why “A First for Everyone” Is a Problem
A fictional team tests two short concentration activities, A and B, with eight volunteers. Each volunteer completes both activities, and the team measures the number of correctly completed items. The team proposes giving every volunteer A in period 1 and B in period 2. Explain the design concern and revise the plan.
Solution. Under the proposed plan, all eight volunteers receive A first and B second. If volunteers learn the task, become tired, or otherwise change between periods, the period effect is inseparable from the treatment comparison: A is always in period 1 and B is always in period 2. A higher score under B, for example, could reflect B, more familiarity with the task, or a combination.
Instead, randomly assign four volunteers to AB and four to BA. In period 1, four volunteers receive A and four receive B; in period 2, four receive B and four receive A. Treatment and period are no longer completely tied together. The balanced random assignment reduces the risk that a general first-versus-second-period change will favor just one activity.
If the response is the number of correct items, a consistent paired difference could be \(d_i=\text{correct under B}-\text{correct under A}\), in items. The result compares the two activities within each volunteer, not simply the first and second scores. The revision improves the order assignment, but the team should still consider whether completing one activity could influence performance on the other.
Worked Example: A Crossover Study With a Possible Carryover Effect
A fictional study compares two ways of reducing muscle soreness after exercise. Fourteen volunteers will try both methods in separate exercise sessions. The methods may have effects lasting several days. The researcher plans to assign seven volunteers to AB and seven to BA, with a week between sessions. Is the sequence assignment balanced, and does it settle the carryover concern?
Solution. The assignment is balanced: seven volunteers receive A in period 1 and B in period 2, while seven receive B first and A second. Therefore, each method is used by seven volunteers in each period. Randomly deciding which volunteers receive each sequence also prevents the researcher from deliberately assigning particular types of volunteers to one order.
The sequence assignment does not, by itself, settle carryover. If soreness reduction from the first method lasts into the next exercise session, a volunteer’s second-period result could reflect the current method and a lingering effect of the first. A week between sessions may be an intended washout period, but the researcher needs a reasonable basis for believing it is long enough in this setting. If lingering effects remain plausible, the crossover comparison may be difficult to interpret despite the balanced random order.
The study should therefore plan for carryover as well as sequence balance. If the effects do not reliably fade, the researcher might need a different design rather than assuming that randomizing AB versus BA has solved the problem. The appropriate conclusion about the design is: the order is balanced, while the possibility of carryover still needs attention.
Common Mistakes and AP Exam Tips
- Giving everyone the same order. If every subject receives A then B, period and treatment are confounded. A full-credit explanation identifies a plausible period effect and explains why it could be mistaken for a treatment effect.
- Confusing the sequence with the treatment. AB and BA are orders of receiving the treatments, not two different treatments. Each subject in the crossover receives both A and B.
- Claiming that randomization guarantees equal groups. Random assignment alone can produce unequal sequence sizes. Say that the design is balanced only when the assignment plan ensures equal numbers, or when the stated group counts show equality.
- Saying balance eliminates period effects. Balance places each treatment equally often in each period; it does not make period-related changes vanish or guarantee that all subjects respond identically.
- Assuming random order eliminates carryover. A lingering effect from the first treatment can influence the second period. Mention a suitable washout period when relevant, and recognize that it may not fully resolve the concern.
- Defining differences by chronological order. In a crossover, “second minus first” compares different treatment orders across subjects. Define differences by treatment, such as B minus A, and keep the order consistent.
For a strong AP response, describe the two sequences, state how chance assigns subjects to them, and explain the specific order concern addressed. If the sequence groups are equal in size, point out that each treatment is represented equally in each period. Then distinguish what the design does not establish, especially whether carryover effects are absent.
Check Your Understanding
Answer each question using the ideas of treatment sequence, period, and carryover.
- In a crossover comparing treatments A and B, what does sequence BA mean for the two treatment periods?
- Why can giving every subject A first and B second make it difficult to interpret a difference between treatments?
- In a balanced study with 18 subjects, how many subjects could be assigned to AB and how many to BA? How many receive each treatment in period 1?
- Does balanced sequence assignment prove that the first treatment has no effect on the second-period response? Explain.
- If paired differences are defined as response under B minus response under A, should a subject who receives BA use a different subtraction order? Why or why not?