What a Three-Period Test-Replicate Schedule Does
A replicate crossover schedule assigns a participant to receive at least one formulation more than once. In a three-period test-replicate schedule, each participant receives the test formulation twice and the reference formulation once. The order matters: each schedule is a sequence of three assignments, one for each period.
Use \(T\) for the test formulation and \(R\) for the reference formulation. Thus \(TTR\) means test in Period 1, test in Period 2, and reference in Period 3. It does not mean that the formulations are given simultaneously. A sequence is the planned treatment order for a participant; a study may assign different participants to different sequences.
The design is called partial replicate because only one of the two formulations is repeated for each participant. The contrasting three-period reference-replicate arrangement repeats \(R\) instead: its sequences are \(TRR\), \(RTR\), and \(RRT\). Both arrangements use three periods per participant, but they allocate the repeated exposure to different formulations.
Constructing the Sequences
There is a direct way to construct every test-replicate sequence: choose which one of the three periods receives the single \(R\). The other two periods must receive \(T\). If \(R\) is in Period 1, the sequence is \(RTT\); if it is in Period 2, the sequence is \(TRT\); if it is in Period 3, the sequence is \(TTR\). This gives all possible ordered arrangements with two \(T\)'s and one \(R\), without adding a sequence that has the wrong treatment count.
Equivalently, choose the position of the single \(T\) in a reference-replicate sequence and switch the repeated and single formulation counts. The labels must still be read by period: changing the formulation assigned in a period changes the sequence, even when the same two formulation labels appear overall.
Worked Example: Listing Every Test-Replicate Order
A hypothetical design team wants three-period sequences in which each participant receives \(T\) twice and \(R\) once. Construct the complete set and verify the counts.
Step 1: Place the singleton formulation. Place the single \(R\) in Period 1, Period 2, and Period 3 in turn. The resulting schedules are \(RTT\), \(TRT\), and \(TTR\).
Step 2: Check each treatment count. In \(RTT\), the counts are one \(R\) and two \(T\)'s. In \(TRT\), they are again one \(R\) and two \(T\)'s. In \(TTR\), they are also one \(R\) and two \(T\)'s.
Step 3: Check completeness. A valid schedule has exactly one position for \(R\), and there are three positions. The three listed sequences account for each possible position once. There is no other three-period ordering with exactly one \(R\).
This verifies the sequence construction itself. It does not establish that participants were allocated to sequence groups in any particular proportions, or that the design is suitable for every study objective.
Counting Exposures Across Participants
The treatment count for one participant is fixed by the sequence: two test exposures and one reference exposure. Counts across a study also depend on how many participants are assigned to each sequence. If there are \(n_{TTR}\), \(n_{TRT}\), and \(n_{RTT}\) participants in the three groups, then the total number of participants is their sum. Each participant contributes two \(T\) exposures and one \(R\) exposure, so the aggregate counts follow directly.
These are counts of planned treatment assignments, not counts of completed pharmacokinetic profiles. Withdrawals, missed periods, or unusable measurements can make the number of observed records differ from the planned number of assignments. The simple arithmetic describes the schedule before such events.
Worked Example: Totals Under Equal Sequence Allocation
In a hypothetical design, 36 participants are allocated equally across \(TTR\), \(TRT\), and \(RTT\), giving 12 participants per sequence. Find the planned treatment counts.
There are \(N=12+12+12=36\) participants, each with three assigned periods. Thus the total number of period assignments is \(3(36)=108\).
Each participant receives \(T\) twice, so the total number of \(T\) assignments is \(2(36)=72\). Each receives \(R\) once, so the total number of \(R\) assignments is \(1(36)=36\). The check \(72+36=108\) agrees with the total number of period assignments.
The allocation gives each sequence 12 participants, but equal sequence-group sizes are not needed to obtain the per-participant count of two \(T\)'s and one \(R\). They are a separate feature of this hypothetical allocation.
For comparison, a reference-replicate schedule gives one \(T\) and two \(R\)'s to every participant. With the same 36 participants, it would plan for \(36\) \(T\) assignments and \(72\) \(R\) assignments, again totaling \(108\) assignments. Repeating \(T\) instead of \(R\) therefore exchanges which formulation receives the additional exposure; it does not change the number of periods.
Worked Example: Unequal Sequence-Group Sizes
A hypothetical test-replicate study plans 15 participants in \(TTR\), 10 in \(TRT\), and 5 in \(RTT\). Calculate the total assignments and check whether unequal group sizes change the formulation totals.
The total number of participants is \(N=15+10+5=30\). Since every participant is assigned to three periods, there are \(3(30)=90\) total assignments.
Every sequence contains two \(T\)'s and one \(R\). Therefore, the total number of \(T\) assignments is \(2(30)=60\), and the total number of \(R\) assignments is \(30\). The check is \(60+30=90\).
The group sizes are unequal, but the treatment totals still follow from the same per-participant pattern. The number assigned to each sequence can matter for other design considerations; it does not change the fact that each participant on any of these three sequences is scheduled for two \(T\) periods and one \(R\) period.
Checking a Candidate Schedule
A candidate schedule can be audited with a few explicit questions: Does every sequence have exactly three periods? Does each sequence contain exactly two \(T\)'s and one \(R\)? If a set of sequences is proposed, are its labels the intended orders? These checks catch both count errors and order-label errors. For example, \(TRR\) has three periods but is a reference-replicate sequence, not a test-replicate sequence. \(TTRT\) repeats \(T\), but it has four periods and is outside a three-period design.
Worked Example: Auditing and Correcting a Proposed Set
A hypothetical planning note proposes \(TTR\), \(TRR\), and \(RTT\) as the three sequences for a three-period test-replicate design. Identify the problem and make the smallest correction that gives the complete test-replicate set.
\(TTR\) has two \(T\)'s and one \(R\), so it meets the treatment-count requirement. \(RTT\) also meets it. But \(TRR\) has one \(T\) and two \(R\)'s, so it repeats the reference formulation instead. The proposed set therefore mixes design types.
Replace \(TRR\) with \(TRT\). The corrected set is \(TTR\), \(TRT\), and \(RTT\); each sequence has three periods and exactly two \(T\)'s and one \(R\). The set now lists each possible position of the single \(R\) once.
This correction establishes the intended treatment counts and ordered sequences. It does not, by itself, determine how many participants should be allocated to each sequence.
How the Alternatives Differ
| Schedule type | Three-period sequences | Per-participant \(T\) count | Per-participant \(R\) count |
|---|---|---|---|
| Test-replicate | \(TTR\), \(TRT\), \(RTT\) | 2 | 1 |
| Reference-replicate | \(TRR\), \(RTR\), \(RRT\) | 1 | 2 |
| Two-period, nonreplicate example | \(TR\) or \(RT\) | 1 | 1 |
The two three-period designs have the same number of periods per participant but repeat different formulations. A two-period, nonreplicate schedule assigns each formulation once, so it does not repeat either within a participant. A four-period full-replicate schedule is another distinct option: it repeats both formulations within a participant, rather than repeating only one. These labels describe treatment counts and schedule structure; they are not, on their own, a ranking of designs or a complete recommendation.
The design choice should be tied to the study objective and the burden of additional periods. A test-replicate schedule allocates more exposures to \(T\) than to \(R\) for each participant; a reference-replicate schedule does the reverse. The implications for estimating treatment-specific variability and for choosing among replicate designs are separate questions. Treatment counts alone do not establish which design is preferable.
Assumptions, Diagnostics, and Limitations
The schedule arithmetic assumes that \(T\) and \(R\) denote the same two formulations throughout the design and that each symbol represents one planned treatment assignment in one period. It also assumes that every participant assigned to a sequence is intended to complete its three scheduled periods. These are planning assumptions, not guarantees about observed data.
An ordered schedule also does not guarantee that periods are free from carryover or that a chosen washout is adequate. Treatment order and washout need to be considered in light of the compound and study conditions. Earlier material on carryover and washout addresses those concerns. Similarly, the count of assignments does not demonstrate that a statistical model can estimate every desired effect; that requires a separate analysis of the design and model.
Useful schedule-level diagnostics are simple but important:
- Count the symbols in every sequence and confirm that it has three periods.
- Count \(T\) and \(R\) separately; a test-replicate sequence must contain two \(T\)'s and one \(R\).
- Read the sequence left to right and confirm that the printed label matches the planned period order.
- Multiply the per-participant counts by the number of participants and check that total \(T\) and \(R\) assignments sum to three times the planned sample size.
A common mistake is to see that a candidate schedule contains both formulations and conclude that it is a test replicate. For instance, \(TRR\) contains \(T\) and \(R\), but it repeats \(R\), not \(T\). The correction is to count each formulation explicitly: the defining test-replicate pattern is \(T=2\) and \(R=1\) per participant. Another mistake is to infer from a total such as 60 \(T\) assignments that every participant received two \(T\)'s. Aggregate counts can conceal a mixed or incorrectly labelled set of sequences, so verify each sequence before using totals as a check.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- List all three-period sequences with two \(T\)'s and one \(R\).
- A hypothetical test-replicate study assigns 18 participants to its sequences. How many \(T\) and \(R\) assignments are planned in total?
- Is \(RTT\) a test-replicate or reference-replicate sequence? State its per-participant counts.
- A proposed set is \(TTR\), \(TRT\), and \(RRT\). Identify and correct the sequence that does not match a test-replicate design.
- For 24 participants, compare the total \(T\) and \(R\) assignments under a test-replicate schedule with those under a reference-replicate schedule. How many total period assignments are there in either case?
Solutions. (1) Place the single \(R\) in each period in turn: \(RTT\), \(TRT\), and \(TTR\). (2) Each participant contributes two \(T\)'s and one \(R\), so there are \(2(18)=36\) \(T\) assignments and \(18\) \(R\) assignments. (3) \(RTT\) is test-replicate: it contains two \(T\)'s and one \(R\). (4) \(RRT\) repeats \(R\); replace it with \(RTT\). The complete test-replicate set is \(TTR\), \(TRT\), \(RTT\). (5) Test replicate gives \(48\) \(T\) assignments and \(24\) \(R\) assignments; reference replicate gives \(24\) \(T\) assignments and \(48\) \(R\) assignments. Either design has \(3(24)=72\) total period assignments.