Why the Assignment Table Matters
In a crossover study, each participant receives treatments according to a sequence, and each treatment is assigned to a period. The assignment table is therefore more than an administrative record: it determines which contrasts the design can support. If treatment and period always move together, their effects cannot be separated from the observed responses. If both orders occur but one is rare, separation is possible in principle but may be imprecise.
The key distinction is between exact aliasing and weak distinguishability. Exact aliasing means that different effect values produce the same fitted values for every observation, so the data cannot identify the effects separately. Weak distinguishability means the design contains information to separate them, but that information is limited, often because the sequence groups are highly imbalanced.
Reading a Two-Period Assignment Table
For a two-treatment, two-period crossover, write AB for participants receiving A in period 1 and B in period 2; write BA for the reverse order. Let \(\theta\) denote the treatment contrast B minus A, and let \(\delta\) denote the period contrast period 1 minus period 2. For participant \(i\), define \(d_i=Y_{i1}-Y_{i2}\), the response in period 1 minus the response in period 2.
The following contrast calculation uses an additive treatment-and-period model: the treatment contrast is common across periods, and the period contrast is common across treatments. It also assumes that any carryover contribution is negligible or otherwise not contaminating the direct-effect contrast. The earlier material on treatment-by-period interaction explains why a common treatment contrast may not be appropriate if treatment effects differ by period; in that case the model and rank check must include the interaction. A schedule alone cannot establish that these scientific assumptions hold.
These relations can be displayed directly from the assignment table.
| Sequence | Period 1 | Period 2 | Treatment contribution to \(d_i\) | Period contribution to \(d_i\) |
|---|---|---|---|---|
| AB | A | B | \(-\theta\) | \(\delta\) |
| BA | B | A | \(\theta\) | \(\delta\) |
The table shows why both orders matter. With both AB and BA, the treatment contribution changes sign while the period contribution does not. That difference in pattern allows the two contrasts to be separated. With only one order, the two contributions appear only as a single sum or difference.
Deriving the Identifiable Contrasts
Let \(\bar d_{AB}\) and \(\bar d_{BA}\) be the sample means of the within-participant differences in the two sequence groups. Under the stated model, their expected values satisfy
Subtracting the AB mean from the BA mean removes the common period contrast and leaves twice the B-minus-A treatment contrast. Adding the means cancels the opposite treatment contributions and leaves twice the period contrast. Thus the corresponding estimates are
This is also a rank argument. The two sequence-group means provide two equations in the two unknown contrasts. The coefficient matrix has rows \((-1,1)\) for AB and \((1,1)\) for BA; its determinant is \(-2\), not zero, so the contrasts are separately identifiable when both sequence patterns are represented. If only AB is observed, there is one equation, \(-\theta+\delta=E(\bar d_{AB})\), for two unknowns. The coefficient matrix then has rank one, and no calculation can recover both contrasts from that schedule alone.
Worked Example: Both Sequences Identify Both Contrasts
Suppose a hypothetical study has complete two-period records for participants in both sequences. The observed mean period-1-minus-period-2 difference is \(5\) response units for AB and \(-3\) response units for BA. Use the contrast formulas to estimate treatment B minus A and period 1 minus period 2.
First check the design condition: both AB and BA occur, so the two equations have distinct treatment patterns. Under the additive model and the carryover condition stated above, the contrasts are identifiable.
For treatment,
For period,
The estimated B-minus-A contrast is \(-4\) response units, while the estimated period-1-minus-period-2 contrast is \(1\) response unit. The negative treatment estimate is not an error: with the specified contrast direction, it means the estimated response under B is 4 units lower than under A, conditional on the model.
Worked Example: One Sequence Aliases Treatment and Period
Suppose a hypothetical schedule assigns every participant to AB. The observed mean of \(d_i\) is \(-6\) response units. Under the model, the data supply the equation
This is one equation for two unknown contrasts. For example, \((\theta,\delta)=(0,-6)\) satisfies it because \(-0+(-6)=-6\). So does \((\theta,\delta)=(2,-4)\), because \(-2+(-4)=-6\). The same observed mean is therefore compatible with different treatment and period effects. Neither effect is separately identified from this schedule.
The assignment table makes the source of the problem visible: treatment A occurs only in period 1 and treatment B only in period 2. A difference attributed to treatment could instead be due, wholly or partly, to the period contrast. Increasing the number of AB participants would estimate the combined contrast \(-\theta+\delta\) more precisely, but would not separate its components.
Imbalance: Identifiable Does Not Mean Precise
A design can contain both sequences and still provide little precision for their contrast. With unequal sequence sizes, the less represented order contributes a less precise estimate of its mean difference. The treatment and period contrasts then depend heavily on information from that smaller group.
Worked Example: Both Sequences, Unequal Precision
Consider hypothetical data with 18 AB participants and 2 BA participants. Suppose the mean differences are \(\bar d_{AB}=-2\) and \(\bar d_{BA}=2\) response units, and the within-sequence sample standard deviation is \(4\) response units in each group. Both sequences are present, so the two contrasts are identifiable under the model.
The point estimates are
For independent participants in the two groups, the estimated standard error of the treatment contrast is
For comparison, if the same total of 20 participants were split equally between sequences, with the same group means’ standard deviations, the standard error would be
Thus the unbalanced schedule still supports a treatment contrast, but its estimated standard error is larger in this comparison. The imbalance does not by itself prove bias or invalidate the point estimate; it limits the precision supplied by the rare sequence. The standard-error calculation assumes independent participants and uses the stated within-sequence standard deviations.
Sequence Effects and Design Diagnostics
Sequence is a between-participant label: each participant follows AB or BA. In a model with a separate fixed effect for every participant, the sequence indicator is exactly determined by those participant indicators. A fixed sequence effect therefore cannot also be estimated separately from fixed participant effects. This is a different issue from whether treatment and period contrasts can be obtained from within-participant differences. Those contrasts use the changing treatment and period assignments within each participant.
A model that treats participant effects as random can estimate a sequence coefficient using between-participant comparisons, but that relies on assumptions about the participant effects and their relation to randomized sequence assignment. It does not make sequence a within-participant contrast. Always state which model is being considered before calling a sequence effect estimable.
A practical diagnostic starts with the assignment table, not a p-value. Count participants in each sequence, then count which treatment appears in each period. In a two-period AB/BA design, both sequences must occur for the two within-participant contrast equations to have full rank. Severe imbalance warns of limited precision. For more complicated schedules, write the model’s design matrix and check whether the columns for the target effects remain linearly independent after nuisance terms are accounted for.
List each sequence, the number assigned to it, and the treatment received in every period.
Specify the treatment, period, sequence, and any interaction terms, with clear contrast directions.
Determine whether the assignment patterns distinguish the target effects; inspect group counts to assess whether the available information is balanced.
Separate a design-based lack of identifiability from uncertainty due to small or imbalanced groups, and assess whether the assumed response model is scientifically credible.
A treatment-by-period interaction or a carryover contribution can change the model columns and the interpretation of the contrasts. Rank checks must use the terms actually included in the model, not just the treatment labels in isolation. In particular, schedule balance cannot establish that carryover is absent, and a rank calculation cannot repair a misspecified model.
Common Pitfalls and Alternatives
| Approach or claim | What it can establish | Important limitation |
|---|---|---|
| Inspecting sequence counts and treatment-by-period counts | Reveals missing or rare assignment patterns and obvious aliasing | Counts alone do not verify model assumptions or quantify all uncertainty |
| Contrasting within-participant differences by sequence | Separates the two additive contrasts when both two-period orders occur | Requires the stated treatment/period model and suitable handling of carryover |
| Fitting a regression model with coded design columns | Provides a general way to assess estimability and incorporate specified terms | A software coefficient is not separately estimable if the design matrix is rank deficient; coding cannot create missing information |
| Using only the more common sequence | May estimate that sequence’s combined mean difference | Cannot separate treatment and period effects from that one contrast |
A common mistake is to see both treatments somewhere in the study and conclude that treatment and period effects must be separable. The relevant question is not whether A and B both occur overall; it is whether the assignment patterns vary in a way that changes treatment while allowing period to be distinguished. If every participant receives A in period 1 and B in period 2, the treatment and period assignments are locked together. Another mistake is to treat a large sample in one sequence as a substitute for the missing sequence. More observations can improve precision for the observed combination, but cannot increase the rank of the design.
Check Your Understanding
Answer each question using the stated contrast definitions: \(\theta\) is B minus A, \(\delta\) is period 1 minus period 2, and \(d_i=Y_{i1}-Y_{i2}\).
- If \(\bar d_{AB}=4\) and \(\bar d_{BA}=-2\) response units, calculate \(\hat{\theta}\) and \(\hat{\delta}\).
- A study contains only BA participants and has mean \(d_i=7\) response units. State the equation supplied by the data and give two distinct \((\theta,\delta)\) pairs that satisfy it.
- Why does a schedule with 12 AB and 12 BA participants identify the two additive contrasts, while a schedule with 24 AB participants does not?
- Why is a fixed sequence effect aliased with fixed participant effects, and does this prevent estimation of within-participant treatment and period contrasts when both sequences occur?
Solutions. (1) \(\hat{\theta}=[-2-4]/2=-3\) response units, and \(\hat{\delta}=[4+(-2)]/2=1\) response unit. (2) BA supplies \(\theta+\delta=7\); for example, \((0,7)\) and \((3,4)\) both satisfy the equation, so the two contrasts are not separately identified. (3) The 12 AB and 12 BA participants provide two distinct equations, with coefficient rows \((-1,1)\) and \((1,1)\), whose determinant is nonzero. The all-AB schedule provides only \(-\theta+\delta\), one equation for two unknown contrasts. (4) Every participant belongs to exactly one sequence, so the sequence indicator is a combination of participant indicators in a fixed-participant model. That prevents a separate fixed sequence coefficient in that model, but does not prevent estimating the treatment and period contrasts from within-participant differences when both assignment orders occur.