Why Carryover Is Not Separately Identifiable in a Two-Period Design

Why Separating the Effects Is a Design Question

A two-period crossover gives each participant two treatment observations, which supports a within-subject comparison. But that advantage does not mean the design can estimate every treatment-related effect independently. In particular, the treatment received in period 1 is also the treatment that could leave a residual effect in period 2. With only two periods and the two sequences AB and BA, the design does not provide independent variation to distinguish the direct treatment contrast from differential first-order carryover.

This is an identifiability issue, not simply a question of sample size or statistical power. If the model contains parameters that the design cannot distinguish, collecting more participants in the same two sequences makes the estimates more precise only for combinations of those parameters. It does not make the separate effects identifiable.

Definition: A parameter is identifiable under a specified model and design if distinct values of that parameter cannot produce exactly the same distribution of the observable data when the other model parameters are allowed to vary. A parameter combination may be identifiable even when its components are not.

The Two-Sequence, Two-Period Model

Let A and B denote the treatments, and let AB and BA denote the assigned sequences. Let \(\tau\) be the direct contrast B minus A. Let \(c_A\) and \(c_B\) be the first-order carryover contributions in period 2 after receiving A and B, respectively. Their differential carryover contrast is \(\kappa=c_A-c_B\). Period 1 has no preceding study treatment, so it has no first-order carryover term.

To focus on what the design can distinguish, consider the expected within-subject changes from period 1 to period 2. A participant's stable baseline contribution cancels in that change. Let \(\lambda\) denote the period 2 minus period 1 effect. The expected changes for the two sequences are:

\begin{aligned} E(\Delta_{AB}) &= \lambda+\tau+c_A,\\ E(\Delta_{BA}) &= \lambda-\tau+c_B. \end{aligned}

For AB, the direct treatment changes from A to B, producing \(+\tau\), and the preceding treatment is A. For BA, the direct treatment changes from B to A, producing \(-\tau\), and the preceding treatment is B. These equations describe the sequence-level expected changes; individual observations also contain subject-to-subject variation and measurement error.

Key result: The usual contrast formed by subtracting the mean change in BA from the mean change in AB and dividing by two has expectation \(\tau+\kappa/2\). Without an assumption or additional design information about \(\kappa\), this contrast does not identify \(\tau\) alone.

Deriving the Confounding

Subtract the expected BA change from the expected AB change. The common period effect cancels, but the differential carryover contribution does not:

\begin{aligned} E(\Delta_{AB})-E(\Delta_{BA}) &=(\lambda+\tau+c_A)-(\lambda-\tau+c_B)\\ &=2\tau+(c_A-c_B)\\ &=2\tau+\kappa. \end{aligned}

Therefore, for the contrast \(T=(\bar{\Delta}_{AB}-\bar{\Delta}_{BA})/2\), where each \(\bar{\Delta}\) is the sample mean change in its sequence,

E(T)=\tau+\frac{\kappa}{2}.

The data may estimate this combined quantity. They do not, from this contrast alone, say how much comes from the direct effect and how much from differential carryover. The same point can be seen directly in the four sequence-by-period cells. Code sequence by \(z\), period by \(p\), and the direct treatment received by \(x\), using values \(+1\) and \(-1\). Define \(q\) as the prior-treatment code in period 2 and 0 in period 1. In the cell order AB period 1, AB period 2, BA period 1, BA period 2, the columns are:

\begin{aligned} z&=(1,1,-1,-1),\\ p&=(-1,1,-1,1),\\ x&=(-1,1,1,-1),\\ q&=(0,1,0,-1). \end{aligned}

For these columns, \(q=(z+x)/2\). Thus, in a cell-mean model containing an intercept, sequence, period, direct treatment, and carryover, the carryover column is an exact linear combination of the sequence and direct-treatment columns. The model matrix cannot have full column rank. If the corresponding coefficients are \(\beta_S,\beta_P,\beta_D,\beta_C\), the observable contributions involving sequence, direct treatment, and carryover reduce to:

\beta_S z+\beta_D x+\beta_C q = \left(\beta_S+\frac{\beta_C}{2}\right)z+ \left(\beta_D+\frac{\beta_C}{2}\right)x.

Different values of the separate coefficients can therefore give the same fitted cell means. This is structural aliasing: the data cannot uniquely resolve the parameters, even if the cell means were known without sampling error.

Worked Calculations

Worked Example: Finding the Identifiable Contrast

Hypothetical data: Suppose the mean period-to-period change is 30 concentration units for sequence AB and \(-20\) concentration units for sequence BA. These are illustrative sequence mean changes, not results from a real study. Compute the usual contrast and identify what it estimates.

The contrast is:

T=\frac{\bar{\Delta}_{AB}-\bar{\Delta}_{BA}}{2} =\frac{30-(-20)}{2} =\frac{50}{2} =25\ \text{concentration units}.

Under the model, the expectation of this contrast is \(\tau+\kappa/2\), not necessarily \(\tau\). Thus 25 concentration units estimates the combined direct-plus-half-differential-carryover contrast. It can be interpreted as the direct contrast only if the model assumption \(\kappa=0\) is justified.

Worked Example: Two Explanations for the Same Changes

Use the same hypothetical changes, 30 for AB and \(-20\) for BA. Show that two different direct and carryover specifications can produce them. Take the period effect to be \(\lambda=5\) concentration units.

One possible specification is \(\tau=20\), \(c_A=5\), and \(c_B=-5\). It gives:

\begin{aligned} E(\Delta_{AB})&=5+20+5=30,\\ E(\Delta_{BA})&=5-20-5=-20. \end{aligned}

A second specification is \(\tau=25\), \(c_A=0\), and \(c_B=0\), with the same \(\lambda=5\):

\begin{aligned} E(\Delta_{AB})&=5+25+0=30,\\ E(\Delta_{BA})&=5-25+0=-20. \end{aligned}

Both specifications reproduce the observed sequence mean changes exactly. The first assigns some of the contrast to differential carryover; the second assigns it all to the direct contrast. The changes alone cannot choose between them.

Worked Example: What an Assumption Changes

Suppose the estimated contrast from hypothetical data is 25 concentration units. If one assumes no differential carryover, then \(\kappa=0\), and the model implies \(\tau=25\) concentration units. If external scientific information instead supported \(\kappa=10\) concentration units, the same contrast would imply:

\tau=T-\frac{\kappa}{2} =25-\frac{10}{2} =20\ \text{concentration units}.

These are different conclusions under different assumptions about carryover. The calculation does not test or establish either assumption. It makes explicit how the direct-effect interpretation depends on information not separated by the two-period contrast.

Assumptions, Diagnostics, and Alternatives

The algebraic result assumes the stated first-order model: a common period effect, direct treatment effects represented by \(\tau\), and carryover in period 2 determined by the treatment given in period 1. The two sequences are the only treatment orders, and the treatment received in each period follows those sequences. The result concerns the model's expected means; it does not require carryover to be present. If carryover is absent, the same algebra applies with \(\kappa=0\).

A practical warning sign is a singular or rank-deficient model matrix when all these effects are included. Statistical software may drop a coefficient, mark it as aliased, or return a non-estimable estimate. A very large standard error can also signal near-dependence in other settings, but exact aliasing is stronger: the separate coefficients are not uniquely determined, so an ordinary finite standard error for each one is not available under that parameterization. Changing software or increasing the number of participants within the same design does not remove the exact dependence.

A useful alternative is to compare analyses under explicit assumptions rather than to claim that the two-period data have separated the effects. For example, a no-carryover model sets the relevant carryover contrast to zero; an externally specified carryover value allows a sensitivity calculation. A first-period-only comparison avoids carryover because no participant has yet received a study treatment in an earlier period. Under randomized sequence allocation, that comparison can estimate a treatment contrast in expectation, but it uses only period 1 and usually gives up the precision benefit of within-subject observations. It also relies on the randomization and analysis being appropriate for that first-period comparison.

These approaches answer different questions and rely on different information. A first-period comparison does not disentangle direct and carryover effects using both periods; it avoids the carryover term by restricting the data used. A no-carryover analysis does use both periods, but its direct-effect interpretation follows from the assumption, not from an independent estimate of carryover. Designs with additional treatment-period information can alter the identifiability problem, but their properties require a separate design analysis.

Common Pitfall and Practical Meaning

A common mistake is to fit a model with both direct treatment and carryover terms, see a non-significant carryover test, and then report the treatment coefficient as though carryover had been ruled out. The correction is that the two-period design itself cannot independently estimate the direct and differential carryover effects under this model. With exact aliasing, the separate model terms may not even have unique estimates to test. A non-significant result from some other carryover assessment is not proof that carryover is absent.

The distinction matters because the treatment contrast can be interpreted too strongly. If differential carryover is plausible, the estimate from the two-period crossover is not automatically a pure direct treatment contrast. The design remains useful for its within-subject comparison, but the meaning of that comparison depends on assumptions about residual treatment effects. State those assumptions and their scientific basis, and do not present an unidentifiable component as though the data estimated it separately.

Key takeaway: In a two-sequence, two-period crossover, the treatment contrast has expectation \(\tau+\kappa/2\). The direct contrast \(\tau\) and differential carryover \(\kappa\) are not independently identifiable without an additional assumption or information supplied by another source or design.

Check Your Understanding

Answer each question, then compare with the solutions below.

If the mean changes are 18 units for AB and \(-6\) units for BA, what is the usual contrast, and what parameter combination does it estimate?
Why does adding more participants in the same two sequences not resolve exact aliasing?
Under what assumption can the usual contrast be interpreted as the direct treatment contrast?
What is one advantage and one limitation of a first-period-only comparison?

Solutions. (1) The contrast is \([18-(-6)]/2=12\) units, estimating \(\tau+\kappa/2\). (2) The carryover design column remains an exact linear combination of the sequence and direct-treatment columns, regardless of sample size; the model matrix remains rank deficient. (3) The interpretation as \(\tau\) requires \(\kappa=0\), or other specified information that determines \(\kappa\). (4) It avoids first-order carryover because it uses only period 1, but it discards period 2 and therefore does not use the within-subject comparison available from the full crossover.