Representing First-Order Carryover

Why Represent Carryover Explicitly?

A crossover response in a given period can reflect more than the treatment administered in that period. The response may also depend on the treatment received immediately beforehand. A model that includes only current treatment can therefore attribute a history-related difference to the wrong component, such as a period effect or a direct treatment effect. Representing treatment history explicitly makes the scientific assumption visible: the response depends on the current treatment and, when applicable, the treatment in the preceding period.

Here, “first-order” refers to the length of treatment history included in the model. It means that the preceding period’s treatment is represented; treatments received earlier than that are not included as separate predictors. This is a model specification, not a claim that any residual influence must actually stop after one period.

Definition: A direct treatment effect is the effect associated with the treatment administered in the current period. A first-order carryover effect is the effect associated with the treatment administered in the immediately preceding period. In period 1 there is no preceding study treatment, so the first-order carryover term is set to zero.

An Additive First-Order Model

Suppose participant $i$ has response $Y_{ij}$ in period $j$, and let $d_{ij}$ denote the treatment administered in that period. Let $q_{ij}$ denote the treatment administered in period $j-1$, when that period exists. For two treatments, label them T and R. Use R as the reference treatment for both direct and carryover effects, and use period 1 as the reference period.

Model: For participant $i$ in period $j$, $$ Y_{ij}=\mu+b_i+\pi_j+\tau_{d_{ij}}+I(j>1)\lambda_{q_{ij}}+\varepsilon_{ij}, $$ where $\pi_1=0$, $\tau_R=0$, and $\lambda_R=0$. The indicator $I(j>1)$ equals 1 after period 1 and 0 in period 1. The parameters $\tau_T$ and $\lambda_T$ are, respectively, the direct effect of T relative to R and the carryover effect of preceding T relative to preceding R. The parameter $\mu$ is the mean for reference treatment R in period 1, before adding a participant deviation.

The model includes a participant-specific random intercept $b_i$, with $E[b_i]=0$ and variance $\sigma_b^2$. The residual $\varepsilon_{ij}$ has mean zero and variance $\sigma^2$; it is independent of $b_i$ and, for the usual simple random-intercept model, residuals are independent across periods and participants. A common likelihood-based analysis additionally assumes normally distributed random effects and residuals. Treatment sequence must be randomized or otherwise independent of participant effects, so the mean of $b_i$ is zero within each sequence in the population.

The mean structure is additive: the direct effect of a treatment does not vary by period, and the effect of a preceding treatment does not vary according to the current treatment. Only the immediately preceding study treatment contributes a carryover term. These assumptions allow a compact model, but they are substantive. If they are implausible, the fitted carryover coefficient may not summarize the history dependence adequately.

Reading the Model by Treatment History

For a two-period, two-sequence design, the two sequences are TR and RT. Taking expectations over the random intercept and residual gives the following sequence-period means:

Sequence and period	Current treatment	Preceding treatment	Expected response
TR, period 1	T	None	$\mu+\tau_T$
RT, period 1	R	None	$\mu$
TR, period 2	R	T	$\mu+\pi_2+\lambda_T$
RT, period 2	T	R	$\mu+\pi_2+\tau_T$

These expressions show why it matters to keep the two treatment roles distinct. In TR period 2, R is the current treatment, while T is the preceding treatment. The mean contains $\lambda_T$, not $\tau_T$. In RT period 2, T is current and the preceding treatment is R, the carryover reference, so the mean contains $\tau_T$ but no nonzero carryover term.

Worked Example: Translating a Treatment Sequence Into Model Terms

Consider a hypothetical participant assigned to sequence T-R-T over three periods. For illustration, let $\mu=100$ response units, $\tau_T=6$ response units, $\pi_2=3$ response units, $\pi_3=5$ response units, and $\lambda_T=2$ response units. Set the participant deviation and residual to zero to calculate the model’s expected responses.

Period 1. T is current and there is no preceding study treatment, so $100+6=106$ response units.

Period 2. R is current and T immediately preceded it, so $100+3+0+2=105$ response units. The zero is $\tau_R$.

Period 3. T is current and R immediately preceded it. Since $\lambda_R=0$, the expected response is $100+5+6+0=111$ response units.

The first-order term in period 3 refers to R in period 2, not T in period 1. This example illustrates how the model encodes the immediately preceding treatment; it does not establish that earlier treatment has no biological influence.

What the Two-Sequence Design Identifies Under This Model

The period-1 means differ by $\tau_T$: the mean in TR is $\mu+\tau_T$, while the mean in RT is $\mu$. Thus, under the stated random-intercept model, with sequence independent of participant effects and no fixed sequence term, the period-1 between-sequence contrast identifies the direct contrast $\tau_T$. This conclusion relies on the random-intercept assumption. It is not a general claim that direct and carryover effects are always separately identifiable in a two-period design.

In period 2, the TR mean identifies the sum $\pi_2+\lambda_T$, while the RT mean identifies $\pi_2+\tau_T+\lambda_R$. With $\tau_T$ identified from period 1, their difference also identifies the difference between carryover effects. But the period effect and the common level of the two carryover effects cannot all be recovered separately. The reason is an invariance in the mean equations.

Write the period-2 means.
For TR the mean is $\mu+\pi_2+\lambda_T$; for RT it is $\mu+\pi_2+\tau_T+\lambda_R$.

Apply a common shift.
For any constant $a$, replace $\pi_2$ by $\pi_2+a$, $\lambda_T$ by $\lambda_T-a$, and $\lambda_R$ by $\lambda_R-a$.

Check what changes.
The TR combination becomes $(\pi_2+a)+(\lambda_T-a)=\pi_2+\lambda_T$. The RT combination becomes $(\pi_2+a)+\tau_T+(\lambda_R-a)=\pi_2+\tau_T+\lambda_R$. Neither period-2 mean changes.

State the consequence.
The data cannot distinguish all three quantities $\pi_2,\lambda_T,\lambda_R$ without an additional constraint or information. The period-1 contrast still identifies $\tau_T$ under the random-intercept assumptions.

For example, the difference between period-2 means is $(\mu+\pi_2+\lambda_T)-(\mu+\pi_2+\tau_T+\lambda_R)=\lambda_T-\lambda_R-\tau_T$. Once $\tau_T$ is identified, this contrast supplies information about $\lambda_T-\lambda_R$, not their shared level. In the chosen reference coding $\lambda_R=0$, the model assigns that shared level by convention; the period-2 means alone do not estimate an unrestricted period effect and both unrestricted carryover effects.

If instead participant-specific fixed intercepts are included, each participant’s intercept is an unrestricted parameter. The between-participant period-1 contrast between TR and RT can no longer be used to identify $\tau_T$ independently of those intercepts. In that formulation, the usual concern about separating direct and carryover effects in the two-period, two-sequence design applies. Therefore, any identifiability statement must specify whether participant effects are modeled as random with the stated mean-zero assumption or as unrestricted fixed effects.

Worked Example: Interpreting Hypothetical Sequence-Period Means

Suppose a hypothetical randomized study has the following observed cell means, in response units: TR period 1, 12.0; RT period 1, 10.0; TR period 2, 13.5; and RT period 2, 12.8. Treat these as illustrative sample means, not as results from a real study. Assume the additive random-intercept mean model is appropriate and that the randomized groups have the same population mean participant effect.

Period-1 contrast. The estimate of the model’s direct contrast is $12.0-10.0=2.0$ response units, because the first-period means are $\mu+\tau_T$ and $\mu$. The estimated reference mean is $\mu=10.0$ response units.

Period-2 combinations. The TR mean gives $\pi_2+\lambda_T=13.5-10.0=3.5$ response units. The RT mean gives $\pi_2+\tau_T=12.8-10.0=2.8$ response units, since $\lambda_R=0$. Subtracting the estimated direct contrast gives $\pi_2=2.8-2.0=0.8$ response units under this reference coding. Then $\lambda_T=3.5-0.8=2.7$ response units.

This calculation uses the constraint $\lambda_R=0$ and the period-1 estimate of $\tau_T$. It is not a demonstration that an unrestricted period effect and two unrestricted carryover levels are separately identified: the reference constraint supplies the carryover origin. A different parameterization can shift the common level between the period and carryover terms without changing fitted period-2 means.

Assumptions, Checks, and Alternatives

Before interpreting estimates, check that the model’s structure corresponds to the design and scientific question. Treatment labels and their order must be recorded correctly; the period-1 carryover term must be zero; and the lag in every later period must refer to the immediately preceding treatment. The response should be analyzed on a scale for which an additive mean model is meaningful. If the response is log-transformed, for example, the additive effects are defined on the log scale, not directly on the original scale.

Residual and influence checks can reveal substantial departures from the assumed error structure, such as markedly different variability by period or unusually influential observations. Sequence-specific patterns in residuals or systematic discrepancies between observed and fitted sequence-period means can indicate that a simple additive model is inadequate. Such checks do not prove that the carryover structure is correct: a small study may have little ability to reveal a misspecified history effect. Estimates can also be unstable when treatment history is poorly represented or when the model includes terms that the design cannot distinguish.

The main alternative is a model with current treatment and period but no carryover term. It is simpler, but assumes preceding treatment contributes nothing to the mean response; if that assumption fails, other estimated terms may absorb the resulting pattern. A more flexible sequence-period means model describes observed cells without imposing the same additive treatment-history structure, but it may not provide a unique decomposition into direct, period, and carryover components. A model with participant-specific fixed intercepts addresses participant-level heterogeneity differently, but in a two-period, two-sequence design it does not use the period-1 between-sequence contrast to identify the direct treatment contrast. These are not interchangeable parameterizations: their assumptions determine what their coefficients mean and what can be estimated.

The first-order model is limited by its additive form and by its restriction to the immediately preceding treatment. It does not itself establish a biological mechanism, determine how long an influence persists, or guarantee that a carryover term is estimable from a particular design. The model is useful when it makes a plausible, explicit representation of response history; it is not a substitute for evaluating whether the design and assumptions support the intended interpretation.

Common Mistake

Common mistake: “A two-period crossover cannot identify the direct treatment contrast if carryover is included.” That statement is too broad. Under the random-intercept model specified here, the period-1 between-sequence contrast identifies $\tau_T$, provided sequence assignment is independent of participant effects and no fixed sequence term removes that comparison. The period effect and unrestricted carryover levels remain confounded in period 2. With participant-specific fixed intercepts, the period-1 between-participant contrast is unavailable, so the identifiability conclusion changes.

Check Your Understanding

Use the reference coding in the model above. The answers are included so you can check both the model terms and the identifiability reasoning.

In sequence R-T, which treatment contributes the period-2 carryover term?
If $\mu=50$, $\tau_T=4$, $\pi_2=2$, and $\lambda_T=3$ response units, what is the expected period-2 response for sequence T-R?
Which contrast identifies $\tau_T$ in the stated random-intercept model, and what assumption is essential to that interpretation?
Explain why shifting $\pi_2$ upward by $a$ and both carryover effects downward by $a$ leaves the period-2 means unchanged.
Why does the first-order term for period 3 in sequence T-R-T refer to R rather than T?

Solutions. (1) R is the preceding treatment, so its carryover term is $\lambda_R=0$ under the reference coding. (2) In T-R, period 2 has current R and preceding T, so the expected response is $50+2+0+3=55$ response units. (3) The period-1 contrast, mean(TR, period 1) minus mean(RT, period 1), identifies $\tau_T$. This relies on the random-intercept model, including mean-zero participant effects within sequence in the population and sequence assignment independent of those effects. (4) Each period-2 mean contains the sum of the period effect and the relevant carryover effect; adding $a$ to one and subtracting $a$ from the other preserves each sum. (5) First-order carryover uses only the immediately preceding period, which is period 2 and contains R.

Sequence and period	Current treatment	Preceding treatment	Expected response
TR, period 1	T	None	\(\mu+\tau_T\)
RT, period 1	R	None	\(\mu\)
TR, period 2	R	T	\(\mu+\pi_2+\lambda_T\)
RT, period 2	T	R	\(\mu+\pi_2+\tau_T\)