Modeling Period And Sequence Effects In Crossover Studies

Why Period and Sequence Need Separate Terms

In a crossover study, each participant receives more than one formulation, but the administrations occur at different times. Concentrations can therefore differ because of the formulation, because the study period differs, or because participants assigned to different treatment orders differ systematically. A model that does not distinguish these sources may attribute a period or sequence pattern to formulation.

This tutorial extends a nonlinear mixed-effects (NLME) model for concentration-time data with period and sequence terms. The central distinction is structural: formulation varies within a participant, period varies within a participant, and sequence is assigned between participants. These terms can all affect the model’s population predictions, but they do not have the same interpretation.

Definition: A period effect is a systematic difference associated with the administration period, shared across formulations after accounting for the model. A sequence effect is a systematic difference between groups assigned different formulation orders. A formulation effect is the contrast associated with receiving one formulation rather than another, holding the modeled period and participant effects fixed.

For clarity, the examples use a two-period, two-sequence design. The sequences are TR (test in period 1, reference in period 2) and RT (reference in period 1, test in period 2). The same modeling principles apply to more complex crossover schedules, although their design matrices and estimable contrasts must be checked for that schedule.

Putting the Effects Into an NLME Model

Concentrations are modeled through a pharmacokinetic structural function, not by adding a period or sequence constant directly to every concentration. Let \(C_{ijpm}\) denote concentration for participant \(i\), sequence \(j\), period \(p\), and sample \(m\), collected at time \(t_{ijpm}\). A general model is

C_{ijpm}=f(t_{ijpm};\theta_{ijp})+\varepsilon_{ijpm}.

Here \(f\) is the selected pharmacokinetic model, \(\theta_{ijp}\) is the vector of parameters for that participant and period, and \(\varepsilon_{ijpm}\) is the residual error. A period or sequence effect can be assigned to a scientifically relevant parameter on a suitable scale. For example, if \(F\) is a positive bioavailability parameter, one possible specification is

\log(F_{ijp})=\beta_{F,0}+\beta_{F,T}I(T_{ijp})+\beta_{F,P}I(p=2)+\beta_{F,S}I(j=TR)+b_{F,i}.

In this coding, reference formulation in period 1 and RT sequence define the reference levels. \(I(\cdot)\) is an indicator equal to 1 when its condition is true and 0 otherwise. The coefficient \(\beta_{F,T}\) is the log-scale formulation contrast, \(\beta_{F,P}\) is the period-2 contrast, and \(\beta_{F,S}\) is the difference associated with assignment to TR rather than RT. The participant effect \(b_{F,i}\) represents participant-level variation; it is typically modeled as a mean-zero random effect. Effects can instead be assigned to other parameters, such as an absorption parameter, when justified by the scientific question and data.

The model assumes that the structural function is suitable for the concentration profiles, that the chosen parameter effects are identifiable under the design, and that the random-effects and residual-error distributions are adequately specified. The sequence coefficient describes a systematic group difference conditional on the model; random assignment does not guarantee that two finite sequence groups have identical participant characteristics. It is not itself a within-participant formulation contrast.

Before interpreting the terms, check the parameterization. A formulation effect on \(\log(F)\), for example, is a multiplicative effect on \(F\), not an additive concentration shift. A common period effect may be plausible for one parameter but not another. Adding period and sequence terms indiscriminately can consume information without improving the scientific model.

How the Two-Period Design Separates the Effects

A useful justification comes from a simpler model for a participant’s log exposure, such as log AUC. It makes the design contrasts visible, while the fitted NLME model can use all concentration observations to estimate its corresponding parameter effects. Consider the fixed-effects model

Y=\mu+\tau I(T)+\pi I(p=2)+\sigma I(j=TR)+b_i+e,

where \(\tau\) is the test-versus-reference formulation effect, \(\pi\) is the period-2 effect, \(\sigma\) is the TR-versus-RT sequence effect, and \(b_i\) is a participant random effect. Assume that the random effects have mean zero and the error has mean zero conditional on the modeled effects. For the fixed-effect cell means, the four expected outcomes are:

\begin{aligned} E(Y\mid TR,p=1,T)&=\mu+\tau+\sigma,\\ E(Y\mid TR,p=2,R)&=\mu+\pi+\sigma,\\ E(Y\mid RT,p=1,R)&=\mu,\\ E(Y\mid RT,p=2,T)&=\mu+\tau+\pi. \end{aligned}

Let \(d_{TR}\) be the TR sequence’s period-1 test mean minus its period-2 reference mean, and let \(d_{RT}\) be the RT sequence’s period-2 test mean minus its period-1 reference mean. Subtracting the relevant cell means gives

d_{TR}=\tau-\pi,\qquad d_{RT}=\tau+\pi.

Adding the equations and dividing by 2 cancels the period effect, while subtracting them and dividing by 2 cancels the formulation effect:

\frac{d_{TR}+d_{RT}}{2}=\tau,\qquad \frac{d_{RT}-d_{TR}}{2}=\pi.

This derivation shows why the balanced two-sequence design can separate average formulation and period effects under the stated model. Sequence is estimated from the between-participant difference between sequence-group means, with the modeled formulation and period contributions accounted for. In an NLME fit, the estimation is based on the likelihood for the concentration data and the specified random-effects structure, rather than simply treating every concentration as an independent exposure measurement.

Worked Example: Separating Formulation and Period Effects

Suppose hypothetical balanced cell means for log AUC are as follows: TR test in period 1, 4.760; TR reference in period 2, 4.700; RT reference in period 1, 4.600; and RT test in period 2, 4.740. Treat these as illustrative model-based cell means, not observations from a real study.

For TR, the within-sequence contrast is \(d_{TR}=4.760-4.700=0.060\). This equals \(\tau-\pi\): the formulation effect is partly offset by the period-2 effect.

For RT, the contrast is \(d_{RT}=4.740-4.600=0.140\). This equals \(\tau+\pi\). Averaging the two contrasts gives the formulation effect: \[ \hat{\tau}=(0.060+0.140)/2=0.100. \] The estimated test-to-reference geometric mean ratio on the log-AUC scale is \(\exp(0.100)=1.105\), or about 1.105.

The period effect is \[ \hat{\pi}=(0.140-0.060)/2=0.040. \] Thus the period-2 shift is 0.040 log units under this model. The two raw within-sequence differences are not interchangeable estimates of formulation effect: one is \(\tau-\pi\) and the other is \(\tau+\pi\).

As a check on the sequence term, the average of the TR cell means is \((4.760+4.700)/2=4.730\); the RT average is \((4.600+4.740)/2=4.670\). Their difference is 0.060, consistent with \(\sigma=0.060\) in the stated parameterization. This group-level difference is not the 0.100 formulation contrast.

Interpreting Effects on a Pharmacokinetic Parameter

An effect on a model parameter should be translated back to that parameter’s scale. For a log-linked positive parameter, exponentiating a coefficient gives a multiplicative ratio. This is not necessarily the same as taking a ratio of concentrations at every time: the structural model determines how a parameter change propagates through the profile and derived exposure.

Worked Example: Applying Effects to Bioavailability

Consider a hypothetical model with reference bioavailability \(F=0.700\) for the reference formulation in period 1 and RT sequence. Suppose the fitted log-scale coefficients are \(\beta_{F,T}=0.100\), \(\beta_{F,P}=0.040\), and \(\beta_{F,S}=0.060\). For a participant in the RT sequence receiving test in period 2, suppose the participant random effect is \(b_{F,i}=0.020\).

The RT sequence indicator is 0, and both the test and period-2 indicators are 1. Relative to the reference value, the modeled log shift is \[ 0.100+0.040+0.020=0.160. \] Therefore, \[ F=0.700\exp(0.160)=0.700(1.1735)=0.8215 \] to four significant figures.

For this same participant and period, the modeled reference-formulation value would be \(0.700\exp(0.040+0.020)=0.7433\). The test-to-reference ratio is \[ 0.8215/0.7433\approx\exp(0.100)=1.105. \] The period and participant terms cancel from this within-participant, same-period contrast. This interpretation depends on the model’s log-link specification and on the formulation contrast being represented by the stated coefficient.

The sequence coefficient does not enter this participant’s calculation because RT is the reference sequence. For a TR participant, it would shift the modeled parameter by \(\exp(0.060)\), conditional on the other terms. Such a shift describes the fitted between-sequence difference, not a formulation effect.

Assumptions, Diagnostics, and Failure Modes

The terms are interpretable only if the model and design support them. At minimum, the analysis should check the following:

Correct treatment and period records. Each profile must be linked to the formulation actually administered, its period, and the participant’s assigned sequence. A coding error can turn a formulation contrast into a misleading period or sequence pattern.
Estimability and overlap. The design must contain the relevant sequence-period-formulation combinations. Severe imbalance, missing periods, or a small number of participants in one sequence can make fixed effects imprecise or confounded in practice.
Parameter-scale plausibility. Inspect whether the estimated effects imply plausible parameter values and profiles. A large coefficient can signal a coding problem, an inadequate structural model, or weak information about that parameter.
Profile and residual fit. Plot observed and predicted concentrations by formulation, period, sequence, and time. Systematic residual patterns by period can indicate a missing period effect or a structural-model problem. A period coefficient cannot repair a wrongly specified absorption or disposition model.
Random-effects behavior. Check whether participant-level variation and the fixed sequence effect are distinguishable with the available participants. A sequence estimate based on few participants can be unstable even when there are many concentration samples per participant.

A sequence term is not a substitute for a carryover model. It represents an average difference between assignment groups; it does not identify which prior formulation caused a later-period change. In a conventional two-period crossover, sequence, period, treatment, and potential carryover patterns require careful design-based interpretation. If a carryover mechanism is scientifically plausible, the analyst should not claim that a sequence coefficient alone has resolved it.

A common mistake is to see a larger mean in the TR group and conclude that test is higher than reference. That comparison mixes sequence assignment, period, and formulation. The correction is to define the formulation contrast using the design and model: in the balanced example, average the two within-sequence contrasts so that the period contribution cancels. Another mistake is to call a sequence difference a formulation effect because both involve treatment order. Sequence is between participants; formulation is compared within participants across administrations.

Choosing the Modeling Level

Period and sequence terms can be included in an NLME model for concentration profiles, or a model can first estimate subject-period exposure summaries and then analyze those summaries. The direct NLME approach can use the concentration-time observations and propagate parameter effects through the structural model. A summary-based approach is easier to inspect but may discard profile information and requires its own assumptions about the summaries and their variability.

Neither approach makes a poorly supported effect identifiable. A more complex parameter model is not automatically better: compare a prespecified, scientifically interpretable model with alternatives using fit, residual behavior, parameter precision, and plausibility. Do not add sequence or period effects merely because a numerical estimate is available. Conversely, omitting a plausible period effect can distort the formulation contrast when period is associated with which formulation is administered.

Key takeaway: Model formulation as the within-participant product contrast, period as the administration-period contrast, and sequence as the between-participant assignment-group contrast. Keep their scales and interpretations explicit, and verify that the design and concentration data support each term.

Worked Example: Interpreting a Sequence Difference

Suppose a hypothetical fitted model reports adjusted mean log exposure of 4.720 for participants assigned to TR and 4.680 for those assigned to RT, with the model accounting for formulation and period. The sequence coefficient for TR relative to RT is \[ \hat{\sigma}=4.720-4.680=0.040. \] Exponentiating gives \(\exp(0.040)=1.041\). Under this model, the adjusted geometric mean exposure for the TR group is about 1.041 times that for the RT group, a between-sequence difference of approximately 4.1%.

This estimate does not show that test is 4.1% higher than reference. It compares assignment groups after the modeled adjustments. It may reflect chance baseline imbalance, unmodeled participant differences, or other model inadequacy; its interpretation depends on the design and assumptions. The formulation effect must be read from the formulation coefficient or corresponding within-participant contrast, not from this sequence comparison.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In the model \(Y=\mu+\tau I(T)+\pi I(p=2)+\sigma I(j=TR)+b_i+e\), what is the interpretation of \(\pi\)?
Suppose \(d_{TR}=0.030\) and \(d_{RT}=0.110\). Calculate the formulation and period effects under the balanced model.
Why is a sequence coefficient not a test-versus-reference formulation contrast?
A fitted coefficient on log bioavailability for test versus reference is 0.080. What multiplicative ratio does it imply, and what does the coefficient not imply about concentration at every time?
What can a sequence term establish about carryover in a two-period crossover?

Solutions. (1) \(\pi\) is the period-2 versus period-1 shift, holding formulation, sequence, and participant effect fixed. (2) \(\hat{\tau}=(0.030+0.110)/2=0.070\), and \(\hat{\pi}=(0.110-0.030)/2=0.040\). These are log-scale effects under the stated model. (3) Sequence is assigned between participants, whereas formulation is compared within participants; the sequence contrast reflects group assignment after modeled adjustments. (4) The ratio is \(\exp(0.080)=1.083\) to four significant figures. It implies an approximately 1.083-fold effect on bioavailability under the model, not a constant 1.083-fold concentration ratio at every time; the structural model determines the profile change. (5) A sequence term estimates an adjusted difference between order groups. By itself it does not identify a carryover mechanism or establish that carryover is absent.