Why Replicate Crossover Data Need a Mixed Model
In a replicate crossover study, each participant receives the test formulation, the reference formulation, or both more than once. A participant’s observations are therefore not independent: measurements from the same person can share stable characteristics, and measurements close in time may also be correlated. Treating every administration as an independent observation ignores the study’s repeated-measures structure.
A mixed-effects model represents that structure by combining fixed effects for the study design with random effects for participant-level variation. On the log scale, it can describe systematic differences among treatments and periods while accounting for repeated measurements within a participant. The focus here is the model specification and what its components mean, rather than how to estimate each variability component.
A Model for Repeated Administrations
Let \(y_{ij}\) be the log-transformed PK endpoint for participant \(i\) in period \(j\), such as log AUC or log Cmax. Let \(g_i\) denote that participant’s assigned sequence and \(d_{ij}\) the formulation received in that period. For two formulations, denote test by \(T\) and reference by \(R\). A basic random-intercept model is
Here, \(\mu\) is an overall mean; \(\alpha_{g_i}\) is the fixed sequence effect; \(\pi_j\) is the fixed period effect; and \(\tau_{d_{ij}}\) is the fixed treatment effect. The random effect \(b_{i(g_i)}\) is the effect of participant \(i\), who is nested within sequence \(g_i\). The term \(\varepsilon_{ij}\) is the period-specific within-participant error. One possible parameterization sets \(\tau_R=0\), so that \(\tau_T\) is the test-minus-reference contrast; another uses a constraint on the two treatment effects and takes their difference.
Sequence is a participant-level factor: an individual remains in the sequence assigned at the start, even though the formulation changes across periods. In the usual crossover setup, sequence is modeled as fixed and participant as random within sequence. Period is fixed because the periods are the specific ordered occasions in the design. Treatment is fixed because the comparison between the named formulations is the target.
The random-intercept version makes a specific covariance assumption. For example, assume \(b_{i(g_i)}\sim N(0,\sigma_b^2)\), \(\varepsilon_{ij}\sim N(0,\sigma_e^2)\), and that all participant effects and errors are mutually independent. Then observations from different participants are independent, and the covariance matrix for one participant’s \(m\) observations is
Thus, each observation has variance \(\sigma_b^2+\sigma_e^2\), while two distinct observations from the same participant have covariance \(\sigma_b^2\). This is a simple shared-intercept structure, not a universal description of repeated PK data. If the design or observed dependence calls for a more general within-participant covariance, the residual covariance can instead be represented by a matrix \(R_i\), with \(\operatorname{Var}(\mathbf y_i)=\sigma_b^2\mathbf 1_m\mathbf 1_m^\mathsf{T}+R_i\). The choice must match the design and be supported by the data.
Assumptions and Interpretation
The basic model is appropriate only under stated assumptions. It assumes that the expected log outcome is described by the included sequence, period, and treatment terms; that the treatment contrast is the same across periods and sequences; and that, conditional on the random effect and fixed effects, the residuals have mean zero. The normal random-effect and residual assumptions specify the probability model used for likelihood-based inference. The simple covariance version further assumes a common random-intercept variance and independent residual errors with a common variance.
As in crossover analysis generally, the model also relies on the design and conduct supporting a direct treatment comparison. In particular, it does not automatically account for carryover or an unmodeled treatment-by-period interaction. These are not solved merely by including a random participant effect. The treatment contrast is \(\tau_T-\tau_R\), and on the original scale its exponential is the ratio of geometric means associated with the two formulations, under the log-scale model.
A short derivation shows why the participant effect matters. For two observations from the same participant, write the model terms other than the participant effect as \(q_{ij}\), so \(y_{ij}=q_{ij}+b_i+\varepsilon_{ij}\). Under independent residuals,
The common participant term induces dependence even when the period-specific errors are independent. Omitting that term, or otherwise failing to model within-participant dependence, can give standard errors and tests that do not reflect the actual design.
Worked Example: Writing the Model for a Four-Period Replicate Design
Suppose a hypothetical study uses the sequences TRTR and RTRT, with four periods per participant. A log-transformed endpoint is recorded after every administration. For a participant in TRTR, the treatment labels by period are \(T,R,T,R\); for a participant in RTRT, they are \(R,T,R,T\).
For either sequence, use \(y_{ij}=\mu+\alpha_{g_i}+\pi_j+\tau_{d_{ij}}+b_{i(g_i)}+\varepsilon_{ij}\), for \(j=1,\ldots,4\). The same participant effect appears in all four observations, while the period effect and treatment effect depend on the observation’s period and formulation. For example, the period-2 observation for a TRTR participant has expected fixed part \(\mu+\alpha_{\mathrm{TRTR}}+\pi_2+\tau_R\), in addition to that participant’s random effect.
This indexing makes the repetition explicit: \(d_{i1}=d_{i3}=T\) and \(d_{i2}=d_{i4}=R\) for the TRTR sequence, with the labels reversed for RTRT. The model estimates a common treatment contrast while accounting for the participant’s repeated outcomes. It does not treat the four measurements as four independent participants.
A necessary design check is that treatment is not perfectly confounded with period across the study. Here both sequences occur, so each formulation is represented in each period. The assumed common treatment effect is also a modeling condition: if treatment effects differ materially by period, this model’s single contrast does not describe that pattern.
Worked Example: A Treatment Contrast From Hypothetical Replicate Data
Consider four hypothetical participants in the same four-period design. The entries below are log-scale outcomes, in natural-log units.
| Participant | Sequence | Period 1 | Period 2 | Period 3 | Period 4 |
|---|---|---|---|---|---|
| 1 | TRTR | 4.8 (T) | 4.7 (R) | 4.9 (T) | 4.8 (R) |
| 2 | TRTR | 5.1 (T) | 4.9 (R) | 5.0 (T) | 4.8 (R) |
| 3 | RTRT | 4.9 (R) | 5.0 (T) | 4.8 (R) | 5.1 (T) |
| 4 | RTRT | 4.7 (R) | 4.9 (T) | 4.8 (R) | 5.0 (T) |
For this balanced illustration, first average each participant’s two observations under each formulation and take the test-minus-reference difference. Participant 1 has \((4.8+4.9)/2-(4.7+4.8)/2=4.85-4.75=0.10\) log units. The corresponding differences for participants 2, 3, and 4 are \(0.20\), \(0.20\), and \(0.20\) log units.
The average of these four differences is
Exponentiating gives \(\exp(0.175)=1.191\), so the illustrative test-to-reference geometric mean ratio is about 1.19. This calculation illustrates the meaning of the contrast; it is not a substitute for fitting the specified mixed model in general. When data are unbalanced or have missing observations, the model uses the full design and its covariance assumptions rather than relying on this simple averaging calculation.
Worked Example: The Covariance Implied by a Random Intercept
Suppose, hypothetically, the fitted random-intercept model has \(\sigma_b^2=0.0400\) and \(\sigma_e^2=0.0100\), in squared log units. The variance of one observation is \(0.0400+0.0100=0.0500\). The covariance between two different observations from the same participant is \(0.0400\).
Under this model, their correlation is
The calculation shows what the assumed covariance structure means: repeated outcomes from one participant share a common component, so they are correlated. It does not establish that this particular structure is adequate for a real data set; that requires checking the design and model fit.
Fitting, Checks, and Alternatives
Before interpreting a fitted treatment contrast, verify that the data are coded at the administration level, that each observation is linked to the correct participant, sequence, period, and formulation, and that the design contains enough information to estimate the intended effects. Confirm that the fixed-effect design is identifiable under the chosen constraints. A missing administration should not be silently treated as an observed value or as a new participant.
After fitting, check whether the model converged and whether variance estimates and standard errors are plausible. Inspect residuals against fitted values and period, and look for patterns within participants that the assumed covariance does not capture. A nearly zero estimated random-intercept variance or convergence warnings deserve attention, but neither alone proves the model is invalid. These checks assess the model; they do not justify automatically removing observations or changing the treatment contrast.
The main alternative is an analysis that first forms a participant-level summary or contrast and then analyzes those summaries. In a balanced, complete design, such an approach can be a useful transparent comparison. It may be less flexible when observations are missing or the design is unbalanced, and it does not by itself specify how all repeated observations are correlated. Treating every administration as independent is not a sound alternative when within-participant dependence is present. A mixed model keeps the observation-level design in view and states the covariance assumptions directly.
Worked Example: Interpreting a Model-Based Contrast
Suppose a hypothetical mixed-model fit estimates the test-minus-reference contrast as \(0.0800\) log units with standard error \(0.0300\) log units. For illustration, use a large-sample normal Wald interval. The test statistic is \(0.0800/0.0300=2.667\), and the approximate two-sided p-value is \(0.0077\).
The approximate 95% interval on the log scale is
Back-transforming the estimate and limits gives \(\exp(0.0800)=1.083\), and \((\exp(0.0212),\exp(0.1388))=(1.021,1.149)\). Thus, under this illustrative model, the estimated test-to-reference geometric mean ratio is 1.083, with an approximate 95% interval from 1.021 to 1.149. The interval describes uncertainty in the model-based ratio; it is not a 95% probability statement about a fixed parameter. This example uses a normal approximation. In an actual analysis, the specified degrees-of-freedom and inference method must be used.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In the model, which term represents the participant’s repeated contribution across periods?
- Under the random-intercept assumptions, what is the covariance between two different outcomes from the same participant?
- Why is period included as a fixed effect in the basic model?
- A model estimates a test-minus-reference contrast of \(-0.0500\) log units. What is the corresponding test-to-reference ratio, to three significant figures?
Solutions. (1) The random participant effect \(b_{i(g_i)}\) is shared across that participant’s periods. (2) The covariance is \(\sigma_b^2\), because the shared random effect is the only common random term when residuals are independent. (3) Period represents systematic differences among the specific ordered occasions, such as common time-related changes, rather than treating period-to-period differences as independent random sampling. (4) The ratio is \(\exp(-0.0500)=0.951\). It is below 1, indicating a lower estimated geometric mean for test than reference under the stated contrast definition.