Why Occasion Structure Matters
Repeated concentration profiles can differ for several reasons. A participant may have a persistent tendency toward higher exposure, their exposure may vary from one dosing occasion to another, and participants measured on the same study occasion may all be affected by a common condition. Those sources of variation imply different patterns of dependence in the data. A model that assigns variation to the wrong source can give misleading estimates of both uncertainty and reproducibility.
Here the comparison is between independent subject-by-occasion effects, which vary separately for each participant and occasion, and a shared occasion effect, which is common to all participants observed on that study occasion. The latter could represent a shared clinic-day or batch-level influence. It is not the same as an individual participant's own deviation on a dosing occasion.
For the basic comparison, assume that \(\eta_i\), \(b_{ij}\), \(g_j\), and \(e_{ij}\) have mean zero, are mutually independent, and are independent across their respective subject, subject-occasion, global-occasion, and profile indices. Assume normal distributions with variances \(\omega_{\mathrm B}^2\), \(\omega_{\mathrm O}^2\), \(\tau^2\), and \(\sigma_e^2\), respectively. The occasion index \(j\) in \(g_j\) must refer to a genuinely shared occasion: participants with the same \(j\) are exposed to the same common occasion-level influence. All candidate structures should use the same fixed-effects specification and residual-error assumptions so that the comparison isolates occasion structure.
Covariance Signatures
The variance of one log-AUC profile includes every random component that varies across profiles. The covariance between two different profiles includes only components they share. This gives a direct way to understand what each occasion structure says about the data.
The second line follows because two occasions from the same participant share \(\eta_i\), but have different independent \(b\), \(g\), and \(e\) terms. The third line follows because two different participants observed on the same global occasion share \(g_j\), but not their subject effects or subject-by-occasion effects. In particular, the diagonal variance includes \(\sigma_e^2\); omitting it would understate the variance of an observed profile.
Worked Example: Reading the Covariance Pattern
Suppose a hypothetical fitted combined model estimates \(\omega_{\mathrm B}^2=0.025\), \(\omega_{\mathrm O}^2=0.012\), \(\tau^2=0.006\), and \(\sigma_e^2=0.020\), all on the log-AUC scale. Calculate the variance and correlations implied for different pairs of profiles.
For one profile, $$ \operatorname{Var}(y_{ij})=0.025+0.012+0.006+0.020=0.063. $$ Its standard deviation is \(\sqrt{0.063}=0.251\) log-AUC units.
Two different occasions from the same participant have covariance \(0.025\), so their correlation is \(0.025/0.063=0.397\). Two different participants measured on the same shared occasion have covariance \(0.006\), giving correlation \(0.006/0.063=0.0952\). Different participants on different occasions have covariance zero under the stated assumptions. These are model-implied correlations, not estimates of causal effects.
Fitting and Comparing Candidate Structures
The independent-occasion model includes \(\eta_i\), \(b_{ij}\), and \(e_{ij}\), but not \(g_j\). The shared-occasion model includes \(\eta_i\), \(g_j\), and \(e_{ij}\), but not \(b_{ij}\). A combined model includes both occasion components. The candidate structures therefore make distinct predictions: in particular, the independent-only model predicts zero covariance between different participants on the same occasion, whereas the shared-only model predicts positive covariance of \(\tau^2\).
Fit each candidate with the same concentration summaries, fixed effects, and residual specification. For Gaussian profile summaries, the model defines a covariance matrix \(V\) for the observed vector \(y\); likelihood fitting chooses parameters that make the observed means and covariance pattern plausible under that model. Model comparisons should use a consistent fitting method and the same observations. AIC is \( \mathrm{AIC}=-2\ell+2k \), where \(\ell\) is the maximized log likelihood and \(k\) is the number of estimated parameters. Lower AIC indicates a better balance of fit and parameter count among the candidate models; it is not proof that the selected structure is true.
Worked Example: Comparing Fitted Occasion Structures
Suppose a hypothetical crossover dataset contains 30 participants, each with log-AUC summaries from three globally indexed study occasions. The same fixed effects and residual assumptions are used for all three fits. A Gaussian likelihood fit reports the following values:
| Structure | Estimated occasion variances | \(-2\ell\) | \(k\) |
|---|---|---|---|
| Independent only | \(\hat{\omega}_{\mathrm O}^2=0.018\) | 42.6 | 4 |
| Shared only | \(\hat{\tau}^2=0.006\) | 48.0 | 4 |
| Combined | \(\hat{\omega}_{\mathrm O}^2=0.010,\ \hat{\tau}^2=0.004\) | 40.2 | 5 |
For the independent-only structure, \(\mathrm{AIC}=42.6+2(4)=50.6\). For the shared-only structure, \(\mathrm{AIC}=48.0+2(4)=56.0\). For the combined structure, \(\mathrm{AIC}=40.2+2(5)=50.2\). The combined model has the lowest AIC, but it improves on the independent-only model by only 0.4 AIC units. These results provide little basis for a strong claim that the combined structure is decisively better; the shared-only structure has substantially less support by this criterion. The fitted shared variance is small, but its uncertainty and the number of independent occasions also matter.
If common treatment or period effects are included in every candidate, those common parameters add to \(k\) in each model and do not change the AIC differences. The table's values are hypothetical fit output, not a claim that the listed estimates can be recovered from the number of participants and occasions alone.
Checking the Occasion Interpretation
The most informative diagnostic is whether the observed dependence resembles the covariance pattern implied by the fitted model. For example, calculate or inspect covariance among different participants who share an occasion index, as well as covariance between different occasions within a participant. These summaries are noisy in finite samples, but systematic discrepancies can point to a missing or misdefined component.
Worked Example: Does the Shared Effect Have a Data Signature?
In a separate hypothetical dataset, covariance summaries of log-AUC profiles are \(0.009\) for different participants on the same occasion, \(0.001\) for different participants on different occasions, and \(0.027\) for different occasions within the same participant. The average profile variance is \(0.065\). Consider whether these summaries are compatible with a shared occasion component.
The same-occasion, different-participant correlation is approximately \(0.009/0.065=0.138\), while the different-occasion, different-participant correlation is approximately \(0.001/0.065=0.0154\). The larger same-occasion covariance is consistent with a shared occasion effect. The within-participant covariance of \(0.027\) is consistent with a persistent between-subject component, although it does not by itself identify the shared occasion variance.
These are descriptive sample summaries, not a formal test. Sampling variation, unequal observations, fixed period effects, and other sources of dependence can affect them. The appropriate next step is to check model-based residuals and uncertainty, and to confirm that the occasion labels truly identify common conditions rather than merely each participant's successive dosing occasions.
A shared random effect is particularly difficult to estimate when there are few independent global occasions. Thirty participants measured on only three shared occasions do not provide thirty independent replications of the shared occasion variance: the variance is learned from variation across the three occasion levels. If occasion is a fixed, named study period, fixed period effects may be more suitable than a random effect, especially when inference concerns only those particular periods. A random occasion effect instead represents variation across a population of possible occasions and relies on having enough independent occasions to characterize that population.
Likewise, if each participant has their own occasion 1, occasion 2, and occasion 3 with no common clinic day, the label \(j\) does not automatically justify a shared \(g_j\). A shared effect requires a real group of observations exposed to the same common occasion-level condition. Confusing these indices can create an artificial covariance structure.
Derivation and Practical Model Choice
The covariance equations follow directly by expanding each profile into its random components. For the same participant on distinct occasions, $$ \operatorname{Cov}(y_{ij},y_{ij'}) =\operatorname{Cov}(\eta_i+b_{ij}+g_j+e_{ij},\ \eta_i+b_{ij'}+g_{j'}+e_{ij'}). $$ By mutual independence, every cross-term is zero except \(\operatorname{Var}(\eta_i)\); hence this covariance is \(\omega_{\mathrm B}^2\). For different participants on the same occasion, the only common random term is \(g_j\), so the covariance is \(\operatorname{Var}(g_j)=\tau^2\). For the same profile, all four independent variances add. This justifies the covariance pattern and shows why the diagonal residual contribution must be retained.
State what occasion index is shared across participants, and why those observations can plausibly share a common influence.
Use the same observations, fixed effects, and residual model; compare independent-only, shared-only, and, when justified, combined structures.
Compare information criteria and inspect the covariance diagnostics. Examine whether estimated variance components are near zero or poorly determined.
Report which dependence pattern is supported and the limits imposed by the number and meaning of the occasions.
An independent-only model is attractive when individual participants vary from one dosing occasion to another but participants measured at the same global time do not show additional shared dependence. A shared-only model is appropriate when a common occasion influence is plausible and the cross-participant, same-occasion covariance is evident. The combined model allows both patterns, but its additional variance component needs enough information to be estimated reliably.
These models are generally not nested alternatives when one replaces the independent effect with the shared effect. A likelihood-ratio test should not be used as if one structure were obtained from the other by setting a single variance to zero. A combined model can be nested against a reduced model by constraining a variance component to zero, but variance parameters lie on the boundary of their allowed range; ordinary reference distributions for likelihood-ratio tests may therefore need adjustment. Information criteria and design-based diagnostics are often more straightforward for the direct independent-versus-shared comparison.
Common Pitfalls and Limitations
- Calling every repeated occasion effect shared. A participant-specific deviation is \(b_{ij}\); a common effect across participants is \(g_j\). They imply different covariances.
- Leaving residual variance off the diagonal. The variance of an observed profile includes the residual contribution \(\sigma_e^2\), even though independent residuals do not contribute to covariance between distinct profiles.
- Treating many participants as many shared occasions. The number of independent shared occasions, not just the number of profiles, controls information about \(\tau^2\).
- Confounding shared random effects with fixed period structure. Mean differences among named periods should not automatically be attributed to a random occasion variance.
- Interpreting a low AIC as proof. AIC compares relative expected predictive adequacy among specified candidates; it cannot rescue misspecified occasion labels, residual assumptions, or fixed effects.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- Under the combined model, what is the covariance between different participants on the same occasion?
- If \(\omega_{\mathrm B}^2=0.020\), \(\omega_{\mathrm O}^2=0.010\), \(\tau^2=0.005\), and \(\sigma_e^2=0.015\), calculate the variance of one profile.
- Why does an independent subject-by-occasion effect not induce covariance between different participants measured on the same occasion?
- Two candidate models have \(-2\ell=30\), \(k=4\), and \(-2\ell=28\), \(k=5\), respectively. Calculate both AIC values.
- What design feature is especially important for estimating a shared occasion variance?
Solutions. (1) The covariance is \(\tau^2\), because the two profiles share \(g_j\) and no other random component. (2) The profile variance is \(0.020+0.010+0.005+0.015=0.050\). (3) The independent effects belong to separate participant-occasion pairs, so they are not a random component shared by the two participants. (4) The AIC values are \(30+2(4)=38\) and \(28+2(5)=38\); by AIC they tie. (5) The number of independent shared occasions matters because \(\tau^2\) describes variation across those occasion levels.