Covariance Between Repeated Formulation Responses

Why Covariance Matters in Replicate Responses

A replicate crossover records more than one response after at least one formulation. Those measurements from the same participant are not generally independent: participant characteristics that influence one response may influence the others as well. Covariance describes the direction and magnitude of this linear relationship. It is needed to represent the joint data correctly and to understand why a within-participant comparison can have a different variance from a comparison of independent observations.

The schedule also matters. A covariance matrix is written for a particular ordering of measurements, usually by period. The sequence tells us which entries correspond to test or reference responses and which repeated responses are separated by one or more periods. The same set of treatment counts can therefore have different covariance representations under different assumptions about how dependence changes with time.

Let \(Y_{ij}\) denote participant \(i\)'s response in period \(j\), with the response defined here as a log-transformed pharmacokinetic measure. Let \(d_{ij}\) identify the formulation given in that period. A useful starting model separates systematic means from shared participant variation and period-specific variation:

Here \(\mu\) is an overall mean, \(\theta_{d_{ij}}\) is the formulation effect, and \(\pi_j\) is the period effect. The participant effect \(b_i\) represents a participant’s tendency to have higher or lower responses across periods. In the simple model, \(b_i\sim N(0,\sigma_b^2)\); the errors \(\varepsilon_{ij}\) are independent across periods, have mean zero and common variance \(\sigma_e^2\), and are independent of \(b_i\). Participants are independent of one another. These are modeling assumptions, not consequences of having a replicate schedule.

Covariance Under a Shared-Participant Model

The fixed mean terms do not contribute to covariance. For two different periods \(j\ne k\) from the same participant, the shared \(b_i\) contributes to both responses, while independent errors contribute no cross-period covariance. Thus the covariance is \(\sigma_b^2\). A response’s variance includes both the shared participant component and its period-specific error variance.

The corresponding correlation between any two distinct periods in this simple model is \(\sigma_b^2/(\sigma_b^2+\sigma_e^2)\). This is a compound-symmetry structure: every period has the same variance and every pair of distinct periods has the same covariance. In a real analysis, the structure must be justified for the response and design at hand; it should not be assumed merely because it is convenient.

Reading the Covariance Matrix Along the Schedule

For a participant with four observed periods, define the period-ordered response vector \(\mathbf{Y}_i=(Y_{i1},Y_{i2},Y_{i3},Y_{i4})^\mathsf{T}\). Under the simple shared-participant model, its covariance matrix is

The diagonal entries are period-response variances; off-diagonal entries are covariances between periods. The matrix is indexed by period, not by formulation. For a T-R-T-R schedule, periods 1 and 3 are the two test responses, and periods 2 and 4 are the two reference responses. For an R-T-R-T schedule, those formulation labels swap. Under compound symmetry, this relabeling does not change the numerical matrix, but it changes which entries are used to describe repeated test responses, repeated reference responses, or cross-formulation pairs.

That distinction becomes more consequential when dependence varies with the separation between periods. For example, one extension retains the shared participant effect but allows period errors to have an autoregressive covariance:

Here the period errors have variance \(\sigma_u^2\), and their correlation at a separation of \(h\) periods is \(\rho^h\). This model assumes a common variance and a correlation that decays geometrically with period distance. It is an alternative structure, not a universal property of crossover data. With it, treatment order identifies the period distance between repeated measurements of a formulation, so the relevant covariance can change when the schedule changes.

Estimating and Checking Dependence

A covariance can also be summarized empirically. If \(x_i\) and \(y_i\) are paired responses from \(n\) participants, the sample covariance is $$ s_{xy}=\frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{n-1}. $$ The pairing is essential: each \(x_i\) and \(y_i\) must come from the same participant and the intended periods or formulations. In a crossover analysis, raw responses may include systematic formulation and period means. A raw covariance can therefore combine shared participant variation with patterns in those means. Model-based covariance estimation accounts for the fixed mean structure that has been specified; a descriptive sample covariance alone does not separate these sources.

Useful checks include plotting paired responses, examining within-participant residuals against period, and comparing residual association at different period separations. Stronger dependence for nearby periods can conflict with compound symmetry. Covariances that differ systematically by sequence or period can also indicate that a single common covariance is inadequate, though mean-model misspecification, missing periods, or carryover may contribute to the pattern. Diagnostics should therefore be interpreted alongside the treatment and period structure rather than as a stand-alone test of a covariance model.

The main alternatives serve different purposes. Compound symmetry is parsimonious and gives a shared covariance for every pair of periods. A time-dependent structure, such as the autoregressive example, distinguishes pairs by period distance. An unstructured covariance estimates each variance and covariance separately, allowing flexibility but requiring more information and parameters. Empirical paired covariances are useful summaries, while a repeated-measures model links the covariance representation to the mean structure and the schedule. None is automatically correct: selection depends on the design, scientific assumptions, amount of data, and adequacy of diagnostics.

Interpretation, Limitations, and a Common Pitfall

Covariance is not the same as a treatment effect. A positive covariance says that paired responses tend to move together; it does not say that one formulation causes a higher response. Nor does covariance alone identify why participants’ responses are related. A shared participant effect is one explanation, while time-dependent biology, measurement conditions, or unmodeled period patterns may also contribute.

A common mistake is to say, “The two test responses are repeated, so their covariance must equal the test-response variance,” or to treat the two responses as independent because they occur in separate periods. Neither follows from replication. The variance includes all variation in one response; the covariance measures only the variation shared by the pair. Under the simple random-intercept model, these are \(\sigma_b^2+\sigma_e^2\) and \(\sigma_b^2\), respectively. The assumed error structure determines whether covariance is constant or depends on period distance.

A second pitfall is to write a covariance matrix in treatment order while silently using period-order assumptions. Always state the vector’s ordering and map each position to the schedule. Also distinguish a mathematical covariance structure from a claim about design quality: treatment order can determine which covariance entries apply, but a covariance calculation by itself does not address period balance, carryover, or the overall choice of design.