Correlated Occasion Effects Across Pharmacokinetic Parameters

Why Occasion Effects Need a Multivariate Description

A pharmacokinetic occasion can shift more than one parameter at once. For example, an occasion-specific change in absorption conditions might coincide with a change in apparent volume. A model that assigns separate occasion effects to clearance, volume, and absorption rate must therefore say not only how variable each effect is, but also whether the effects tend to move together. That information is represented by a between-occasion covariance matrix.

The key distinction is between two questions: are occasion effects on different parameters correlated within the same occasion, and are deviations on different occasions correlated across occasions? These are not the same correlation. The first is described by off-diagonal entries in the between-occasion matrix. The second can arise because the same subject effect persists over multiple occasions, even when occasion effects themselves are independent.

Definition: For subject $i$, occasion $j$, and pharmacokinetic parameter $k$, write a log-scale hierarchical parameter equation as $$ \log(\theta_{ijk})=\log(\theta_{\mathrm{pop},k})+\eta_{ik}+\kappa_{ijk}. $$ Here $\theta_{\mathrm{pop},k}$ is the population parameter, $\eta_{ik}$ is the subject-level random effect, and $\kappa_{ijk}$ is the between-occasion random effect. For $K$ parameters, collect the occasion effects for one subject and occasion in $\boldsymbol{\kappa}_{ij}=(\kappa_{ij1},\ldots,\kappa_{ijK})^\mathsf{T}$. The between-occasion covariance matrix is $\Omega_{\mathrm{BO}}=\operatorname{Cov}(\boldsymbol{\kappa}_{ij})$.

In this tutorial, occasion effects have mean zero and a common covariance matrix across subjects and occasions. Unless stated otherwise, occasion-effect vectors for distinct occasions are independent of one another, and the subject-effect vector $\boldsymbol{\eta}_i$ is independent of the occasion-effect vectors. These are modeling assumptions, not consequences of writing down a covariance matrix. The log-scale formulation means that an occasion effect acts multiplicatively on the natural parameter scale: a value $\kappa$ multiplies the parameter by $\exp(\kappa)$.

Constructing and Interpreting the Matrix

For three parameters, such as clearance (CL), volume (V), and absorption rate constant ($k_a$), the matrix has a variance on each diagonal and a covariance for each parameter pair off the diagonal:

\Omega_{\mathrm{BO}}= \begin{pmatrix} \omega^2_{\mathrm{CL,BO}} & \omega_{\mathrm{CL,V,BO}} & \omega_{\mathrm{CL},k_a,\mathrm{BO}}\\ \omega_{\mathrm{CL,V,BO}} & \omega^2_{\mathrm{V,BO}} & \omega_{\mathrm{V},k_a,\mathrm{BO}}\\ \omega_{\mathrm{CL},k_a,\mathrm{BO}} & \omega_{\mathrm{V},k_a,\mathrm{BO}} & \omega^2_{k_a,\mathrm{BO}} \end{pmatrix}.

The subscripts identify the two effects whose covariance is being described. The matrix is symmetric because $\operatorname{Cov}(A,B)=\operatorname{Cov}(B,A)$. On this log scale, the entries are variances and covariances of dimensionless log deviations. A positive covariance indicates that above-average occasion deviations in one parameter tend to accompany above-average deviations in the other; a negative covariance indicates a tendency in opposite directions. Neither statement says that every occasion follows that pattern.

Covariance magnitude depends on the variability of both effects, so correlation is often easier to compare across parameter pairs. For parameters $a$ and $b$, the occasion-effect correlation is

\rho_{ab,\mathrm{BO}}= \frac{\operatorname{Cov}(\kappa_{ija},\kappa_{ijb})} {\sqrt{\operatorname{Var}(\kappa_{ija})\operatorname{Var}(\kappa_{ijb})}}.

This correlation is defined for effects from the same occasion when both variances are positive. It is bounded between $-1$ and $1$, and it is not itself a covariance or a percentage change in either pharmacokinetic parameter.

Worked Example: Building a Three-Parameter Occasion Matrix

Suppose a hypothetical model estimates the following variances and covariances for log-scale occasion effects on CL, V, and $k_a$:

\Omega_{\mathrm{BO}}= \begin{pmatrix} 0.130 & 0.018 & -0.012\\ 0.018 & 0.250 & 0.020\\ -0.012 & 0.020 & 0.040 \end{pmatrix}.

The diagonal entries are the occasion-effect variances: $0.130$ for CL, $0.250$ for V, and $0.040$ for $k_a$. For the CL–V pair, the correlation is

\rho_{\mathrm{CL,V,BO}}= \frac{0.018}{\sqrt{0.130(0.250)}}= \frac{0.018}{\sqrt{0.0325}} \approx 0.09985.

Thus, the hypothetical occasion effects on CL and V have a small positive correlation. For CL and $k_a$, the correlation is $\frac{-0.012}{\sqrt{0.130(0.040)}}\approx -0.1664$. For V and $k_a$, it is $\frac{0.020}{\sqrt{0.250(0.040)}}=0.2000$. These are three pairwise associations within an occasion; they do not imply that one parameter causes another to change.

The matrix must also be mathematically admissible. Its leading principal minors are $0.130$, $0.130(0.250)-0.018^2=0.032176$, and determinant $0.0011904$, all positive. By the positive-definiteness criterion for a symmetric matrix, this matrix is positive definite and is a valid covariance matrix.

Why Positive Definiteness Matters

A covariance matrix must produce a nonnegative variance for every linear combination of its random effects. In matrix form, for every vector $\mathbf{a}$, the variance of $\mathbf{a}^\mathsf{T}\boldsymbol{\kappa}_{ij}$ is

\operatorname{Var}(\mathbf{a}^\mathsf{T}\boldsymbol{\kappa}_{ij}) =\mathbf{a}^\mathsf{T}\Omega_{\mathrm{BO}}\mathbf{a}\geq 0.

A positive-definite matrix satisfies the strict inequality for every nonzero $\mathbf{a}$. This condition rules out combinations of variances and covariances that would imply an impossible negative variance. For a two-parameter matrix with variances $v_1,v_2$ and covariance $c$, positive definiteness requires $v_1>0$ and $v_1v_2-c^2>0$. Equivalently, the correlation must have magnitude less than 1 when both variances are positive.

For larger matrices, checking each pairwise correlation is not enough: every pair can have a correlation within $[-1,1]$ while the complete matrix is still not positive definite. A factorization such as a Cholesky factorization is a practical way to check or parameterize admissible matrices. The principle is to ensure that the model represents a valid joint distribution, rather than a collection of individually plausible pairwise summaries.

Worked Example: From Covariance to Correlation

Consider hypothetical log-scale occasion effects on CL and V with variances $0.130$ and $0.250$, and covariance $0.018$. The standard deviations are $\sqrt{0.130}\approx 0.3606$ and $\sqrt{0.250}=0.5000$. The covariance is converted to a correlation by dividing by the product of these standard deviations:

\rho_{\mathrm{CL,V,BO}}= \frac{0.018}{(0.3606)(0.5000)} \approx 0.09983.

The covariance matrix is positive definite because its determinant is $0.130(0.250)-0.018^2=0.032176>0$, and both diagonal variances are positive. The positive correlation means that, in the model, occasion deviations above the subject's typical level for CL tend weakly to accompany above-typical deviations for V on the log scale. It does not mean that a 1% increase in CL produces a 0.09985% increase in V.

Within-Occasion Correlation Versus Across-Occasion Dependence

Let $\Omega_{\mathrm{BS}}=\operatorname{Cov}(\boldsymbol{\eta}_i)$ denote the between-subject covariance matrix. Under the independence assumptions stated above, the vector of total log deviations on occasion $j$ is $\boldsymbol{\eta}_i+\boldsymbol{\kappa}_{ij}$. Its covariance within one occasion is the sum of the two component matrices:

\operatorname{Cov}(\boldsymbol{\eta}_i+\boldsymbol{\kappa}_{ij}) =\Omega_{\mathrm{BS}}+\Omega_{\mathrm{BO}}.

For distinct occasions $j\ne l$, independence of the occasion effects gives $\operatorname{Cov}(\boldsymbol{\kappa}_{ij},\boldsymbol{\kappa}_{il})=0$. The same subject effect remains in both occasions, so

\operatorname{Cov}(\boldsymbol{\eta}_i+\boldsymbol{\kappa}_{ij}, \boldsymbol{\eta}_i+\boldsymbol{\kappa}_{il}) =\Omega_{\mathrm{BS}}.

Consequently, the correlation between total CL deviations and total V deviations on one occasion is not generally the same as the correlation between total deviations on different occasions. The latter can be nonzero even when occasion effects are independent across occasions: shared subject effects create that dependence. This is distinct from the within-occasion CL–V correlation in $\Omega_{\mathrm{BO}}$.

Worked Example: Separating Same- and Different-Occasion Correlations

Suppose a hypothetical two-parameter model has

\Omega_{\mathrm{BS}}= \begin{pmatrix} 0.250&0.150\\ 0.150&0.360 \end{pmatrix}, \qquad \Omega_{\mathrm{BO}}= \begin{pmatrix} 0.090&0.018\\ 0.018&0.160 \end{pmatrix},

with the first parameter representing log CL and the second log V. Assume subject and occasion effects are independent, and occasion effects are independent across occasions. On the same occasion, the total variances are $0.250+0.090=0.340$ for CL and $0.360+0.160=0.520$ for V. Their total covariance is $0.150+0.018=0.168$, so their same-occasion total correlation is

\frac{0.168}{\sqrt{0.340(0.520)}}= \frac{0.168}{\sqrt{0.1768}} \approx 0.3995.

Across two different occasions, the total covariance is only the shared subject covariance, $0.150$. The marginal variances on each occasion remain $0.340$ and $0.520$, so the cross-occasion correlation between CL on one occasion and V on the other is $\frac{0.150}{\sqrt{0.1768}}\approx 0.3567$. Thus, $0.3995$ is a same-occasion total correlation, while $0.3567$ is a cross-occasion, cross-parameter correlation. Neither value is the occasion-effect correlation alone, which is $0.018/\sqrt{0.090(0.160)}=0.1500$.

Modeling Choices, Diagnostics, and Limitations

A diagonal $\Omega_{\mathrm{BO}}$ allows each parameter to have its own occasion variance but assumes zero occasion-effect covariance between parameters. An unstructured matrix estimates all variances and pairwise covariances, allowing flexible relationships but requiring information to estimate many quantities. Structured alternatives impose a more restricted pattern, for example a common correlation or a small number of covariance parameters. The appropriate choice depends on the scientific model and on whether the data can support the additional parameters; an elaborate matrix is not automatically a better model.

Useful checks include verifying symmetry, positive variances, positive definiteness, and whether estimated correlations lie within their allowed range. Also examine the covariance and correlation scales together: a seemingly large covariance can reflect large variances, while a small covariance can still represent substantial correlation if the variances are small. Near-zero estimated variances or correlations near $1$ or $-1$ can signal a boundary or a poorly determined covariance structure. These patterns warrant investigation rather than an automatic conclusion that the biological effects are absent or perfectly linked.

The matrix describes the assumed distribution of random effects, not every possible dependence in observed concentrations. Its interpretation depends on the parameter scale and the hierarchical model. For example, a covariance on the log scale is not a covariance in natural units such as volume or clearance. If occasion effects are allowed to be correlated across occasions, then the zero cross-occasion occasion-effect covariance used above no longer applies; the model would need to specify that additional dependence explicitly.

Common pitfall: Seeing a nonzero CL–V entry in $\Omega_{\mathrm{BO}}$ and calling it the correlation between CL on occasion 1 and V on occasion 2 is incorrect. That entry is a covariance for the two occasion effects on the same occasion. Under independent occasion effects, the cross-occasion covariance of total deviations instead comes from $\Omega_{\mathrm{BS}}$; the denominator for a correlation must also use the relevant total variances.

Specify the effects.
State which parameters have occasion effects and whether the effects are on the natural or log scale.

Construct the covariance matrix.
Put variances on the diagonal and symmetric pairwise covariances off the diagonal.

Check admissibility and interpretation.
Verify positive definiteness, convert covariances to correlations when useful, and distinguish occasion-only effects from total deviations.

Check Your Understanding

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

A two-parameter occasion covariance matrix has variances $0.16$ and $0.25$, with covariance $-0.10$. What is the correlation, and is the matrix positive definite?
Under independent occasion effects and a shared subject effect, what component contributes to the covariance of total deviations on two different occasions?
What does a positive off-diagonal entry in $\Omega_{\mathrm{BO}}$ say, and what does it not establish?
Why does a diagonal between-occasion matrix differ scientifically from an unstructured one?

Solutions. (1) The correlation is $-0.10/\sqrt{0.16(0.25)}=-0.10/0.20=-0.5000$. The determinant is $0.16(0.25)-(-0.10)^2=0.0300>0$, and the diagonal variances are positive, so the matrix is positive definite. (2) The shared subject effect contributes; the occasion-effect contribution is zero under the stated independence assumption. (3) It says that above-average occasion deviations in one parameter tend to accompany above-average deviations in the other, on the modeled scale. It does not establish causation, nor does it describe cross-occasion correlation by itself. (4) The diagonal matrix constrains all between-occasion cross-parameter covariances to zero; the unstructured matrix allows them to be estimated rather than fixed at zero.