Designing Profiles to Estimate Random-Effect Covariance

Why Profile Design Matters for Random-Effect Correlations

A population design can estimate the typical clearance and volume reasonably well yet provide little information about whether subjects with above-typical clearance also tend to have above-typical volume. A correlation concerns joint variation within subjects. Its estimation therefore depends not only on the number of subjects, but also on whether each subject’s profile contains enough information to distinguish the random effects being related.

For a pharmacokinetic model with random effects on clearance, volume, and absorption, profile timing affects how well those effects can be separated. Early samples may be especially informative about absorption, while later samples help characterize elimination. Replication can reduce uncertainty in a subject’s profile, but it does not automatically compensate for sampling at uninformative times or for too few independent subjects.

Definition: Let \(\eta_i=(\eta_{CL,i},\eta_{V,i},\eta_{ka,i})^\top\) be the random effects for subject \(i\), on the log-parameter scale, with \(E(\eta_i)=0\) and covariance matrix \(\Omega\). The correlation between two random effects is \(\rho_{jk}=\Omega_{jk}/\sqrt{\Omega_{jj}\Omega_{kk}}\). A design for random-effect covariance is a choice of sampling times and replication intended to provide information about one or more entries of \(\Omega\), and hence about the corresponding correlations.

The focus here is local design reasoning: evaluate a candidate schedule around specified typical parameter values and a specified \(\Omega\). The standard assumptions are independent subjects; a correctly specified structural model; a stated distribution for the random effects, commonly multivariate normal; and a specified residual-error model. The calculations also rely on the design’s observations being informative enough for the local approximation to be useful. If these conditions are doubtful, information calculations should be treated as approximations rather than guarantees.

Sampling Times and Parameter Sensitivity

To make the timing argument concrete, consider a hypothetical single-dose oral one-compartment model with first-order absorption and elimination, no lag time, and linear pharmacokinetics:

C(t)=\frac{D k_a}{V(k_a-k)} \left(e^{-kt}-e^{-k_a t}\right), \qquad k=\frac{CL}{V}.

Here \(D\) is dose, \(CL\) is clearance, \(V\) is volume, and \(k_a\) is the absorption rate constant. Suppose individual parameters are \(CL_i=CL_{\mathrm{typ}}\exp(\eta_{CL,i})\), with analogous expressions for \(V_i\) and \(k_{a,i}\). The effect of a sampling time on a parameter is described locally by the sensitivity of the predicted concentration to its random effect, such as \(\partial C(t)/\partial\eta_{CL}\). Sensitivities depend on the typical parameters as well as on time.

Early observations can capture the rising portion of the profile and provide information about absorption. Samples around the peak and distribution from absorption toward elimination can help distinguish \(V\), \(k_a\), and \(CL\). Later observations can be valuable for the elimination phase and thus for \(CL\). These are tendencies, not rules that hold regardless of the parameter values: for example, when \(k_a\) and \(k\) are close, their effects can be difficult to separate. A schedule concentrated in one phase may leave two sensitivity patterns nearly indistinguishable.

Worked Example: Checking Phase Coverage in a Hypothetical Profile

Consider a hypothetical oral dose of \(D=100\) mg, with typical \(V=50\) L, \(CL=5\) L/h, and \(k_a=1\) h\(^{-1}\). Thus \(k=CL/V=5/50=0.10\) h\(^{-1}\). Compare predicted concentrations at 0.25, 2, and 12 hours.

The concentration multiplier is \(Dk_a/[V(k_a-k)]=100(1)/[50(1-0.10)]=2.222\) mg/L. At 0.25 h, \(C(0.25)=2.222(e^{-0.025}-e^{-0.25})=2.222(0.9753-0.7788)=0.4367\) mg/L.

At 2 h, \(C(2)=2.222(e^{-0.2}-e^{-2})=2.222(0.8187-0.1353)=1.519\) mg/L. At 12 h, \(C(12)=2.222(e^{-1.2}-e^{-12})=2.222(0.3010-0.000006)=0.6689\) mg/L.

The three times span an early rising observation, a near-peak observation, and a later elimination-phase observation. This coverage is more promising for separating the effects than three observations all taken close together. It does not by itself establish that all three random effects, or their correlations, will be estimable: that requires examining the sensitivities and the information in the full design.

From Profile Information to Covariance Information

A useful local approximation makes the link between profile precision and covariance estimation explicit. Linearize the subject’s predicted observations around the typical parameters. For subject \(i\), write \(y_i\approx f_i+J_i\eta_i+\epsilon_i\), where \(f_i\) is the vector of typical predictions, \(J_i\) is the matrix of sensitivities at that subject’s sampling times, and \(\epsilon_i\) is residual error with covariance \(R_i\). Under the stated normal assumptions, the marginal distribution is approximately normal with mean \(f_i\) and covariance \(V_i=J_i\Omega J_i^\top+R_i\).

For a covariance parameter \(\theta_a\), define \(V_{i,a}=\partial V_i/\partial\theta_a\). The expected Fisher information for covariance parameters in this normal approximation is:

\mathcal{I}_{ab} =\frac{1}{2}\sum_i \operatorname{tr}\left( V_i^{-1}V_{i,a}V_i^{-1}V_{i,b} \right).

This expression is local: it is evaluated at specified parameter values. It shows why timing and replication matter. Sampling times change the rows of \(J_i\); replication changes the amount and pattern of information in the observations relative to residual error. Both changes affect \(V_i\), its derivatives, and therefore the information for covariance parameters.

Justification: For one subject with a normal observation vector and fixed mean, the covariance-dependent log likelihood is \(-\tfrac12\{\log|V|+(y-f)^\top V^{-1}(y-f)\}\), apart from a constant. Differentiating with respect to \(\theta_a\) gives score \(\tfrac12\{(y-f)^\top V^{-1}V_aV^{-1}(y-f)-\operatorname{tr}(V^{-1}V_a)\}\). Taking the expected product of two such scores under \(y-f\sim N(0,V)\), and using the normal fourth-moment identity, gives \(\tfrac12\operatorname{tr}(V^{-1}V_aV^{-1}V_b)\). Independent subjects contribute additively, giving the sum above.

Designs are usually judged on correlations, while the information calculation may be parameterized in terms of variances and covariances. For example, \(\Omega_{CL,V}=\rho_{CL,V}\sigma_{CL}\sigma_V\). The derivatives used in the information calculation must match the parameterization being optimized. If variances or other model parameters are also estimated, information about a correlation must be assessed after allowing for that uncertainty. In a Fisher information matrix partitioned into target parameters and nuisance parameters, the adjusted information for the target is the Schur complement \(\mathcal{I}_{tt}-\mathcal{I}_{tn}\mathcal{I}_{nn}^{-1}\mathcal{I}_{nt}\), when the nuisance block is invertible. A large unadjusted entry alone can overstate practical precision.

Replication and a Transparent Information Approximation

A simple approximation helps quantify the value of replication. Suppose each subject provides noisy estimates \(z_C\) and \(z_V\) of two random effects, and the estimation errors are independent of the random effects and of each other. Then \(z_C=\eta_C+e_C\) and \(z_V=\eta_V+e_V\). Let the error variances be \(q_C\) and \(q_V\). When the true cross-covariance is zero and the marginal random-effect variances are treated as known, the covariance matrix of \((z_C,z_V)\) is diagonal with entries \(a=\sigma_C^2+q_C\) and \(b=\sigma_V^2+q_V\). The information per subject for the cross-covariance \(c=\Omega_{CV}\) is \(1/(ab)\). This approximation is not a substitute for a full NLME design calculation, but it isolates the role of profile precision.

Worked Example: Improving Paired Profile Precision

Suppose, hypothetically, that \(\sigma_C=0.20\) and \(\sigma_V=0.30\) on the log-random-effect scale, with \(\rho_{CV}=0\). An initial schedule gives proxy-estimation error variances \(q_C=0.04\) and \(q_V=0.04\). Then \(a=0.20^2+0.04=0.08\), and \(b=0.30^2+0.04=0.13\).

The approximate information per subject for \(c=\Omega_{CV}\) is \(1/(ab)=1/(0.08\times0.13)=96.15\), in inverse squared covariance units. For 40 independent subjects, the information is \(40(96.15)=3846\).

Suppose a revised schedule reduces the error variances to \(q_C=0.01\) and \(q_V=0.04\), while the number of subjects and the model assumptions stay fixed. Now \(a=0.20^2+0.01=0.05\), \(b=0.13\), and the information per subject is \(1/(0.05\times0.13)=153.8\). For 40 subjects the total is \(6154\), about \(6154/3846=1.60\) times the original information. In this approximation the gain comes from better estimation of the clearance random effect within each subject; it is not a guarantee of a 1.60-fold gain in a nonlinear model with nuisance parameters.

Worked Example: Comparing Replication With More Subjects

Use the same hypothetical random-effect variances, \(\sigma_C^2=0.04\) and \(\sigma_V^2=0.09\), and suppose each paired profile occasion gives proxy-error variance \(q=0.12\) for each random effect. Compare two designs with the same total number of occasions: 40 subjects with two occasions each, versus 20 subjects with four occasions each. Assume the occasion-level errors are independent, and average the replicate proxies within each subject.

With two occasions, the error variance of each averaged proxy is \(q/2=0.06\). Thus \(a=0.04+0.06=0.10\) and \(b=0.09+0.06=0.15\). Information for the cross-covariance is \(1/(0.10\times0.15)=66.67\) per subject, or \(40(66.67)=2667\) across subjects.

With four occasions, each averaged proxy has error variance \(q/4=0.03\). Thus \(a=0.04+0.03=0.07\), \(b=0.09+0.03=0.12\), and information is \(1/(0.07\times0.12)=119.0\) per subject. Across 20 subjects this is \(20(119.0)=2381\). Under these simplifying assumptions, the design with more subjects has higher total covariance information, even though each subject’s random effects are measured less precisely.

This comparison depends on the proxy model and the assumed error behavior. In an NLME design, a replicate can also help distinguish residual variation from between-subject variation, and a sample at a new time can add a new sensitivity direction rather than simply improve precision. The appropriate comparison therefore uses the actual candidate schedules and the same target information criterion.

How to Evaluate Candidate Profiles

For a design focused on a particular correlation, such as \(\rho_{CL,V}\), the calculation should reflect the joint information for the clearance and volume effects and the uncertainty in all estimated nuisance parameters. A useful sequence is:

Specify the target.
State which correlation or covariance is important, and specify whether its marginal variances and other model parameters are treated as known or estimated.

Choose plausible model values.
Set typical parameters, random-effect variances and correlations, and a residual-error model. The calculation is conditional on these values unless a range is explicitly evaluated.

Calculate profile sensitivities.
For every proposed sampling time, evaluate how predictions change with each random effect. Look for schedules that distinguish the effects rather than repeatedly measuring the same profile feature.

Calculate and adjust information.
Form the marginal information for covariance parameters, then account for nuisance-parameter uncertainty. Compare the resulting target precision or criterion across schedules.

A candidate design is concerning when the sensitivity columns for two random effects are nearly proportional over the sampled times. In that situation, changes in one effect can resemble changes in the other, so their separate contributions and covariance are hard to identify. Other warning signs include information matrices that are nearly singular, large uncertainty in the estimated correlation, unstable estimates across plausible parameter values, or repeated fitting failures in simulation. These signs call for redesign or a more cautious interpretation, not merely a different optimizer.

Common Pitfall and Practical Limits

Common mistake: “A larger number of samples per subject must give better information about random-effect correlations, because each individual profile becomes more precise.” Replication can improve within-subject precision, but correlations are properties of variation across independent subjects. Under a fixed total sampling budget, extra replicates can reduce the number of subjects and thereby reduce information about that population-level relationship. The correction is to compare complete designs, including both the number of subjects and the times and number of samples per subject.

No single sampling schedule is best for every random-effect covariance. A design that is informative for \(\rho_{CL,V}\) may be less informative for \(\rho_{CL,ka}\), because the relevant sensitivities differ. Correlations are also difficult to estimate when a random-effect variance is near zero: the correlation then has little practical meaning and can be poorly determined even if the corresponding covariance is small. Finally, local calculations can be optimistic if the structural model, typical parameter values, or random-effect distribution is wrong. Candidate designs should be checked over scientifically plausible values, and conclusions should be reported in terms of the target and assumptions used.

Check Your Understanding

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

Why can a schedule with many late samples be weak for estimating the correlation between absorption and volume random effects?
In the proxy approximation, suppose \(a=0.10\) and \(b=0.20\). What is the information per subject for the cross-covariance at zero covariance, with the marginal variances treated as known?
Why is information for a target correlation adjusted for nuisance parameters rather than read directly from one diagonal entry of the information matrix?
A design has excellent estimates of each subject’s clearance effect but very few subjects. What aspect of correlation estimation may remain weak?

Solutions. (1) Late samples mainly characterize the elimination phase; if the schedule contains little information about the absorption phase, the absorption sensitivity may be weak or difficult to distinguish from other effects. The joint information needed for the correlation can therefore remain low. (2) The information is \(1/(ab)=1/(0.10\times0.20)=50\) per subject, in inverse squared covariance units. (3) Uncertainty in variances and other parameters can be confounded with the target correlation. The Schur complement accounts for the information lost when those nuisance parameters are estimated rather than known. (4) A correlation describes how effects co-vary across subjects. With few independent subjects, the population-level relationship can remain uncertain even when individual clearance effects are measured precisely.