Comparing Candidate NLME Sampling Schedules

Why Compare Complete Schedules?

A candidate sampling schedule is not useful merely because it includes a time at which the model changes rapidly. A schedule must provide information about the parameters that matter after accounting for the other parameters that must also be estimated. Two schedules can have similar information for a target parameter when all other parameters are treated as known, yet differ substantially once uncertainty in nuisance parameters is included.

This tutorial compares schedules using local Fisher information from an NLME model. The comparison combines parameter-specific precision, a determinant criterion, and checks for weak identification. The resulting ranking is conditional on the assumed model, nominal parameter values, residual and random-effect distributions, and the design objectives.

Information for a Candidate Schedule

Consider a nonlinear mixed-effects model with subject-specific observations \(Y_i\), a marginal mean vector \(\mu_i(\theta)\), and marginal covariance matrix \(V_i(\theta)\). The schedule specifies the observation times and, where relevant, the allocation of observations among subjects. For a Gaussian marginal model, the expected Fisher information for parameters \(\theta\) is the sum of contributions from subjects:

The first term uses sensitivities of the marginal mean; the second represents information in covariance changes when the covariance depends on estimated parameters. If the covariance parameters are treated as fixed, the second term is zero. Within-subject dependence must still be represented in \(V_i\); treating correlated observations as independent generally gives the wrong information.

The formula is a local calculation at specified nominal parameter values. For a nonlinear model, sensitivities and hence information can change with those values. The inverse information approximates the covariance matrix of an efficient estimator under regularity conditions, a correctly specified model, adequate information, and an interior parameter value. It is not a guarantee that a finite-sample fit will converge or that the model is correct.

Adjusting for Nuisance Parameters

Partition the parameter vector into a target parameter \(\psi\) and nuisance parameters \(\lambda\). Write the information matrix in corresponding blocks. If the matrix is nonsingular, the information about \(\psi\) after accounting for estimation of \(\lambda\) is the Schur complement:

For a scalar target, the large-sample variance approximation is \(1/\mathcal I_{\psi\mid\lambda}\), and its standard error is \(1/\sqrt{\mathcal I_{\psi\mid\lambda}}\). By contrast, \(1/\sqrt{\mathcal I_{\psi\psi}}\) is a conditional standard error that treats the nuisance parameters as known. The subtraction in the Schur complement is nonnegative when the information matrix is positive semidefinite, so accounting for nuisance estimation cannot increase the effective information for the target.

A short justification follows from the block inverse identity: the target-parameter block of \(\mathcal I^{-1}\) is \(\mathcal I_{\psi\mid\lambda}^{-1}\). Thus the nuisance-adjusted precision is not an ad hoc penalty; it is the target's variance approximation from the joint estimation problem.

Worked Comparison of Three Schedules

Suppose the target is an elimination rate \(k\), measured in \( \mathrm{h}^{-1}\), and the nuisance parameter is an amplitude \(A\), measured in concentration units. For illustration, consider three candidate schedules for a hypothetical NLME study: an early-heavy schedule, a balanced schedule, and a late-heavy schedule. Assume the marginal model and nominal parameter values have been specified, the variance components are treated as known, and the displayed matrices are the local information calculated for the two fixed effects \(A\) and \(k\). The numbers are hypothetical; they are not results from a real study.

Comparing Criteria and Checking Identification

Different criteria answer different design questions. For the two-parameter matrices above, the determinant summarizes joint information volume. The determinants are \(400(180)-240^2=14{,}400\) for the early schedule, \(600(514.4)-360^2=179{,}040\) for the balanced schedule, and \(200(250)-180^2=17{,}600\) for the late schedule. Thus the balanced schedule also has the largest determinant and would be preferred by a local D-optimality comparison among these candidates. A determinant is a joint criterion, however: it does not replace reporting the adjusted precision of a particular target parameter.

The off-diagonal entries reveal overlap between parameter sensitivities. For a two-parameter information matrix, the normalized information correlation is \(\mathcal I_{Ak}/\sqrt{\mathcal I_{AA}\mathcal I_{kk}}\). Its values are approximately \(-0.8944\), \(-0.6477\), and \(-0.8050\) for the early, balanced, and late schedules, respectively. A magnitude near one indicates that changes in the two parameters can produce similar patterns in the observations, making their separate estimation difficult. The balanced schedule has the least severe overlap among these three, though its correlation is still substantial.

Worked Example: Updating a Candidate Schedule

A design comparison can also assess whether an additional observation is useful. In this example, treat the following increment as a hypothetical information contribution on a common, whitened scale. It adds information in a direction that is not aligned with the existing schedule's sensitivity to both parameters.

Assumptions, Diagnostics, and Limitations

A numerical ranking is meaningful only under the assumptions used to calculate it. The model structure and nominal values should be plausible for the intended population; the residual and random-effect covariance must represent the dependence in the observations; and the local information should be positive definite for the parameters being compared. If variance components are estimated, they belong in the joint information matrix as nuisance parameters rather than being silently treated as known. When integrating a nonlinear random-effects model does not yield a Gaussian marginal distribution, the Gaussian formula above is not exact; information should be calculated from the appropriate marginal likelihood or clearly identified as an approximation.

Useful checks include verifying that the information matrix is symmetric, that its eigenvalues are nonnegative up to numerical tolerance, and that its rank is sufficient for the parameters of interest. A small eigenvalue, a near-unit normalized information correlation, a very small Schur complement, or a large condition number can signal weak identification. Compare adjusted standard errors and relevant eigen-directions across schedules, not just raw diagonal entries. If results change sharply under plausible nominal parameter values, the ranking is fragile and should not be presented as a universal property of the schedules.

A common mistake is to rank schedules by \(\mathcal I_{kk}\) alone and conclude that the largest value guarantees the best estimate of \(k\). That reasoning assumes nuisance parameters are known. The correction is to use the full information matrix, calculate the Schur complement for \(k\), and inspect whether the joint matrix is well conditioned. Conversely, a favorable determinant does not guarantee the smallest standard error for every chosen parameter: the determinant rewards joint volume, whereas a target-specific objective focuses on one parameter's adjusted variance.

Fisher information is an expected, local, model-based summary. It does not diagnose model misspecification by itself, account automatically for operational failures or missing samples, or ensure an estimator will attain the asymptotic variance in a finite sample. Simulation under plausible parameter and error scenarios can reveal issues beyond the local calculation, but the information comparison remains a transparent way to understand why candidate schedules differ.