One study, different estimation goals
A sampling schedule is informative only with respect to a stated target. In a nonlinear mixed-effects model, a study may aim to estimate a population parameter, such as typical clearance, or to estimate a particular subject’s random effect and thereby characterize that subject’s parameters. These are related goals, but they are not interchangeable. A time that is highly sensitive to a population mean parameter need not reveal much about an individual deviation from that mean. Conversely, repeated observations that help characterize one subject may add less population information than observations spread across more independent subjects.
The distinction matters when choosing when to sample and how to distribute a fixed measurement budget. The calculations below use Fisher information as a local measure of expected precision. They assume a correctly specified model and a design evaluated at specified parameter values. Information is not a guarantee that an estimate will be precise in a realized study; it describes expected curvature of the relevant likelihood or conditional distribution under those assumptions.
Two information calculations
Consider subjects indexed by \(i\), each observed at planned times \(t_{ij}\). Write the population parameter vector as \(\theta\), the subject-specific random-effects vector as \(\eta_i\), and the observations for subject \(i\) as \(y_i\). In an NLME model, the conditional distribution of \(y_i\) depends on both \(\theta\) and \(\eta_i\), while the random effects have a population distribution, often written \(\eta_i\sim N(0,\Omega)\). The within-subject residual error is described by its own model.
For population estimation, the random effects are integrated out to obtain the marginal distribution of each subject’s data. If that distribution is normal with mean \(\mu_i(\theta)\) and covariance \(V_i(\theta)\), the expected Fisher information for population parameters has contributions from changes in both the mean and covariance. For a scalar parameter \(\theta_k\), one standard expression is:
When the target affects only the marginal mean and \(V_i\) is fixed, the second term is zero. For independent subjects, their marginal information contributions add. Crucially, \(V_i\) includes the variability induced by random effects as well as residual error. Treating all observations as independent residual-error observations would generally overstate population information.
For individual estimation, consider the conditional likelihood for one subject when the population parameters are treated as known. If \(J_i\) is the sensitivity matrix of the conditional mean with respect to \(\eta_i\), and \(R_i\) is the residual covariance, the local data information about \(\eta_i\) is \(J_i^{\mathsf T}R_i^{-1}J_i\). Under a normal random-effects distribution, the prior contribution to local posterior precision is \(\Omega^{-1}\). Thus the approximate conditional posterior precision is:
This expression describes information for a subject’s random effects, not information for the population parameters. It conditions on the population model being known; uncertainty in estimated population parameters can add uncertainty to individual estimates or predictions. The distinction between marginal population information and individual conditional information is therefore both a distinction in target and a distinction in probability calculation.
Worked Example: A sampling time that favors one target
Worked Example: A Sampling Time That Favors One Target
Suppose a local approximation to a pharmacometric model at two candidate times is \(Y=\theta f+\eta g+\varepsilon\), where \(\theta\) is a population parameter, \(\eta\sim N(0,1)\) is a subject-specific random effect, and \(\varepsilon\sim N(0,1)\) is independent residual error. Treat the variances as known and compare one observation at each candidate time. Let the sensitivities be \((f,g)=(1,0.2)\) at the early time and \((f,g)=(0.3,1)\) at the late time.
Population target. After integrating out \(\eta\), a single observation has variance \(1+g^2\) and mean \(\theta f\). Since the variance does not depend on \(\theta\), its information for \(\theta\) is \(f^2/(1+g^2)\). At the early time this is \(1^2/(1+0.2^2)=1/1.04=0.9615\). At the late time it is \(0.3^2/(1+1^2)=0.09/2=0.0450\). The early time provides more marginal information about \(\theta\) in this example.
Individual target. Conditional on known \(\theta\), the data information about \(\eta\) from one observation is \(g^2\), because the residual variance is 1. At the early time it is \(0.2^2=0.0400\); at the late time it is \(1^2=1.000\). Including the prior precision of 1, the approximate posterior variances are \(1/(1+0.0400)=0.9615\) and \(1/(1+1)=0.500\), respectively. The late time is more informative about \(\eta\).
The two rankings differ because the time sensitivities differ: early observations respond strongly to \(\theta\), while late observations respond strongly to \(\eta\). These are hypothetical local sensitivities, not a universal rule that early samples estimate population parameters and late samples estimate individual effects.
Why subject allocation changes the comparison
A repeated observation from the same person shares that person’s random effect with the other observations. It can help distinguish the subject’s deviation from residual noise, but it does not create a new independent draw from the population distribution of random effects. In contrast, adding a subject adds another random-effects draw and another marginal data contribution. The relative value of these options depends on the target, the model, and the sampling times.
The following derivation makes this difference explicit for a simple repeated-measures model. Suppose a subject has \(m\) observations with the same sensitivities \(f\) and \(g\), and independent residual errors with variance \(\sigma^2\). The random intercept has variance \(\omega^2\). The marginal covariance matrix for that subject is \(V=\sigma^2 I_m+\omega^2 g^2\mathbf{1}\mathbf{1}^{\mathsf T}\), where \(I_m\) is the identity matrix and \(\mathbf{1}\) is a vector of \(m\) ones. The mean sensitivity to scalar \(\theta\) is \(f\mathbf{1}\). Since the covariance is fixed with respect to \(\theta\), marginal information is \(f^2\mathbf{1}^{\mathsf T}V^{-1}\mathbf{1}\). Because \(V\mathbf{1}=(\sigma^2+m\omega^2g^2)\mathbf{1}\), it follows that:
The same subject’s conditional data information for \(\eta_i\), when \(\theta\) is known, is \(mg^2/\sigma^2\). Thus the two quantities grow differently as \(m\) increases: population information can be limited by the shared random effect, whereas individual information can continue to accumulate from repeated residual-error observations.
Worked Example: Twelve Measurements, Two Allocations
Use the early-time sensitivities from the previous example, \(f=1\) and \(g=0.2\), with \(\omega^2=\sigma^2=1\). Compare twelve subjects receiving one observation each with three subjects receiving four observations each. The total number of observations is twelve in both plans.
For twelve independent subjects with one observation each, the population information for \(\theta\) is \(12\{1/(1+0.2^2)\}=12/1.04=11.54\). Each subject’s individual posterior variance for \(\eta_i\), conditional on known \(\theta\), is \(1/(1+0.2^2)=0.9615\).
For each of three subjects with four repeated observations, the marginal population information per subject is \(m f^2/(\sigma^2+m\omega^2g^2)=4/(1+4(0.2^2))=4/1.16=3.448\). Across three independent subjects, the total is \(3(3.448)=10.34\), less than 11.54. For each of those subjects, the individual posterior variance is \(1/(1+mg^2/\sigma^2)=1/(1+4(0.2^2))=1/1.16=0.8621\), lower than 0.9615.
Under these stated assumptions and this specific target, the first allocation gives more information about the population parameter, while the second gives better information about the random effect for each enrolled subject. The comparison does not say which allocation is preferable overall: that depends on whether the study needs population estimates, individual estimates, or both, as well as on practical constraints.
Worked Example: Complementary observations for two individual effects
Worked Example: Complementary Observations for Two Individual Effects
Suppose an individual has two independent random effects, \(\eta_i=(\eta_{i1},\eta_{i2})^{\mathsf T}\), with \(\Omega=I_2\), and residual variance 1. Treat the population parameters as known. At one candidate time, the observation’s sensitivity to these effects is \(g=(1,0)\); at another, it is \(g=(0,1)\). These are hypothetical local sensitivities.
With only the first observation, \(J_i^{\mathsf T}J_i\) is the matrix with diagonal entries 1 and 0. Posterior precision is therefore \(\begin{pmatrix}2&0\\0&1\end{pmatrix}\), and posterior covariance is \(\begin{pmatrix}1/2&0\\0&1\end{pmatrix}\). The first random effect is informed by the data; the second retains its prior variance.
With only the second observation, posterior covariance is \(\begin{pmatrix}1&0\\0&1/2\end{pmatrix}\). With one observation at each time, data information is \(I_2\), posterior precision is \(2I_2\), and posterior covariance is \(\frac{1}{2}I_2\). The two-time schedule informs both individual effects. This illustrates why a schedule judged by one scalar summary can conceal a weakly informed individual direction.
Assumptions, diagnostics, and limits
These comparisons rely on local information calculations. The sensitivities are evaluated at specified parameter values, and the result can change at other values. The model must represent the mean response, random-effects distribution, and residual error adequately for the information calculation to be useful. The examples also assumed independent subjects, independent residual errors, known variance components, and no missing observations or deviations from the planned schedule.
Before using a design comparison, check whether the assumed model and sensitivities are plausible across the range of parameters and subject profiles relevant to the study. In simulation, inspect whether the planned data can recover the intended target without systematic bias, excessive uncertainty, or frequent estimation failures. Warning signs include estimates near parameter bounds, poor separation of parameters with similar time profiles, high sensitivity of conclusions to assumed variance components, and individual estimates that remain close to the population distribution because the subject’s data are weak.
Fisher information is an expected, model-based precision measure. It does not by itself account for every practical objective, such as prediction at a particular future time, estimation of random-effects covariance, treatment comparisons, assay costs, or the consequences of missing samples. Nor does conditional information for \(\eta_i\) include all uncertainty in estimated population parameters. Those limitations matter if the design will be used for individual decisions rather than only for a model-based precision calculation.
Common pitfall and comparison of targets
Common mistake: “A schedule gives many observations per person, so it must be best for estimating the population model.” The reasoning ignores dependence induced by shared random effects. Repeated observations can improve the estimate of a person’s random effect, yet their marginal information about a population mean may be limited by between-subject variability. The correction is to calculate information for the stated target using the appropriate marginal or conditional model, and to compare allocations under the same resource constraints.
A population-based calculation integrates over random effects and asks how well the study estimates parameters common to the population. An individual-based calculation asks how much a subject’s own observations update information about that subject, given the population model. A prediction objective instead concerns uncertainty in a future outcome and can combine population uncertainty, individual uncertainty, and residual variability. These objectives can favor different schedules; selecting one requires stating the scientific decision the design is meant to support.
Check Your Understanding
Answer each question, then compare with the worked solutions below.
- In the one-observation model \(Y=\theta f+\eta g+\varepsilon\), with \(\eta\sim N(0,1)\) and residual variance 1, what are the population information for \(\theta\) and individual data information for \(\eta\) if \(f=0.5\) and \(g=0.8\)?
- Why can four observations from one subject provide less information about a population mean than one observation from each of four subjects?
- If a sampling time has zero sensitivity to one component of a two-dimensional random effect, what does the data contribute to information about that component at that time?
- What important uncertainty is omitted when calculating individual posterior precision conditional on known population parameters?
Solutions. (1) Marginal variance is \(1+0.8^2=1.64\), so population information is \(f^2/(1+g^2)=0.25/1.64=0.1524\). Individual data information is \(g^2=0.6400\). (2) Repeated observations share a subject’s random effect, so they are not independent population draws; new subjects add independent marginal contributions. The actual comparison depends on the model and schedule. (3) Its data-information contribution for that component is zero at that time, though its posterior distribution still reflects its prior and may be informed by other times. (4) It omits uncertainty in estimated population parameters, including uncertainty in the population mean and random-effects distribution.