Allocating Subjects Versus Samples Per Subject

Why Allocation Changes Information

A design with many subjects and few samples per subject is not generally equivalent to one with fewer subjects and denser profiles. Additional independent subjects replicate the population-level information contributed by a subject. Additional samples can improve the precision of that subject’s profile, but repeated observations from the same person are usually correlated through shared random effects. As a result, adding samples may yield diminishing returns for some parameters, while remaining essential for others.

The comparison depends on the estimand, the within-subject covariance, the sampling times, and the budget. Fisher information gives a local measure of how much precision a design can provide under a specified model and parameter value. It is not a universal rule that more subjects or more samples are always preferable.

For a multivariate normal observation vector from one subject, with mean \(\boldsymbol{\mu}(\boldsymbol{\theta})\) and covariance \(V(\boldsymbol{\theta})\), the information between parameters \(\theta_a\) and \(\theta_b\) is

This expression assumes a correctly specified multivariate normal model, a differentiable mean and covariance, and a nonsingular \(V\). The first term describes information carried by changes in the mean; the second describes information carried by changes in the covariance. If covariance parameters are known and only the mean parameters are of interest, the second term is absent. The formula also makes clear why a sample count alone is insufficient: information depends on parameter sensitivities and on the covariance among observations.

A Random-Intercept Result: Diminishing Returns for a Mean

Consider a simple model that isolates the effect of repeated measurements. For subject \(i\), sample \(j\) is

Here \(\theta\) is a population mean response, \(b_i\) is a subject-specific random intercept, and \(\varepsilon_{ij}\) is residual error. Assume the random intercepts are independent across subjects, residuals are independent across samples and subjects, and the two error terms are independent. Treat \(\omega^2\) and \(\sigma^2\) as known and positive. These assumptions are important: this calculation isolates information about \(\theta\), not the information for estimating variance components.

The \(m\) observations for one subject have mean \(\theta\mathbf{1}\) and covariance \(V=\sigma^2 I_m+\omega^2\mathbf{1}\mathbf{1}^{T}\), where \(\mathbf{1}\) is an \(m\)-vector of ones. Since \(V\mathbf{1}=(\sigma^2+m\omega^2)\mathbf{1}\), it follows that \(V^{-1}\mathbf{1}=\mathbf{1}/(\sigma^2+m\omega^2)\). Applying the mean-information formula gives

Thus \(N\) identical independent subjects contribute \(Nm/(\sigma^2+m\omega^2)\). At a fixed total sample count \(K=Nm\), this is \(K/(\sigma^2+m\omega^2)\): increasing \(m\) reduces the total information when \(\omega^2>0\), because it replaces independent subjects with more correlated observations on existing subjects. The gain from repeatedly sampling one subject approaches a limit: as \(m\) grows, that subject’s information approaches \(1/\omega^2\). New subjects continue to provide independent information about the population mean.

Allocation Under a Cost Budget

A fixed sample count is only one possible constraint. If enrolling a subject has a cost in addition to collecting each sample, the preferred allocation can change. Suppose each subject costs \(c_0>0\), each sample costs \(c_1>0\), and the total budget is \(B\). For a common integer \(m\), the approximate number of subjects is \(N=B/(c_0+c_1m)\). In the random-intercept mean example, total information per budget unit is therefore proportional to

Treating \(m\) temporarily as continuous, differentiate this expression. Its denominator is positive, so its derivative has the sign of the numerator obtained by the quotient rule:

The function increases while \(c_0\sigma^2-c_1\omega^2m^2>0\) and decreases after it becomes negative. Its positive continuous maximum is consequently at \(m=\sqrt{c_0\sigma^2/(c_1\omega^2)}\). Actual designs require integer \(m\), feasible subject counts, and any minimum sampling schedule; the continuous value is a guide, not the final allocation.

When Denser Profiles Are Necessary

The random-intercept example has a single target parameter. In a nonlinear PK model, a profile may need to distinguish several population parameters. If the sensitivity patterns at the sampled times are similar, adding subjects at those same times can increase information without resolving the weakly distinguished parameter combination. A denser or differently timed profile can change the information matrix’s rank or improve its weak directions.

The following local example illustrates the distinction. Let the mean response near a nominal parameter value be \(\mu(t)=\theta_0+\theta_1t\), with time \(t\) in hours. Each subject also has a random intercept with variance 1 response-unit-squared, and each residual has variance 1 response-unit-squared; both components are known. Compare 20 subjects sampled once at \(t=0\) with 10 subjects sampled at \(t=0\) and \(t=1\). Both designs use 20 samples. The mean sensitivity row is \((1,t)\).

How to Compare Designs and Detect Failure

Before recommending an allocation, specify the target parameters and the assumptions under which its information is calculated. A sound comparison should examine the whole information matrix, not just the number of observations or one diagonal entry. For a locally identifiable parameter vector, the inverse information approximates the covariance matrix of an efficient estimator in a large-sample regime. A singular matrix indicates a locally unidentified direction; a nearly singular matrix signals that some parameter combination will be imprecise.

A useful comparison sequence is:

The main alternative to a simple information calculation is direct evaluation of a design criterion, such as a determinant-based or average-variance criterion. Such criteria can summarize a multivariate trade-off, but they still depend on the model, nominal parameter values, covariance assumptions, and chosen parameter scaling. A criterion cannot rescue a design whose observations contain no sensitivity to a target parameter. Simulation can also compare expected estimation performance, but its results depend on the data-generating and analysis models and should not be confused with a model-free guarantee.

Common Pitfalls and Limits

• Counting repeated samples as independent. This treats correlated observations as though each were a new subject and overstates information. Use the within-subject covariance in the calculation.
• Assuming information always scales with total samples. It does so only under restrictive independence and design conditions. A random intercept can make repeated samples increasingly redundant for a population mean.
• Optimizing the wrong target. More subjects may improve population-level precision, while repeated within-subject measurements may be necessary to distinguish parameters that affect the profile differently over time.
• Using known-variance results when variances must be estimated. The scalar formula above assumes \(\sigma^2\) and \(\omega^2\) are known. Estimating them changes the joint information problem and may alter allocation preferences.
• Treating a local calculation as a guarantee. NLME information can depend on nominal parameter values. Model misspecification, poor numerical conditioning, or a different true parameter value can make realized precision worse than predicted.

The practical conclusion is conditional: independent subjects are especially valuable when the target is a population quantity affected by between-subject variation, whereas denser profiles are valuable when within-subject time patterns are needed to separate parameters. The allocation should be judged against the actual estimand, covariance assumptions, and resource constraint.