Allocating Subjects Versus Within-Subject Measurements

Why the Allocation Matters

A study can often spend resources in two different ways: recruit another independent subject, or collect another measurement from someone already enrolled. These choices are not interchangeable. Measurements from the same subject can share random effects, so they are generally correlated; a new subject adds another independent realization of those effects. Which choice is more informative depends on the parameter of interest, the measurement schedule, and the costs.

The comparison below uses a specified approximate mixed-effects model to make these distinctions explicit. The calculation is local: for a nonlinear response model, its mean is understood as linearized around a nominal parameter value, and the resulting marginal distribution is approximated as Gaussian. The goal is to compare allocations under that approximation, not to claim that every nonlinear mixed-effects model has the same allocation rule.

A Random-Intercept Approximation

Consider the following linear mixed-effects model for subject \(i\) at measurement occasion \(j\):

Assume that the random intercepts \(b_i\) are independent across subjects, the residuals are independent across subjects and occasions, and random intercepts and residuals are independent. For the calculations that follow, \(\omega^2\) and \(\sigma^2\) are treated as known, positive variances, and the target is the population mean \(\beta_0\). These assumptions make it possible to isolate the effect of allocation on information for the mean.

If subject \(i\) contributes \(m\) measurements, their marginal mean vector is \(\beta_0\mathbf{1}\) and their marginal covariance matrix is \(V_m=\omega^2\mathbf{1}\mathbf{1}^{\mathsf T}+\sigma^2 I_m\), where \(\mathbf{1}\) is an \(m\)-vector of ones and \(I_m\) is the \(m\)-by-\(m\) identity matrix. The covariance has diagonal entries \(\omega^2+\sigma^2\) and off-diagonal entries \(\omega^2\). Thus, repeated measurements are correlated even though the residual errors are independent.

For a Gaussian model with known covariance and a mean that depends on \(\beta_0\), the Fisher information for \(\beta_0\) from one subject is the mean derivative transposed, multiplied by the inverse covariance, and multiplied by the mean derivative. Here that derivative is \(\mathbf{1}\), so the subject contributes \(\mathbf{1}^{\mathsf T}V_m^{-1}\mathbf{1}\). A short calculation gives a useful closed form.

Why the Formula Has Diminishing Returns

To verify the result, note that \(V_m\mathbf{1}=(\sigma^2+m\omega^2)\mathbf{1}\). Therefore \(V_m^{-1}\mathbf{1}=\mathbf{1}/(\sigma^2+m\omega^2)\), and the information is

When \(\omega^2=0\), observations within a subject are independent and each adds \(1/\sigma^2\) information for the mean. When \(\omega^2>0\), the information from one subject approaches \(1/\omega^2\) as \(m\) grows: increasingly many readings estimate that subject's level more precisely, but cannot eliminate the subject-to-subject deviation \(b_i\). For \(N\) independent subjects, the total information continues to increase with \(N\), because each new subject supplies another independent random intercept.

For a Gaussian mean parameter with known covariance, the variance of the maximum-likelihood estimator of \(\beta_0\) is the reciprocal of total information. Thus, if two candidate designs differ only in their allocation, this variance provides a direct comparison of their local precision under the model. It is not a guarantee about performance if the covariance structure or other assumptions are wrong.

Adding a Cost Constraint

An allocation comparison is incomplete when subject recruitment and measurement costs differ. Let \(c_s\) be the cost of enrolling a subject and \(c_m\) the cost of each measurement. A balanced design with \(N\) subjects and \(m\) measurements per subject costs \(N(c_s+m c_m)\) and has the information given above. With a fixed budget, a larger \(m\) may be worthwhile if it avoids expensive recruitment, even though each additional measurement has diminishing information for the mean.

The Target Parameter Can Change the Answer

The random-intercept calculation is specifically about \(\beta_0\). If the target is a within-subject change or slope, repeated measurements at informative times can be especially valuable. Consider the extension \(Y_{ij}=\beta_0+\beta_1t_{ij}+b_i+\varepsilon_{ij}\), with the same random-intercept and residual assumptions. Two measurements at times \(-1\) and \(+1\) have a difference in which \(b_i\) cancels. This is a different information question from estimating the population intercept.

Diagnostics, Alternatives, and Limitations

These comparisons are useful for planning only when the model and the allocation being compared are stated clearly. In practice, inspect whether estimated within-subject correlations are compatible with the assumed covariance, whether residual variability changes over time or with the mean, and whether the proposed schedule samples the parts of the response where the target parameter affects the mean. In a nonlinear model, a local linearization can be poor when the response changes sharply over the scheduled times or nominal parameter values are uncertain. Information calculations should then be treated as local approximations rather than definitive predictions.

A key limitation of the closed-form mean calculation is that it treats \(\omega^2\) and \(\sigma^2\) as known. If variance components are also targets, measurements within a subject can help distinguish residual variation from between-subject variation; the formula above does not quantify that benefit. Similarly, unequal schedules, missing observations, covariates, and random slopes require their own subject-level information calculations. The broad principle remains that independent subjects and repeated measurements contribute through different covariance structures, not as interchangeable independent records.

A main alternative to a balanced allocation is an unequal or adaptive schedule, for example assigning more measurements to subjects at times expected to be informative for a slope. Such a design may improve information for a particular target but can complicate comparisons and may be sensitive to model misspecification. The balanced calculations here are transparent benchmarks, not a substitute for evaluating the actual candidate schedules under the intended model.

• Pitfall: counting every measurement as an independent subject. This ignores the shared random intercept and overstates information for the population mean. Use the subject's marginal covariance, then sum contributions across independent subjects.
• Pitfall: concluding that more subjects always dominate repeated measurements. That conclusion does not follow for slope or variance-component targets; paired measurements can remove shared subject effects from contrasts.
• Pitfall: comparing allocations without specifying cost. Equal numbers of measurements need not have equal recruitment and measurement costs. State the budget and feasible schedules before ranking designs.