Why More Samples per Patient Cannot Fully Replace More Patients

Two Different Kinds of Information

A repeated concentration profile can contain many measurements, but those measurements do not represent as many independent patients. Adding samples to the same patient can show more clearly how that patient’s concentration changes over time. Adding patients gives more independent examples of how profiles differ across the population. These are related but distinct design goals.

The distinction matters when choosing a study design. A dense schedule in a handful of patients may describe their observed profiles well, while providing little basis for assessing how much patients differ from one another. Conversely, a larger group measured at fewer times may better represent population diversity but give less detail about each individual profile. The design should reflect which information is needed; a large count of concentration measurements alone does not settle that question.

A Simple Variance Decomposition

A basic model makes the distinction precise. Suppose each of \(n\) independent patients is measured \(m\) times at a common setting, and let \(Y_{ij}\) be the measurement for patient \(i\) on occasion \(j\). For this illustration, use a random-intercept model:

Here \(\mu\) is the population mean at the setting being considered, \(b_i\) is patient \(i\)’s deviation from that mean, and \(e_{ij}\) is measurement-level variation. Assume \(E(b_i)=0\), \(\operatorname{Var}(b_i)=\tau^2\), \(E(e_{ij})=0\), and \(\operatorname{Var}(e_{ij})=\sigma^2\). Assume the patient effects are independent across patients, the measurement errors are independent across occasions and patients, and the two sources of variation are independent. This is a deliberately simple illustration: real profiles can change with time, and measurements within a patient may remain correlated after accounting for a patient effect.

The average across all measurements is \(\bar{Y}_{..}=(1/n)\sum_{i=1}^n\bar{Y}_i\), where \(\bar{Y}_i=(1/m)\sum_{j=1}^mY_{ij}\). The patient average is \(\bar{Y}_i=\mu+b_i+\bar{e}_i\). Thus, its variance is \(\tau^2+\sigma^2/m\): repeated observations reduce the measurement-level contribution to the uncertainty in that patient’s average, but they do not reduce the patient’s deviation \(b_i\). Averaging the \(n\) independent patient averages gives the result below.

The first term, \(\tau^2/n\), reflects uncertainty from sampling patients who differ from the population mean. The second, \(\sigma^2/(nm)\), reflects measurement-level variation averaged over all observations. More occasions per patient reduce the second term. They do not create additional independent patient effects, so they do not replace a larger \(n\) for learning about between-patient variation.

Worked Examples

Why the Patient Count Sets a Limit

The variance formula also gives a short justification for the limit of dense sampling. If \(n\) is fixed and the number of measurements per patient increases, then the measurement contribution \(\sigma^2/(nm)\) approaches zero. The remaining variance approaches \(\tau^2/n\):

If patients truly vary, so that \(\tau^2>0\), the uncertainty does not disappear merely by taking more measurements from the same finite group. This is not a universal formula for every repeated-measures design; it follows from the assumptions stated for the random-intercept example. But the underlying design principle is broader: repeated observations from a patient share that patient’s characteristics, so they cannot fully stand in for independent patients when the target is population diversity.

Choosing and Checking a Design

A design should allocate sampling effort according to the question. If the priority is to trace the time course within an individual, more measurement times can improve coverage. If the priority is to characterize between-patient differences, recruit more independent patients. Many studies need both, so a practical compromise is to sample enough patients to represent the target population and collect enough observations per patient to address the profile questions.

The simple calculation is useful only when its assumptions are plausible. Check whether measurements at the same nominal setting are comparable, whether patients are sampled independently from the population of interest, and whether occasions within a patient have additional dependence. If within-patient errors are correlated, the \(\sigma^2/m\) reduction does not apply as written. Unequal schedules, dropout, or sampling patterns related to patient characteristics can also change what the pooled measurements represent; the earlier material on informative sampling and dropout addresses those concerns.

A common mistake is to count every concentration as a new independent patient when judging how much evidence a study has about population variability. That reasoning confuses the measurement count with the number of independent units. The correction is to identify the target first: repeated samples inform a patient’s observed trajectory, while independent patients supply evidence about how patients differ. A mixed-effects analysis can represent both sources of variation, but it cannot manufacture patient diversity that the study did not sample.

Another mistake is to conclude that more patients always dominate more samples. That is also too broad. For example, when the main goal is to describe a single patient’s profile, one sample per patient may be insufficient regardless of the number enrolled. The trade-off depends on the estimand and on the sources of variation; the variance decomposition is a guide to the distinction, not a substitute for specifying the study objective.