Participants Versus Samples: Allocating Study Effort

Two Ways to Spend a Study’s Sampling Effort

A pharmacokinetic study can collect more information in two basic ways: enroll more participants, or collect more concentration samples from each participant. These choices are not interchangeable. More participants can better represent differences across people; more samples from one person can reveal more of that person’s concentration-time profile. The useful allocation depends on the question, the expected variability, the sampling times, and the practical burden of each collection.

For example, a study with many participants and one sample per person may show a broad range of concentrations at a selected time, but cannot trace each person’s profile over time. A study with few participants and many samples per person may trace those profiles well, but provide limited coverage of differences across the target population. Counting samples alone misses this distinction.

The earlier material on repeated samples and within-subject information distinguishes observations from different participants from repeated observations within one participant. Here the design question is how to allocate a limited collection effort between those two sources. In population pharmacokinetics, the aim is generally to learn about a population while also using concentration-time data to describe pharmacokinetic behavior. A design must support both aims to a degree appropriate to its purpose.

What More Participants and More Samples Can Contribute

Adding participants creates additional opportunities to observe different body sizes, covariates, clinical characteristics, and pharmacokinetic responses. This can improve representation of the target population and provide more information about between-person variability. It does not guarantee representativeness: a large group selected from an unrepresentative source may still fail to cover the population of interest.

Adding samples to an existing participant can clarify that participant’s concentration-time course. For example, appropriately timed observations may help distinguish a rising phase from a declining phase, or reveal whether the observed concentrations are consistent with a smooth profile. Repeated samples can also reduce the influence of measurement noise on a within-person summary, provided the observations contribute relevant information and their errors are not perfectly correlated.

However, repeated observations from one person do not create additional independent people. If a study has many samples from a small number of participants, its concentration record can be extensive while its coverage of population heterogeneity remains limited. Conversely, a large participant count does not automatically provide rich individual profiles, particularly if each person contributes only one observation.

A Simple Model Shows Why Repeats Do Not Replace People

A simplified model makes one part of this trade-off explicit. Suppose repeated measurements are taken under comparable conditions, and write the \(j\)th observation from participant \(i\) as

Here \(\mu\) is a population mean, \(b_i\) is the deviation for participant \(i\), and \(e_{ij}\) is measurement-level variation. Assume the participants are independent draws from the target population, \(E(b_i)=0\), \(\operatorname{Var}(b_i)=\tau^2\), \(E(e_{ij})=0\), and \(\operatorname{Var}(e_{ij})=\sigma^2\). Also assume the \(e_{ij}\) values are independent across observations and independent of the \(b_i\) values. These are simplifying assumptions, not a complete pharmacokinetic model.

With \(N\) participants and \(m\) measurements per participant, the grand mean is

Substituting the model into this expression gives

The participant effects are independent across \(N\) people, so their contribution to the variance of \(\bar{Y}\) is \(\tau^2/N\). Each of the \(Nm\) independent measurement errors has weight \(1/(Nm)\), so their total contribution is \(Nm\sigma^2/(Nm)^2=\sigma^2/(Nm)\). Thus,

The first term decreases when there are more participants; increasing the number of repeated measurements alone does not remove it. The second term decreases as the total number of independent measurements increases. This result explains the basic allocation trade-off for this simple mean, but it should not be mistaken for a general formula for the precision of pharmacokinetic parameter estimates.

Profile Information Depends on Timing, Not Just Count

The simple variance calculation assumes comparable observations with independent measurement errors. Pharmacokinetic measurements usually occur at different times, and concentrations at different times do not carry identical information. A sample taken during a rapidly changing part of a profile may help describe that change; another sample close in time to an existing observation may add less distinct information. Samples at poorly chosen times can leave important parts of the profile unclear even when there are many of them.

This means the relevant comparison is not always “more samples” versus “more people” in the abstract. It is the comparison between the specific opportunities each design creates: how many different participants are observed, what parts of their concentration-time courses are measured, and whether measurements from the same person complement one another. A single measurement per participant can contribute to a population picture at one time, but it cannot by itself show that person’s trajectory.

Practical Checks, Alternatives, and Limits

Before allocating effort, identify what the study most needs to learn. If the main uncertainty is whether the enrolled people reflect important population differences, prioritize recruiting participants who represent the target group. If the main uncertainty concerns how concentrations evolve within a person, consider whether additional appropriately timed observations would fill meaningful gaps. Often the design must balance both priorities rather than maximize only one.

Useful checks include the number of distinct participants, the range of relevant participant characteristics, the number of observations contributed by each person, and the spread of actual sampling times. A nominal schedule may not describe the data collected if visits are missed or samples are taken at different times than planned. Also check whether a few participants contribute most of the observations: a high total count can disguise limited population coverage.

When a study needs population representation and individual profiles, one alternative to concentrating all effort in either direction is to distribute effort across participants while still collecting multiple informative measurements per person. The best allocation cannot be selected from the total number of samples alone. More formal design comparison can evaluate candidate schedules against particular parameter or precision goals, but that requires a specified model and assumptions beyond the simple calculation here.

The random-intercept calculation has important limits. It describes a common mean under a simple variance structure; it does not represent a changing concentration-time curve, different variances at different times, correlated errors, or all features of a population pharmacokinetic model. If repeated measurements are correlated beyond the shared participant effect, their contribution is not equivalent to adding independent measurements. If participants are selectively enrolled or observations are missing in a systematic way, the independent, representative-sampling assumptions may also fail.

Common Mistake

The opposite oversimplification is to conclude that more participants are always better. If the question depends on how an individual profile changes over time, adding people who each contribute only an uninformative observation may not answer it. Study effort is best evaluated by the distinct information the design can provide, not by either count in isolation.