Why Counting Records Can Mislead
A pharmacokinetic data set may contain many concentration records but relatively few people. For example, each participant may contribute a concentration measurement after several standardized doses. The records are not automatically independent: measurements from the same person can resemble one another because that person tends to have higher or lower concentrations than others. Treating every record as if it came from a different person is pseudoreplication. It can make the sample size appear larger than the number of independent subjects and make uncertainty look too small.
To isolate this issue, consider the population mean concentration at one prespecified post-dose time. Suppose each subject contributes measurements at that same nominal time on several standardized occasions. The target is the mean concentration in the subject population at that time, not a mean that mixes different points on the concentration-time curve. In actual PK analyses with measurements at different times, time must be accounted for; raw concentrations at different times cannot simply be treated as repeated measurements of one common mean.
A Simple Model for Repeated Concentrations
For a concentration measurement \(Y_{ij}\) from subject \(i\) on occasion \(j\), use the random-intercept model
Here \(\mu\) is the population mean concentration at the specified time, \(b_i\) is subject \(i\)'s departure from that mean, and \(e_{ij}\) is the occasion-specific deviation. Assume \(E(b_i)=0\), \(\operatorname{Var}(b_i)=\tau^2\), \(E(e_{ij})=0\), and \(\operatorname{Var}(e_{ij})=\sigma^2\). Subject effects are independent across subjects; residuals are independent across occasions and subjects and independent of the subject effects. These assumptions imply that measurements from different subjects are independent, while measurements from the same subject share \(b_i\).
The model gives \(E(Y_{ij})=\mu\), and the marginal variance of an individual record is \(\tau^2+\sigma^2\). Two measurements from the same subject have covariance \(\tau^2\), because they share the same subject effect. Their within-subject correlation is therefore \(\rho=\tau^2/(\tau^2+\sigma^2)\). When \(\tau^2>0\), that correlation is positive: one subject's records tend to move together.
With the same number \(m\) of measurements for each of \(n\) subjects, the pooled mean is
It is an unbiased estimator of \(\mu\) under the stated mean assumptions. Its being unbiased does not make an independence-based standard error valid. The issue is its sampling variance.
Why the Naive Standard Error Is Too Small
The pooled mean can also be written as the average of the \(n\) subject means, where \(\bar{Y}_{i.}=m^{-1}\sum_jY_{ij}\). Under the model, \(\bar{Y}_{i.}=\mu+b_i+\bar{e}_{i.}\), so its variance is \(\tau^2+\sigma^2/m\). Because subject means are independent across subjects, averaging them gives
An analysis that falsely treats all \(nm\) records as independent uses the marginal record variance divided by \(nm\), namely \((\tau^2+\sigma^2)/(nm)\). The ratio of the correct variance to that independence-based variance is
Thus, when \(m>1\) and \(\rho>0\), the independence-based variance is smaller. The true standard error is larger by the square root of this ratio. This derivation assumes a balanced design and the stated random-intercept covariance structure; it is not a universal correction for every repeated-measures data set.
Worked Example: Four Subjects With Three Concentrations Each
Suppose a hypothetical study measures concentration at 2 hours after each of three standardized doses for each of four subjects. The values below are in ng/mL. The simplified analysis targets the population mean at that sampling time.
| Subject | Occasion 1 | Occasion 2 | Occasion 3 | Subject mean |
|---|---|---|---|---|
| A | 8 | 9 | 10 | 9 |
| B | 12 | 13 | 14 | 13 |
| C | 18 | 19 | 20 | 19 |
| D | 22 | 23 | 24 | 23 |
There are 12 concentration records, but only four independent subjects. The record-level mean is \((8+9+\cdots+24)/12=192/12=16.0\) ng/mL. The mean of the four subject means is \((9+13+19+23)/4=64/4=16.0\) ng/mL, the same point estimate because each subject contributed three records.
Incorrect independence-based calculation. The sum of squared deviations of the 12 records from 16 is 356. The record-level sample variance is \(s^2=356/(12-1)=32.36\;(\text{ng/mL})^2\), so the naive standard error is \(SE=s/\sqrt{12}=\sqrt{32.36/12}=1.642\) ng/mL. An ordinary \(t\) interval would use \(df=11\). With \(t^*=2.201\), the nominal 95% interval is \(16.0\pm2.201(1.642)\), or 12.39 to 19.61 ng/mL.
Subject-level calculation. The subject means are 9, 13, 19, and 23 ng/mL. Their sample variance is \([(-7)^2+(-3)^2+3^2+7^2]/3=116/3=38.67\;(\text{ng/mL})^2\). The standard error across four independent subject means is \(\sqrt{38.67/4}=3.109\) ng/mL. Using \(df=3\) and \(t^*=3.182\), the 95% \(t\) interval is \(16.0\pm3.182(3.109)\), or 6.11 to 25.89 ng/mL.
These intervals differ substantially because the first calculation counts within-subject similarity as if it were independent information. The subject-level interval is appropriate for this simple analysis if subject means are independent and approximately normally distributed. With only four subjects, the interval is necessarily imprecise. Neither interval is a general substitute for a carefully specified longitudinal PK model.
Worked Comparisons and Practical Checks
Worked Example: Quantifying the Uncertainty Increase
Suppose, hypothetically, that the marginal standard deviation of concentration at the chosen time is 5 ng/mL, there are three records per subject, and the within-subject correlation is \(\rho=0.60\). Assume the balanced random-intercept model above applies.
An independence calculation for three records would give an SE of \(5/\sqrt{3}=2.887\) ng/mL for their mean. The variance ratio is \(1+(3-1)(0.60)=2.20\), so the appropriate standard error under the model is \(2.887\sqrt{2.20}=4.282\) ng/mL. Equivalently, using the variance components, \(\tau^2=0.60(25)=15\) and \(\sigma^2=0.40(25)=10\); the subject-mean variance is \(15+10/3=18.33\), whose square root is 4.282 ng/mL. The records still contribute to estimating the mean, but their precision is not the same as three independent subjects.
Worked Example: Same Record Count, Different Subject Count
Compare two hypothetical designs with 20 total concentration records and marginal variance \(25\;(\text{ng/mL})^2\), with within-subject correlation \(\rho=0.50\). Assume the same balanced random-intercept model and the same target mean at a fixed time.
In Design A, five subjects each contribute four records. The variance of the mean relative to the variance that would apply to 20 independent records is multiplied by \(1+(4-1)(0.50)=2.50\); the standard error is multiplied by \(\sqrt{2.50}=1.581\). In Design B, 20 subjects each contribute one record, so there is no within-subject repetition and that multiplier is 1. The number of records is equal, but the designs do not provide equal independent-subject information when repeated records are correlated.
This comparison is conditional on the model and fixed-time target. It does not mean that one design is always preferable: sampling times, subject burden, the PK question, and how well the model can be estimated also matter.
Several checks help reveal pseudoreplication. First, identify the subject identifier and count distinct subjects, not just rows. Second, inspect a subject-by-occasion table or a plot connecting records from the same subject; visible clustering is a warning that independence is implausible. Third, compare a record-level analysis with a subject-level or repeated-measures analysis. A dramatic reduction in the estimated standard error under the record-level method deserves investigation. Finally, confirm that the records being averaged target a common quantity. A concentration at 1 hour and one at 8 hours are not exchangeable replicates of the same mean merely because both are concentration measurements.
Choosing an Analysis and Recognizing Its Limits
For a simple fixed-time population mean with equal numbers of repeated observations, a subject-level analysis can average the repeated values within each subject and then calculate the mean and standard error across subjects. This respects the independent sampling unit, though it compresses the repeated records into one summary per person. A mixed-effects model can instead represent subject-level variation and within-subject residual variation separately, and is often more useful when the design is unbalanced or the scientific target involves a trajectory. The record-level independent analysis is only defensible when the observations truly are independent under the sampling design and model; repeated observations from one subject generally do not meet that condition.
The simple derivation has limits. It assumes the same number of observations per subject, a common mean at the selected time, a random intercept with a particular covariance structure, and residuals that are independent after accounting for that intercept. Real PK profiles have systematic changes over time, may have different residual variability at different concentrations, and may include occasion effects or missing samples. A random-intercept calculation is a teaching model, not a complete analysis plan for every PK study. The central lesson remains: uncertainty calculations must reflect how the data were collected and which observations share a subject.
Specify whether the question concerns a mean at one time, a subject-level summary, or a concentration-time trajectory.
Count subjects and identify which records belong to each subject.
Use a subject-level summary or a repeated-measures model rather than treating clustered records as independent without justification.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- A hypothetical study measures a concentration at the same nominal time on four occasions for each of 12 subjects. How many records and how many independent subjects are there?
- In the random-intercept model, why are two measurements from the same subject correlated?
- With \(m=4\) and \(\rho=0.25\), by what factor does the model-based standard error exceed the independence-based standard error, under the balanced model?
- Why is it generally invalid to pool concentrations measured at different post-dose times as repeated observations of one common mean?
Solutions. (1) There are \(12(4)=48\) records and 12 independent subjects. (2) Both measurements contain the same subject effect \(b_i\), so their covariance is \(\tau^2\); the shared subject tendency makes them dependent when \(\tau^2>0\). (3) The variance ratio is \(1+(4-1)(0.25)=1.75\), so the standard-error factor is \(\sqrt{1.75}=1.323\). (4) The population concentration generally changes with post-dose time, so different-time records do not share a common mean; time must be part of the target and analysis.