Why Incomplete Profiles Need Endpoint-Specific Decisions
A missing concentration does not have one universal consequence for a bioequivalence (BE) analysis. Its impact depends on where the gap occurs, which endpoint is being calculated, and whether the missing information belongs to one scheduled sample or an entire subject-period profile. A missing point near the peak may compromise \(C_{\max}\), while a gap in a well-sampled middle portion of the profile may have a smaller effect on total AUC. A missing late sample can also affect estimation of the terminal phase and \(AUC_{\infty}\).
The first task is to describe what is absent. A concentration that was not collected is missing; it is not automatically zero and is not the same as a below-quantification result. The earlier material on BLQ concentrations treats those results as censored analytical observations under specified handling rules. For an uncollected sample, there is no measured concentration to classify as BLQ. Likewise, a subject with no usable profile is different from a subject who has a profile with one missing time point.
Before calculating an endpoint, verify that the dose time and sampling times are known, the available concentrations are valid records, and the gap has been described accurately. Endpoint calculations also rely on the usual noncompartmental assumptions: the recorded concentrations represent the sampled subject and period, the sampling schedule is adequate for the endpoint, and the chosen numerical integration rule is appropriate for the available points. Missingness does not itself justify changing the endpoint definition or inventing a concentration.
What a Gap Does to AUC
With measured concentrations \(C_i\) and \(C_{i+1}\) at times \(t_i\) and \(t_{i+1}\), the linear trapezoidal area for that interval is
When a scheduled interior point is missing, the ordinary calculation cannot be performed for the two intervals that would have used that point. One computational option is to connect the available measurements on either side of the gap with a single trapezoid. This gives an area for the longer interval, but it is an approximation based on the two observed endpoints. It does not reconstruct the missing concentration or guarantee the same AUC as integration through the unobserved profile.
Worked Example: Bridging a Missing Interior Sample
In a hypothetical period, concentrations are measured at 0, 1, 2, 4, and 8 hours. The values are 0, 4, 7, missing, and 2 mg/L, respectively. Use the linear trapezoidal rule on the available points and state what the result means.
The observed areas from 0 to 1 hour and from 1 to 2 hours are:
If the available observations at 2 and 8 hours are bridged directly, their area is:
The calculated \(AUC_{0-8}\) is therefore \(2+5.5+27=34.5\) mg·h/L. This is an observed-data calculation with a six-hour bridge, not a recovered complete-profile AUC. For illustration, if the unobserved concentration at 4 hours had been 5 mg/L, the two areas from 2 to 4 and 4 to 8 hours would instead be \(12\) and \(14\) mg·h/L, for a total \(AUC_{0-8}\) of \(2+5.5+12+14=33.5\) mg·h/L. The actual missing value is unknown, so that alternative is not evidence about the subject’s true concentration.
A longer bridge may be reasonable to report as a clearly identified observed-data calculation when the endpoint and analysis plan permit it, but it may conceal profile curvature. Linear interpolation to create an unmeasured concentration is a different choice: it imposes a straight-line assumption, and any resulting endpoint is partly based on an imputed value. A model-based estimate imposes still other assumptions. Neither approach turns a missing measurement into an observed one. Any imputation or alternative calculation should be prespecified when possible, justified, documented, and examined in sensitivity analyses rather than selected because it gives a preferred BE result.
Why the Location of the Gap Matters
\(C_{\max}\) is the largest observed concentration in the profile, and \(T_{\max}\) is the time at which that observed maximum occurs, subject to the study’s rules for ties. If a scheduled sample near the peak is missing, the largest remaining observation may not capture the peak that would have been recorded with complete sampling. Thus an observed \(C_{\max}\) can understate the maximum across the planned sampling times, and the observed \(T_{\max}\) may shift to an earlier or later time. The degree of concern depends on the gap’s position and the sampling schedule; it cannot be corrected reliably by assuming the peak’s shape from distant observations alone.
Worked Example: A Missing Sample Near the Peak
Consider a hypothetical profile with sampling at 0, 3, 6, 9, and 12 hours. The concentrations are 0, 8, missing, 7, and 3 mg/L. Calculate an observed-data \(AUC_{0-12}\) by bridging the gap, and assess \(C_{\max}\).
The areas from 0 to 3 and 9 to 12 hours are \( (0+8)(3)/2=12\) and \((7+3)(3)/2=15\) mg·h/L. Bridging from 3 to 9 hours gives \((8+7)(6)/2=45\) mg·h/L. The resulting \(AUC_{0-12}\) is \(12+45+15=72\) mg·h/L.
The largest measured concentration is 8 mg/L at 3 hours, so the observed-data \(C_{\max}\) is 8 mg/L and the observed-data \(T_{\max}\) is 3 hours. These are not proof that the complete scheduled profile would have had its maximum at 3 hours: the unmeasured 6-hour concentration could have been higher or lower than 8 mg/L. For example, if it had been 12 mg/L, the two areas from 3 to 6 and 6 to 9 hours would be 30 and 28.5 mg·h/L, respectively, giving \(AUC_{0-12}=12+30+28.5+15=85.5\) mg·h/L. That hypothetical value illustrates uncertainty; it is not a replacement for the missing result.
Terminal samples raise a related but distinct concern. Under the usual noncompartmental calculation, \(AUC_{\infty}\) is estimated as \(AUC_{0-t}+C_{\text{last}}/\lambda_z\), where \(C_{\text{last}}\) is the last quantifiable concentration and \(\lambda_z\) is the estimated terminal log-linear elimination rate constant. Estimating \(\lambda_z\) requires an adequate set of suitably spaced terminal observations that reasonably represent a log-linear decline. If a missing sample removes a point needed to characterize that decline, the slope and extrapolated area may become less reliable. Report the observed \(AUC_{0-t}\) separately where relevant, and do not assume that an extrapolation is dependable merely because software returns a value. The fraction extrapolated and the adequacy of the terminal fit help judge whether \(AUC_{\infty}\) is usable.
Worked Example: A Missing Terminal Measurement
In a hypothetical profile, measured terminal concentrations are 4 mg/L at 8 hours and 2 mg/L at 12 hours; the planned 16-hour concentration is missing. Suppose, solely to illustrate the calculation, the available terminal decline supports \(\lambda_z=0.1733\) per hour. The last measured concentration is 2 mg/L at 12 hours, and the calculated \(AUC_{0-12}\) is 40 mg·h/L.
Using the stated slope, the estimated residual area is:
The resulting \(AUC_{\infty}\) estimate is \(40+11.54=51.54\) mg·h/L, and the extrapolated fraction is \(11.54/51.54=0.224\), or 22.4%. These arithmetic results do not establish that the estimate is reliable: only two terminal observations were supplied, so this example does not demonstrate a well-supported terminal regression. The missing 16-hour point could have informed the slope. The appropriate conclusion is that the numerical extrapolation is conditional on an assumed \(\lambda_z\); whether \(AUC_{\infty}\) is evaluable requires adequate terminal data under the analysis rules.
Missing Samples Versus Missing Subject-Period Profiles
A missing sample leaves some information for that subject-period; a missing profile leaves no concentration-derived endpoint for that period. In a crossover study, the usual within-subject comparison requires usable endpoints under both treatments for the relevant subject. If one period’s endpoint is unavailable, that subject cannot contribute a complete paired treatment contrast in a simple complete-case paired analysis. Other analysis approaches may use incomplete records under their specified statistical assumptions, but they do not recover the unobserved treatment-period endpoint. The choice of analysis population and method must follow the protocol and prespecified analysis plan.
Missingness also deserves a substantive description. It may be unrelated to concentrations, related to observed information, or related to unobserved concentrations—for example, if a high concentration is associated with an event that prevents later sampling. These possibilities can affect how plausible a missing-data assumption is. They are not determined merely by counting missing values. State how many samples and profiles are missing, their timing and treatment periods, and the documented reasons where known.
Worked Example: One Missing Period Endpoint in a Crossover
Suppose four hypothetical subjects have log-transformed AUC endpoints for test (T) and reference (R). The available paired log differences, defined as \(\log(AUC_T)-\log(AUC_R)\), are 0.05 for subject 1, −0.02 for subject 2, missing for subject 3 because the test-period profile is unusable, and 0.01 for subject 4. For a simple complete-pair summary, calculate the mean log difference and its exponentiated ratio.
Subject 3 contributes no paired difference because the test-period endpoint is absent; the missing value is not zero. The mean of the three available paired differences is:
Exponentiating gives \(\exp(0.01333)=1.013\), so the geometric mean ratio represented by this limited summary is approximately 1.013, or 101.3%. This calculation is not a BE conclusion: it omits a subject, has no confidence interval, and does not assess the required acceptance limits. It shows why a missing subject-period endpoint changes the available paired analysis rather than supplying a neutral contribution.
Diagnostics, Alternatives, and Limitations
Before fitting the BE model, make a subject-by-period inventory of scheduled, collected, valid, and missing samples. Check whether gaps cluster near \(T_{\max}\), in the terminal phase, or in one treatment or period. Confirm which endpoints can be calculated under the prespecified rules, and flag endpoints that rely on long bridges, weak terminal fits, or a large extrapolated fraction. These are endpoint-usability checks, not evidence that a particular treatment effect is present. The later data-integrity diagnostics address broader record-level checks; here the focus is what the missing observations mean for exposure endpoints.
The main options are not interchangeable:
| Approach | What it does | Main limitation |
|---|---|---|
| Calculate from available observations, including a longer trapezoid | Uses recorded concentrations and times to produce an observed-data endpoint. | A long interval may miss profile curvature and change the numerical approximation. |
| Interpolate or impute a missing concentration | Fills a gap using an explicit rule or model before endpoint calculation. | Relies on additional assumptions; the filled value is not an observation and may understate uncertainty. |
| Exclude an unusable endpoint or subject-period | Avoids presenting an unsupported endpoint as reliable. | Can reduce information and, in a crossover, remove a paired comparison; exclusions can also affect representativeness. |
There is no universally best option for every gap. Follow the prespecified endpoint and missing-data rules, record any deviations, and avoid choosing a method after seeing which one favors BE. When justified, a sensitivity analysis can compare the primary rule with a defensible alternative, such as a specified gap-handling rule or analysis set. Report how the endpoint values and BE estimates change, and distinguish that sensitivity analysis from the primary result. A sensitivity analysis cannot establish the missing concentrations or remove uncertainty due to informative missingness.
Common Pitfall
Correction: Zero asserts that the concentration at that time was zero, which is not implied by an uncollected sample and may distort AUC and \(C_{\max}\). Use the prespecified approach for the gap, identify the endpoint as incomplete or potentially unreliable when warranted, and assess the impact transparently. A missing subject-period endpoint is not a zero treatment response either.
Check Your Understanding
Answer each question, then compare with the concise solutions below.
- A sample at an interior time is not collected. Why is setting its concentration to zero not a neutral way to handle the gap?
- A sample near the expected peak is missing. Which endpoints are most directly at risk, and why?
- What additional concern arises when a missing sample is in the terminal phase and \(AUC_{\infty}\) is an endpoint?
- In a simple paired crossover summary, what does a missing test-period profile mean for that subject’s paired log difference?
- Why should an alternative gap-handling rule be reported as a sensitivity analysis rather than silently substituted after results are seen?
Solutions. (1) Zero is an unobserved concentration claim, not a missing-data rule; it can distort the integrated area. (2) \(C_{\max}\) and \(T_{\max}\), because the missing observation could have changed the largest sampled concentration or its time. (3) The terminal slope \(\lambda_z\), last concentration, and extrapolated area may be less reliable, making \(AUC_{\infty}\) uncertain or unevaluable. (4) The subject has no paired difference in that simple analysis; the missing value is not zero. (5) Prespecification limits outcome-driven choices, while a sensitivity analysis makes the dependence of the result on the alternative assumption visible.