Why Intermittent Missing Samples Matter
A planned concentration sample that was not collected or could not be quantified removes information at a particular time. It does not, by itself, invalidate a PK/PD analysis. The main questions are whether the remaining concentrations still identify the individual exposure adequately, and whether the chance of a sample being missing is related to its unobserved value or to other outcome-relevant information.
These questions matter for staged and joint estimation in different ways. In a staged analysis, PK parameters or individual exposure estimates are obtained first and then supplied to the PD analysis. A missing sample can make those estimates less precise, and treating estimated exposure as known can conceal that added uncertainty. In a joint analysis, the PD observations may contribute information about shared latent quantities, but only through the specified PK/PD model. A joint fit cannot recover information that is absent without assumptions, and it does not automatically correct informative missingness.
Here, let \(C_{ij}\) denote the concentration for individual \(i\) at planned time \(t_{ij}\), when observed, and let \(R_{ij}=1\) indicate that the sample was observed (\(R_{ij}=0\) indicates it was missing). Let \(\theta_{\mathrm{PK}}\) and \(\theta_{\mathrm{PD}}\) denote PK and PD parameters. A missing value is distinct from a below-quantification-limit value, which has a different observation model, as discussed in the earlier material on handling BLQ PK data.
What a Missing Sample Does to Exposure Information
The effect of a gap depends on its location, the sampling schedule, the PK model, and what other data are available. Missing a sample near a rapid distribution phase, a peak, or a change in elimination can matter more than missing one in a region where the concentration-time curve is already well characterized. A missing point can also weaken estimates of individual parameters even when population-level PK parameters remain reasonably precise.
A useful distinction is between an observed-data gap and a substituted value. If a concentration was not measured, it should not be entered as zero or treated as an ordinary observed concentration. A PK model can predict concentration at that time, but the prediction is not a measurement and has uncertainty. Simple interpolation may be useful for a rough descriptive calculation, but it is not a general substitute for modeling the concentration-time profile.
Worked Example: A Missing Interior Sample and Trapezoidal Exposure
Suppose a hypothetical individual has concentrations of 10, 6, and 4 mg/L at 0, 2, and 4 hours after a dose. The 2-hour sample is then considered missing. Compare the trapezoidal area under the concentration-time curve (AUC) using all three values with the value obtained by joining only the observed endpoints.
With all three concentrations, the trapezoidal calculation is:
If the interior sample is unavailable and the two observed endpoints are joined directly, the corresponding trapezoid is:
The endpoint-only calculation is 2 mg·h/L higher than the calculation using the observed interior point. This difference arises from the shape represented by the observations; it does not establish that the missingness was informative. In a PK/PD analysis, a model-based exposure estimate could use the available concentrations and the PK model, but should reflect uncertainty from the missing measurement. The example also shows why a crude interpolation or endpoint calculation can misrepresent exposure when the concentration curve is not linear between samples.
Staged and Joint Likelihoods With Missing Concentrations
In staged estimation, the PK stage uses observed concentrations to estimate population PK parameters and, where applicable, individual PK effects. A subsequent PD stage uses estimated concentrations, exposure summaries, or individual PK estimates as inputs. When samples are missing, this first-stage information is reduced. If the second stage uses point estimates as though they were known exactly, the resulting PD uncertainty may be too small; this is a form of uncertainty-transfer problem, not necessarily bias in the missingness mechanism.
A joint analysis instead estimates PK and PD quantities together from their likelihood contributions. A schematic observed-data likelihood is:
Here, \(\eta_i\) represents individual-level latent effects, \(f_C\) is the concentration model, and \(f_Y\) is the PD model; the PD model may depend on the PK-predicted concentration profile. Only observed concentration values contribute concentration observation terms. The PK model nevertheless defines a profile at unobserved times, and uncertainty about that profile is carried through the joint likelihood. A fuller likelihood may also integrate over latent concentrations or other unobserved quantities, depending on the model.
A joint fit can use PD data to inform an individual exposure-related quantity when the PK/PD model links that quantity to the response and the data identify the link adequately. It cannot do so reliably if the PD relationship is misspecified, weakly identified, or confounded with individual effects. Earlier material on sparse PK sampling and PD parameter identifiability is relevant here: the presence of PD observations does not guarantee that exposure and PD parameters can be separated.
When Can the Missingness Process Be Ignored?
The key statistical distinction is whether the observation process can be omitted from the likelihood for the parameters of interest. Let \(Z\) represent all planned data, including unobserved values, and let \(Z_{\mathrm{obs}}\) and \(Z_{\mathrm{mis}}\) be its observed and missing parts. Let \(\psi\) denote parameters governing the observation process. Under missing at random (MAR), the probability of the observed missingness pattern may depend on observed data, but not on the unobserved values after conditioning on those observed data. If, in addition, \(\theta\) and \(\psi\) are distinct parameter sets, the missingness factor can be separated from the likelihood for \(\theta\):
The second equality follows because MAR makes \(f(R\mid Z_{\mathrm{obs}},Z_{\mathrm{mis}},\psi)\) depend only on \(Z_{\mathrm{obs}}\) and \(\psi\), so it can be taken outside the integral. With distinct parameter sets, this factor does not change likelihood-based inference about \(\theta\). Thus, under these conditions, the PK/PD model can use the observed data without explicitly modeling the missingness process. This result does not say that missingness is harmless: fewer observed concentrations can still mean wider uncertainty.
If the probability of a missing sample depends on its unobserved concentration after conditioning on observed information, the mechanism is not MAR. This is often called missing not at random (MNAR). Ignoring such a process can bias parameter estimates because the observed concentrations may not represent the concentrations that would have been observed under complete sampling. Whether a particular mechanism is MAR or MNAR cannot generally be settled from the observed data alone; the reason for missingness and scientific knowledge about the sampling process matter.
Worked Example: Combining PK and PD Information About Exposure
Consider a simplified hypothetical Gaussian approximation for one individual's latent exposure summary \(x\), measured in mg·h/L. The PK data, including an intermittent missing sample, provide an estimate represented by \(x\sim N(8,4)\), where 8 is the mean and 4 is the variance. Suppose, only for illustration, that the PD likelihood contributes information represented by a normal likelihood centered at 10 mg·h/L with variance 1 \((L(x)\propto \exp[-(x-10)^2/2])\). Assume the normal approximation is adequate, the PK and PD information are conditionally compatible under the model, and relevant PD parameters are sufficiently identified.
The combined distribution is proportional to the product of the PK information and the PD likelihood. Normal precisions add, so its variance is:
The combined mean is the precision-weighted mean:
The PD information shifts the exposure summary from 8 to 9.6 mg·h/L and reduces its variance from 4 to 0.8 in this idealized calculation. This illustrates how a joint analysis can retain information that a staged point-estimate approach discards. It is not a general recipe for combining PK and PD estimates: a real joint model must specify the likelihood and account for uncertainty in other parameters and individual effects. If the PD model does not identify \(x\), or is incorrect, the apparent feedback can instead distort the PK/PD estimates.
Worked Example: Concentration-Dependent Missingness Can Bias the Observed Mean
Imagine a hypothetical population in which planned concentrations at one sampling time are equally likely to be 2, 4, or 8 mg/L. The true mean is:
Suppose the probability that a sample is observed is 0.9 when its concentration is 2 mg/L, 0.9 at 4 mg/L, and 0.3 at 8 mg/L. The expected observed concentration, weighted by the probabilities of observation, is:
Thus the mean among observed samples is lower than the population mean because high concentrations are less likely to be observed. This is not just a loss of precision: it is selection bias from a concentration-dependent observation process. If the reason for missingness is known or measured, a suitable model may account for it. If the dependence is on the unobserved concentration itself, inference generally requires explicit assumptions and sensitivity analysis, not simply a standard PK fit to the observed values.
Diagnostics, Alternatives, and Limitations
A practical assessment begins by describing missingness at the planned-sample level. Tabulate the fraction missing by nominal time, individual, study occasion, site, and recorded reason. Check whether missingness is associated with observed covariates, prior observed concentrations, dose timing, or observed PD outcomes. Such patterns can indicate that a simple assumption of completely random loss is implausible. They cannot prove that missingness is MAR, because the missing concentration itself is unavailable for comparison.
Then examine how much the missing observations affect the PK information. Compare the distribution of sampling times and the number of observed concentrations across individuals; inspect whether gaps cluster around phases important for estimating the model. Assess individual exposure uncertainty and parameter precision, not just the number of samples. If estimates change substantially after excluding a small set of individuals with sparse profiles, the analysis may be sensitive to the way those profiles are represented.
The main alternatives answer different questions:
| Approach | What it does | Main concern |
|---|---|---|
| Observed-data staged fit | Fits PK to observed concentrations, then supplies PK estimates to a PD fit. | May understate PD uncertainty when estimated exposure is treated as fixed. |
| Joint PK/PD fit | Estimates PK and PD parameters together, using both observed PK and PD likelihood contributions. | Depends on a defensible joint model and adequate identification; it does not itself correct MNAR. |
| Missingness-mechanism model | Models the probability of observation along with PK/PD data when the process may be informative. | Requires assumptions and information about why samples are missing; MNAR conclusions may be sensitive to untestable choices. |
| Sensitivity analysis | Compares estimates under plausible alternative assumptions about unobserved concentrations or observation probabilities. | Shows dependence on assumptions rather than identifying the true mechanism from incomplete data alone. |
Common failure modes include coding a missing value as zero, silently replacing it with a point prediction, and interpreting a more precise joint estimate as proof that missingness was corrected. A model-based prediction at an unobserved time is conditional on the model and available data; it is not observed evidence. A joint fit can reduce uncertainty when its cross-model information is valid, but it can also propagate misspecification between PK and PD. When missingness is plausibly informative, report the assumed mechanism and assess how conclusions change under alternatives.
Check Your Understanding
Answer the questions, then compare your reasoning with the solutions.
- An individual's 2-hour PK sample is missing, but samples before and after 2 hours are observed. Does this fact alone establish informative missingness?
- State the two conditions used above that permit the missingness process to be omitted from likelihood inference about the data-model parameters.
- In the Gaussian illustration, what is the combined exposure variance, and why is the PD contribution not automatically reliable in every joint fit?
- Observed concentrations are lower on average than the known or scientifically plausible population distribution, and high-concentration samples are less likely to be recorded. Is this only a precision loss?
Solutions. (1) No. It establishes an intermittent gap, but not why the sample is missing or whether its missingness depends on its unobserved concentration. (2) The missingness must be MAR, and the missingness-process parameters must be distinct from the parameters of interest. (3) The combined variance is \(0.8\ (\mathrm{mg\cdot h/L})^2\). The PD contribution is reliable only if the joint model is suitable and the exposure-related quantity is adequately identified; otherwise it can transmit misspecification or weak identification. (4) No. If high concentrations are selectively less likely to be observed, the observed distribution can be biased downward, so this is informative missingness rather than merely reduced precision.