Why Missing Concentrations Matter
A concentration-time profile is often observed at planned sampling times, but a missed sample or early withdrawal can leave part of that profile unavailable. Model-based analyses can use the observed measurements without requiring every subject to have a complete profile. That does not mean missingness is harmless: the validity of an estimated formulation contrast depends on how missingness relates to observed and unobserved concentrations.
An isolated missed sample and study dropout are related but not identical. A missed sample may be followed by later measurements, whereas dropout usually removes all later measurements in that period or study. The pattern of missing records matters, as does the reason they are missing. For example, a sample missed for a documented logistical reason may be plausibly unrelated to concentration; withdrawal after symptoms or poor response may be related to a participant’s unobserved profile.
These labels describe assumptions about the missingness process, not properties that can generally be established from the observed data alone. A mechanism that is MAR after conditioning on prior concentrations and a recorded withdrawal reason may not be MAR if those variables are omitted. MCAR is the strongest and often least plausible assumption; MAR allows missingness to depend on information that was observed.
Observed-Data Likelihood and Ignorability
Suppose a pharmacokinetic model specifies the distribution of concentration records \(Y\) given model parameters \(\theta\). The parameters of interest may include formulation effects that determine model-predicted profiles and exposure contrasts. Let \(\psi\) denote parameters governing missingness. A joint model for data and missingness has the factorization
The observed-data likelihood is obtained by summing or integrating over the unobserved concentrations. Under MAR, the missingness probability for a given pattern depends only on observed data and included covariates, not on the missing values themselves. If, in addition, the parameters \(\theta\) and \(\psi\) are distinct, the likelihood factors into a term for the observed concentrations and a term for the missingness pattern:
The second equality follows because the missingness term does not vary with \(Y_{\mathrm{mis}}\), so it can be taken outside the integral; integrating the concentration model over the missing values gives the marginal model for observed concentrations. The missingness factor contains no \(\theta\), so it can be omitted when estimating \(\theta\) by likelihood. This is the standard justification for using the observed-data likelihood under an ignorable MAR mechanism.
Omitting missing records from a likelihood does not mean replacing them with zeros, treating them as measured at the last observation, or fitting the model as if the observed sample were necessarily representative. Rather, the likelihood contribution for a subject is based on the measurements actually observed, with their times, formulation, period, sequence, and relevant covariates represented correctly. The model must also be adequately specified, and the missingness assumptions must be defensible.
Worked Example: Likelihood Contribution With a Missed Sample
Hypothetical data. At three planned times, a subject has concentrations of 4.0, a missing value, and 2.0 mg/L. For illustration, suppose the fitted model predicts means of 4.2, 3.0, and 2.1 mg/L, respectively, and assumes independent normal residuals with standard deviation \(\sigma=0.5\) mg/L at each time. Assume the missing sample is MAR conditional on recorded covariates and the observed record, with distinct missingness parameters.
For an observed concentration \(y\) with model mean \(\mu\), its normal likelihood contribution is \( (1/(\sqrt{2\pi}\sigma))\exp(-(y-\mu)^2/(2\sigma^2)) \). The subject’s likelihood contribution includes the first and third observations, not the unobserved second concentration:
The two squared standardized residuals are \(0.2^2/0.5^2=0.16\) and \(0.1^2/0.5^2=0.04\). Thus the likelihood contribution is \((1/(2\pi(0.5)^2))\exp[-(0.16+0.04)/2]\), approximately \(0.576\) in the corresponding concentration-density units. No residual is calculated for the missing sample. This treatment is justified by the stated model and MAR assumptions, not by the numerical likelihood value itself.
Dropout Can Make Formulation Contrasts Unrepresentative
When dropout depends on an unobserved concentration or on a latent response related to it, the observed records can overrepresent some profiles. For example, if participants with low exposure on one formulation are more likely to withdraw before later sampling, a contrast estimated from retained profiles may be shifted upward. The direction and size of bias depend on the missingness process and the model; they cannot be inferred from the fraction missing alone.
Worked Example: Selection Bias in a Formulation Contrast
Hypothetical data. Consider six subjects with true paired log-exposure differences \(d_i=\log(AUC_{T,i})-\log(AUC_{R,i})\), where \(T\) and \(R\) denote test and reference. The complete-data differences are \(-0.30,-0.20,-0.10,0.10,0.20,0.30\), so the full-sample mean contrast is
Suppose, however, that the three subjects with negative differences withdraw before enough measurements are obtained to estimate their test exposure, while all three with positive differences remain. The available-case mean is \( (0.10+0.20+0.30)/3=0.20 \) log units. Exponentiating gives a geometric mean ratio of \(\exp(0.20)\approx1.221\), or about 122.1%. The complete-data contrast corresponds to \(\exp(0)=1\), or 100%.
This is an illustrative MNAR mechanism: dropout depends on the unobserved contrast that would have been estimated for the withdrawing subjects. The observed data alone do not reveal the three missing values or establish that they were negative. The example shows why a precise model fitted to retained records cannot, by itself, correct selection caused by unobserved outcomes.
The estimation target should be stated explicitly. A formulation contrast among participants who remain observed is not automatically the same as the contrast that would have been estimated for all randomized participants under complete follow-up. Dropout may also affect the estimability of a profile-derived AUC or \(C_{\max}\), particularly when missing times occur in regions that strongly influence those metrics. Model-based prediction fills in a profile only through the structural and statistical assumptions of the model; it does not turn unobserved concentrations into measured data.
Approaches When Missingness Is Plausibly MAR
Under a credible MAR assumption, likelihood-based analysis can use all observed concentrations without requiring complete profiles. The missingness model may be omitted from the likelihood for the concentration parameters when the missingness and concentration parameters are distinct. For inference to be valid, the analysis should include relevant predictors of both concentration and missingness, and the model should represent the observed-data structure adequately.
Inverse-probability weighting is another option under MAR. Each observed record is weighted by the inverse of its probability of being observed, conditional on observed history and covariates. The aim is to give greater influence to observed records that represent similar records with a lower chance of being observed. This requires a correctly specified observation-probability model and adequate positivity: every relevant covariate-history group must have a nonzero chance of remaining observed. Very small probabilities produce large, unstable weights.
Worked Example: Weighting for Observed-History-Dependent Dropout
Hypothetical data. In a cohort of eight subjects, four have low and four have high observed early concentrations. Dropout by the next sampling time depends on this observed early-concentration group. Of the low group, two are observed at the next time, each with a concentration of 8 mg/L; the estimated probability of remaining observed is 0.50. Of the high group, three are observed, each with a concentration of 12 mg/L; the estimated probability is 0.75. Assume the two observed values represent their respective groups for this illustration.
The unweighted mean among the five observed concentrations is \( (2(8)+3(12))/5=52/5=10.4 \) mg/L. Inverse-probability weighting gives weight \(1/0.50=2\) to each low-group record and \(1/0.75=1.333\) to each high-group record. The weighted mean is
In this constructed example, the weighted estimate equals the full-cohort mean if the four low-group concentrations would all be 8 mg/L and the four high-group concentrations would all be 12 mg/L. Weighting addresses dropout explained by observed early concentration, not dropout driven by the unobserved next concentration after conditioning on observed history. The group probabilities and the outcome model are illustrative assumptions, not guarantees supplied by weighting.
Diagnostics, Alternatives, and Limitations
No diagnostic test can prove that data are MAR rather than MNAR, because the missing concentrations needed to distinguish the mechanisms are unavailable. Diagnostics can nevertheless identify weaknesses in the analysis and help assess whether its assumptions are plausible.
- Describe the pattern. Tabulate missingness by formulation, period, sequence, planned time, and reason for withdrawal. Distinguish isolated missed samples from monotone dropout after withdrawal.
- Compare observed histories. Examine whether prior concentrations, covariates, adverse events, or observed exposure summaries differ between participants who do and do not remain observed. Such differences do not prove MNAR; they indicate which observed predictors may need to be represented.
- Check model behavior on observed records. Inspect residual patterns by time and formulation, and compare fitted predictions with observed concentrations. Poor fit among observed records undermines the concentration model regardless of the missingness assumption.
- Assess observation probabilities. For weighted analyses, look for probabilities near zero, extreme weights, and lack of overlap across histories. These indicate that the weighted estimate may be unstable or depend heavily on extrapolation.
- Vary unverifiable assumptions. For plausible MNAR mechanisms, use sensitivity analyses that specify how missing concentrations differ from MAR predictions, or fit a joint model in which dropout and concentration share subject-level features. Report how the formulation contrast changes across reasonable assumptions.
The main alternatives answer different needs. Observed-data likelihood under MAR is often efficient when the concentration model is well specified and missingness is plausibly ignorable. Inverse-probability weighting makes the observation process explicit and can be useful when its probabilities are estimable, but can be unstable with sparse histories. Joint models specify both concentrations and dropout, often linking them through shared random effects or an explicit dropout hazard; they can represent informative dropout but depend on additional, generally untestable modeling assumptions. Complete-case analysis discards incomplete subjects and is generally inefficient; it is unbiased for the intended target only under restrictive conditions, such as missingness unrelated to the contrast after the analysis design is taken into account.
| Approach | Key assumption | Main limitation |
|---|---|---|
| Observed-data likelihood | Correct concentration model, MAR, and distinct parameters | Does not address dependence on unobserved concentrations |
| Inverse-probability weighting | Correct observation model and positive observation probabilities | Extreme weights can make estimates unstable |
| Joint concentration-dropout model | Correct specification of both processes and their dependence | Dependence on unobserved outcomes is difficult to verify |
| Complete-case analysis | Missingness does not select subjects in a way that biases the target contrast | Can discard information and change the effective target population |
A concentration recorded as missing because it is below a quantification threshold is not necessarily an ordinary missed sample: it may convey that the concentration lies within a known range. That measurement situation requires its own likelihood treatment and is distinct from the missing-record methods discussed here.
Check Your Understanding
Answer each question, then compare with the solutions.
- A sample is missed because a courier failed to collect it, and this event is unrelated to observed or unobserved concentrations. Which missingness category is plausible?
- Dropout depends on the participant’s recorded early concentration, with no further dependence on later unobserved concentrations after conditioning on that history. Which category is plausible, and what should be included in the analysis?
- In one sentence, explain why the missingness factor can be omitted from the concentration likelihood under MAR with distinct parameters.
- Why can inverse-probability weighting fail when estimated observation probabilities are close to zero?
- Can the observed data alone establish that dropout is MNAR? What is one suitable response to uncertainty about this assumption?
Solutions. (1) MCAR is plausible if the courier failure is genuinely unrelated to all concentration data and relevant covariates. (2) MAR is plausible; include early concentration and other relevant observed predictors in the concentration and, when using weighting, observation-probability models. (3) Under MAR, the missingness probability depends only on observed information, so it factors from the integral over missing concentrations and contains no concentration-model parameter. (4) The inverse probabilities become very large, making estimates highly sensitive to a small number of observations and potentially unstable. (5) No; the missing concentrations are unavailable. Use sensitivity analyses or a joint model under explicit plausible assumptions and report how conclusions depend on them.