Why a Missing Dose Time Matters
An individual concentration forecast depends on both the patient’s pharmacokinetic parameters and the dose history used to drive the model. If a dose is known to have been administered but its time is missing, treating an imputed time as exact can make the concentration data appear more informative about the patient’s parameters than they really are. The resulting forecast may also be too precise or shifted in either direction.
The central distinction is between conditioning on a dose history and integrating over uncertain dose histories. Conditioning uses one specified history as if it were known. Integration represents several plausible histories and averages their contributions, with weights updated by the observed concentrations. Here, the issue is missing timing information for an administered dose; uncertainty about whether doses were taken is a separate modeling question.
Posterior Inference With an Uncertain Dose History
Write the observed concentrations as \(y\). A PK model specifies the concentration predictions given parameters and history; a residual-error model specifies the distribution of observations around those predictions. For example, under additive normal residual error with known standard deviation \(\sigma\), the conditional likelihood is \(p(y\mid\theta,H)=\prod_j \phi(y_j; f_j(\theta,H),\sigma^2)\), where \(f_j(\theta,H)\) is the model prediction at observation \(j\), and \(\phi(\cdot;m,v)\) is a normal density with mean \(m\) and variance \(v\). This example assumes conditionally independent residuals; a different residual model requires its corresponding likelihood.
Suppose the available dose record and other information give a prior distribution \(p(H)\) over plausible histories, and the individual parameter prior is \(p(\theta)\). If these are independent before observing concentrations, the joint posterior is proportional to their priors times the likelihood. The marginal posterior for \(\theta\) must account for each possible history:
For continuous dose times, the sum is replaced by an integral over time. The posterior probability of a discrete history is proportional to its prior probability multiplied by the concentration-data likelihood, with the parameter uncertainty also integrated out:
This expression shows why it is generally not enough to compute the PK posterior assuming one guessed history and then report it as though the history were known. Different histories can explain the same concentrations using different parameter values. They can therefore produce different parameter posteriors and different future concentration predictions.
Worked Examples
Worked Example: Updating Probabilities for Three Possible Dose Times
Hypothetical data. A dose is confirmed, but its administration time is missing. For simplicity, take the individual PK parameters as fixed and suppose the record supports three candidate histories, \(H_1,H_2,H_3\), with prior probabilities 0.50, 0.30, and 0.20. At a sample time, the model predicts concentrations of 2.500, 1.914, and 3.500 mg/L under these histories. The observed concentration is 3.000 mg/L. Assume additive normal residual error with \(\sigma=0.500\) mg/L, independent of the history conditional on its predicted mean.
For comparing histories, the common normalizing constant in the normal density cancels. The likelihood kernel for a prediction \(m\) is \(\exp[-(y-m)^2/(2\sigma^2)]\). For \(H_1\), this is \(\exp[-(3.000-2.500)^2/(2(0.500)^2)]=\exp(-0.5)=0.6065\). For \(H_2\), it is \(\exp[-(3.000-1.914)^2/(2(0.500)^2)]=\exp(-2.3588)=0.09453\). For \(H_3\), it is \(\exp[-(3.000-3.500)^2/(2(0.500)^2)]=\exp(-0.5)=0.6065\).
Multiplying each likelihood kernel by its prior probability gives unnormalized weights: \(0.50(0.6065)=0.3033\), \(0.30(0.09453)=0.02836\), and \(0.20(0.6065)=0.1213\). Their sum is \(0.4529\). Dividing each weight by this sum gives posterior history probabilities of approximately 0.6696, 0.0626, and 0.2678. The observation makes \(H_2\) less plausible, but it does not establish a unique dose time.
Suppose the model predicts future concentrations of 1.600, 1.200, and 2.100 mg/L under the three histories. The history-averaged forecast is \(0.6696(1.600)+0.0626(1.200)+0.2678(2.100)=1.709\) mg/L, to rounding. Conditioning on \(H_1\) alone would instead give 1.600 mg/L. These are different summaries because the integrated forecast retains uncertainty about which history occurred.
Worked Example: Quantifying Timing-Induced Forecast Variability
Hypothetical local approximation. Consider a confirmed dose whose time \(t_d\) is unknown within a one-hour window. Suppose the dose-time record supports a uniform distribution over \(0\leq t_d\leq1\) hour. With the individual parameters fixed, approximate the model-predicted concentration at a specified forecast time by \(C(t_d)=1.4699+0.9263t_d\) mg/L. This linear function is a local approximation over the short admissible timing window, not a general PK model. Assume the timing distribution is uniform and the model parameters are held fixed for this calculation.
For a uniform random variable on \([0,1]\), \(E(t_d)=0.5\) h and \(\operatorname{Var}(t_d)=1/12\) h\(^2\). Therefore the mean model prediction is \(E[C]=1.4699+0.9263(0.5)=1.933\) mg/L. The variance attributable to timing is \(\operatorname{Var}(C)=(0.9263)^2(1/12)=0.0715\ (\mathrm{mg/L})^2\), so the corresponding standard deviation is \(\sqrt{0.0715}=0.2674\) mg/L.
The second moment provides a check: \(E[C^2]=\operatorname{Var}(C)+(E[C])^2=0.0715+(1.933)^2=3.808\ (\mathrm{mg/L})^2\), to rounding. Thus a forecast at the mean dose time has the same mean in this linear example, but it omits the 0.2674 mg/L standard deviation due to the unknown time. With a nonlinear PK model, the prediction at the mean time need not even equal the average prediction across possible times.
Worked Example: Comparing a Mean-Time Imputation With History Integration
Hypothetical one-compartment illustration. Assume a confirmed bolus dose has an amount-to-volume ratio of 4.000 mg/L and a known elimination rate constant \(k=0.200\ \mathrm{h}^{-1}\). The concentration after dose time \(t_d\) is \(C(t)=4.000\exp[-0.200(t-t_d)]\) mg/L for \(t\geq t_d\). The dose time is equally likely to be hour 0 or hour 2. A concentration of 3.200 mg/L is measured at hour 2. Assume additive normal residual error with \(\sigma=0.400\) mg/L and equal prior probabilities for the two histories.
At hour 2, the predicted concentrations are \(4.000e^{-0.200(2-0)}=2.681\) mg/L for the hour-0 dose and \(4.000\) mg/L for the hour-2 dose. The likelihood kernels are \(\exp[-(3.200-2.681)^2/(2(0.400)^2)]=\exp(-0.8408)\approx0.4313\) and \(\exp[-(3.200-4.000)^2/(2(0.400)^2)]=\exp(-2)=0.1353\). Because the prior probabilities are equal, the posterior probability of the hour-0 history is \(0.4313/(0.4313+0.1353)=0.7612\); the hour-2 history has probability 0.2388.
At hour 4, the two predictions are \(4.000e^{-0.200(4-0)}=1.797\) mg/L and \(4.000e^{-0.200(4-2)}=2.681\) mg/L. Integrating over the updated history probabilities gives \(0.7612(1.797)+0.2388(2.681)=2.008\) mg/L. By contrast, imputing the mean time, hour 1, and treating it as known gives \(4.000e^{-0.200(4-1)}=2.195\) mg/L at hour 4. This difference is not a rounding effect: the observation changed the relative plausibility of the two histories, and the model’s response to dose time is nonlinear.
Why the Integration Formula Works
For discrete histories, the future prediction is a function of both \(\theta\) and \(H\). Its posterior mean is obtained by averaging first over the joint posterior, or equivalently by averaging conditional posterior means over the posterior distribution of histories:
This follows by partitioning the joint expectation according to the mutually exclusive histories. For any possible future concentration value \(c\), the same partition gives \(p(c\mid y)=\sum_H p(c\mid y,H)p(H\mid y)\). Thus integration is not an optional correction to a forecast already calculated under a known history; it is the marginal prediction when the history itself is uncertain.
The law of total variance also separates two sources of forecast variability:
The first term averages uncertainty remaining within each history, such as parameter and residual uncertainty. The second is variation among the history-specific conditional means. Ignoring the second term generally understates forecast variance when different plausible dose histories produce meaningfully different predictions.
Assumptions, Diagnostics, and Alternatives
The calculations require plausible histories and a probability model for them. Candidate times should be consistent with the available record, such as the documented administration window or a reliable report. The likelihood must reflect the appropriate residual-error assumptions, and the PK model must be applicable to the patient and dosing conditions. If the dose amount is also uncertain, that uncertainty belongs in \(H\) as well. The examples above hold dose amount fixed to isolate timing uncertainty.
Useful checks include comparing history-specific predictions against the observed concentration data, checking that posterior history weights are not driven by an implausibly narrow timing prior, and examining forecasts over the full supported timing range. If every plausible history gives a poor fit, missing timing may not be the only problem: the PK model, parameter prior, residual model, or dose record may also be inadequate. A posterior history probability near one can be reasonable when the data are highly discriminating, but it can also signal that the prior excluded other credible times.
| Approach | What it does | Main concern |
|---|---|---|
| Condition on a verified time | Uses a sufficiently reliable recorded time as fixed. | Not appropriate when the record leaves material timing uncertainty. |
| Impute one time | Replaces the missing time with a chosen value, such as the midpoint. | Can distort nonlinear predictions and omit between-history uncertainty. |
| Integrate over plausible times | Uses a time distribution or candidate histories and averages with posterior weights. | Requires defensible history probabilities and adequate computation. |
| Discard affected concentrations | Avoids using a record whose dose history cannot be established. | Wastes information and may be biased if exclusion is related to observed data. |
When the dose time is verified accurately relative to the relevant PK timescale, conditioning on that time is usually sensible; adding arbitrary timing uncertainty would not improve the model. When only a small, well-defined range is plausible, numerical integration or a suitably fine set of time points can be practical. A single imputation can be useful as a sensitivity analysis, but it should not be confused with uncertainty propagation.
Common Pitfall and Limitations
Wrong reasoning: “The record is missing, so I will use the midpoint of the allowed interval and proceed as though it were exact. The model has now accounted for the missing time.” The correction is that a midpoint substitution represents one history; it does not average across histories or retain their contribution to uncertainty. If the forecast is nonlinear in dose time, even the average prediction need not equal the prediction at the average time.
Integration cannot recover information that the record and concentrations do not contain. If different combinations of dose time and PK parameters fit the data similarly, the posterior can remain broad or sensitive to the prior over histories. A narrow set of candidate times can also give false confidence if the true time lies outside that set. Accordingly, document the source of the timing distribution, inspect sensitivity to reasonable alternatives, and report when the forecast depends materially on that choice.
Check Your Understanding
Try each problem, then check the solution immediately below it.
- A confirmed dose has two possible times with equal prior probability. The likelihoods of the observed concentrations under the two times are 0.20 and 0.60. What are their posterior probabilities?
- Under two possible dose histories, future mean concentrations are 1.0 and 3.0 mg/L, with posterior history probabilities 0.75 and 0.25. Find the integrated forecast mean.
- Why can a prediction at the midpoint of a missing-time window differ from the average of predictions over that window?
- What is the difference between the within-history and between-history terms in the law of total variance?
Solutions. (1) The unnormalized weights are \(0.5(0.20)=0.10\) and \(0.5(0.60)=0.30\), with sum 0.40. The posterior probabilities are \(0.10/0.40=0.25\) and \(0.30/0.40=0.75\). (2) The mean is \(0.75(1.0)+0.25(3.0)=1.50\) mg/L. (3) PK predictions can be nonlinear in dose time, so in general \(f(E[t_d])\ne E[f(t_d)]\). (4) The within-history term averages uncertainty conditional on each history; the between-history term measures variation among the conditional forecast means and represents uncertainty from not knowing which history occurred.