Robustness to Missed Visits and Timing Deviations

Why Robustness Matters

A sparse PK schedule can look adequate on paper yet leave a parameter weakly informed if an important visit is missed or a sample arrives well outside its planned time. A planned early sample may be intended to characterize absorption, for example, but a substantial delay could place it after the most informative part of the concentration-time profile. A terminal sample collected too early may similarly contribute little information about the terminal decline.

Robustness is the extent to which a schedule retains useful information under plausible departures from its plan. It is not a guarantee that every participant will complete every visit, nor does it mean that one schedule is best for every objective. The practical task is to compare candidate schedules under stated assumptions about missed visits and timing deviations, and to see which design features preserve coverage of the phases relevant to the study question.

The earlier material on sampling windows for absorption, distribution and clearance, terminal-phase samples, and peak and trough characterization explains why different times can contribute different kinds of information. Here the focus is what happens when planned opportunities are not realized exactly. A schedule should be assessed against the phases and parameters that matter for its intended use; a count of all samples, without regard to their timing, is not enough.

A Simple Framework for Missed Visits

Divide the planned opportunities into relevant phases, such as early absorption, distribution, and terminal decline. For phase \(j\), let \(m_j\) be the number of planned opportunities that could provide useful coverage if collected. Suppose opportunity \(i\) in that phase is missed with probability \(p_{ij}\), and assume for this calculation that the miss events are independent. The phase retains at least one sample unless all \(m_j\) opportunities are missed.

If each opportunity in the phase has the same miss probability \(p_j\), this becomes \(1-p_j^{m_j}\). The independence assumption matters: several visits cancelled by the same transport disruption or illness may fail together. In that case, the product formula can overstate the protection provided by repeated opportunities.

For a compact comparison across phases, assign nonnegative weights \(w_j\) to the phases, with \(\sum_j w_j=1\). The weights express the design priorities for the question at hand. One simple summary of expected retained coverage is

This summary is useful for comparing explicit scenarios, but it is not a universal objective function. It treats each phase as either covered or not covered, and does not distinguish a barely useful observation from a highly informative one within a phase. It also cannot, by itself, account for parameter correlations or the full concentration-time model.

Evaluating Timing Deviations

A collected sample can still fail to provide the intended phase coverage if its actual time falls outside the useful region. Start by specifying, for each phase and design objective, a time window that will count as useful. Then describe plausible deviations from the planned time. A deviation can be represented as the difference between actual and planned collection time, but the key question is whether the resulting time remains in the relevant window.

For a simple calculation, suppose the timing deviation is uniformly distributed between \(-d\) and \(+d\) hours, and a sample is useful only if its actual time lies in a specified window. The probability of useful timing is the length of the overlap between the possible actual-time interval and the useful window, divided by the length of the possible interval. This calculation assumes the stated uniform distribution; it is a scenario assumption, not a general law of clinic timing.

When multiple opportunities are available, timing and missingness can be considered together. If an opportunity is missed with probability \(p\), and, conditional on collection, it lands in the useful window with probability \(q\), its probability of providing useful coverage is \((1-p)q\). If opportunities fail independently under both mechanisms, the chance that at least one of \(m\) opportunities provides useful coverage is \(1-[1-(1-p)q]^m\). Independence should be examined rather than assumed: a shared clinic delay can shift several samples in the same direction.

A Practical Review of Candidate Schedules

A useful comparison begins with the scientific priorities and the disruptions that are plausible in the setting. It then asks which design features give useful coverage more than one route to survive. Staggering times across participants can spread opportunities across phases; repeated coverage of a phase can reduce dependence on one visit; and a planned time comfortably inside a useful window can be less sensitive to small deviations than a time near its boundary. These features have costs and trade-offs, so none is automatically preferable.

A scenario table can make the comparison transparent even when reliable miss probabilities are unavailable. For example, examine the schedule with one early visit missed, with all visits delayed by a common amount, and with one participant's visit times shifted in either direction. Record which phases remain covered and which design objective becomes vulnerable. Such stress tests are not claims about how often each event will happen; they reveal dependence on particular observations.

Diagnostics, Alternatives, and Limitations

Warning signs include a phase represented by only one visit, a design whose useful coverage disappears after one plausible cancellation, and planned times whose small shifts move them outside the intended window. A further warning is a robustness calculation that assumes independent disruptions even though visits share staff, transport, clinic days, or other causes of delay. Check these assumptions explicitly and, where possible, compare results under both independent and shared-disruption scenarios.

A nominal schedule comparison asks how candidate times perform if every sample is collected as planned. A disruption-based comparison asks how often, or under which scenarios, useful coverage remains. The former is simpler and may be sufficient when deviations are negligible; the latter is more informative when missed or delayed visits are credible. Neither replaces evaluation of parameter precision. In particular, a schedule can preserve at least one sample in each broad phase while still providing weak information about a parameter or producing strong parameter correlations. The earlier material on parameter precision and parameter correlations addresses those distinct diagnostics.

The simple coverage calculations also have limitations. They depend on the selected useful windows, the assumed miss and timing distributions, and the independence structure. A binary covered/not-covered summary ignores how informative a sample is within a window, how concentration variability affects the observation, and how information is shared across participants. If the decision depends on precise parameter estimates, robustness scenarios should be carried into model-based design evaluation rather than treated as a substitute for it.

Common Pitfall

Wrong reasoning: “The schedule has several samples, so a missed visit cannot matter.” The total sample count does not show whether those samples are concentrated in one phase or whether the only terminal-phase sample is vulnerable.

Correction: Evaluate coverage by phase and by plausible disruption scenario. Repeated observations help only if they provide useful, sufficiently independent opportunities for the information at risk. If all opportunities share a cause of failure, nominal redundancy may provide little protection.