Detecting Influential Concentrations in Patient Fits

Why Examine the Influence of an Individual Concentration?

A patient-specific fit combines measured concentrations with a population-based prior and a model of the dosing history. A concentration can fit the model reasonably well and still have substantial influence on the fitted individual parameters or a future forecast. Conversely, a large residual need not move the fit much if the observation is imprecise or redundant with other measurements. Influence analysis asks a focused question: how much would the patient’s estimate or forecast change if this observation were omitted or slightly changed?

This is a sensitivity analysis, not a test of whether a concentration is valid. It can identify measurements that deserve review, reveal that a forecast depends heavily on one sample, or show that several observations contribute similar information. It cannot, by itself, establish that an assay result is wrong or that the PK model is misspecified.

Define the Influence Quantities

Let \(y=(y_1,\ldots,y_n)\) denote the concentrations for one patient, together with their sampling times and known dosing history. Let \(\eta\) denote an individual parameter, such as a deviation in log clearance from its population-prior value. Write \(\hat{\eta}(y)\) for a chosen posterior summary, such as the posterior mean or posterior mode, and \(F(y)\) for a specified forecast summary at a specified future time.

The signs matter: a positive \(\Delta_i^{(\eta)}\) means the full-data estimate is higher than the estimate after deletion. For positive concentrations, a forecast ratio is often easier to interpret than a difference because the ratio expresses a proportional change. If forecasts are analyzed on the log scale, the log ratio is \(\log F(y)-\log F(y_{-i})\).

A perturbation analysis changes one observation by a small amount while keeping the other data fixed. For example, one can compare the fit using \(y_i\) with the fit using \(y_i+\delta\), where \(\delta\) is a prespecified concentration change or a fraction of the observation’s measurement-error scale. Unlike deletion, perturbation asks about sensitivity to a change in value rather than to removing the observation altogether.

A Conjugate Example and a Short Derivation

A simple normal model gives an exact derivation of deletion and perturbation effects. It is not a full PK model; it isolates the information-weighting mechanism that also helps interpret more complex patient fits. Suppose transformed measurements \(z_i\), such as log concentration summaries after accounting for a fixed structural prediction, satisfy

with conditionally independent measurement errors and known variances \(s_i^2\). The normal prior and likelihood give a normal posterior. Define prior precision \(P_0=1/V_0\), observation precision \(w_i=1/s_i^2\), and total precision \(P=P_0+\sum_{j=1}^n w_j\). Completing the square in the prior-times-likelihood density gives

After deleting observation \(i\), the denominator becomes \(P-w_i\) and its numerator loses \(w_i z_i\). Subtracting the two posterior means yields

This identity shows two contributors to deletion influence in this model: the observation’s precision relative to the total precision, \(w_i/P\), and its discrepancy from the estimate based on the remaining data. A precise observation that differs from the deleted-data estimate can have substantial influence. A large discrepancy paired with very low precision may have less. Also, influence depends on the prior: the prior precision is part of \(P\), so a stronger prior generally reduces the effect of any one measurement.

For a small perturbation \(\delta\) to \(z_i\), the posterior mean changes by \(w_i\delta/P\). This follows directly because \(z_i\) enters the numerator linearly and does not change the denominator. In nonlinear PK models, this exact expression generally does not apply; the patient fit must be recalculated, or a suitable local approximation used.

Perturbation and Comparison Across Observations

Case deletion is deliberately a large change: the observation is removed entirely. A perturbation can show whether an estimate responds smoothly to a smaller plausible change in a measured value. In the conjugate model, the exact change is available from the precision weight. In a PK fit, the same idea can be implemented by changing one concentration on a meaningful scale, rerunning the individual estimation, and recalculating the same forecast endpoint.

The following comparison uses the same hypothetical data. Each deletion fit retains the same prior and the two remaining observations.

Applying the Analysis to Individual PK Fits

For an individual Bayesian PK fit, a case-deletion calculation should hold the model, prior, dosing history, covariates, and estimation target fixed. Remove one concentration at a time, recompute the posterior summary of interest, and recalculate the same future forecast at the same time and dose. In nonlinear models, deleting a sample can alter the posterior shape as well as its center, so record which summary is being compared: posterior mean, posterior mode, median, or another prespecified quantity.

An efficient workflow is:

For nonlinear or multivariate models, a useful comparison is to examine a vector of changes, for example in clearance, volume, and a forecast. A single scalar influence score can conceal that one observation increases the forecast while decreasing a parameter estimate. Reporting the target-specific changes is more transparent than treating all parameters as interchangeable.

Diagnostics, Alternatives, and Limitations

Influence results are meaningful only if the full and reduced fits are numerically reliable and comparable. Check convergence, posterior summaries, and forecast definitions in every refit. A large apparent change caused by different local optima, an unstable optimizer, or altered data-processing rules is not a trustworthy measure of observation influence. With very sparse data, deleting a sample can leave a weakly identified patient fit; a large change may then reflect dependence on limited information rather than a defective measurement.

Case deletion, perturbation, and residual review answer different questions. Case deletion asks how much the analysis depends on retaining a particular sample. Perturbation asks how sensitive it is to a specified small change. A residual measures discrepancy between an observation and a model prediction under a particular fit; it does not directly measure the impact of the observation on that fit. These approaches are complementary, not substitutes. In complex models, full case-deletion refits are usually the clearest direct assessment, while local derivative approximations are faster but can be inaccurate for large changes or strongly nonlinear posterior geometry.

Influence depends on the model and prior used. A point can be influential because it is precise, because its timing provides information unavailable elsewhere, or because the fit is weakly constrained—not only because it is extreme. Deleting it also changes the information available and may widen uncertainty. Conversely, a small change in a posterior mean does not guarantee that tail probabilities or prediction intervals are unchanged. The target and uncertainty summary should match the clinical decision being considered.

A common mistake is to reason, “Deleting this concentration changes the forecast a lot, so the concentration must be an outlier and should be removed.” The conclusion does not follow. Influence is a property of the observation in relation to the chosen model, prior, other data, and forecast target; it is not a probability that the observation is invalid. The correction is to treat a large change as a prompt to inspect the sample and the fit, document the sensitivity, and justify any data handling using information beyond influence alone.