Influential Observations and PD Outlier Assessment

Why a Large Residual Is Not the Same as an Influential Observation

A PD observation can disagree sharply with a model prediction and yet have little effect on the fitted model. Conversely, a less dramatic residual can matter greatly if it occurs where the model has little other information or strongly informs an important parameter. Residual size describes disagreement at an observation; influence describes how much the fitted result changes when that observation is perturbed or omitted. These are related, but they are not interchangeable.

This distinction matters in repeated-measures PK/PD data. A single measurement may be unusual because of recording or assay error, ordinary residual variation, a short-lived biological fluctuation, or a model limitation. A whole subject's trajectory may also be influential even when no one measurement stands out. The purpose of an assessment is to understand and report these possibilities, not to remove inconvenient data until a preferred result appears.

For a continuous PD response, write the residual as \(r_i=y_i-\hat{y}_i\), where \(y_i\) is the observed response and \(\hat{y}_i\) is its model prediction. The raw residual has the response's units, but its magnitude is hard to compare across observations when residual variability differs. A standardized residual divides the residual by an appropriate estimate of its standard deviation. The relevant predictive variability depends on the diagnostic: for example, a prediction conditional on a subject's estimated effects does not represent the same uncertainty as a population prediction.

Here \(s_i\) is the standard deviation appropriate to the chosen residual or predictive diagnostic. A value of \(z_i\) is a scale for investigation, not a universal rule for deleting data. Its interpretation depends on the model, the residual distribution, dependence among repeated observations, and how the prediction was constructed. Residual-versus-time, residual-versus-concentration, and observed-predicted displays provide complementary context; none alone establishes that a measurement is erroneous.

Assessment Before Exclusion

Begin with the data and the observation's context. Confirm subject and visit identifiers, units, assay and transcription records, dose and sampling times, and any documented sample-handling or protocol issues. Check whether the response lies within a plausible measurement range and whether the same pattern appears in source data. A confirmed data error should be corrected according to a documented rule; it should not be silently discarded. If the value is valid, retain it in the primary analysis unless a prespecified analysis rule says otherwise.

Next inspect the subject's full trajectory and the observations around the flagged time. Ask whether the measurement is an isolated spike, part of a sustained departure, or consistent with a known phase of response. Check the sampling schedule: a single measurement in a sparsely sampled region can have disproportionate influence because nearby observations provide little constraint. Also examine whether the residual is part of a systematic pattern by time, concentration, dose, baseline, or covariates. A pattern shared by several observations may point to model misspecification rather than one bad datum.

Finally, define what “influence” means for the scientific question. A point may change one parameter while barely changing predictions over the concentration range of interest. A subject may affect a population parameter yet have little effect on typical-subject predictions. Therefore, assess changes in interpretable targets: key parameter estimates, their uncertainty, and clinically relevant response predictions. When observations are repeated within subjects, compare omitting one measurement with omitting the entire subject; these are different perturbations and answer different questions.

Why Influence Depends on Information as Well as Residual Size

A local approximation makes the distinction clear. Suppose a model with parameter vector \(\theta\) is fitted by minimizing weighted squared residuals, and \(f_i(\theta)\) is the prediction for observation \(i\). Let \(J_i\) be the row vector of derivatives of that prediction with respect to the parameters, evaluated near the fitted value, and let \(w_i\) be the observation's weight. The approximate information matrix is \(J^\mathsf{T}WJ\). It reflects how strongly and in what directions the observations constrain the parameters.

At the full-data fit, the weighted residual contributions balance approximately in the estimating equations. Removing observation \(i\) removes its contribution \(J_i^\mathsf{T}w_i r_i\). Linearizing the prediction around the full-data estimate gives the following approximate change in the fitted parameter vector:

The equation is a local approximation, not a substitute for refitting a nonlinear PK/PD model. It shows why a large \(r_i\) alone does not determine influence: the derivative \(J_i\) describes which parameters the observation informs, and the inverse information matrix describes how well those parameter directions are constrained by the data as a whole. A large residual in a well-supported direction may have limited impact; a smaller residual can matter if it aligns with a weakly identified direction. In nonlinear models, random-effects structures, or correlated residual models, use the approximation as intuition and use actual refits for the principal sensitivity assessment.

Refit-Based Sensitivity Checks

A direct and interpretable influence assessment is to fit the specified model to the full data, then refit it after omitting the flagged measurement. Compare the same quantities from both fits. For a parameter \(\theta_j\), a simple scale-free summary is the absolute parameter change divided by the full-data standard error:

This is a descriptive sensitivity measure, not a test statistic with a universal cutoff. It is meaningful only when the parameter has the same definition and scale in both fits, and the standard error is estimable and suitable for comparison. Report the estimates and their units alongside \(D_j\); a ratio alone can conceal whether the change matters scientifically. Also compare predictions at prespecified, relevant exposure or time values, because parameter changes can offset one another and leave predictions stable—or small parameter changes can substantially alter a prediction in a sensitive region.

For repeated data, repeat the procedure at the subject level when the concern is a potentially influential trajectory. Omit all observations from that subject and refit, clearly distinguishing this result from point deletion. A subject-level change does not imply that the subject is erroneous: the subject may represent legitimate between-subject variability or provide unusually informative data.

Interpreting Diagnostics and Choosing a Response

Influence checks are most useful when they are tied to a clear question and interpreted alongside the model diagnostics already used for continuous PD. A flagged point might expose a transcription error, unusual but valid variation, or a structural pattern that the model does not capture. Repeated positive or negative residuals over time, for example, call for examination of the trajectory and residual dependence rather than an isolated-point explanation. Likewise, systematic residual patterns against concentration or predicted effect are not resolved merely by deleting the largest residual.

The main alternatives to deletion address different concerns. Correct a documented data error using a traceable correction. If the value is valid but the residual distribution is heavier-tailed than assumed, a robust or heavy-tailed error model may reduce the leverage of extreme residuals while retaining the observation; that is a different model and should be scientifically justified. If several observations show a coherent pattern, reconsider whether the model structure or variability description is adequate. A leave-one-out refit is a diagnostic comparison, not a replacement primary analysis. Each alternative should be reported transparently, including the data changed or model altered and the resulting estimates and predictions.

There is no universal numerical threshold that makes an observation “influential” in every PK/PD application. A change should be judged against the precision of the estimate, the scale and clinical relevance of the prediction, the model's stability, and the analysis objective. In a nonlinear model, omission can also affect convergence or produce a different local solution; inspect fit status and parameter plausibility before interpreting a change as a data effect. If the refit fails, report that limitation rather than treating an unavailable estimate as evidence of extreme influence.

Common Pitfall

Wrong reasoning: “This response has the largest residual, so it is an outlier and should be removed.” A large residual establishes neither a data error nor a material effect on the fitted result. It also ignores the possibility that a valid point reveals an inadequate model.

Correction: Verify the data, evaluate the residual in its predictive and trajectory context, and assess influence by refitting for the specific parameter or prediction of interest. Keep valid observations in the primary analysis unless a prespecified or scientifically defensible exclusion rule applies. Present any exclusion or alternative-model analysis as a sensitivity analysis, not as an automatic consequence of a diagnostic flag.