Why an Extreme Residual Starts an Investigation
A very large residual can signal a data problem, a feature of the patient’s concentration-time profile that the model does not capture, or ordinary variation in a tail of the residual distribution. Those possibilities have different implications. A mistyped sampling time may be corrected from source records; a verified concentration that is poorly predicted is evidence about model performance and should not be erased merely because it is inconvenient.
The practical aim is therefore not to make every residual small. It is to determine whether the observation is accurately recorded, understand how the model and the data produced the discrepancy, and document the consequences of any justified correction or sensitivity analysis. Residual quantile plots and tail behavior help identify unusual values for review, but a plot alone cannot establish that a value is erroneous.
Set the Model and the Question Before Reviewing a Point
Let \(y_{ij}\) be the observed concentration for subject \(i\) at observation \(j\), and let \(\hat y_{ij}\) be the model prediction used for the chosen diagnostic. The unweighted residual is \(e_{ij}=y_{ij}-\hat y_{ij}\), in concentration units. A weighted residual expresses the discrepancy relative to an estimated residual standard deviation. In a simple additive-error illustration, if the residual standard deviation is \(\hat\sigma_{ij}\), the weighted residual is \(r_{ij}=e_{ij}/\hat\sigma_{ij}\). Conditional weighted residuals use the corresponding model-based conditional prediction and variance.
Before interpreting a large value, establish which prediction and residual definition produced it. A residual based on a population prediction and one based on a conditional prediction need not be the same. Nor is a raw discrepancy of 2 concentration units necessarily as unusual at a concentration of 3 as it is at a concentration of 100. Use the residual scale and model assumptions that generated the diagnostic; do not treat different residual types as interchangeable.
For the simple standardized illustration, suppose the model is correctly specified, the observation is aligned to its true dose and sampling time, and the residual is approximately normal with mean zero and standard deviation \(\hat\sigma_{ij}\). Then \(r_{ij}\) is approximately standard normal. This provides a reference for identifying values worth reviewing. In a fitted population model, however, residuals can be affected by estimated parameters, repeated observations, model structure, and the specific residual calculation. They are not guaranteed to be independent standard normal draws.
Thus, \(|r_{ij}|>3\) can be a useful review flag under an approximately standard-normal reference, but it is not a universal cutoff. Even under that ideal reference, checking many observations makes it unsurprising that at least one crosses the flag. For 100 independent standard-normal values, the probability of at least one exceeding 3 in absolute value is approximately \(1-(1-0.0027)^{100}=0.237\), or 23.7%. Real residuals may not meet the independence assumption in this calculation, which is another reason not to turn a threshold into a deletion rule.
A Traceable Review of an Unusual Residual
A useful investigation proceeds from the recorded observation back to its source, rather than beginning by changing the model or removing the point. The checks below are not a mechanical checklist that proves an observation valid; they organize the evidence and make the final decision reproducible.
Identify the observation, residual type, prediction, residual scale, units, and sign. Check that the plotted record matches the analysis data and that no display or data-processing issue created the apparent extreme.
Compare the dose amount, route, date and time, missed or delayed doses, infusion details where relevant, and dose changes with source records. Confirm which dose history the analysis actually used.
Compare the analysis time with collection records and the relevant dose time. Check units, date rollover, time-zone conventions, and whether the recorded time is clock time or time since dose.
Check the reported result, units, dilution or rerun records, sample identity, and any documented assay issue. Review subject identifiers and relevant clinical or dosing notes for a plausible explanation, without assuming that an unusual result must be erroneous.
Correct a value only when evidence supports a correction under the data-handling rules. Preserve the original value and its provenance, rerun the analysis, and compare conclusions. If records support the value, retain it in the primary analysis and consider a clearly labelled sensitivity analysis.
Worked Example: A Dose-Record Discrepancy
Suppose a hypothetical subject has an observed concentration of 8.4 mg/L. The fitted model predicts 5.0 mg/L at the recorded time, and the estimated residual standard deviation is 1.0 mg/L. The diagnostic uses the simple weighted residual defined above.
Initial calculation. The raw residual is \(8.4-5.0=3.4\) mg/L. The weighted residual is \(3.4/1.0=3.40\). This is a large positive discrepancy on the stated scale and merits review, but the calculation does not say why it occurred.
The analysis dose record lists 400 mg. In this hypothetical example, the medication administration record documents that 600 mg was actually given. After the analysis dose history is corrected and the model is rerun, its prediction for this observation is 7.6 mg/L. The revised raw residual is \(8.4-7.6=0.8\) mg/L, and the revised weighted residual is \(0.8/1.0=0.800\), assuming the residual standard deviation remains 1.0 mg/L.
The source record supports changing the dose history; the residual alone would not. The correction should be recorded with its source and the analysis rerun using the corrected history. The original dose entry should remain traceable, and the reported analysis should make clear that the change was based on verified documentation, not a desire to reduce the residual.
Worked Example: A Sampling-Time Error
In a second hypothetical case, the recorded observation is 12.0 mg/L, the recorded sampling time is 2.0 hours after dose, and the model predicts 8.0 mg/L at that time. The residual standard deviation is 1.5 mg/L.
At the recorded time, the raw residual is \(12.0-8.0=4.0\) mg/L, and the weighted residual is \(4.0/1.5=2.67\). The sampling log then confirms that the blood sample was collected at 1.5 hours after dose, not 2.0 hours. At 1.5 hours, the model prediction is 11.2 mg/L.
Using the verified collection time, the raw residual is \(12.0-11.2=0.8\) mg/L. The corresponding weighted residual is \(0.8/1.5=0.533\). This recalculation shows why time alignment matters: a valid concentration can look poorly fitted when compared with a prediction at the wrong time. The corrected time must be supported by the collection record, and the model should be rerun with that time rather than substituting a prediction by hand in the final analysis.
Valid but Influential Observations
An audit may find no support for changing the dose, sampling time, assay result, or subject identity. That does not prove the model is correct, but it changes the appropriate response: the observation should not be discarded merely for having a large residual. A valid extreme value can reflect unexplained variability, model misspecification, an uncommon but real concentration profile, or an observation in a region where the model performs poorly.
Influence is a separate question from extremeness. A large residual may have little effect on population estimates, while a less extreme observation can matter substantially if it is informative about a sparsely represented part of the data. One practical assessment is to compare the primary fit with a clearly labelled sensitivity refit that omits the observation. This refit is an assessment of dependence on the point, not a replacement primary analysis or proof that the point should be excluded.
Worked Example: A Verified Observation Changes Clearance
Consider a hypothetical subject with a measured concentration of 1.20 mg/L. At the observation time, the fitted prediction is 4.20 mg/L and the residual standard deviation is 0.75 mg/L. Dose and collection records agree with the analysis, the assay result is confirmed in the laboratory record, and the subject identifier is verified.
The raw residual is \(1.20-4.20=-3.00\) mg/L. Its weighted value is \(-3.00/0.75=-4.00\). The negative sign indicates that the observed concentration is below the prediction; the magnitude indicates a large discrepancy on this residual scale.
In this hypothetical analysis, the estimated population clearance is 5.0 L/h with the observation included and 4.6 L/h in a sensitivity refit without it. The absolute change is \(5.0-4.6=0.4\) L/h. Relative to the estimate without the observation, the change is \((0.4/4.6)\times100=8.70\%\). The valid observation is therefore influential for this estimate in this particular comparison.
The appropriate conclusion is not that the observation is wrong, nor that the model with it is automatically preferable in every respect. It is that the estimated clearance is sensitive to a verified observation, and that sensitivity should be reported and investigated scientifically. The primary analysis retains the observation unless an independent, justified data-handling rule supports exclusion.
Justifying the Review Rule
The logic behind separating a review flag from a deletion decision follows directly from the residual calculation. Under the simple model \(Y_{ij}=\hat y_{ij}+\epsilon_{ij}\), with \(\epsilon_{ij}\sim N(0,\hat\sigma_{ij}^{\,2})\), standardization gives
This result says that, if the model and scale assumptions hold, a standardized residual far from zero is unusual relative to the reference distribution. It does not identify the cause of the discrepancy. The same large value can arise from an incorrect recorded time, an incorrectly specified prediction, or a valid draw from the tail. Therefore, the statistic justifies looking more closely; source evidence and scientific context are needed to decide whether a record should be corrected.
The derivation is intentionally limited to the simple normal-error illustration. In practice, conditional weighted residuals and other model-based residuals may only be approximately standard normal after fitting. Use the relevant diagnostic definition and its limitations, as in the earlier material on when conditional weighted residuals are approximately standard normal. Residual quantile plots and tail behavior provide distributional context, but neither replaces record verification.
Common Pitfalls, Alternatives, and Limits
- Deleting because a residual exceeds a threshold. A cutoff identifies a value for review under a reference model; it does not establish an error. Retain a valid observation in the primary analysis.
- Changing a value to make the fit look better. Correct a dose, time, or assay entry only when traceable source evidence supports the change. Preserve the original and document the reason.
- Treating every unusual observation as a data error. A verified observation may reveal a model limitation or genuine variability. Investigate those possibilities instead of relabelling the point.
- Confusing residual size with influence. The residual describes a discrepancy for an observation; influence concerns how analysis results change when that observation is included. Assess them separately.
- Using a with-and-without refit as a deletion rule. Sensitivity analysis reveals how dependent a conclusion is on a point. It does not establish that excluding the point is scientifically justified.
Compared with automatic deletion, source-record review preserves information and can identify correctable input errors. Compared with leaving every value unquestioned, it creates an opportunity to detect transcription, timing, identity, or assay problems. A sensitivity refit adds information about influence, while robust or alternative error models may be considered when their assumptions have a scientific basis. Such alternatives should not be selected solely to suppress an inconvenient residual, and they do not replace data verification.
This approach also has limits. Source records can be incomplete, assay confirmation may not settle every biological question, and a sensitivity refit can be unstable when data are limited. A verified record does not guarantee that the model captures the relevant biology. Document what was checked, what evidence was available, and what uncertainty remains; do not claim that an observation is definitively valid or erroneous when the records cannot decide the matter.
Check Your Understanding
Answer each question before reading the solutions.
- An observation has a weighted residual of 3.2. Does this value alone justify deleting it? What should happen next?
- A recorded concentration is 6.0 mg/L, the prediction is 5.0 mg/L, and the residual standard deviation is 0.5 mg/L. Calculate the raw and weighted residuals.
- A collection log verifies that a sample was taken 30 minutes earlier than the analysis record states. What is the appropriate way to reassess its residual?
- A verified observation changes a parameter estimate in a sensitivity refit. Does that establish that the observation is erroneous?
- Why can a threshold that is unusual for one standardized residual still be crossed when many observations are inspected?
Solutions. 1. No. It is a review signal, not proof of error; check the diagnostic and trace dose, time, assay, and subject records, then document the result. 2. The raw residual is \(6.0-5.0=1.0\) mg/L; the weighted residual is \(1.0/0.5=2.00\). 3. Verify the time source, update the analysis record under the applicable data rules, rerun the model, and calculate the residual against the prediction at the corrected time. Preserve the original entry and correction history. 4. No. It establishes sensitivity of that estimate to the observation in that comparison. Error status depends on evidence about the record, not the size of the parameter change. 5. Each observation has some chance of falling in the tail under the reference model; with many observations, the chance that at least one does so increases. The calculation also depends on assumptions such as independence and the stated distribution.