Residual Diagnostics When Values Are Censored

Why Censored Records Need Different Residuals

Residual plots are useful only when the residual represents the difference between the observation and what the model predicts. For a quantified concentration, the recorded value can be compared with a model prediction, with appropriate attention to the residual-error distribution. A below-quantification-limit (BQL) record is different: it says the concentration fell below a threshold, not that it equalled zero, half the threshold, or any other chosen substitute. Treating it as an exact value creates an artificial residual and can make an apparently systematic pattern that is only a consequence of the substitution.

A BQL record still contains diagnostic information. Its threshold and censoring status tell us whether the model assigned reasonable probability to the observed event. A quantified record also has information about its position above the threshold. Censoring-aware diagnostics use these facts while respecting the observation model used to fit the nonlinear mixed-effects model.

Definition: Let \(Y_{ij}\) denote the latent concentration for individual \(i\) at observation \(j\), and let \(L_{ij}\) be its applicable quantification threshold. Define \(C_{ij}=1\) when the record is BQL, meaning \(Y_{ij}<L_{ij}\), and \(C_{ij}=0\) when it is quantified, meaning \(Y_{ij}\geq L_{ij}\). For a BQL record, the observed information is the event \(Y_{ij}<L_{ij}\), not an exact value of \(Y_{ij}\).

The threshold may vary by observation, for example when the assay or its validated quantification limit changes. Use the threshold that applies to that record; do not compare a BQL indicator with a single threshold if the actual analysis used different ones. The observation model must also match the model being diagnosed. For example, additive error is commonly specified on the concentration scale, whereas a lognormal observation model specifies a distribution for log concentration.

Model-Based Probabilities and Residual Summaries

Write \(F_{ij}(y)\) for the model-implied cumulative distribution function (CDF) of \(Y_{ij}\), conditional on the model predictions and the conditioning information used for the diagnostic. That conditioning might use population predictions or individual-specific predictions; the choice should be reported because the two answer different diagnostic questions. Under a continuous observation model, the predicted probability of a BQL result is

p_{ij}^{\mathrm{BQL}}=P(Y_{ij}<L_{ij})=F_{ij}(L_{ij}).

For a quantified record, a useful conditional probability position is

u_{ij}^{\mathrm{Q}}= \frac{F_{ij}(y_{ij})-F_{ij}(L_{ij})}{1-F_{ij}(L_{ij})}, \qquad y_{ij}\geq L_{ij}.

This is the conditional CDF of the measured value given that it is quantifiable. If the model is correct and the observation distribution is continuous, these values follow a uniform distribution on \([0,1]\) across observations with the same conditioning scheme. Transforming them with the inverse standard normal CDF gives a conditional normal-score residual, \(r_{ij}^{\mathrm{Q}}=\Phi^{-1}(u_{ij}^{\mathrm{Q}})\). It is not the ordinary residual \((y_{ij}-\hat y_{ij})/SE\); it accounts for the specified observation distribution and the fact that only values above the threshold appear as quantified.

For a BQL observation there is no exact value to insert into \(F_{ij}\). One option is to retain its predicted BQL probability alongside its observed censoring indicator. Another is a randomized probability-integral-transform residual: draw \(U_{ij}\) uniformly from \([0,F_{ij}(L_{ij})]\) for a BQL record, and set \(U_{ij}=F_{ij}(y_{ij})\) for a quantified record. Then \(r_{ij}=\Phi^{-1}(U_{ij})\). This randomization accounts for the fact that all latent values below the threshold were recorded as the same BQL category. A fixed substitute does not.

Key result: Under a correctly specified continuous observation model and the conditioning scheme used to calculate \(F_{ij}\), the randomized \(U_{ij}\) is uniform on \([0,1]\). Thus its normal-score transform has a standard normal reference distribution. This is a diagnostic reference, not a guarantee that residuals are independent: repeated observations on an individual and estimated parameters can affect their dependence and apparent spread.

The uniformity result follows directly from the construction. Let \(a=F_{ij}(L_{ij})\). For any \(u\) between 0 and \(a\), a randomized value is at most \(u\) only when the record is BQL and its draw falls in the first fraction \(u/a\) of the interval. Therefore \(P(U_{ij}\leq u)=a(u/a)=u\). For \(a<u\leq1\), all BQL records contribute probability \(a\), and quantified values with \(F_{ij}(Y_{ij})\leq u\) contribute \(u-a\). Hence \(P(U_{ij}\leq u)=a+(u-a)=u\), as required.

Worked Example: Conditional Residual for a Quantified Lognormal Observation

Suppose, hypothetically, that a model predicts a lognormal concentration with median \(1.5\) mg/L and standard deviation \(0.40\) on the log scale. The assay threshold is \(L=0.50\) mg/L, and the observed quantified concentration is \(y=1.8\) mg/L. Under this model, \(\log Y\) is normal with mean \(\log(1.5)\) and standard deviation \(0.40\).

First calculate the model-predicted BQL probability:

F(L)=\Phi\left(\frac{\log(0.50/1.5)}{0.40}\right) =\Phi(-2.74653)\approx0.00301.

The model therefore assigns about a 0.301% probability to a concentration below this threshold. For the quantified observation,

F(y)=\Phi\left(\frac{\log(1.8/1.5)}{0.40}\right) =\Phi(0.455804)\approx0.675735.

Its conditional probability position among quantifiable values is, using the unrounded CDF values,

u^{\mathrm{Q}}= \frac{0.675735-0.00301}{1-0.00301} \approx0.67476, \qquad r^{\mathrm{Q}}=\Phi^{-1}(0.67476)\approx0.453.

The conditional normal-score residual is about \(0.453\), with no concentration units because it is a standardized score. It indicates a value somewhat above the conditional median of the quantifiable distribution, not a concentration error of \(0.453\) mg/L. The calculation uses the stated lognormal model and threshold; replacing the BQL threshold by a concentration would change the diagnostic.

Using BQL Indicators and Thresholds in Diagnostics

A useful diagnostic display keeps both the event actually observed and its model-predicted probability visible. For each record, retain \(C_{ij}\), \(L_{ij}\), and \(p_{ij}^{\mathrm{BQL}}\). Plot predicted BQL probability against time, predicted concentration, or another relevant covariate, marking whether the record was BQL. Examine whether BQL records occur in regions where the model assigns appreciable probability to censoring, and whether quantified records occur where the model predicts nearly certain censoring. A single event with a small predicted probability is possible; a pattern of such discrepancies is more concerning.

For quantified records, plot conditional normal-score residuals against time, prediction, and relevant covariates. A trend, changing spread, or repeated structure within individuals can signal a misspecified time course, residual-error model, covariate relationship, or assay description. For BQL records, plot the predicted probability or the randomized residual, but label randomized displays as such: their exact positions vary from draw to draw. The probability itself is often the clearest way to communicate how compatible a BQL event is with the model.

Worked Example: Interpreting a BQL Probability Under Additive Error

Consider a hypothetical observation model with additive normal error: \(Y\) is normally distributed with mean prediction \(2.0\) mg/L and residual standard deviation \(0.50\) mg/L. At this sample time, the threshold is \(1.0\) mg/L and the observed record is BQL. The normal model assigns

P(Y<1.0)= \Phi\left(\frac{1.0-2.0}{0.50}\right) =\Phi(-2)\approx0.02275.

The predicted BQL probability is about 2.275%. This individual BQL record is less expected under the model than a BQL record with a 30% predicted probability, but it is not impossible. It should be recorded as an event with probability 0.02275, not converted to \(1.0\) mg/L or \(0.5\) mg/L and plotted as an ordinary residual. If several BQL observations in comparable circumstances each have very small predicted probabilities, investigate the predictions, error model, thresholds, and assay information rather than explaining the pattern by arbitrary substitutions.

Worked Example: Diagnosing a Pattern Across Different Thresholds

Suppose a hypothetical lognormal model predicts a median concentration of \(1.0\) mg/L and log-scale standard deviation \(0.30\) at two sampling times. At the first time the threshold is \(0.80\) mg/L and the record is BQL; at the second, the threshold is \(0.50\) mg/L and the record is quantified at \(0.55\) mg/L. Calculate the BQL probability at the first time and the conditional score at the second.

At the first time,

P(Y<0.80)= \Phi\left(\frac{\log(0.80/1.0)}{0.30}\right) =\Phi(-0.74380)\approx0.2285.

Thus the BQL result has a model-predicted probability of approximately 22.85%. At the second time, the threshold differs, so use \(0.50\) mg/L for the quantified record:

F(0.50)=\Phi\left(\frac{\log(0.50)}{0.30}\right) =\Phi(-2.31049)\approx0.01044,

and

F(0.55)=\Phi\left(\frac{\log(0.55)}{0.30}\right) =\Phi(-1.99258)\approx0.02314.

Therefore,

u^{\mathrm{Q}}= \frac{0.02314-0.01044}{1-0.01044} \approx0.01283, \qquad r^{\mathrm{Q}}=\Phi^{-1}(0.01283)\approx-2.23.

The first record is a BQL event with a moderate predicted probability; the second is a quantified value in the low tail of the model's conditional quantifiable distribution. The two diagnostics are not interchangeable. The calculation also illustrates why applying the first sample's threshold to the second would give the wrong residual summary.

Assumptions, Alternatives, and Failure Modes

These diagnostics are meaningful only if the CDF represents the observation process used in the analysis. Check that the residual-error distribution and scale are correct, each threshold is correctly assigned, and the censoring rule is represented accurately. The usual continuous-distribution formulas assume that exact equality to the threshold has probability zero. If the assay has rounding, interval reporting, or a different censoring convention, the observation categories and their probabilities must reflect that process.

The conditional quantified residuals assess the distribution among values above the threshold; they do not by themselves diagnose whether the model predicts the right fraction of BQL records. Conversely, a BQL probability alone does not show whether the quantified concentrations are well fit. Use both sources of information. Population-prediction diagnostics and individual-prediction diagnostics can also differ: individual predictions use information from that individual's observations, so their residuals may look less extreme. State which predictions were used and avoid interpreting reduced residual spread as independent confirmation of fit.

Main alternatives have distinct trade-offs:

Approach	What it uses	Main limitation
Substitute a value for each BQL record	An invented exact concentration	Residuals depend on the chosen substitute and can show artificial trends or spread.
Plot quantified residuals only	Exact quantified values	Useful for the quantified distribution, but omits the diagnostic information in BQL events.
Use BQL probabilities and censoring-aware residuals	The event \(Y<L\), the applicable threshold, and quantified values	Depends on a suitable observation model and can be sensitive to prediction and parameter uncertainty.

Randomized residuals have additional practical limitations: their plotted values vary with the random draw, especially when many records are censored. Repeating the randomization can reveal whether a visual pattern is stable or an artefact of one draw. Residual plots are diagnostic tools, not formal proof of model adequacy. Sparse data, many BQL records, correlated repeated measures, or imprecise parameter estimates can make patterns difficult to distinguish. Avoid treating a collection of probabilities as a formal assessment of the overall frequency of BQL observations; that is a separate predictive-check question.

Common Mistake

Pitfall: “The threshold is \(0.5\) mg/L, so every BQL concentration equals \(0.5\) mg/L; I can calculate an ordinary residual from that value.” This is incorrect because the record only establishes \(Y<0.5\) mg/L. Substitution invents a concentration, and the resulting residual can be distorted by the threshold and the number of censored observations. Instead, use the model probability \(P(Y<0.5)\) for that record and use censoring-aware residuals for quantified values and, if desired, randomized probability-integral-transform residuals for the censored record.

Key takeaway: A quantified concentration is an exact observation subject to residual error; a BQL record is an inequality with a threshold. Diagnose them accordingly: use the specified observation CDF, preserve each record's censoring indicator and threshold, and interpret a predicted BQL probability as a model probability for that event.

Check Your Understanding

Answer each question using the censoring-aware definitions above. Brief solutions follow the questions.

Why is a BQL record not an ordinary concentration residual?
A model assigns a 0.08 probability to a BQL event, and that record is observed as BQL. What does 0.08 mean, and what does it not mean?
Why must the actual threshold for a quantified record be used when calculating its conditional probability position?
What is the purpose of randomizing the probability-integral-transform value for a BQL observation?
Name one limitation of using individual-specific predictions for residual diagnostics.

Solutions. (1) The record states \(Y<L\), not an exact value of \(Y\), so a numeric residual would require inventing a concentration. (2) The fitted observation model assigns an 8% probability to a concentration below that record's threshold, conditional on the predictions used; it is not an 8% probability that the model is true or that the observation is erroneous. (3) The conditional distribution of quantified values depends on the lower truncation point \(L\), so a different threshold changes both \(F(L)\) and the conditional probability position. (4) Randomization spreads the probability mass for the censored event across \([0,F(L)]\), allowing a uniform reference distribution under the specified continuous model. (5) Individual-specific predictions use that individual's data and can make residuals appear less extreme; this dependence means they are not independent confirmation of population-level fit.