Why Below-Quantification Data Need Special Handling
A concentration reported as below the lower limit of quantification (LLOQ) is not an observed concentration equal to zero. It tells us that the assay did not quantify the concentration and that the result fell below a known threshold. That distinction matters in population pharmacokinetic (PK) analysis: treating such a record as an exact value can distort the information supplied to the model, while discarding it can throw away useful information.
The same care applies to extreme or implausible records. An unusual value may reflect genuine variability, an assay or data-processing issue, or an incorrect sample time or dose history. Removing a record simply because it does not fit the current model risks hiding model misspecification; keeping a verified data error as if it were valid can distort estimates. The first task is to understand what the record means and how it was produced.
Likelihood Contribution for a Below-Quantification Record
Suppose the model predicts concentration \(C_{\mathrm{pred}}\) at a sampling time. Under an additive normal residual model, assume the measured concentration \(Y\), conditional on the model prediction and individual effects, satisfies \(Y\sim N(C_{\mathrm{pred}},\sigma^2)\). Here \(\sigma\) is the residual standard deviation in concentration units. If the assay reports only that \(Y<L\), where \(L\) is the LLOQ, the appropriate likelihood contribution is the probability of that event:
The model does not pretend that the concentration equals the LLOQ, half the LLOQ, or zero. Instead, it asks how likely the observed classification “below \(L\)” would be under each candidate model prediction. With a lognormal residual model, a common choice for positive PK concentrations, assume \(\log(Y)\sim N(\log(C_{\mathrm{pred}}),\sigma^2)\). The corresponding contribution is:
For a quantified value \(y\), the likelihood instead uses the model’s density at that measured value. Thus a data set containing both quantified and below-quantification results has a likelihood with density contributions for the former and probability contributions for the latter. For a population model, these contributions are combined with the model for individual and residual variability; the censoring probability is conditional on the individual’s model prediction.
Worked Example: Different Assay Limits, Different Censoring Information
Worked Example: Different Assay Limits, Different Censoring Information
Suppose a hypothetical model predicts a concentration of \(0.700\) mg/L. For illustration, use an additive normal residual model with \(\sigma=0.150\) mg/L. One assay has an LLOQ of \(0.500\) mg/L, while another has an LLOQ of \(0.300\) mg/L. Calculate the likelihood contribution for a below-quantification result under each assay.
For the assay with \(L=0.500\) mg/L:
Its contribution is about \(0.09121\), or \(9.121\%\). This is the probability of a below-quantification assay result under the stated prediction and residual model; it is not a probability that the model prediction itself is below the LLOQ.
For the assay with \(L=0.300\) mg/L:
The probability is about \(0.003830\), or \(0.3830\%\). The same predicted concentration and residual standard deviation therefore imply different likelihood contributions because the assay limits differ. The residual standard deviation and both limits are expressed in mg/L, so the standardized \(z\)-values are dimensionless.
Why Deletion and Substitution Can Mislead
Deleting a below-quantification record treats it as if it supplied no information. But the record does distinguish predictions under which a below-limit result is likely from predictions under which it is unlikely. Conversely, replacing the record with zero, \(L/2\), or \(L\) treats a threshold classification as an exact concentration. The resulting analysis can be sensitive to the arbitrary replacement rule, especially when many observations are below the limit or occur in the parts of the profile that inform elimination.
| Approach | How it treats the record | Main concern |
|---|---|---|
| Censored likelihood (M3) | Uses the probability of a result below the record-specific LLOQ | Depends on a suitable residual model and correct censoring information |
| Deletion | Removes the record from the likelihood | Discards information and can change which parts of the profile contribute |
| Substitution | Replaces the record with a fixed value, such as zero or half the LLOQ | Imposes an exact value not observed and can bias estimates |
| Separate model for assay results | Models the quantified/below-limit classification directly | May be useful in specific settings but requires a correctly specified observation model |
Substitution is not always equally damaging, and a small number of late below-quantification samples may have little influence on a particular parameter. The important point is that deletion and substitution make different assumptions from censored likelihood. Their consequences should be assessed in context rather than assumed harmless or universally catastrophic.
Worked Example: How Censoring Changes a Simple Estimate
Consider a hypothetical, deliberately simple model in which measurements are independent and normally distributed with unknown mean concentration \(\theta\) and known \(\sigma=0.200\) mg/L. The data are one quantified result, \(y=0.800\) mg/L, and one result reported as \(Y<0.500\) mg/L. This example illustrates the likelihood calculation; it is not a full population PK model.
The likelihood, apart from constants, is the density for the quantified observation multiplied by the probability for the censored observation:
Taking the derivative of the log-likelihood gives the score:
The first term is the contribution from the quantified value; the second is the contribution from the probability of falling below the limit. Setting the score to zero and solving numerically gives \(\hat{\theta}\approx0.584\) mg/L. For example, near \(\theta=0.584\), \(z=-0.420\), and the two score terms are approximately \(5.400\) and \(5.414\) per mg/L, respectively, so the score is close to zero.
If the below-limit record is deleted, the maximum-likelihood estimate is just \(0.800\) mg/L, the only remaining quantified value. If it is replaced by \(L/2=0.250\) mg/L, the arithmetic mean of the two treated values is \((0.800+0.250)/2=0.525\) mg/L. Replacing it with zero gives a mean of \((0.800+0)/2=0.400\) mg/L. These different results arise from different treatments of the same threshold information; none of the substituted values was measured.
The calculation assumes independent normal measurements with a common known standard deviation and a correctly recorded LLOQ. Those assumptions are intentionally simple for transparent arithmetic. A real population PK analysis must use an appropriate model for concentration and residual variability, and account for the population structure.
Derivation: Why the Censoring Probability Is the Right Contribution
A likelihood contribution is the probability density of what was observed, viewed as a function of model parameters. When an exact concentration \(y\) is observed, the contribution is its density \(f(y\mid\theta)\). When the only observation is \(Y<L\), every value below \(L\) is consistent with the recorded result. The probability of the recorded event is therefore the integral of the density over all values that could have produced it:
For a normal model with mean \(C_{\mathrm{pred}}\) and standard deviation \(\sigma\), standardizing by subtracting the mean and dividing by \(\sigma\) gives \(F(L\mid\theta)=\Phi((L-C_{\mathrm{pred}})/\sigma)\). This follows because \((Y-C_{\mathrm{pred}})/\sigma\) has a standard normal distribution. For positive concentrations modeled on the log scale, apply the same standardization to \(\log(Y)\), yielding the lognormal expression above. This derivation also shows why using a density evaluated at zero or at \(L/2\) does not represent the observed event.
Investigating Implausible Records and Dosing Errors
A value that is surprising under a fitted model is a reason to investigate, not by itself a reason to delete. Check source records and data processing for units, decimal placement, sample identifiers, assay flags, duplicate samples, collection and processing times, and the concentration’s relationship to the dose history. Confirm whether a value was actually quantified, whether an LLOQ changed across assays or visits, and whether dosing was observed, reported, or assumed.
Dosing and sampling times are especially consequential because the model prediction depends on their relationship. A correct concentration attached to the wrong time can resemble a different concentration-time pattern. The size and direction of the resulting parameter distortion depend on the model, data, and error; no universal adjustment can be inferred from one discrepant record.
Worked Example: A Sample-Time Entry Error
In a hypothetical one-compartment intravenous bolus example, let the concentration follow \(C(t)=C_0e^{-kt}\), with \(C_0=10.0\) mg/L and \(k=0.100\) per hour. A sample was actually collected 8 hours after dosing, but the data file records it as 2 hours after dosing. Calculate the predictions at both times and explain the risk.
At the true time of 8 hours:
At the incorrectly recorded time of 2 hours:
The ratio of the predictions is \(C(2)/C(8)=e^{0.6}\approx1.822\). Thus, under this illustrative model, the entered time makes the predicted concentration about 1.822 times the prediction at the actual time. Fitting the record at the wrong time may pull model estimates toward a pattern that explains the data error rather than the underlying PK. The remedy is to investigate and correct the time if the source data support a correction; otherwise, document the uncertainty and assess its influence rather than silently changing or deleting the record.
Diagnostics, Alternatives, and Limitations
A censored likelihood is not a substitute for checking the assay and data pipeline. Review the number and proportion of below-quantification records by sampling time, assay, site, dose group, and relevant study period. A sudden change in the fraction below the limit may reflect an assay or recording change, a design difference, or a model that predicts the profile poorly. Compare observed and model-predicted proportions below the limit, as well as the quantified concentration data. Ordinary residual plots based only on quantified values cannot diagnose every mismatch in the censoring pattern.
For an implausible quantified value, verify the original record and units first. If it is confirmed, investigate whether the structural or residual model can describe it and examine its influence on estimates. If it is a verified error, correct or exclude it according to a documented rule that does not depend on whether the value supports the preferred model. Sensitivity analyses—with and without a questionable record, or under justified alternative residual assumptions—can reveal whether conclusions depend heavily on that decision.
The censored-likelihood approach is most informative when the assay threshold is known and the model adequately describes measurements around that threshold. A large fraction of censored records can leave some parameters weakly informed, particularly if few quantified samples remain in informative parts of the profile. A likelihood method cannot recover exact values that were never measured, repair incorrect dose histories, or make an unsuitable model correct. Where a likelihood is difficult to specify or fit, deletion or substitution may appear simpler, but those alternatives do not preserve the threshold information and should be justified and accompanied by sensitivity assessment.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A model predicts \(0.600\) mg/L with additive normal residual standard deviation \(0.100\) mg/L. An assay reports a result below \(0.500\) mg/L. What is the likelihood contribution?
- Why does deleting a below-quantification result differ from using its censored-likelihood contribution?
- A laboratory changes the LLOQ partway through a study. What should be checked before fitting the model?
- A concentration is implausibly high relative to the recorded sample time. Name two checks that should precede excluding it.
Solutions. 1. Standardize: \(z=(0.500-0.600)/0.100=-1\). The contribution is \(\Phi(-1)\approx0.1587\), the probability of a result below \(0.500\) mg/L under the stated model. 2. Deletion contributes no likelihood information for that record, while the censored likelihood contributes the probability that the observed below-limit event would occur. 3. Verify the correct assay and LLOQ for each sample, and check that assay identifiers and below-limit flags were recorded and mapped correctly. Use the applicable threshold record by record. 4. Check the source sample and concentration records, units and decimal placement, and the dosing and collection times; any two of these are appropriate. Exclusion should not be based solely on disagreement with the fitted model.