Why BQL Strategy Is a Modeling Decision
A below-quantification-limit (BQL) result is not a measured concentration of zero. It indicates that the concentration was not quantified and is known only to fall below an assay limit. How that partial information enters an NLME analysis can affect estimated exposure, elimination, variability, and uncertainty. The appropriate strategy therefore depends on more than the overall percentage of BQL observations: their timing, distribution across subjects, assay limits, and the observation model all matter.
This tutorial compares three broad choices: use the censoring information in the likelihood, substitute a fixed value for each BQL result, or omit BQL records. The decision framework is not a substitute for checking model adequacy. It makes explicit what each choice assumes and which features of the PK data make those assumptions more or less credible.
What Should Guide the Choice?
Begin by mapping BQL records to their actual sampling times, subjects, doses, and assay limits. A single overall BQL percentage hides important patterns. BQL results concentrated at late times may carry information about the terminal part of the concentration-time profile, while BQL results near a sampling period intended to characterize early exposure may affect different features of the model. If nearly all observations in a time region or for a subgroup are BQL, the data may provide little direct information there, regardless of the chosen method.
Record whether the assay limit was constant. If limits changed across samples or study periods, use the limit applicable to each individual record. Also check whether BQL status arose from the assay limit alone, or whether samples were missing, mishandled, or excluded for other reasons. Those are not interchangeable data states: a missing sample supplies no concentration inequality, whereas a valid BQL sample supplies the information \(Y_{ij}<L_{ij}\).
For a measured concentration \(Y_{ij}\), let \(f_{ij}\) denote the model-predicted concentration for subject \(i\) at time \(j\), conditional on that subject’s model parameters. Under an additive normal observation model, a BQL record contributes the probability that the assay result is below the limit. Under a lognormal observation model, the corresponding probability is evaluated on the log-concentration scale. In an NLME model, the individual parameters are random; the population likelihood must account for their distribution rather than treating an estimated individual prediction as known without uncertainty.
Three Strategies and Their Assumptions
| Strategy | What enters the analysis | Key concern |
|---|---|---|
| Censored likelihood | The probability that each BQL assay result falls below its applicable limit | Requires a credible residual-error model, correct limits, and adequate numerical integration over random effects |
| Substitution | A fixed value, such as \(L_{ij}/2\), is entered as if observed | Creates artificial exact values and can distort the distribution and parameter estimates |
| Omission | BQL records are removed from the likelihood | Can discard time-dependent information and induce selection if BQL status is related to concentration |
A censored likelihood retains the information that a result lies in an interval. For a left-censored record, that interval is below the limit. For example, if the observation model is \(Y_{ij}\sim N(f_{ij},\sigma^2)\), its conditional contribution is \(P(Y_{ij}<L_{ij})\). Under a lognormal model with median \(f_{ij}\) and log-scale standard deviation \(\sigma_{\log}\), the probability is \(P(\log Y_{ij}<\log L_{ij})\). These expressions rely on the stated residual model and a valid assay limit for that record.
Substitution is easy to implement, but a value such as \(L_{ij}/2\) is a convention, not an observed concentration or an estimate of the unobserved value. It treats all BQL results as though they were measured at the same point, even though their true values may differ. Omission avoids inventing values, but it discards the inequality information and can leave a selected set of quantified observations. Neither approach is automatically harmless just because the resulting software run is stable.
Worked Example: Choosing a Method From the BQL Pattern
Consider a hypothetical study with 10 subjects and six scheduled samples per subject, for 60 planned observations. The assay has a constant LLOQ of \(0.20\) mg/L. A data review finds 18 valid BQL results: 0 of 10 at 0.5 hours, 1 of 10 at 1 hour, 2 of 10 at 2 hours, 4 of 10 at 4 hours, 6 of 10 at 8 hours, and 5 of 10 at 12 hours. No sample is missing, and the same assay limit applied throughout.
The overall BQL proportion is \(18/60=0.300\), or 30.0%. More importantly, 15 of the 20 observations at 8 and 12 hours are BQL, compared with 0 of 10 at 0.5 hours. Thus censoring is concentrated at later times rather than being an isolated handful of records. If those later observations help characterize the decline in concentration, deleting them would remove potentially relevant information.
A censored likelihood is the leading choice here, assuming the residual-error model is reasonable and each BQL record is correctly identified with its \(0.20\) mg/L limit. The constant assay limit simplifies the data specification but does not by itself validate the residual model. Substitution at \(0.10\) mg/L would impose the same exact value on all 18 records, and omission would remove 30.0% of the planned observations, disproportionately at later times. Neither alternative preserves the same information as a censoring likelihood.
This choice does not establish that every PK parameter is estimable. The concentration data may still be insufficient to identify a parameter, and the model’s predictions and diagnostics must be checked. It does establish that a method using the below-limit information is more defensible than pretending the later samples were exact values or did not exist.
Why the Censored Contribution Has This Form
For a BQL record, the observed data do not specify one value of \(Y_{ij}\); they specify the event \(Y_{ij}<L_{ij}\). The likelihood contribution must therefore be the probability of that event under the model. Under a lognormal observation model, suppose \(\log Y_{ij}\) is normal with mean \(\log f_{ij}\) and standard deviation \(\sigma_{\log}\). Standardizing the log observation gives the conditional probability below.
Here \(\Phi\) is the standard normal cumulative distribution function. In the population likelihood, this conditional probability is combined with the distribution of subject-specific random effects. Conceptually, if \(\eta_i\) denotes those effects with density \(p(\eta_i)\), the subject’s likelihood contribution integrates the conditional likelihood over that density:
In this expression, \(Q_i\) indexes quantified observations and \(B_i\) indexes BQL observations for subject \(i\). The density \(p(y_{ij}\mid\eta_i)\) is used for an exact quantified result, while the cumulative probability is used for a BQL result. The products assume the observation errors are conditionally independent given the model parameters; if the model specifies dependence, the joint observation likelihood must reflect it instead. This is the essential justification for using a probability contribution rather than an invented concentration.
Worked Example: Calculating a Lognormal BQL Probability
Suppose a hypothetical subject’s predicted median concentration at one sample time is \(f_{ij}=0.15\) mg/L, the sample’s LLOQ is \(L_{ij}=0.20\) mg/L, and the log-scale residual standard deviation is \(\sigma_{\log}=0.40\). Under the lognormal model, the probability of a BQL result is
The ratio is \(0.20/0.15=1.3333\ldots\), so \(\log(0.20/0.15)\approx0.28768\). Therefore,
The model assigns about a 76.4% probability to a BQL result at this time, conditional on the stated prediction and residual error. Its likelihood contribution is \(0.764\), and its log-likelihood contribution is approximately \(\log(0.764)=-0.269\). The quantified probability is the complement, about \(1-0.764=0.236\). This calculation depends on the lognormal residual assumption; it is not a direct estimate of the subject’s unobserved concentration.
When Simpler Methods May Be Considered
Worked Example: Sparse BQL Records With Limited Influence
In a second hypothetical analysis, 80 samples are scheduled across subjects and times. Only one valid sample is BQL, at the final sampling time, with an assay limit of \(0.05\) mg/L. The other observations are quantified, the assay limit is stable, and the final sample is not intended to be the sole basis for estimating a key feature of the profile.
A censored likelihood remains a principled option: it can use the statement that the final result is below \(0.05\) mg/L. If the software and model support it, there is little statistical reason to replace that inequality by an exact value. However, because only 1 of 80 records is BQL, an analysis that omits this record may be a defensible practical approximation if the omission is documented and its effect is checked.
Substitution at \(0.025\) mg/L is less attractive: the value was not measured, and the choice of one-half the limit does not follow from the data. Whether omission is adequate depends on the analysis objective and the influence of that record. If the final-time observation is especially important for a parameter or if omitting it materially changes estimates, the low overall frequency does not justify ignoring it.
Diagnostics, Failure Modes, and Limitations
After selecting a method, check whether it behaves consistently with the observed data. Compare observed BQL frequencies with model-predicted frequencies by sampling time, subject, dose, covariate group, and assay limit. A model that predicts almost no BQL observations where many were observed, or predicts frequent BQL results where nearly all samples were quantified, may have an inadequate concentration or residual-error model. This check should be interpreted in light of the sampling design and the uncertainty in predicted frequencies.
Also inspect whether estimates change sharply when the BQL handling assumptions change. Large shifts can indicate that the observations contain limited information about a parameter, that an assumption matters materially, or that the model is sensitive to how censoring is represented. A small overall BQL percentage does not guarantee small impact if those records occur at influential times or in a particular subgroup. Conversely, many BQL results in a region where the predictions are far below the limit may contribute little information for distinguishing among parameter values.
- Incorrect limits: Using one study-wide limit when records had different limits assigns the wrong event to affected samples.
- Wrong residual model: A likelihood can be correctly coded but still misleading if its assumed observation distribution does not describe the data.
- Confusing BQL with missing: A valid below-limit result contributes an inequality; a sample that was not obtained does not.
- Overinterpreting substitution: A substituted value is a computational convention, not a measured value or a precise recovery of the latent concentration.
- Ignoring concentration range: If most observations are BQL, the data may not identify important parameters well under any handling method.
Censored likelihoods also have limitations. They rely on correctly specified observation and random-effects models, correctly recorded assay limits, and reliable numerical evaluation. If data collection produces additional features such as rounding, assay changes, or non-random missingness, the observation model may need to represent those features. A software option labelled “BQL likelihood” does not remove the need to verify what probability it uses and how it handles subject-level random effects.
A Practical Selection Sequence
Separate valid BQL results from missing samples, check units, and attach the correct assay limit to every record.
Tabulate BQL frequency by time, subject, dose, covariate group, and assay period; do not rely on the overall percentage alone.
Assess whether the residual-error distribution and random-effects model can support a censored likelihood, and whether the data contain enough quantified observations to inform the target parameters.
Use censored likelihood when its assumptions are credible and censoring is meaningful; consider simpler handling only with a clear rationale and checks of predictive fit and parameter stability.
Common Mistake
A common mistake is to say, “Only 10% of records are BQL, so replacing each with half the limit cannot matter.” This reasoning uses the overall percentage but ignores where the records occur and what they inform. If those observations are concentrated in a particular time region or subject group, they may affect a parameter or comparison even when they are few. The correction is to examine the pattern and the analysis objective, then evaluate whether a simpler method changes the conclusions. The percentage alone is not a decision rule.
Check Your Understanding
Answer each question, then compare with the short solutions.
- A study has 20% BQL observations overall, but 70% of samples at the final two times are BQL. Why is the overall percentage insufficient for choosing a method?
- Under a lognormal observation model, what probability contribution corresponds to a BQL result with limit \(L_{ij}\)?
- A valid sample is reported BQL, while another scheduled sample was not collected. Do both records contribute the same information?
- What additional check is important before treating omission as harmless when only one observation is BQL?
Solutions. (1) The BQL records are concentrated in time and may inform profile features differently from scattered records; the pattern and analysis objective matter. (2) The contribution is \(P(Y_{ij}<L_{ij}\mid f_{ij})\), which under the stated lognormal model is \(\Phi\bigl(\log(L_{ij}/f_{ij})/\sigma_{\log}\bigr)\), conditional on the individual prediction. (3) No. The valid BQL sample supplies the inequality \(Y_{ij}<L_{ij}\); the uncollected sample supplies no measured result or such inequality. (4) Check where that observation occurs and whether omitting it changes estimates, uncertainty, or conclusions; a small count alone does not establish negligible influence.