Avoiding Log Transformation of Substituted BQL Values

Why a BQL record is not a small measured concentration

A below-quantification-limit (BQL) record says that a concentration fell below an assay’s lower limit of quantification (LLOQ). It does not say that the concentration equalled zero, half the LLOQ, or any other chosen replacement value. This distinction matters especially when the observation model is lognormal: logging a replacement turns an inequality into an apparently exact number, changing the information supplied to the likelihood.

The correct contribution for a BQL observation is the probability of falling below the LLOQ under the model. That probability is evaluated on the concentration scale, or equivalently as a cumulative probability after a monotone log transformation. It is not the density evaluated at a substituted concentration.

The distinction is between a point and a region. A quantified value such as \(y_{ij}=1.4\) mg/L identifies a point at which a density is evaluated. A BQL report with LLOQ \(1.0\) mg/L identifies the whole interval \(0<Y_{ij}<1.0\) mg/L. Substitution selects one point from that interval without evidence that it was the actual concentration.

The probability-scale contribution under a lognormal model

Suppose, conditional on a subject’s random effects, the observation model is

where \(\mu_{ij}>0\) is the model-predicted median concentration and \(\sigma\) is the residual standard deviation on the log scale. This specifies a lognormal concentration distribution. It assumes that the residual is normal on the log scale with the stated variance; a BQL result is interpreted as censoring at the known LLOQ, not as a separate exact concentration.

Because the logarithm is strictly increasing for positive concentrations, \(Y_{ij}<L_{ij}\) is equivalent to \(\log Y_{ij}<\log L_{ij}\). Standardizing the normal variable gives the conditional BQL probability:

Here \(\eta_i\) denotes the subject-specific random effects, and \(\Phi\) is the standard normal cumulative distribution function. The conditional mean structure may depend on time, dose, fixed effects, and \(\eta_i\); the formula applies once that structure and the residual model are specified. A quantified observation instead has conditional density

This is a density per concentration unit. If the likelihood is expressed using the log-concentration \(z=\log y\), its density is the normal density per log-concentration unit. The two density forms are related by the change-of-variables factor \(1/y\). For BQL data, however, the relevant quantity is a probability, so no point-density Jacobian is needed.

Why substitution followed by logging changes the likelihood

Suppose a BQL concentration is replaced by \(c^*\), such as \(L/2\), and then transformed to \(\log c^*\). A conventional point-observation likelihood treats that number as if the measured concentration were exactly \(c^*\). The resulting contribution is a density at \(c^*\), or a normal density at \(\log c^*\) if the analysis is written on the log scale. Neither is the probability of the event \(Y<L\).

This replacement changes more than the numerical value used in a calculation. It imposes a false observation: it locates the censored value at a particular point and gives it the precision of an exact observation. The chosen replacement can therefore alter parameter estimates, residual patterns, and uncertainty. Logging does not restore the information lost by replacing an inequality with a point.

Including subject random effects

In a nonlinear mixed-effects model, observations from the same subject share random effects. The appropriate subject likelihood first forms contributions conditional on \(\eta_i\), then integrates over the random-effects distribution. If quantified and BQL observations are conditionally independent given \(\eta_i\), a subject’s conditional likelihood is the product of density contributions for quantified records and CDF contributions for BQL records. The marginal subject likelihood is the expectation of that product over \(\eta_i\).

Here \(Q_i\) and \(B_i\) are the sets of quantified and BQL records for subject \(i\), respectively, and \(p(\eta_i)\) is the random-effects density. Conditional independence is an assumption of this expression; if the observation errors have a different dependence structure, the appropriate joint conditional probability or density must be used. Crucially, the integral is over the product of conditional contributions. Shared random effects generally make repeated observations dependent after integration.

Practical checks and common pitfalls

The probability-scale method relies on a clearly specified observation model and threshold. Check that the LLOQ is correct for each record, that the concentration distribution has support above zero when a log model is used, and that the residual distribution and conditional-dependence assumptions are appropriate. If the assay reports a different censoring event, such as a value below a limit that varies by sample, use that event and its corresponding threshold. A record known only to be BQL is not automatically a missing-at-random observation; its likelihood contains information through the threshold event.

Useful diagnostics include comparing observed and model-predicted BQL frequencies across time, dose, and relevant subgroups; checking whether BQL records cluster in regions where the model predicts very few; and inspecting whether residual or prediction summaries improperly treat substituted values as exact. Predictions for the frequency of BQL samples should be calculated from the model’s event probabilities, not by counting replacement values as if they were observed concentrations. A mismatch can indicate problems in the structural model, residual model, random-effects distribution, assay thresholds, or data handling.

The main alternative is substitution-based analysis: replace each BQL value with a fixed amount and fit an ordinary observation likelihood. It can be easy to implement, but its result depends on the chosen replacement and it does not represent censoring in the likelihood. A censored likelihood instead retains the actual event and can use the observation model’s probability for that event. It requires a suitable probability calculation and, in a mixed-effects model, integration over random effects. That computational requirement is not a reason to interpret a substituted point as measured data.

Substitution may also fail at the most basic mathematical level: replacing a BQL value by zero makes \(\log(0)\) undefined. Replacing it by a positive number avoids that computational problem but not the inferential one. More generally, censoring likelihoods depend on the assumed model below the LLOQ; if that model is misspecified or the data contain assay features not represented in it, the resulting probabilities may still be unreliable. Probability-scale treatment correctly represents the stated censoring event, but it does not guarantee that every model assumption is true.