Diagnosing Bias From BQL Handling

What a Change in Estimates Can—and Cannot—Tell You

A BQL record says that a concentration was below a known quantification limit; it does not say that the concentration equalled zero or any particular replacement value. If an analysis deletes that record or assigns it a fixed value, the fitted model can change because the record’s information has been discarded or altered. A useful diagnostic is therefore to compare parameter estimates and predicted concentration-time profiles from that analysis with results from a censored-likelihood analysis that uses the BQL status.

The comparison is a diagnostic of patterns consistent with bias, not a direct measurement of bias. A parameter shift may also reflect limited information, model misspecification, instability, or differences in implementation. The comparison is most interpretable when all analyses use the same subjects, quantified observations, structural model, random-effects structure, residual-error model, covariates, and estimation settings, changing only how BQL records are handled.

The term M-method can refer to different operational rules. In this tutorial, it means a BQL-handling approach that deletes records, substitutes fixed values, or otherwise does not use the BQL event as a censored probability. Always identify the actual rule used before interpreting a method comparison. In particular, deletion and substitution need not have the same consequences.

Why the Censored Likelihood Is the Comparator

Suppose that, conditional on \(\eta_i\), a concentration \(Y_{ij}\) has cumulative distribution function \(F_{ij}(y\mid\theta,\eta_i)\). For a quantified result \(y_{ij}\), the likelihood uses its observation density. For a BQL record, the relevant information is the event \(Y_{ij}<L_{ij}\), whose conditional likelihood contribution is

For example, if the observation model is lognormal, then \(\log Y_{ij}\) is normally distributed conditional on the model parameters and random effects. The BQL contribution is the normal probability that \(\log Y_{ij}<\log L_{ij}\), not the density evaluated at \(L_{ij}/2\). In a mixed-effects likelihood, the contributions are combined across records and subjects while accounting for the distribution of the random effects.

A substitution instead treats a chosen value—perhaps \(L_{ij}/2\)—as if it were an observed concentration. Deletion uses neither the event probability nor a concentration value for that record. Both approaches can discard information about where a model’s predicted distribution lies relative to the limit. Their effect depends on the timing and frequency of BQL records, the residual model, and how model parameters jointly determine the profile.

Why Deletion and Substitution Can Shift a Profile

A simple derivation shows why deleting BQL values can affect a fit. Consider a log-concentration \(Z\sim N(\mu,\sigma^2)\) and a log-limit \(\ell=\log L\). If values below \(\ell\) are deleted, the retained values follow a truncated distribution. With \(a=(\ell-\mu)/\sigma\), its mean is

where \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution function. The inequality holds because \(\sigma>0\), \(\phi(a)>0\), and \(1-\Phi(a)>0\). Thus the retained values tend to be higher than the unconditional mean at that time. In a declining profile, deletion often removes more late, low concentrations and can make the apparent decline shallower. This is a direction to watch for, not a guaranteed direction for every fitted NLME parameter: correlations, nonlinear parameterization, and the pattern of available samples can alter the result.

Substitution can create a different pattern. Replacing several BQL values with the same positive number can create an artificial plateau near that number. A fitted model may then predict concentrations that decline too slowly at late times or shift parameters controlling disposition. If the replacement is zero, log-scale analyses may be undefined, and even on the original scale zero is not the information conveyed by a BQL result. These mechanisms motivate checking both estimates and predictions rather than interpreting a parameter shift in isolation.

Comparing Estimates and Predictions in Practice

A practical comparison starts with a prespecified censored-likelihood analysis and one clearly defined M-method analysis. For each, record the same population parameter estimates, their uncertainty, and model-predicted concentrations at clinically relevant times. A relative parameter change can be summarized as \((\hat\theta_{\mathrm{M}}-\hat\theta_{\mathrm{cens}})/\hat\theta_{\mathrm{cens}}\times100\%\), provided the censored estimate is not near zero and the parameter scale makes a percentage meaningful. For a profile, report absolute differences in concentration with units as well as relative differences when appropriate.

Consider the following hypothetical NLME fits. Both use the same structural model and quantified observations; only BQL handling differs. The reported clearance and volume are positive, so percentage comparisons are interpretable.

Assumptions, Diagnostics, and Limits

Interpret method differences as evidence about BQL handling only when the comparison is controlled. The censoring limit and BQL status must be correctly recorded; the observation model must describe the quantified values and the censoring mechanism; and the compared fits must otherwise use the same model and data. In particular, if a different residual-error model or random-effects structure is used in one fit, the parameter shift cannot be attributed specifically to BQL handling.

Useful checks include the following:

• Map BQL records in time. A concentration of BQL records in the terminal phase makes changes in terminal predictions especially relevant; BQL records at other times may affect different profile features.
• Compare the full profile, not just one parameter. Inspect predictions at times with quantified and BQL records, and note whether changes are concentrated where the handling methods differ.
• Check estimate uncertainty and stability. Small or poorly identified data sets can yield substantial parameter changes across fits even when no single handling rule is the sole explanation.
• Check numerical fit quality. Failed convergence, boundary estimates, or markedly different optimization behavior weaken a direct interpretation of the comparison.
• Check whether the change is coherent. A parameter shift that accompanies a profile change in the time region containing many BQL records is more suggestive than an isolated shift with little predicted-profile change.

The comparison has important limitations. A censored likelihood is not automatically correct if its structural or residual model is wrong, or if the assay limit has been misrecorded. Likewise, agreement between two methods does not show that either is unbiased: the data may contain too little information to distinguish their consequences. A large difference is a warning to investigate, not a formal test of bias. This diagnostic also does not, by itself, select a preferred method or resolve identifiability when most samples are BQL.

Choosing the Right Interpretation

Deletion and substitution alter the information used by the fit in distinct ways, while a censored likelihood directly represents the event that a concentration fell below the limit. The main comparison is therefore not “which estimate changed?” alone. Ask whether parameter changes and profile changes are coherent with the BQL pattern, whether the analysis conditions were held constant, and whether uncertainty and fit stability support the comparison.

A concise report should name the handling rule, give the censored and M-method estimates with units, quantify important differences, and describe the predicted-profile changes in context. Use language such as “consistent with a flatter late profile under substitution,” rather than claiming that a single comparison proves bias. That distinction keeps a useful diagnostic from being mistaken for a definitive causal explanation.