Covariate Comparisons With Unequal BQL Rates

Why Unequal BQL Rates Need Careful Interpretation

A higher fraction of below-quantification-limit (BQL) records in one covariate group can be consistent with lower concentrations in that group. But it can also arise because that group was measured with a higher lower limit of quantification (LLOQ). The observed BQL rate is therefore not, by itself, evidence of a biological group difference. The comparison must account for the threshold that determined whether each latent observation was reported as quantified or BQL.

This distinction is especially important in a nonlinear mixed-effects model (NLME), where observations depend on both population parameters and individual random effects. The model should represent a covariate effect on the latent concentration process separately from the observation rule that classifies a result as BQL. The earlier material on censored likelihoods supplies the basic likelihood contributions; here the emphasis is on comparing groups when their thresholds, or the resulting censoring rates, differ.

Model the Latent Process and the Observation Rule Separately

Let \(g_i\) denote an individual's covariate group, such as a category of renal function or a treatment group. A population model may allow a parameter such as clearance to depend on \(g_i\), while individual random effects describe variation between individuals. For example, a structural prediction can be written \(f(t_{ij},D_i,\theta_i)\), where \(t_{ij}\) is time, \(D_i\) is dose, and \(\theta_i\) contains individual parameters. A covariate model specifies how a population parameter in \(\theta_i\) changes with \(g_i\). The observation model then describes variation around that prediction and the censoring rule.

For a simple illustration, suppose the latent concentration is normally distributed conditional on the individual effects, with mean \(\mu_{ij}\) from the structural model and residual standard deviation \(\sigma\). Write \(F_{ij}\) and \(f_{ij}\) for its conditional cumulative distribution function and density. If the observation is quantified at value \(y_{ij}\), its likelihood contribution is \(f_{ij}(y_{ij})\). If it is BQL, the contribution is the probability of the event \(Y_{ij}^{*}<L_{ij}\), namely \(F_{ij}(L_{ij})\). For this normal example, that probability is \(\Phi((L_{ij}-\mu_{ij})/\sigma)\), where \(\Phi\) is the standard normal cumulative distribution function.

A lognormal observation model instead applies the normal distribution to the log concentration, so a BQL contribution is the probability that the log concentration is below \(\log L_{ij}\). The correct distribution and residual-error form depend on the model. Whichever is used, the threshold must be the one relevant to that record; a BQL observation is not a measured concentration equal to zero or to the LLOQ.

Here \(\eta_i\) denotes the individual random effects and \(p(\eta_i)\) their population distribution. The integral makes the likelihood marginal with respect to those effects. In comparing models, both the covariate-dependent concentration process and the record-specific thresholds belong in the likelihood. If the thresholds are known and correctly assigned, a difference in observed BQL rates does not have to be attributed to a covariate effect: the likelihood can predict different rates from different \(L_{ij}\) values alone.

When Can a Covariate Effect Be Distinguished?

A covariate effect is supported when the observed data contain information that separates changes in the latent concentration distribution from changes in the censoring threshold. Several conditions help:

• The thresholds are recorded correctly for the individual observations, and the observation model describes how quantified values and BQL records arise.
• The covariate groups have sufficient overlap in dose, sampling time, and other important predictors. Otherwise, differences in those predictors may be mistaken for a covariate effect.
• There is useful information about the concentration distribution, such as quantified observations in relevant regions as well as BQL records. A group with almost all observations BQL may contribute little information about the location and shape of its latent distribution.
• The covariate effect is not perfectly confounded with threshold assignment. If one group is observed only with a higher threshold and there is no suitable overlap or external information, separating a group shift from threshold-driven censoring may depend heavily on model assumptions.

Unequal thresholds do not automatically make a covariate effect unidentifiable: known thresholds are included directly in each observation's likelihood. But known thresholds cannot create information that the design did not collect. When groups have little overlap, or most records are censored, the model may rely on extrapolation from its structural and residual assumptions. A numerically estimated covariate coefficient is not, by itself, proof that the data identify it well.

A practical model comparison starts with a model that allows the known LLOQ to vary by observation but does not include the covariate effect. The alternative adds the proposed covariate relationship to a population parameter while retaining the same observation model, thresholds, and other structural features. For a single added parameter, the likelihood-ratio statistic is twice the difference in maximized log likelihoods. Under the null model and standard regularity conditions, it is approximately distributed as a chi-square random variable with one degree of freedom. Those conditions include a correctly specified likelihood, an identifiable null model, and a parameter that is not on a boundary under the null.

Standardizing Comparisons and Reading the Data

One useful interpretive check is to compare groups at a common threshold, at common sampling times and doses, using the fitted latent distributions. This is a model-based standardization: it asks what BQL probabilities or concentration distributions the model predicts if the groups faced the same observation threshold and comparable design conditions. It helps distinguish a threshold-driven difference in observed censoring from a difference attributed to the latent process. It is not a substitute for adequate data or a correctly specified model.

Diagnostics should focus on whether the model reproduces both parts of the observation process: the quantified concentrations and the frequency and pattern of BQL records. Check predictions separately by covariate group, threshold, time, and dose where the data permit. A model that predicts the overall BQL rate but systematically misses rates within groups or threshold strata may be masking a problem. Also inspect whether the covariate estimate changes substantially when a small number of quantified records are removed or when weakly informed groups are examined; such instability is a warning that the comparison may be fragile.

Alternatives, Limitations, and a Common Pitfall

A censored likelihood with record-specific thresholds is generally preferable to comparing raw BQL percentages because it uses the actual thresholds and retains information in quantified values. A simple comparison of BQL proportions ignores differences in \(L_{ij}\) and in time or dose. Substituting a fixed value for each BQL record, such as zero or half the LLOQ, treats an unobserved concentration as if it had been measured and can distort both group contrasts and parameter estimates. Such substitutions do not resolve confounding between covariate group and threshold.

A likelihood-based covariate test also has limitations. It depends on a reasonable structural model, residual-error distribution, random-effects distribution, and threshold information. Heavy censoring can leave parameters weakly identified even when the likelihood is coded correctly. If group and threshold are nearly aligned, conclusions may be sensitive to assumptions about the latent distribution. A non-significant test does not establish absence of a covariate effect, and a significant test does not establish causation. Report the estimated effect and its uncertainty in context, alongside the degree of censoring and the design limitations.

• Wrong reasoning: “Group B has a higher BQL rate, so its concentrations must be lower.”
• Correction: First account for each record's LLOQ and other design factors. Different thresholds alone can produce different BQL rates. Then assess whether the censored likelihood supports a covariate effect on the latent concentration process and whether the design identifies that effect.