Subjects With Only BQL Samples

Why an All-BQL Subject Still Contributes Data

A subject whose every sample is below the limit of quantification has no quantified concentration value to contribute as an ordinary concentration density. The records are not, however, equivalent to having no samples. Each one reports an event: the measured concentration fell below a known limit. Under a specified PK and residual-error model, that event has a probability, and those probabilities contribute to the likelihood.

The information is indirect. An all-BQL profile can make predictions that are too high less plausible, or distinguish parameter combinations that predict concentrations on different sides of the assay limits. But it does not reveal the exact concentrations, and it may not identify the subject’s individual PK parameters separately. How much it says depends on the sampling times, applicable limits, residual variability, PK model, and the range of plausible individual effects.

Conditional Likelihood for BQL Records

For a concrete formulation, suppose concentrations are positive and the residual error is lognormal: \(\log Y_{ij}=\log C_{ij}(\eta_i,\theta)+\epsilon_{ij}\), where \(\epsilon_{ij}\sim N(0,\sigma^2)\). Here \(\sigma\) is the standard deviation on the log-concentration scale. This model assumes that, conditional on the model prediction and random effects, each residual has this normal distribution. To multiply probabilities across samples, we also assume the residuals are conditionally independent.

Because the logarithm is increasing, the BQL event \(Y_{ij}<L_{ij}\) is equivalent to \(\log Y_{ij}<\log L_{ij}\). Standardizing that normal variable gives:

where \(\Phi\) is the standard normal cumulative distribution function. Define the standardized distance \(z_{ij}=(\log L_{ij}-\log C_{ij})/\sigma\). A positive \(z_{ij}\) means the predicted concentration is below the limit; a negative value means it is above the limit. The sign and size of \(z_{ij}\), not a fabricated concentration value, determine the event probability.

If subject \(i\) has \(m_i\) BQL records and no quantified records, conditional independence gives the subject’s conditional likelihood contribution:

The product is conditional on the same \(\eta_i\) across that subject’s records. The predictions and thresholds may differ from sample to sample. If residual errors are dependent after conditioning on the model, multiplying marginal event probabilities is not valid; the appropriate joint probability of all BQL events is required instead.

Why the Event Probability Carries Information

The probability is not constant as the predicted concentration changes. For one record, write \(z=(\log L-\log C)/\sigma\), treating \(L\) and \(\sigma\) as fixed. Differentiating the log-likelihood contribution with respect to \(\log C\) shows the direction of the information:

where \(\phi\) is the standard normal density. This derivative is negative: increasing the predicted concentration lowers the probability of observing a BQL result. The derivative with respect to a PK parameter \(\psi\) follows by the chain rule, \(\partial\log\Phi(z)/\partial\psi= [-\phi(z)/\{\sigma\Phi(z)\}]\, \partial\log C/\partial\psi\). Thus, a BQL record informs a parameter only insofar as that parameter changes the predicted concentration at that sample.

The strength of the information also depends on where the prediction lies relative to the limit. If the predicted concentration is far below the limit, \(z\) is large and positive, so the BQL probability is near one and changes little with modest parameter changes. If the prediction is above the limit, the BQL probability is smaller, so the observed event can weigh strongly against those predictions. Neither situation turns the BQL result into an exact concentration.

Combining Several BQL Samples

An all-BQL subject’s records must be combined using each record’s own prediction and limit. This matters when samples were measured by different assays or when limits vary over time. A record with a relatively high BQL probability can coexist with another whose probability is much lower. Under conditional independence, their product is the subject’s likelihood contribution at the specified random effects and population parameters.

What an All-BQL Subject Can Identify

A BQL profile can constrain the predictions at its sampling times. If a candidate parameter value predicts concentrations well above the applicable limits, the all-BQL contribution may be small. Alternative values that bring predictions below the limits may receive larger contributions. This is information about model predictions, translated into information about parameters through the structural model.

That translation is often not unique. A single event at one time supplies one probability constraint, and different PK parameter combinations can produce the same prediction at that time. Even several BQL events may fail to distinguish parameters if their predicted concentrations change together, if the predictions are all far below the limits, or if residual variability is large relative to the differences between predictions. A profile can therefore inform a combination of parameters without identifying each one separately.

The following simple one-compartment example isolates this limitation. It is illustrative rather than a claim that this model is suitable for every drug or sampling design.

In population analysis, an individual’s random effects are not ordinarily free-standing estimates obtained from that subject alone. The population model and the other subjects’ data also matter. In a marginal likelihood framework, a subject’s conditional contribution is integrated over the random-effects distribution. An all-BQL subject can therefore affect population estimates while still providing weak or non-unique information about that subject’s individual parameters. Any individual estimate should be interpreted in light of this model-based pooling, not as a direct reconstruction of unobserved concentrations.

Assumptions, Diagnostics, and Alternatives

Before using the product of BQL probabilities, verify that the likelihood represents how the records were generated. In particular, confirm that each record is classified against the correct assay-specific limit, that the residual distribution and scale match the model, and that conditional independence is a defensible assumption. When records share residual dependence not represented by the model, their joint probability cannot generally be obtained by multiplying separate probabilities.

Useful checks include inspecting the model predictions relative to each record’s limit, confirming that all-BQL probabilities are calculated from those predictions rather than substituted values, and checking that the resulting likelihood changes in the expected direction when predictions change. Very high BQL probabilities at every plausible parameter value indicate that the subject may contribute little discrimination. Very low probabilities can flag a conflict between the observed BQL pattern and the fitted predictions, the assumed residual model, or the recorded limits. Numerical underflow can also cause trouble when many small probabilities are multiplied; summing log probabilities is usually more stable than forming a tiny product directly.

The main alternatives handle the observation differently:

The M3 method is the censored-likelihood approach for BQL records; in this setting, its essential contribution is an event probability rather than a concentration density. Deletion and substitution may be convenient, but they do not express the observed event in the same way. A censored likelihood is not automatically reliable: misspecified PK predictions, residual distributions, dependence assumptions, or limits can still yield misleading inference. Nor can it recover concentration detail that was never measured.

Common Pitfalls

• Calling an all-BQL subject uninformative. The subject does supply threshold-event information. Its strength may be small, but it is not necessarily zero.
• Calling the event probability an individual concentration estimate. A value such as 0.80 is a model probability for a BQL event, not a predicted observation or a statement that the concentration equals 80% of the limit.
• Assuming all individual PK parameters are identified. BQL events constrain predictions at observed times. Different parameter combinations may yield the same predictions, as the one-sample example demonstrates.
• Multiplying marginal probabilities without checking dependence. The product formula requires conditional independence of residual events given the model quantities. If that assumption fails, a joint probability is needed.
• Using a common limit when limits differ. Each event must be evaluated using the limit applicable to that specific sample and assay.