Likelihood Contributions for Quantified and BQL Observations

Why the Observation Status Determines the Likelihood Term

A concentration reported as quantified and one reported as below the limit of quantification (BQL) do not provide the same kind of information. A quantified record supplies an observed concentration value. A BQL record supplies an event: the concentration was below the known lower limit of quantification (LLOQ). The likelihood must represent the information actually recorded, rather than treating both records as if they were exact numbers.

The earlier material on left censoring and quantification limits establishes the interpretation of a BQL result as a censored observation. Here the emphasis is the corresponding likelihood: a density evaluated at an exact quantified value, versus a cumulative probability for the BQL event. The distinction matters because a density and a probability are different mathematical objects, even though both can appear as factors in a likelihood.

For a continuous distribution, \(P(C=y)=0\) at any single point. Thus an exact quantified result contributes a density, not the probability of observing precisely that value. A BQL record contributes a probability because it identifies a range of possible concentrations. The density has inverse-concentration units, while the BQL probability is dimensionless. Both are valid likelihood contributions; a likelihood for mixed continuous and event data can contain both types.

Likelihood for a Set of PK Records

Let \(i\) index individuals and \(j\) index records within an individual. Write \(y_{ij}\) for a quantified concentration and \(L_{ij}\) for the known LLOQ associated with a BQL record. Let \(\theta\) denote population and residual-model parameters, and let \(\eta_i\) denote the individual's random effects. Conditional on \(\eta_i\), the model specifies the concentration distribution at each sampling time. Suppose residual observations are conditionally independent across records given \(\theta\) and \(\eta_i\). Then the individual conditional likelihood is

where \(Q_i\) is the set of quantified records and \(B_i\) is the set of BQL records. Taking logarithms turns the products into sums:

The conditional-independence assumption is important: it justifies multiplying the separate terms. If residual observations are dependent after conditioning on the modeled effects, the joint density and event probabilities must represent that dependence instead. In a population likelihood with random effects, the individual's conditional likelihood is integrated over the random-effects distribution; the quantified and BQL terms are first assembled conditional on the same individual effects.

The exact-value contribution is the density under the observation model, evaluated at the recorded concentration. It is not automatically a density conditional on being quantified. In a dataset that records both quantified and BQL outcomes, the event “quantified” is already represented by the observed exact value lying in the quantified region, while BQL records contribute the probability below the threshold. If, instead, an analysis deliberately samples or retains only quantified records and conditions on that selection, the appropriate density is conditional on exceeding the LLOQ, with a normalizing probability in the denominator. That is a different data-collection likelihood.

Deriving the Contributions Under Lognormal Residual Error

A common concentration model is lognormal residual error. Suppose that, conditional on the model prediction \(m\), the natural logarithm of concentration is normally distributed:

Here \(m\) is the conditional median concentration, in concentration units, and \(\sigma\) is the standard deviation on the log scale, which is dimensionless. The arithmetic mean is \(m\exp(\sigma^2/2)\), so \(m\) should not be described as the arithmetic mean unless the model has been parameterized accordingly.

To obtain the density on the concentration scale, set \(Z=\log C\). The normal density of \(Z\) is evaluated at \(\log c\), and the change-of-variable factor is \(\left|d(\log c)/dc\right|=1/c\). Therefore,

For a BQL record, the event \(C<L\) is equivalent to \(\log C<\log L\), because the logarithm is increasing for positive concentrations. Standardizing the normal variable gives

where \(\Phi\) is the standard normal cumulative distribution function. These expressions are specific forms of the general density and CDF contributions. They rely on positive concentrations, the stated lognormal residual distribution, a known threshold, and the appropriate conditional-independence assumptions for multiplying records.

Combining Quantified and BQL Records

When multiple observations are conditionally independent given the individual's effects and model parameters, the joint contribution is the product of their respective terms. Equivalently, the individual log-likelihood contribution is the sum of the log densities for quantified values and the log CDF probabilities for BQL records. This distinction should be preserved record by record; using a density for every record or a CDF for every record discards the meaning of one of the observation types.

Assumptions, Diagnostics, and Common Errors

The contribution formulas are straightforward, but their validity depends on the data-generating and recording process. Check that the observation model describes the measured concentration scale, that the residual distribution is appropriate, that the recorded LLOQ is the threshold used for that sample, and that the BQL indicator means the concentration fell below that threshold. Confirm which records are exact and which are censored before constructing the likelihood. The product form additionally assumes conditional independence of observation errors given the modeled effects.

Useful diagnostics include comparing observed BQL frequencies with model-predicted BQL probabilities across relevant sampling times and concentration ranges, and checking whether quantified residuals are systematically inconsistent with the assumed error distribution. A model that predicts almost no BQL outcomes when many are observed, or predicts BQL outcomes at times when few occur, may have a misspecified structural model, residual distribution, or parameter values. These checks do not by themselves identify which component is wrong, but they flag incompatibility between the fitted model and the observation pattern.

A frequent mistake is to replace every BQL result by the LLOQ and then evaluate a density there. The reasoning is that the threshold is the only available number, so it can stand in for the missing concentration. The correction is that the threshold is not the observed concentration: it defines the event \(C<L\). The contribution is \(F_C(L)\), which accounts for all concentrations below the threshold under the model.

Another mistake is to treat a BQL result as no data and omit its term. A BQL record is informative whenever the model assigns different probabilities to being below the LLOQ under different parameter values. Omitting it discards that information. Conversely, treating a continuous exact value as a probability mass at that point is also incorrect: use its density. If residual errors are correlated, multiplying marginal record contributions is not justified without conditioning or modeling that dependence. Likewise, if only quantified records were selected for analysis, the selection mechanism must be considered rather than automatically using the unconditioned density.

The main conceptual alternative to the event-probability likelihood is to alter or discard the recorded information by substituting a value or deleting BQL records. Those operations do not generally preserve the likelihood implied by the stated observation process. The density/CDF formulation is limited, in turn, by the adequacy of the chosen concentration distribution and by whether the censoring threshold and recording mechanism are known. A mathematically correct contribution cannot compensate for an inappropriate structural or residual model.