Interspersed BQL and Quantified Observations

Why the Sequence Must Be Built Record by Record

A subject’s concentration profile can include both reported numerical concentrations and samples reported as below the limit of quantification (BQL). When the types alternate, it is tempting to summarize the subject as having “some BQL data” and apply one rule to the whole sequence. That loses information: each record has its own time, its own observation type, and potentially its own lower limit of quantification (LLOQ). The likelihood must preserve those record-level details.

The task here is to form the likelihood contribution for one subject’s ordered sequence. A quantified value contributes a probability density evaluated at the reported value. A BQL result contributes the probability that the unobserved concentration fell below that record’s LLOQ. These are different kinds of contributions, and they must not be interchanged.

For a continuous observation model, the probability of being exactly equal to the LLOQ is zero, so using \(Y_{ij}\leq L_{ij}\) rather than \(Y_{ij}<L_{ij}\) gives the same probability. The key distinction is still between a density at a numerical observation and a cumulative probability for a censored observation.

Constructing the Conditional Sequence Likelihood

Suppose the model gives the conditional density \(f_{ij}\) and distribution function \(F_{ij}\) for each record, given the subject’s random effects. Assume that, conditional on \(\eta_i\) and the model parameters, residual errors at different records are independent. Also assume that the BQL indicator accurately records whether the latent result fell below its stated LLOQ, and that the LLOQ used for each record is known. Under these assumptions, the conditional likelihood for the subject is the product of the contributions in record order:

This compact product does not mean that all records use the same expression. It means that each record supplies the appropriate factor, determined by its own status. The record order is useful for checking the data and the computation, even though multiplication itself is commutative. A quantified record contributes no BQL probability factor, and a BQL record contributes no density at an invented concentration.

The usual NLME likelihood for subject \(i\) is marginal over the random effects, whose population distribution has density \(p(\eta_i\mid\Omega)\), where \(\Omega\) denotes the random-effects distribution parameters. The marginal individual contribution is

The product over records is formed before this integration. A subject’s random effect is shared across that subject’s records; it is not a new independent draw at every sample. For independent subjects, the population likelihood is the product of these individual marginal contributions.

Why Product-Then-Integrate Is Necessary

Conditional independence gives a product after conditioning on the shared random effect. Marginalization then averages that product over the possible random effects. In general, averaging each record separately and multiplying those averages gives a different result, because it effectively allows the random effect to vary independently from record to record.

Calculating Each Type of Contribution

The contribution formulas depend on the observation model. Earlier material on below-LLOQ probabilities establishes how to obtain the appropriate distribution function under a specified residual error model. Once that model is specified, the same record-by-record construction applies to an interspersed sequence.

For example, under an additive normal residual model with conditional mean \(\mu_{ij}\) and standard deviation \(\sigma\), a quantified value contributes the normal density \[ f_{ij}(y_{ij}\mid\eta_i)= \frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac{(y_{ij}-\mu_{ij})^2}{2\sigma^2}\right], \] and a BQL record contributes \[ F_{ij}(L_{ij}\mid\eta_i)= \Phi\left(\frac{L_{ij}-\mu_{ij}}{\sigma}\right), \] where \(\Phi\) is the standard normal cumulative distribution function. These expressions use the same conditional mean and residual model, but evaluate different features of its distribution. Use the residual model appropriate to the analysis rather than assuming these normal formulas apply universally.

Implementation Checks, Alternatives, and Limitations

A reliable implementation keeps each record’s time, status, reported value or LLOQ, and model prediction together. Before evaluating a subject likelihood, check that the status selects exactly one contribution for every record; that quantified records have valid numerical observations; and that each BQL record uses its own applicable LLOQ and scale. If an LLOQ is recorded on the original concentration scale while the observation model is on a transformed scale, convert the limit consistently before evaluating the probability.

The main alternatives change how BQL records are handled, not the record-level logic of a censored likelihood. Deleting BQL records discards their information. Substituting a value such as half the LLOQ or the LLOQ itself treats a censored result as if it were an exact concentration, changing the observation model. A censored likelihood instead uses the probability below the threshold. These approaches can yield different estimates; they are not interchangeable computational shortcuts.

Diagnostics should focus on whether the assembled likelihood matches the actual records and assumed model. Useful checks include independently recomputing selected record contributions, confirming that probabilities lie between zero and one, and checking that densities have the expected units and scale. A probability above one, for example, signals an error; a density above one is possible and is not by itself an error. Also inspect whether status and LLOQ fields were aligned correctly after sorting or merging records. Incorrect alignment can produce plausible-looking calculations with the wrong likelihood.

This construction relies on the assumed observation model, known limits, and conditional independence of residual errors. If residual errors are dependent across records after conditioning on the modeled effects, the product factorization needs an appropriate joint observation model instead. The likelihood also cannot recover information absent from the records: when many observations are BQL, estimates may be weakly informed. The construction itself does not resolve that identifiability issue.