Censoring Under Additive Residual Error

Why Additive Error Determines the BQL Probability

A below-quantification-limit (BQL) record is not an exact concentration. It reports that the assay result fell below a known threshold, the lower limit of quantification (LLOQ). To use that information in a likelihood, we need the probability of the reported event under the observation model. With additive residual error, that probability depends on the absolute distance between the model prediction and the LLOQ, measured in units of the residual standard deviation.

For observation \(j\) from individual \(i\), let \(F_{ij}\) denote the model-predicted concentration conditional on that individual’s random effects. In an NLME model, \(F_{ij}\) is obtained from the structural model and individual parameters. The observation model considered here adds a normal residual error to this prediction.

The assumptions are that the stated additive normal model is appropriate for the assay result, that the residual standard deviation is \(\sigma\) for the observation under consideration, and that the censoring threshold is known. In a typical likelihood, residual errors are also taken to be conditionally independent across observations given the model parameters and individual random effects. A quantified record is treated as an observed value; a BQL record supplies only the inequality \(Y_{ij}<L_{ij}\).

Deriving the Censored Probability

Let \(\Phi(z)=P(Z\leq z)\) for \(Z\sim N(0,1)\). Standardizing the observation by subtracting its conditional mean and dividing by its standard deviation turns the BQL event into a standard normal event:

The strict inequality is appropriate to the reporting rule, but for a continuous normal distribution the probability of an observation being exactly equal to the LLOQ is zero. The same probability therefore results if the boundary is written with a non-strict inequality.

For one individual with conditionally independent observations, the conditional likelihood is the product of the contributions for that individual’s records. If the individual random effects are not conditioned on but integrated out, the individual likelihood integrates that product over the random-effects distribution. The CDF formula is still the conditional BQL probability; it is not, by itself, the population-averaged probability after that integration.

Behavior Near Zero and Near the LLOQ

The formula makes the behavior of the BQL probability explicit. With \(L\) and \(\sigma\) fixed, increasing \(F\) decreases \((L-F)/\sigma\), so it decreases the BQL probability. A prediction far below the LLOQ gives a probability near one; a prediction far above it gives a probability near zero. At \(F=L\), the standardized gap is zero and the BQL probability is \(\Phi(0)=0.5\).

An important edge case occurs as the predicted concentration approaches zero. If \(L>0\) and \(\sigma>0\), then \(P(\mathrm{BQL}\mid F)\to\Phi(L/\sigma)\) as \(F\to0\). This limit is less than one for finite \(L/\sigma\). Therefore, an additive normal model does not generally imply that a near-zero prediction has a BQL probability of one: the model allows positive measurement errors large enough to exceed the LLOQ. The limit is closer to one when the LLOQ is many residual standard deviations above zero.

The rate of change also helps describe behavior close to the threshold. Differentiating \(p(F)=\Phi((L-F)/\sigma)\), using \(\Phi'(z)=\phi(z)\), gives \(p'(F)=-\phi((L-F)/\sigma)/\sigma\). The derivative is negative everywhere. Its magnitude is largest at \(F=L\), because the standard normal density is largest at zero. Thus, with \(L\) and \(\sigma\) fixed, small changes in the prediction alter the BQL probability most rapidly when the prediction is near the LLOQ.

Using the Probability in a Likelihood

The observation status determines which contribution is used. For a BQL record, the exact assay result is unknown; the likelihood sums over all possible values below the LLOQ, which is why the normal CDF appears. For a quantified record with observed value \(y\), the likelihood uses the density at that value. Replacing a BQL record with the LLOQ or another chosen number changes the information being used: it treats an unobserved value as though it had been measured.

This is the essential distinction between a censored likelihood and substitution or deletion rules. The introductory material on censored likelihoods and the earlier material on BQL handling establish the broader motivation; here the key point is that an additive error assumption specifies the CDF and density used for each record. The calculation does not require an invented concentration for a BQL observation.

The formula is conditional on \(F_{ij}\). If \(F_{ij}\) depends on an individual random effect, then each individual’s conditional BQL probability depends on that effect as well. In a population likelihood, one must account for random effects through the model’s integration; substituting a population-typical prediction into the formula generally does not equal the marginal BQL probability.

Checks, Limitations, and Common Mistakes

Several quick checks can reveal calculation or implementation errors. First, the probability must lie between zero and one and must decrease as \(F\) increases. At \(F=L\), it must equal 0.5. If the prediction is several \(\sigma\) below the LLOQ, the probability should be close to one; several \(\sigma\) above it, close to zero. These checks do not establish that the model fits the data, but they do test whether the censoring calculation is consistent with its assumptions.

• Using the variance instead of the standard deviation. The denominator is \(\sigma\), not \(\sigma^2\). The standardized difference must be dimensionless.
• Reversing the subtraction. For a left-censoring rule \(Y<L\), use \(L-F\). Using \(F-L\) without changing the tail produces the probability of the opposite event.
• Treating the prediction as the measured result. The BQL probability is not a deterministic comparison of \(F\) with \(L\); it accounts for residual error.
• Using a density at the LLOQ for a BQL record. A BQL result identifies an interval of possible measurements, so its likelihood contribution is the probability over that interval, not the density at its boundary.
• Assuming zero prediction guarantees a BQL result. Under additive normal error, positive errors remain possible, and the limiting probability is \(\Phi(L/\sigma)\), not necessarily one.

The additive model has constant absolute residual standard deviation. Near zero, that can imply large relative measurement variability, and a normal observation distribution can assign some probability to negative assay values. The BQL probability formula follows from that specified model; whether those features are scientifically adequate depends on the assay and concentration range. It is not valid to change the error model silently while retaining the same probability formula.