Using Below-Quantification Results in Individual Forecasts

Why a Below-Quantification Result Still Contains Information

A concentration reported as below the assay’s quantification limit is not an exact concentration of zero, nor is it an exact concentration equal to the limit. It tells us that the unobserved assay result fell below a known threshold. That event can still distinguish between individual parameter values: a model predicting a low concentration may assign it a higher probability than a model predicting a high one.

The key is to use the probability of the reported event, rather than the probability density at an invented replacement value. This matters when an individual forecast combines observed concentrations with a population-based prior. If a below-quantification result is treated as an exact number, the likelihood—and therefore the balance between patient data and prior information—can change.

Constructing the Censored Likelihood

Suppose the observation model for a positive concentration is lognormal: \(\log Y\mid\theta\sim N(\log m(\theta),\sigma^2)\), where \(m(\theta)\) is the model-predicted median concentration and \(\sigma\) is the residual standard deviation on the log scale. This assumes that the log concentration is normally distributed around the log model prediction, with the stated residual variability. The resulting probability of a below-limit result is

Here \(\Phi\) is the standard normal cumulative distribution function. The formula follows by standardizing \(\log Y\), which has a normal distribution with mean \(\log m(\theta)\) and standard deviation \(\sigma\). It applies when the recorded result means that the assay result was below \(L\), and the lognormal observation model is appropriate. If the assay or reporting rule implies a different event or observation distribution, the probability must be adapted to that rule and model.

For comparison, if an exact positive result \(y\) is reported under the same model, its likelihood contribution is a density:

A density is appropriate for an observed exact value; a probability is appropriate for an observed interval or event. With conditionally independent observations given \(\theta\), multiply the contributions for all observations. Thus, for a set containing both exact results and below-limit results,

where \(E\) indexes exact observations and \(C\) indexes censored observations. Conditional independence means the observation errors are independent given the individual parameters and the model; it should not be assumed if the observation process or residual structure implies dependence.

In an individual Bayesian fit, the posterior is proportional to the prior multiplied by this likelihood. For a finite set of candidate parameter states \(\theta_k\), the normalized posterior weight is

A forecast for a future concentration \(Y_{\mathrm{new}}\) averages over these posterior parameter values and the future observation model. In particular, if \(\mu_k\) is the conditional mean future concentration under state \(\theta_k\), the posterior predictive mean is \(\sum_k P(\theta_k\mid\text{data})\mu_k\). This average reflects both the evidence from the patient’s observations and the prior weights.

Worked Forecast Comparisons

Assumptions, Diagnostics, and Alternatives

The likelihood calculation depends on more than the algebra. Check that \(L\) is the relevant limit for the particular sample and that the recorded category really means the result fell below it. Confirm that the residual distribution and its scale are plausible for the assay and concentration range. In the lognormal example, \(\sigma\) is on the log scale, not in concentration units. If residual error varies with concentration or the assay reports a different censoring interval, use an observation model that represents those features.

Useful diagnostics include checking whether the predicted below-limit probabilities are plausible at the observed sample times and whether the resulting posterior is highly sensitive to the assumed \(L\), residual error, or prior. A very small probability for a censored observation under every plausible individual state may indicate conflict between the data and the model, an incorrect threshold, or an observation-process problem. If nearly all candidate states assign probabilities close to zero or close to one, the data may provide little discrimination between those states. For several censored samples, inspect whether the conditional-independence assumption is defensible; shared assay effects or other dependence can make a product of marginal probabilities inappropriate.

Fixed-value substitution is sometimes used as a simple approximation, but it changes the evidence: it treats an interval event as though a particular exact concentration had been measured. Substituting \(L\), \(L/2\), or zero can produce different likelihoods and forecasts, and none is generally the censored likelihood. Deleting the result avoids inventing a value but discards the information that the result fell below \(L\). When the censoring threshold and observation model are available and adequate, using the event probability preserves that information directly.

Censored-likelihood forecasts are not automatically reliable if the underlying model is poor. They depend on a defensible observation model, known or appropriately represented thresholds, and a suitable prior and individual pharmacokinetic model. If all available observations are censored, or the candidate states predict similar event probabilities, the data may not identify the individual well; the posterior can remain largely determined by the prior. The likelihood handles what was observed, but it cannot create information that the assay did not provide.