Incorporating Censored and Below-Quantification Data

What a Censored Assay Result Tells the Model

An assay may establish that a biomarker concentration is below its lower limit of quantification (LLOQ) without providing a reliable numerical measurement. Similarly, an instrument may report that a result exceeds its upper limit of quantification (ULOQ), without specifying how far above the limit it lies. These are not exact observations at the limits. They are observations of events: the measurement fell below a threshold or above one.

In QSP calibration, the model predicts a quantity that is linked to the assay through an observation model. The likelihood should represent what the assay actually reported. For a censored record, that means using the probability of the observed event under the model, rather than pretending that the limit or a substituted value was measured exactly.

Likelihood Contributions Under Normal Assay Error

Suppose the QSP model, together with its observation model, predicts mean measurement \(m_i(\theta)\) for record \(i\), where \(\theta\) denotes the model parameters. Assume that, conditional on the model and any relevant covariates, the assay measurement follows \(Y_i\sim N(m_i(\theta),\sigma_i^2)\). The standard deviation \(\sigma_i\) describes assay-scale error in the same units as \(Y_i\). This tutorial uses the normal observation model for clarity; the same event-probability principle applies with other specified measurement distributions.

The assumptions for the simple formulas are that the threshold is known, the recorded censoring category accurately describes the measurement relative to that threshold, and the conditional measurement distribution is correctly specified. When multiplying contributions across records, also assume conditional independence. If replicate errors are correlated, their joint likelihood must account for that dependence rather than multiplying marginal contributions as if the replicates were independent.

For an exact observation \(y_i\), the contribution is the normal density evaluated at \(y_i\). For a below-threshold record, integrate that density over all values compatible with the report. For an above-threshold record, integrate over the corresponding upper tail.

Here \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution function. An exact continuous measurement contributes a density, not the probability of observing exactly one point (which is zero for a continuous distribution). A censored category contributes a probability, obtained by integrating the density over the reported range. The distinction is important: densities have inverse-measurement units, while event probabilities are dimensionless.

For conditionally independent records, the likelihood is the product of the appropriate contribution for each record. Equivalently, the log likelihood is the sum of their logarithms. An interval-censored record, reported only as \(A<Y_i<B\), contributes \(\Phi((B-m_i(\theta))/\sigma_i)-\Phi((A-m_i(\theta))/\sigma_i)\). These formulas use the model prediction and assay error on the scale where the normal assumption is made.

Why Substitution Is Not the Same Likelihood

A common shortcut replaces a censored result with a fixed value, such as zero, half the LLOQ, or the LLOQ itself, and then treats that value as an exact observation. This changes the information represented by the record. The censored report establishes only which side of a threshold the measurement fell on; substitution asserts a particular numerical value and, under a continuous observation model, gives that artificial value an exact-data density contribution.

The difference can be seen directly for the below-LLOQ example. With \(L=5\) ng/mL and \(\sigma=2\) ng/mL, compare candidate means of 4 and 6 ng/mL. The censored probabilities are \(\Phi(0.5)\approx0.6915\) and \(\Phi(-0.5)\approx0.3085\), respectively. If instead the result is replaced by \(L/2=2.5\) ng/mL and treated as exact, the normal density contributions are different quantities:

The density ratio is about \(0.1505/0.04314=3.49\), whereas the ratio of the censored probabilities is about \(0.6915/0.3085=2.24\). The substitution has therefore changed the relative support for these two candidate means, as well as replacing a dimensionless event probability with a density. The issue is not that half the LLOQ is always too high or too low; the issue is that the chosen value is not what the assay observed.

Implementation, Diagnostics, and Limitations

In a calibration implementation, retain each record’s status—exact, below threshold, or above threshold—and use the matching contribution at each parameter value. If \(\sigma_i\) varies by assay, concentration, or record, use that record-specific error model consistently in both the exact and censored contributions. If the observation model is specified on a transformed scale, express the threshold on that scale too. For example, a lognormal measurement model applies normal-error formulas to \(\log(Y_i)\) and the corresponding log threshold; it does not justify applying a normal model to the original scale automatically.

Check the data-processing and model assumptions before interpreting an optimization result. Verify units and threshold values for each assay, and confirm that “below” and “above” flags refer to the intended LLOQ and ULOQ. A numerical value recorded at the threshold might mean an exact measurement, rounding, or a censored report; those cases should not be conflated. Also check whether samples outside the quantification range were retained as censored records. If they were discarded and only samples inside a range were included, the data are truncated rather than merely censored: the likelihood must account for the selection condition.

Censored records do not have ordinary point residuals, because their exact values are unknown. Useful checks include comparing observed counts below and above limits with the model-implied probabilities, examining whether censoring is concentrated in particular conditions or time ranges, and checking whether fitted means and assay errors place plausible probability mass on the reported side of each threshold. A large number of censored records can leave parameters weakly informed: many different predicted means may yield similar event probabilities, especially when the predictions lie far into the same tail.

In computation, extreme tail probabilities may underflow numerically to zero. Evaluate log cumulative probabilities using stable log-CDF or log-survival-function routines rather than taking the logarithm of a rounded probability. A zero contribution caused only by numerical underflow can disrupt optimization even when the mathematical likelihood is positive. Conversely, stable computation does not repair a misspecified error distribution or an incorrectly entered threshold.

Substitution is simple but imposes an artificial exact value; discarding censored records avoids that artificial value but discards the information that the measurement crossed a threshold. The censored likelihood uses that directional information while respecting its uncertainty. It is still only as reliable as its observation model and assumptions. If the probability of a result being missing or censored depends on unrecorded factors beyond the stated measurement process, or if the assay error distribution is poorly specified, an appropriate model of that mechanism or distribution may be needed. When censored replicates are correlated, their joint probability is required; multiplying marginal tail probabilities would overstate the information as though each replicate were conditionally independent.