Auditing Likelihoods for Censored Observations

Why Audit Censored-Observation Likelihoods?

A below-quantification (BLQ) record does not provide an exact concentration. It tells us that the result was below a stated limit, but not how far below. That distinction changes the likelihood: an exact concentration contributes a probability density, whereas a BLQ result contributes the probability of falling below its limit. If model code uses the wrong contribution, reverses a censoring flag, or applies the wrong limit, the fitted model is answering a different statistical question from the one specified.

An audit therefore follows each record from its data representation to the likelihood factor it selects. This tutorial focuses on left-censored observations reported as below a quantification limit. The calculations use a normal observation model to make the distinction explicit; the same audit logic applies with another specified observation distribution, using that distribution’s density and cumulative distribution function.

Likelihood Contributions for Exact and Censored Records

For a normal observation model, suppose that, conditional on the model quantities represented by \(\theta\), \(Y_i\sim N(\mu_i,\sigma_i^2)\), where \(\mu_i\) is the model-predicted mean and \(\sigma_i>0\) is the residual standard deviation. A quantified value contributes the normal density. A BLQ record contributes the normal probability from the lower tail through its stated limit:

Here \(\Phi\) is the standard normal cumulative distribution function. The second expression follows by standardizing \(Y_i\): \((Y_i-\mu_i)/\sigma_i\sim N(0,1)\), so the event \(Y_i\leq L_i\) is equivalent to \((Y_i-\mu_i)/\sigma_i\leq(L_i-\mu_i)/\sigma_i\).

When observations are conditionally independent given the model quantities, the likelihood is the product of their individual contributions. The product may combine densities for quantified records and probabilities for censored records. A density has units inverse to the measurement units, while a probability is dimensionless; this is expected for a likelihood involving both exact and censored data. Changing measurement units changes the numerical density, so comparisons require consistent units and data representations.

A Practical Audit Sequence

Before evaluating the likelihood, establish what each data field means. In particular, determine the software’s censoring-flag convention rather than assuming that a value of 1 always means BLQ. Confirm that each censored record has the limit that applies to that record, in the same units as the model prediction. Then trace each record through the code path: the flag should select a cumulative probability for a censored record and a density for an exact record.

Record-Specific Limits and Indicator Checks

A dataset can contain different quantification limits across records, for example because the applicable limit varies by assay, sample, or analytical batch. The correct probability is then calculated with the limit for that particular record. Replacing record-specific limits with one default value changes the event whose probability is being evaluated.

The censoring indicator must also select the correct branch. A useful audit is to make a small truth table from the model’s documented convention: for each possible flag value, write whether the code should use a density or a cumulative probability. Then check representative records against both the raw data status and the implemented branch. This is especially important when a flag is inverse-coded or when a stored concentration field contains a placeholder for BLQ observations.

Assumptions, Diagnostics, and Alternatives

The calculations above rely on a specified observation distribution and correct parameters for its location and variability. They also assume that a BLQ report means the underlying modeled observation fell on the stated left-censored side of the limit, that the limit is known and represented on the same scale as the observation, and that the specified censoring rule is applied consistently. Multiplying row contributions additionally requires conditional independence for the observations being multiplied. If that assumption does not hold under the model, the joint likelihood must reflect the dependence rather than simply multiplying marginal factors.

A normal model is only an example. If the specified observation model is not normal, use its own CDF at the limit. If it is formulated on a transformed scale, the event and threshold must be expressed on that scale; for a monotone increasing transformation, the threshold transforms in the same direction. An auditor should not substitute a normal CDF merely because it is convenient. The distribution and scale must match the written model and its implementation.

Common warning signs include a censored row entering the exact-observation density branch, a quantified row entering a CDF branch, use of one limit for records with different limits, a mismatch between indicator coding and the code’s condition, or a unit mismatch between the limit and prediction. Recalculating selected row contributions is a direct diagnostic. When likelihoods are accumulated on the log scale to avoid numerical underflow, the corresponding censored term should be \(\log F_i(L_i\mid\theta)\), not \(\log f_i(L_i\mid\theta)\). The log calculation does not alter which event the likelihood must represent.

Substitution methods, such as replacing each BLQ value by zero or by a fraction of the limit, are not the same likelihood. They treat an imputed value as though it were an exact observation and can change both the contribution and the information attributed to that record. Excluding BLQ observations discards the information that the result fell below its limit. A censored likelihood retains that event information without pretending to know the unobserved exact value. These alternatives may be used for other purposes under explicit methods, but they should not be mistaken for the likelihood of a left-censored observation.

The censored likelihood also has limits. It cannot recover the exact unobserved concentration from a BLQ record, and its contribution depends on the assumed observation distribution and the correctness of the limit. If the reporting process is more complicated than a known threshold, or the data represent a different kind of censoring, the likelihood must represent that process appropriately. The audit establishes whether the implemented contribution matches the stated censoring model; it does not by itself establish that the model or assay assumptions are scientifically adequate.