The M3 Method for Censored PK Data

Why censored concentrations need a likelihood contribution

A concentration reported as below the lower limit of quantification (LLOQ) is not an exact concentration value. It tells us that the concentration fell below a known threshold, but not where below that threshold it lay. The M3 method uses that information directly: an exact quantified value contributes a probability density, while a below-quantification (BQL) record contributes the probability of falling below its limit under the model.

This distinction matters in population pharmacokinetics because concentrations near or below the assay limit can carry information about the predicted concentration and the residual variability. M3 retains that information in the likelihood without pretending that a BQL concentration was measured exactly. The earlier material on likelihood contributions for quantified and BQL observations establishes the two types of contributions; here they are assembled into the M3 method for a PK profile.

Model and likelihood construction

For illustration, assume a lognormal residual model. Conditional on the individual’s random effects, the concentration is positive and its logarithm is normally distributed around the logarithm of the model prediction:

Here \(\eta_i\) denotes the individual random effects, and \(\sigma_{ij}\) is the residual standard deviation on the log scale under the specified residual-error model. The notation allows the model to specify different residual standard deviations for different observations, for example through an established heteroscedastic error model. The lognormal assumption is an example used to make the contributions explicit; another residual distribution would require its own density and cumulative distribution function.

For a quantified concentration, the lognormal density on the concentration scale follows by a change of variables. If \(Z=\log Y_{ij}\), then \(Y_{ij}=\exp(Z)\) and \(dZ/dY_{ij}=1/Y_{ij}\). Thus the density of \(Y_{ij}\) is the density of \(Z\) multiplied by this Jacobian:

In this expression, \(\phi\) is the standard normal density. The factor \(1/y_{ij}\) is essential: it converts a density on the log-concentration scale into a density on the concentration scale. For a BQL record, standardizing the log concentration gives the event probability:

where \(\Phi\) is the standard normal cumulative distribution function. Because the lognormal distribution is continuous, using \(<\) or \(\leq\) at the threshold gives the same probability. The likelihood treats the BQL record as this event, not as an exact observation at zero, at the LLOQ, or at any other substituted value.

Let \(Q_i\) be the set of quantified records and \(C_i\) the set of BQL records for individual \(i\). Conditional on \(\eta_i\), the M3 likelihood contribution for the individual is the product of the appropriate density and probability terms:

The population likelihood does not generally treat the random effects as known. If their density is \(p(\eta_i\mid\Omega)\), where \(\Omega\) denotes the random-effects parameters, the individual’s marginal contribution is obtained by integrating the conditional likelihood over that distribution:

This is the standard marginal-likelihood treatment of individual random effects. M3 specifies how each record enters the conditional likelihood; it does not remove the need to account for random effects in a population analysis. A quantified density has units of inverse concentration, whereas a censoring probability is dimensionless. Their product therefore has the appropriate density units for the exact observations in that profile.

Worked calculations

Derivation, diagnostics, and interpretation

The M3 construction follows from what the observation process actually reports. A quantified value identifies a point \(y_{ij}\), so its likelihood contribution is the density at that point. A BQL report identifies a set of possible values, \(0<Y_{ij}<L_{ij}\), so its contribution is the probability mass over that interval. Integrating the density over the reported set yields the cumulative probability:

This also clarifies why inserting an arbitrary value for a BQL record changes the likelihood: it replaces the probability of an interval by the density at one chosen point. M3 preserves the distinction between a measured value and a censored event.

Useful checks start with the model-implied BQL probability. If \(f_{ij}=L_{ij}\), the lognormal model gives probability 0.5. If \(f_{ij}\) is much larger than \(L_{ij}\), the probability should be small; if \(f_{ij}\) is much smaller, it should be large. A computed probability outside \([0,1]\), or one that moves in the opposite direction as \(f_{ij}\) changes with the other terms fixed, signals a coding or sign error. Checking a few records by hand against the formula can expose incorrect limits, inconsistent concentration units, or a missing logarithm.

At the fitted-model level, compare observed BQL frequencies with predicted probabilities across relevant time or concentration ranges. Large systematic discrepancies can indicate a misspecified structural model, residual distribution, or censoring mechanism. This comparison is a diagnostic, not proof that any one component is at fault. Heavy censoring can also leave limited information about parameters: a BQL event says the concentration lies below a threshold, not how far below. Detailed identifiability and model-diagnostic questions require analyses beyond the construction of the M3 likelihood.

M3 is commonly contrasted with approaches that omit BQL records or substitute a fixed value for them. Those approaches do not use the same interval-probability contribution and can therefore change the information supplied to the fit. The M4 method and deletion or substitution approaches are distinct alternatives; their detailed formulations and trade-offs are separate topics. M3’s own advantage is that it uses the reported censoring event directly, while its limitation is dependence on the chosen structural, residual, and censoring assumptions. A formally correct contribution cannot compensate for a poor model or an incorrectly recorded LLOQ.

Common mistakes and practical limitations

• Using a density for a BQL record. A BQL result does not give an exact concentration at which to evaluate the density. Use the cumulative probability below the LLOQ.
• Using a probability for a quantified record. An exact quantified concentration contributes a density, not the probability that the concentration is above the LLOQ.
• Dropping the density Jacobian. Under a lognormal model, the concentration-scale density includes \(1/y_{ij}\). Omitting it gives the wrong density.
• Reversing the censoring event. For a BQL record, the event is below the LLOQ. The lognormal contribution uses \(\Phi(\log(L_{ij}/f_{ij})/\sigma_{ij})\), not its upper-tail complement.
• Interpreting a likelihood contribution as an observed concentration. A BQL contribution is a probability under the model, not a replacement value or an estimate of the latent measurement.

M3 is most useful when the assay limit and the censoring status are known, and when the model can assign meaningful probabilities to values around that limit. If these conditions are doubtful, interpretation of the fitted parameters should be cautious. Sensitivity to residual-distribution assumptions and to the pattern of censoring matters because different plausible models can assign different probabilities to the same BQL event.