Parameter Identifiability Under Heavy Censoring

Why Heavy Censoring Raises an Identifiability Question

A BQL record is not an exact concentration value. Under the censored likelihood, it contributes the probability that the observation fell below its quantification limit. This uses more information than deleting the record, but it does not guarantee that every model parameter is well determined. If many records are BQL, the data may support only broad features of the concentration profile, or a combination of parameters, while providing little information about individual parameters.

The practical question is therefore not simply how many observations are quantified. It is whether plausible changes in each parameter, and in combinations of parameters, produce detectable changes in the likelihood. Local likelihood curvature, correlations implied by that curvature, and changes after removing quantified observations provide complementary ways to investigate this question. These are diagnostics of the fitted model and dataset, not proofs that the model is biologically correct.

For a fitted parameter vector \(\theta\), write the log-likelihood as \(\ell(\theta)\). In censored PK analysis, this likelihood includes density contributions for quantified observations and probability contributions for BQL observations, with random effects handled according to the model. The earlier material on marginalizing censored likelihoods over random effects is relevant here: the curvature to diagnose is that of the likelihood actually used for estimation, not a conditional likelihood that ignores the random effects.

Curvature and a Single BQL Contribution

The observed information matrix is the negative Hessian of the log-likelihood evaluated at the estimate \(\hat{\theta}\). Its diagonal and off-diagonal elements describe local curvature and coupling among parameters. A small curvature in some direction means the log-likelihood changes little for small movements in that direction. A singular or nearly singular matrix indicates a locally unsupported parameter direction or a direction with very weak support.

For a simple illustration, suppose a measured concentration \(Y\), conditional on the model prediction \(m\), has a normal residual distribution with known standard deviation \(\sigma\). A BQL record at limit \(L\) contributes \(P(Y<L)=\Phi(z)\), where \(z=(L-m)/\sigma\), and \(\Phi\) is the standard normal cumulative distribution function. Define \(\lambda(z)=\phi(z)/\Phi(z)\), where \(\phi\) is the standard normal density. Differentiating the BQL log-likelihood with respect to \(m\) gives:

The first derivative is negative: a BQL result favors a lower predicted concentration. The last expression is the local curvature with respect to \(m\). It is positive for this normal model, but its size depends on the location of the prediction relative to the limit and on residual variability. A quantified normal observation, by contrast, has curvature \(1/\sigma^2\) with respect to its predicted mean. These are curvatures on the concentration scale, not automatically information about any particular PK parameter.

From Prediction Curvature to Parameter Identifiability

Parameters affect the likelihood through model predictions. If a prediction \(m\) depends on parameters \(\theta_1,\ldots,\theta_p\), its local sensitivity is represented by the gradient \(g=\partial m/\partial\theta\). For a single observation, the part of the information attributable to prediction curvature is approximately the curvature with respect to \(m\) multiplied by \(gg^\mathsf{T}\). This chain-rule result explains why curvature in concentration space does not by itself establish that all parameters are identified: the model sensitivities determine which parameter directions the observation informs.

If many BQL observations have similar sensitivity gradients, they can add substantial curvature in one direction while contributing little or none in a different direction. For example, if the prediction depends locally on the sum of two parameters, observations may constrain that sum while allowing the parameters to trade off. Quantified observations at informative times may contribute different gradients and resolve the trade-off. Conversely, if all observations have nearly parallel gradients, adding more records may not resolve the weak direction.

Parameter Correlations and Sensitivity to Quantified Records

When the fitted model is locally regular, the estimate is away from problematic boundaries, and the observed information matrix is positive definite, its inverse approximates the covariance matrix of the parameter estimates. The corresponding correlation matrix can reveal trade-offs: a large magnitude correlation indicates that changes in two estimates tend to occur together under this local approximation. In the previous example, the inverse of the combined matrix gives an approximate correlation of \(+0.4595\) between \(a\) and \(b\). The positive sign is consistent with uncertainty in the sum-like direction: increasing both parameters can partly preserve the predicted concentration.

This correlation is not a standalone measure of identifiability. It depends on parameterization and on the validity of the local quadratic approximation. A singular information matrix has no ordinary inverse, and a nearly singular matrix can make the estimated covariance and correlations unstable. Very high correlations can flag a trade-off, but moderate correlations do not rule out weak identification in a larger parameter combination. Inspecting eigenvectors or profiles can reveal directions that pairwise correlations miss.

Diagnostics, Alternatives, and Limits

Curvature is a local diagnostic. It is most interpretable when the fitted model is differentiable, the estimate is interior, the likelihood has a well-behaved local maximum, and the residual and random-effect assumptions are reasonable. For a censored model, the censoring limit and the likelihood contribution assigned to each record must also be correct. If these assumptions fail, a sharply curved likelihood may still provide misleading precision.

Useful checks include the following:

• Inspect the eigenvalues and eigenvectors of the observed information, not only its diagonal. Very small eigenvalues identify weak local directions; their eigenvectors describe the parameter combinations involved.
• Examine parameter correlations from the inverse information only when that inverse is stable. Treat extreme or erratic correlations as evidence to investigate, not as a formal test.
• Refit after removing quantified observations one at a time or in sensible groups. Compare estimates and likelihood profiles, and identify whether particular sampling times or subjects provide most of the support.
• Compare local curvature with profile likelihoods. A profile fixes one parameter at successive values and re-optimizes the others. A broad or asymmetric profile can expose weak or nonlinear identification that a local Hessian misses.
• Check whether estimates are near parameter bounds and whether the Hessian is positive definite. A boundary estimate or non-positive curvature may reflect a boundary, a saddle point, a flat direction, or numerical problems rather than a meaningful standard error.

These approaches answer related but different questions. The Hessian is fast and summarizes local curvature, but relies on a quadratic approximation near the estimate. Parameter correlations summarize pairwise local trade-offs, but can obscure multi-parameter directions. Profile likelihood assesses the likelihood over a wider range while accounting for re-optimization, but can be computationally demanding. Leave-one-quantified-observation checks assess data dependence, not identifiability in isolation: a change can arise from ordinary influence, model misspecification, or weak support. No one diagnostic replaces examination of the fitted model, sampling design, and concentration-time data.

Heavy censoring alone does not prove that a parameter is unidentifiable. BQL records can be informative, especially when their timing and model sensitivities differ. Conversely, a high number of records does not ensure that parameters are separately estimated. If the likelihood is weak in a relevant direction, possible responses include reporting the uncertainty clearly, reconsidering whether the model is more complex than the data support, or obtaining data at times or concentrations that distinguish the parameters. Do not treat a numerically converged optimizer as evidence of practical identifiability.

Common Mistake

Wrong reasoning: “There are many BQL records, so the parameters must be precise; the optimizer also converged, so identifiability is not a concern.” The number of records and optimizer convergence do not establish curvature in every parameter direction. Many BQL records with similar sensitivities can constrain the same combination repeatedly while leaving another combination weakly informed.

Correction: Examine the likelihood’s local directions, parameter trade-offs, and dependence on quantified observations. Confirm that the Hessian is usable, and compare it with profiles or refits when necessary. Distinguish numerical convergence from evidence in the data.