Why Sensitivity Analysis Matters
A population PK analysis with below-quantification (BQL) records depends on assumptions that cannot always be established from the observed data alone. The residual distribution determines the probability that a concentration falls below an assay limit. Different assays may have different lower limits of quantification (LLOQs), and the chosen treatment of BQL records determines how much information those records contribute. When many observations are censored, plausible alternatives can produce meaningfully different parameter estimates.
Sensitivity analysis asks whether the scientific interpretation is robust to those alternatives. It is not a search for the scenario that gives the preferred answer, nor does it establish which assumption is true. Instead, it makes the dependence of the result on modeling choices visible. A useful analysis specifies a defensible primary model, varies assumptions one at a time to locate sources of change, and then evaluates a small number of plausible combinations.
The comparisons are interpretable only when the scenarios target the same population and use the same PK structure, dosing information, covariates, and data inclusion rules, except where a stated sensitivity question requires changing one of them. If several choices change together, a difference between results cannot be attributed to any one choice.
What to Vary and What to Hold Fixed
A practical scenario set begins with the primary analysis and changes assumptions that are both uncertain and consequential. Examples include a justified alternative residual distribution, using the correct LLOQ for each assay rather than a common limit, or comparing a censored likelihood with a specified alternative treatment. Keep a record of every scenario and the reason it is plausible; indiscriminately trying many alternatives makes results difficult to interpret.
For example, under an additive normal residual model, a concentration \(Y\) conditional on typical prediction \(F\) may be represented as \(Y\mid F\sim N(F,\sigma^2)\). A BQL observation with limit \(L\) contributes the probability \(P(Y<L\mid F)\), not a fabricated measured concentration. For a lognormal model, the corresponding assumption is \(\log(Y)\mid F\sim N(\log(F),\sigma_{\log}^2)\), with positive \(F\) and \(Y\); its BQL contribution is also a probability below \(L\). These models can imply different probabilities even at the same \(F\) and \(L\).
A change in handling method is a distinct sensitivity dimension. For instance, an M3 analysis uses a censored likelihood contribution for a BQL record, while an M4-style analysis omits such records from the likelihood. Substitution instead assigns a chosen value to a BQL record. These are not equivalent statistical operations, and substitution values do not become observed concentrations merely because they are used in a computation.
The target estimand and analysis population should remain clear. If an alternative changes which records contribute, say so, and interpret any resulting change as a comparison of analyses with different information, not merely a change in numerical optimization. If the question is whether a treatment effect, dose adjustment, or terminal-phase conclusion changes, define that conclusion before comparing scenarios.
Worked Example: Comparing Residual-Model BQL Probabilities
Suppose a hypothetical model predicts \(F=0.20\) mg/L at a sampling time, the assay LLOQ is \(L=0.10\) mg/L, and the residual standard deviation under an additive normal model is \(\sigma=0.05\) mg/L. Compare its BQL probability with that from a lognormal residual model having \(\sigma_{\log}=0.25\).
Under the additive model, standardize the limit relative to the prediction:
Thus, under these hypothetical additive-error assumptions, the model assigns about a 2.28% chance to a value below the LLOQ at this prediction.
Under the lognormal model, the calculation is on the log scale:
This model assigns about a 0.278% BQL probability, substantially lower than the additive model. The arithmetic illustrates why residual-distribution sensitivity may matter; it does not establish which model is appropriate. That judgment requires checking the measurement scale and residual behavior as well as the fit to quantified and BQL records.
Assay-Specific LLOQs and a Simple Justification
When samples are measured by different assays, the BQL event is defined by the limit applicable to that record. Let \(L_i\) be the LLOQ for observation \(i\), and let \(F_i\) be its model prediction. Under additive normal residual error, the probability contribution for a BQL record is \(\Phi((L_i-F_i)/\sigma)\). Under lognormal residual error, it is \(\Phi(\log(L_i/F_i)/\sigma_{\log})\). Replacing \(L_i\) by a common limit when the assays actually have different limits changes the likelihood contribution, even if the PK prediction and residual parameters are unchanged.
This follows directly from the definition of a censored record. Under the additive model, write \(Y_i=F_i+\epsilon_i\), where \(\epsilon_i\sim N(0,\sigma^2)\). Then:
The model therefore uses the event \(Y_i<L_i\); it does not claim to know the unobserved concentration. Assay identity and its applicable LLOQ should be linked to each record before fitting. A sensitivity analysis can compare the correctly assigned limits with a plausible alternative only when there is a genuine uncertainty about the assay-limit assignment or its value.
Worked Example: Effect of Applying the Wrong Common LLOQ
Consider two hypothetical assays. Assay A has \(L_A=0.05\) mg/L, and assay B has \(L_B=0.20\) mg/L. For a sample predicted at \(F=0.15\) mg/L, suppose the additive residual standard deviation is \(\sigma=0.05\) mg/L. Calculate the BQL probabilities under each assay limit.
For assay A:
For assay B:
The same prediction has a 2.28% BQL probability under assay A and an 84.13% probability under assay B because the limits differ. Assigning assay A's limit to an assay B record would therefore misrepresent the event recorded as BQL. In a fitted analysis, the resulting changes in the full likelihood can alter parameter estimates as well as these individual probabilities.
Quantifying Stability Across Scenarios
A sensitivity analysis should compare more than whether a model converged. For each scenario, report the parameter estimates that matter to the scientific question, their uncertainty summaries, key derived quantities, and relevant diagnostics. Useful derived quantities may include exposure measures, terminal half-life, or a treatment contrast, provided they are defined consistently in each fit.
For a parameter estimate \(\hat\theta_j\), a simple descriptive measure of relative change from the primary estimate is:
This percentage is meaningful when the primary estimate is nonzero and the parameter scale makes relative differences interpretable. For parameters that can be near zero, for contrasts that can change sign, or for estimates on a log scale, an absolute difference or a ratio on the appropriate scale may be more informative. Always include units for dimensional parameters.
A parameter may shift while the substantive conclusion remains the same, or a modest shift may matter if it crosses a decision threshold. Therefore, define stability in relation to the question: for example, whether a dose recommendation changes, whether a treatment contrast retains its direction and clinically relevant magnitude, or whether a terminal-phase interpretation is maintained. There is no universal percentage change that proves robustness.
Worked Example: Reporting Estimate Changes and Their Meaning
In a hypothetical analysis, the primary censored-likelihood fit estimates clearance at \(5.00\) L/h and terminal half-life at \(8.0\) h. A prespecified alternative residual model gives clearance \(5.30\) L/h and half-life \(8.8\) h. Calculate the relative changes and interpret them.
For clearance:
For terminal half-life:
In this hypothetical comparison, clearance is 6.00% higher and half-life is 10.0% higher under the alternative. Those differences alone do not establish whether either conclusion is stable. If the analysis question is whether half-life exceeds 6 h, both point estimates are above that threshold; uncertainty intervals and the prespecified decision rule still need consideration. If the question concerns a narrow dose adjustment based on clearance, even a 6.00% shift may or may not matter depending on the adjustment rule and clinical context.
Design, Diagnostics, and Failure Modes
A clear scenario plan separates the primary assumptions from alternatives and explains why each alternative is plausible. A useful sequence is to first vary one assumption at a time, then fit selected combinations if there is reason to believe assumptions interact. For example, compare residual distributions while holding assay-specific LLOQs and the BQL method fixed; then compare handling methods while retaining the primary residual distribution and correct assay limits. This makes the source of sensitivity easier to identify.
Specify the PK parameter, contrast, derived quantity, or decision that will be judged for robustness.
State the residual models, assay-limit assignments, and BQL treatments before reviewing their results.
Check convergence, parameter plausibility, uncertainty, and predicted BQL frequencies in addition to objective-function or fit summaries.
Report estimate changes, uncertainty, and whether the predefined scientific conclusion or decision changes.
A scenario that fails to converge or produces implausible estimates is not evidence of robustness. It may signal weak information, poor parameterization, or an implementation problem. Inspect whether estimates are at bounds, whether standard errors or uncertainty summaries are unstable, and whether predicted BQL frequencies are systematically inconsistent with observed patterns. Residual diagnostics among quantified records are also relevant, but they cannot by themselves validate the distribution in the censored tail.
Check that each BQL record is paired with the correct assay limit and that the model's prediction and residual scale are on compatible scales. Under a lognormal model, for example, using an additive-scale formula for the BQL probability is an implementation error. A model may produce a plausible objective value while still using the wrong censoring threshold or probability calculation, so record-level checks are valuable.
Sensitivity results should be compared with uncertainty, not only point estimates. Overlapping intervals do not prove that two assumptions are equivalent, and non-overlap is not a substitute for defining the scientific decision. If intervals are wide, apparent agreement may simply reflect weak information. In heavily censored settings, some parameters may be weakly identified, and no reasonable re-analysis can create information that the data do not contain.
Alternatives, Limitations, and Interpretation
| Approach | What it changes | Interpretive caution |
|---|---|---|
| Censored likelihood, such as M3 | Uses the probability of being below the applicable LLOQ | Depends on the residual distribution and correct limit assignment |
| Omission of BQL records, such as M4-style handling | Removes those records from the likelihood | May discard information and can change the effective observation set |
| Substitution | Replaces BQL records with selected values | Introduces a chosen value in place of an unobserved concentration |
| Alternative residual distribution | Changes the modeled probability in the censored tail | Must be scientifically and empirically defensible, not selected only for a preferred estimate |
The censored likelihood is often a natural reference because it directly represents the information that a value lies below a limit. Sensitivity comparisons with omission or substitution can nevertheless be informative when they represent methods used in practice or a specific analysis concern. They should not be portrayed as equally valid merely because they are compared side by side.
A sensitivity analysis has limits. Results depend on which alternatives were considered; stability across a narrow or implausible set does not demonstrate general robustness. Different residual models may be hard to distinguish when few quantified observations inform the lower tail. Uncertainty summaries may themselves depend on approximation quality, especially near parameter boundaries or under weak identification. Report the scenario set and these limitations rather than claiming that assumptions have been proven.
Common Mistake
A common mistake is to say, “The estimates changed by less than 10%, so the result is robust.” That reasoning applies an arbitrary cutoff without asking whether the parameter change affects the scientific conclusion, whether uncertainty is large, or whether the alternatives were plausible. The correction is to report the actual changes with units, compare uncertainty and the prespecified decision or interpretation, and explain why the chosen scenarios are credible. A small numerical change can be consequential near a decision boundary; a larger change can be immaterial for a broad qualitative conclusion.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- Under an additive normal residual model, let \(F=0.30\) mg/L, \(L=0.20\) mg/L, and \(\sigma=0.10\) mg/L. What is the modeled BQL probability?
- Why should a record measured with an assay LLOQ of \(0.15\) mg/L not automatically use another assay's LLOQ of \(0.05\) mg/L?
- A primary estimate is \(4.0\) L/h and a sensitivity estimate is \(4.4\) L/h. Calculate the relative change and state what it does not establish.
- Name two fit or data checks that should accompany comparisons of BQL scenarios.
Solutions. (1) \(z=(0.20-0.30)/0.10=-1\), so the probability is \(\Phi(-1)=0.1587\), or 15.87%. (2) The BQL event is defined relative to the limit applicable to that record; using the other limit changes the event probability and misrepresents the assay information. (3) \(100(4.4-4.0)/4.0=10.0\%\). This gives a relative point-estimate change, but does not by itself establish whether the scientific conclusion changes or account for uncertainty. (4) Examples include checking convergence and parameter bounds, verifying assay-to-LLOQ assignments, comparing predicted with observed BQL frequencies, and reviewing uncertainty summaries.