Sensitivity Analysis Across BQL Methods

Why Compare BQL Methods?

A below-quantification-limit (BQL) record says that the measured concentration was below a stated assay threshold; it does not provide an exact concentration. How that record enters an analysis can affect the information available about the concentration-time profile, especially when many observations are BQL or their timing is informative. A sensitivity analysis asks a focused question: how much do the estimates and predictions change when plausible BQL conventions are used instead of a censored likelihood?

The goal is not to run many analyses and choose the one with the most appealing result. Each comparison should keep the model, data, covariates, estimation settings, and prediction targets as consistent as possible. Then differences can be interpreted as method-dependent changes, while instability common to all fits can be recognized as a separate problem.

Definition: A BQL sensitivity analysis is a planned comparison of fits that differ in how BQL records contribute to the analysis, while holding other modeling and computational choices fixed where feasible. The reference analysis is often a censored-likelihood fit: a BQL record contributes the probability of a measurement falling below its quantification limit, rather than a fabricated numerical concentration.

Set Up a Fair Comparison

Start by defining the reference model and the scope of the sensitivity analysis. State the structural model, residual error model, random-effects structure, assay limits, and which records are BQL. The censored-likelihood analysis must use the appropriate threshold for each record, including any assay or study-period changes. A BQL designation is interpreted relative to its own limit, not as though every record shared one universal threshold.

For a given individual, let \(f(t;\theta,\eta)\) be the model-predicted concentration at time \(t\), where \(\theta\) denotes population parameters and \(\eta\) denotes that individual's random effects. Under additive normal residual error with standard deviation \(\sigma\), the measured concentration conditional on \(\eta\) is modeled as normal with mean \(f(t;\theta,\eta)\) and variance \(\sigma^2\). A quantified value contributes its density; a BQL record with limit \(L\) contributes the probability that the measurement is less than \(L\). The likelihood calculation must also account for the random-effects distribution, rather than treating each individual's \(\eta\) as known.

Model condition: Under additive normal residual error, conditional on the individual random effects, a quantified observation \(y\) has density \(f_Y(y\mid\eta)=\frac{1}{\sigma}\phi\!\left(\frac{y-f(t;\theta,\eta)}{\sigma}\right)\). A BQL observation with limit \(L\) contributes \(P(Y<L\mid\eta)=\Phi\!\left(\frac{L-f(t;\theta,\eta)}{\sigma}\right)\). Here \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution functions. The likelihood contribution is conditional on the specified error model and threshold; the population likelihood integrates over the random-effects distribution.

The M1–M7 labels refer to commonly used conventions for handling BQL data. The operational details of historical implementations can vary, so record the precise rule used rather than relying on a label alone. A commonly used description is:

M1: treat BQL observations as missing and omit them from the likelihood.
M2: replace each BQL observation with zero.
M3: use the censored-likelihood contribution for each BQL observation.
M4: replace each BQL observation with one-half of its quantification limit.
M5: exclude an individual's entire profile if its first observation is BQL; otherwise treat BQL observations in that profile as missing.
M6: treat BQL observations before the first quantified observation as missing and replace later BQL observations with zero.
M7: replace BQL observations before the first quantified observation with zero and treat later BQL observations as missing.

These are analysis conventions, not probability models that make substituted values observed concentrations. When a software implementation uses a different M5–M7 rule, document its exact record-level behavior. A useful sensitivity plan usually includes M3 and a small, scientifically motivated subset of alternatives. It need not fit every convention if some are inappropriate for the sampling design or would discard essential data.

Worked Example: A Censored-Likelihood Contribution

Suppose a hypothetical observation has a BQL limit of \(0.500\) mg/L. At that time, the model predicts \(1.00\) mg/L for an individual, and additive residual error has standard deviation \(0.500\) mg/L. Calculate the contribution of this record under the censored likelihood.

First standardize the assay limit relative to the prediction:

z=\frac{L-f(t;\theta,\eta)}{\sigma} =\frac{0.500-1.00}{0.500}=-1.00.

The BQL contribution is the probability of an observation below the limit:

P(Y<0.500\mid\eta)=\Phi(-1.00)=0.1587.

Thus, conditional on these values and the assumed normal error model, this record contributes a probability of about \(0.1587\), not a point concentration of zero or \(0.250\) mg/L. Its log-likelihood contribution is \(\log(0.1587)\approx-1.841\). For comparison, if a quantified value were exactly \(0.500\) mg/L, its conditional density at that value would be

\frac{1}{0.500}\phi\!\left(\frac{0.500-1.00}{0.500}\right) =2\phi(-1)\approx 0.4839\;(\mathrm{mg/L})^{-1}.

The density has units of inverse concentration, whereas the BQL probability is dimensionless. They are different types of contributions and should not be equated.

Derive the Censored Contribution

The censored-likelihood rule follows directly from what the assay reported. Let \(Y\) be the measurement and suppose that, conditional on the individual random effects, \(Y\) has a continuous density \(f_Y(y\mid\eta)\). If the assay reports a quantified value \(y\), the likelihood uses the density at \(y\). If it reports only that \(Y<L\), the possible values are all values below \(L\). Their probabilities add, which for a continuous variable is an integral:

P(Y<L\mid\eta)=\int_{-\infty}^{L} f_Y(y\mid\eta)\,dy.

For the additive normal model, substitute the normal density and standardize with \(u=(y-f(t;\theta,\eta))/\sigma\). Then \(dy=\sigma\,du\), and the upper limit becomes \((L-f(t;\theta,\eta))/\sigma\). Therefore,

\begin{aligned} P(Y<L\mid\eta) &=\int_{-\infty}^{L} \frac{1}{\sigma}\phi\!\left(\frac{y-f(t;\theta,\eta)}{\sigma}\right)\,dy\\ &=\int_{-\infty}^{(L-f(t;\theta,\eta))/\sigma}\phi(u)\,du\\ &=\Phi\!\left(\frac{L-f(t;\theta,\eta)}{\sigma}\right). \end{aligned}

This derivation also shows what must be checked: the error distribution and its scale must be suitable for the observations, and \(L\) must be the applicable limit. With proportional, combined, or lognormal observation models, the corresponding probability must be calculated under that model, not by reusing this additive-normal formula.

Worked Example: Applying M1–M7 to a Profile

Consider a hypothetical profile with an assay limit of \(0.10\) mg/L and measurements at four successive times: BQL, \(0.40\) mg/L, BQL, BQL. Apply the conventions defined above. Assume the same limit at all four times.

M1 omits observations 1, 3, and 4; only the quantified \(0.40\) mg/L value contributes. M2 analyzes the sequence \(0,\ 0.40,\ 0,\ 0\) mg/L. M3 uses a censored probability for observations 1, 3, and 4, and a quantified-value contribution for observation 2. M4 analyzes \(0.05,\ 0.40,\ 0.05,\ 0.05\) mg/L because half the limit is \(0.10/2=0.05\) mg/L.

For M5, the first observation is BQL, so this convention excludes the entire profile. Under M6, the first BQL occurs before the first quantified value and is omitted; the two later BQL values are replaced by zero. Under M7, the first BQL is replaced by zero, while the later BQL values are omitted. Thus, the approaches differ not only in numerical values but also in which parts of the profile inform the fit.

This example illustrates why the rule must be reported at the record level. If a different implementation defines M6 or M7 differently, its actual transformation must be shown instead of assuming that the label guarantees identical handling.

Compare Estimates and Predictions

A practical comparison uses a primary censored-likelihood fit and a limited set of alternatives chosen in advance. For example, M1 can show the consequence of discarding BQL records, while M4 can show the consequence of substituting half the limit. Add M5–M7 only when their special treatment of profile timing is relevant. Keep the structural and random-effects model unchanged; if a fit fails, report that failure rather than silently changing the model until the methods appear comparable.

For each successful fit, compare key population parameters, uncertainty, and predictions at clinically relevant times or regimens. Report parameter values with units and relative changes from the reference, calculated as \(100(\hat\theta_{\mathrm{alternative}}-\hat\theta_{\mathrm{M3}})/\hat\theta_{\mathrm{M3}}\). Compare predictions on a common scale and over the same population or individual-prediction target. The estimated difference is descriptive of the fitted analyses; it is not automatically evidence that one convention is closer to truth.

Worked Example: Summarizing Method-Dependent Changes

Suppose hypothetical refits of the same one-compartment model produce the following estimates. The values are illustrative outputs, not observations from a real study. Assume the reported clearance and volume estimates are population typical values.

Analysis	Clearance (L/h)	Volume (L)	Predicted concentration at 12 h (mg/L)
M3 censored likelihood	5.00	40.0	0.300
M4 half-limit substitution	5.40	39.0	0.260

The clearance change is \(5.40-5.00=0.400\) L/h, or

100\frac{5.40-5.00}{5.00}=8.00\%.

The volume change is \(39.0-40.0=-1.00\) L, or \(100(-1.00/40.0)=-2.50\%\). At 12 h, the predicted concentration changes by \(0.260-0.300=-0.0400\) mg/L, which is \(100(-0.0400/0.300)=-13.3\%\). The prediction shift is larger in relative terms than either parameter shift, so it deserves attention in the intended prediction range.

For an additional derived comparison, suppose the same hypothetical dose is 100 mg and the reported clearance is used to calculate dose divided by clearance. Under M3 this is \(100\text{ mg}/5.00\text{ L/h}=20.0\) mg·h/L; under M4 it is \(100\text{ mg}/5.40\text{ L/h}=18.5\) mg·h/L. The relative difference is \(100(18.5-20.0)/20.0\approx-7.41\%\), using the unrounded M4 value \(18.5185\) mg·h/L. This calculation is a comparison of that derived quantity, not a replacement for predictions from the fitted model.

Diagnostics, Limitations, and Common Pitfalls

A sensitivity analysis is informative only if the fits and comparisons are credible. Check that each run converged under the same estimation settings, that parameter estimates are not at inappropriate boundaries, and that standard errors or uncertainty summaries are available and interpretable. Compare objective-function values only for likelihoods fitted to the same observed-data definition; values from likelihoods that use different substitutions or omit different records are not a straightforward ranking of methods.

Use the earlier material on residual diagnostics with censored values and predictive checks for BQL frequencies to assess whether the model reproduces both quantified concentrations and the observed pattern of BQL records. Also examine changes in parameter correlations, uncertainty, and individual profiles. Large shifts accompanied by weak precision or poor BQL-frequency predictions may indicate that the data provide limited information about parts of the profile, rather than establishing a precise alternative estimate.

Incorrect threshold: applying one limit to records from assays with different limits changes the censoring probabilities. Use the limit associated with each record.
Overinterpreting substitution: a substituted zero or half-limit value is an analysis convention, not a measured concentration. Do not describe it as though the assay observed that exact value.
Changing too many things: changing the error model, random-effects structure, or covariates along with BQL handling prevents attribution of differences to the BQL method.
Ignoring failed fits: a nonconverged alternative is not a valid numerical comparison. Report the failure and investigate it; do not present its estimates as stable results.
Equating stability with validity: similar estimates across methods do not prove the model or assay assumptions are correct. Conversely, a difference does not alone identify which method is preferable.

The main limitation is that sensitivity analyses compare assumptions; they do not supply information absent from the study. If most samples are BQL, or if key profile regions are sparsely sampled, several methods may yield unstable estimates or materially different predictions. A censored likelihood also depends on a correctly specified observation model and threshold. The comparison should therefore include uncertainty, relevant predictions, and diagnostics, not just a table of point estimates.

A common mistake is to choose the method whose estimate best matches an expected value and then call that method validated. That reasoning uses the desired answer to select the analysis. Instead, define the reference and alternatives on scientific grounds before comparing results, report the method-dependent changes transparently, and interpret them alongside model diagnostics and the intended use of predictions.

Key takeaway: A useful BQL sensitivity analysis preserves a common model framework, changes only explicitly stated BQL rules, and reports how those rules affect both estimates and predictions. The censored likelihood respects the information actually reported by a BQL assay, while M1–M7 conventions provide comparisons whose exact record handling must be documented.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A BQL result has limit \(0.20\) mg/L, predicted concentration \(0.30\) mg/L, and additive normal residual standard deviation \(0.10\) mg/L. What is its censored probability contribution?
Why is it misleading to rank M3 and M4 by their objective-function values when each analysis defines a different data likelihood?
In the hypothetical profile BQL, \(0.40\) mg/L, BQL, BQL, which observations are retained as numerical values under M1?
An alternative method changes a parameter estimate by 2% but a prediction at a decision-relevant time by 15%. What should the analyst report and investigate?

Solutions. (1) Standardize: \(z=(0.20-0.30)/0.10=-1.00\). Thus the contribution is \(\Phi(-1.00)=0.1587\), a dimensionless probability under the stated model. (2) M3 uses probabilities for censored events, whereas M4 substitutes values; the resulting likelihoods do not represent the same observed-data definition, so their objective values are not a direct method ranking. (3) M1 omits all three BQL observations and retains only the quantified \(0.40\) mg/L observation. (4) Report both the parameter and prediction changes with units and uncertainty, then check predictions across the relevant time range, convergence, precision, and predictive diagnostics. A modest parameter shift does not guarantee a modest prediction shift.