What M2 Does
A below-quantification-limit (BQL) record says that the assay did not quantify the sample at or above its lower limit of quantification (LLOQ). It does not report the sample concentration as zero, nor does it report a particular value below the limit. M2, or half-LLOQ substitution, replaces each BQL record with one-half of its applicable LLOQ and then analyzes that number as though it were an exact observed concentration.
This is a simple data-handling rule, not a model for how an assay produces a BQL result. Its central consequence is important: after substitution, the analysis no longer knows that the record was censored. It sees an ordinary numerical value. In an NLME fit, that value can contribute a residual, affect individual parameter estimates, and influence population parameters just like a quantified concentration.
For context, let \(f(t_{ij};\theta,\eta_i)\) denote the model-predicted concentration for individual \(i\) at time \(t_{ij}\), where \(\theta\) represents population parameters and \(\eta_i\) the individual random effects. Under a conditional additive normal residual model, a quantified concentration is represented as
This statement describes the model assumption for the numerical observations supplied to the fit. Under M2, a BQL record is supplied as \(Y^*_{ij}=L_{ij}/2\), so the model assigns it the same exact-value residual likelihood as any other supplied concentration. The approach presumes that the LLOQ is correctly recorded for each sample, units are consistent, and the analysis is otherwise appropriate for its residual-error scale. Those conditions do not make the substitution itself an accurate representation of censoring.
Implementation and the Implied Observation
A practical implementation starts with the assay’s BQL flag and the LLOQ applying to that sample. Retain quantified concentrations as reported. For each BQL record, calculate half of that record’s LLOQ in the concentration units used by the model. Then use the resulting numeric value as the observation in the ordinary fitting procedure. If different records have different limits, apply the rule record by record rather than using one global replacement.
The substitution’s implied observation is exact: it tells the model that the observed concentration was \(L_{ij}/2\). But the assay result only tells us that the measurement was below \(L_{ij}\). In particular, it does not identify the unobserved value as \(L_{ij}/2\). This distinction remains even if half the LLOQ seems like a reasonable representative value for a data summary.
Worked Example: Applying the Rule Record by Record
Suppose a hypothetical assay reports concentrations in mg/L. Three samples have LLOQs of 0.08 mg/L. One is quantified at 0.12 mg/L, while two are flagged BQL. Apply M2.
The quantified result remains 0.12 mg/L. For each BQL record, the replacement is \(0.08/2=0.04\) mg/L. The analysis therefore receives the three values 0.12, 0.04, and 0.04 mg/L, with the last two treated as exact concentrations.
The units are preserved because the LLOQ is divided by a dimensionless number. The two replacement values are identical because their LLOQs are identical; if either sample had a different LLOQ, its replacement would be half of that sample-specific limit.
On a log-concentration analysis, the transformation must be applied consistently. If M2 is defined as replacement on the original concentration scale, the value entered after taking logs is \(\log(L_{ij}/2)\). It is not \(\log(L_{ij})/2\), which corresponds to a different concentration, \(\sqrt{L_{ij}}\) when the logarithm is natural and the units are treated consistently. The transformation does not turn a BQL record into an exact observation of the true concentration; it merely transforms the substituted value.
How Substitution Affects the Likelihood
The difference between a censored record and an M2 record can be stated precisely. With the additive normal model above, a BQL event at limit \(L_{ij}\) means \(Y_{ij}<L_{ij}\). Conditional on the individual’s random effects, that event has probability
where \(\Phi\) is the standard normal cumulative distribution function. In contrast, M2 supplies the exact value \(L_{ij}/2\), whose normal density contribution is
where \(\phi\) is the standard normal density. One contribution describes the probability of falling below the limit; the other describes the density at one particular substituted value. They encode different information. In the hierarchical model, contributions are combined over observations and the individual random effects are integrated according to the model. Substituting a value can therefore affect both an individual’s fit and the population-level fit.
Worked Example: A BQL Event Versus an Exact Substitution
Consider a hypothetical observation with LLOQ \(L=0.10\) mg/L. Suppose the model-predicted mean is 0.12 mg/L and the additive residual standard deviation is \(\sigma=0.02\) mg/L. Compare the probability of a BQL event with the exact-value density assigned by M2.
For the BQL event, the standardized limit is \((0.10-0.12)/0.02=-1\). Thus \(P(Y<0.10\mid\eta_i)=\Phi(-1)=0.1587\). Under this model, a BQL result is not impossible even though the predicted mean is above the LLOQ.
M2 instead substitutes \(0.10/2=0.05\) mg/L. Its standardized residual is \((0.05-0.12)/0.02=-3.5\). The normal density at that value is \[ \frac{1}{0.02\sqrt{2\pi}}\exp\!\left(-\frac{(-3.5)^2}{2}\right) \approx 0.0436\;(\mathrm{mg/L})^{-1}. \] This is a density, not a probability, so it should not be compared numerically to 0.1587 as though they were the same kind of quantity. The key point is that the BQL event has probability 0.1587 under the model, while M2 asks the model to account for an exact concentration of 0.05 mg/L, far below its predicted mean.
Why Half-LLOQ Values Can Distort a Profile
In a declining concentration profile, successive BQL results can be replaced by the same number when they share an LLOQ. That creates a sequence of identical observations even if the underlying concentrations continue to fall. A fitted model may then show an artificial flattening, or it may alter other parameters to compromise between the substituted values and quantified observations. Which parameters move depends on the sampling times, residual model, random-effects structure, and information elsewhere in the data.
Worked Example: An Artificial Terminal Plateau
Suppose, hypothetically, that a subject has two consecutive samples at 12 and 18 hours, both BQL with an LLOQ of 0.08 mg/L. M2 replaces each with 0.04 mg/L. If an exponential decline is estimated from these two substituted values, the estimated elimination rate \(k\) satisfies \[ \frac{0.04}{0.04}=\exp[-k(18-12)]. \] Taking logarithms gives \(0=-6k\), so \(k=0\) h\(^{-1}\). The two substituted observations alone imply no decline between 12 and 18 hours.
That conclusion is about the values supplied by M2, not evidence that the true concentration stopped declining. For illustration, hypothetical underlying concentrations of 0.06 and 0.02 mg/L would both be below the 0.08 mg/L limit, yet would represent a three-fold decline. The BQL records do not reveal those concentrations; the point is that equal replacements conceal possible variation below the limit and impose an artificial flat segment on the data.
In a full NLME fit, quantified samples and other individuals may prevent the estimated rate from being zero. Nevertheless, repeated substituted values can pull the fitted terminal profile toward a plateau or otherwise change estimated disposition parameters and variability. The magnitude and direction of the effect cannot be determined from this pair alone.
A short derivation also shows why half the LLOQ is not generally the expected assay result among BQL samples. Under the additive normal model, write \(\mu=f(t_{ij};\theta,\eta_i)\), \(a=(L-\mu)/\sigma\), and let \(\phi\) and \(\Phi\) be the standard normal density and distribution functions. Truncating the normal distribution below \(L\) gives
For the last equality, \(E(Z\mid Z<a)=\int_{-\infty}^{a}z\phi(z)\,dz/\Phi(a)\). Since \(\phi'(z)=-z\phi(z)\), the numerator is \(-\phi(a)\), yielding the stated expression. This conditional mean depends on the predicted concentration and residual variability; there is no general identity setting it equal to \(L/2\). Moreover, an assay’s reported BQL status may be defined by its measurement and reporting process, so this calculation is an illustration under a specified model, not a universal assay correction.
Diagnostics, Alternatives, and Limitations
M2 does not fail only when an optimizer reports an error. A fit may converge cleanly while the substitutions still shape the profile in an undesirable way. Examine the BQL times in relation to the fitted concentrations, inspect runs of repeated replacements, and check whether the fitted curve develops a flat segment or an abrupt change near the LLOQ. Residuals at substituted records are especially informative: a cluster of large, similarly directed residuals can reveal conflict between the imposed exact values and the model predictions.
Also verify the data handling itself. Check that every BQL flag has an applicable LLOQ, that the replacement uses the correct units and sample-specific limit, and that quantified observations were not inadvertently replaced. If concentrations are modeled on a transformed scale, confirm that the substitution was made on the intended scale. Finally, compare estimates, fitted profiles, and uncertainty summaries with a scientifically suitable alternative. Sensitivity to the handling rule is evidence that conclusions depend on an assumption about unobserved values.
M2 is easy to implement and retains a numeric row for each BQL record. Its limitation is that it converts an inequality into a point observation, often adding more specific information than the assay provided. Deleting BQL records, as in M1, avoids inventing a concentration but discards the information that each sample fell below its limit and may leave a selected subset of observations. A censored likelihood instead uses the probability of the observed below-limit event under the residual model. These approaches answer the data problem differently; half-LLOQ substitution should not be treated as equivalent to either one.
- Do not interpret the replacement as a measured concentration. It is a chosen value created for analysis.
- Do not infer a plateau from repeated replacements alone. Equal values can result from the rule, even when concentrations below the limit differ.
- Do not assume the bias has one direction for every parameter. The effect depends on the model and the rest of the data.
- Do not rely on successful convergence as evidence that the rule is appropriate. Numerical convergence does not validate the observation model.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A BQL sample has an LLOQ of 0.06 mg/L. What value does M2 supply, and what does that value claim in an ordinary exact-observation analysis?
- Two BQL samples six hours apart have the same LLOQ of 0.10 mg/L. What exponential elimination rate follows from fitting only their M2 values? What does this calculation fail to establish?
- Under an additive normal residual model, why is a BQL likelihood contribution not the density at half the LLOQ?
- Name two diagnostics that could reveal that M2 is influencing a fitted concentration profile.
- How does deletion differ from half-LLOQ substitution in the information supplied to a model?
Solutions. (1) M2 supplies \(0.06/2=0.03\) mg/L and treats it as an exact observed concentration, although the assay reported only that the sample was below 0.06 mg/L. (2) The rate is \(-\log(0.10/0.10)/6=0\) h\(^{-1}\). This does not establish that the true concentration was constant; it reflects the identical substituted values. (3) A BQL record indicates the event \(Y<L\), whose contribution is the probability \(P(Y<L)\), not an exact value \(Y=L/2\) with a density at that point. (4) Examples include checking for a flat fitted terminal profile, inspecting residuals at substituted records, and comparing estimates under an alternative handling method. (5) Deletion omits the BQL record, including its below-limit information; substitution creates a numeric value and treats it as an exact observation.