Handling Different LLOQs Across Assays

Why the LLOQ Belongs to Each Record

A concentration profile may be analyzed using more than one assay, or a single assay may have different lower limits of quantification (LLOQs) across runs or time periods. In either case, a BQL report does not mean the same thing for every sample: it means that the measured concentration fell below the applicable threshold for that particular record. The likelihood must therefore use the threshold attached to that record, rather than a common value assigned to the whole subject or study.

The threshold and the residual-error model play different roles. The residual model describes the distribution of a possible measured concentration around the model prediction. The LLOQ determines which part of that distribution is consistent with a BQL report. Changing the LLOQ changes the probability assigned to that report, even when the predicted concentration and residual model are unchanged.

Definition: For record \(j\) from subject \(i\), let \(C_{ij}\) be the model-predicted concentration, \(Y_{ij}\) the potential measured concentration, \(L_{ij}\) the applicable LLOQ, and \(a_{ij}\) the assay used. A quantified record reports a value \(y_{ij}\) at or above its applicable threshold; its likelihood contribution is the observation-model density evaluated at \(y_{ij}\). A BQL record reports the event \(Y_{ij}<L_{ij}\); its likelihood contribution is the probability \(P(Y_{ij}<L_{ij})\) under the observation model for that record.

The threshold \(L_{ij}\) can vary with assay, laboratory run, sample processing, or another documented feature of the measurement. The assay label \(a_{ij}\) identifies which residual-error or calibration model applies. A threshold change alone does not imply that the residual distribution changed; conversely, using a different assay may require a different observation model even if the two assays share an LLOQ.

Record-Specific Likelihood Contributions

Write the conditional distribution of \(Y_{ij}\), given the PK prediction and the model parameters, as \(F_{ij}\), with density \(f_{ij}\). The distribution may depend on the assay and on the predicted concentration. For a quantified record, the contribution is \(f_{ij}(y_{ij})\). For a BQL record, integrate the density over all values compatible with the report:

\mathcal{L}_{ij} = \begin{cases} f_{ij}(y_{ij}), & \text{if the record is quantified},\\[4pt] F_{ij}(L_{ij}) = \displaystyle\int_{-\infty}^{L_{ij}} f_{ij}(y)\,dy, & \text{if the record is BQL}. \end{cases}

For a continuous measurement distribution, it makes no probability difference whether the reporting rule is written \(Y_{ij}<L_{ij}\) or \(Y_{ij}\leq L_{ij}\). If records are conditionally independent given the model’s individual effects and parameters, their conditional likelihood is the product of their record-level contributions. Independence is an assumption, not a consequence of having different rows in a data set. If replicate measurements or assay procedures create residual dependence, the joint observation model must represent it.

Two common residual models make the role of the threshold explicit. Under an additive normal model, \(Y_{ij}\sim N(C_{ij},\sigma_{a_{ij}}^2)\), so a BQL contribution is \(\Phi((L_{ij}-C_{ij})/\sigma_{a_{ij}})\), where \(\Phi\) is the standard normal cumulative distribution function. Under a lognormal model, \(\ln(Y_{ij})\sim N(\ln(C_{ij}),\sigma_{a_{ij}}^2)\) for positive \(C_{ij}\). The BQL contribution is then \(\Phi((\ln L_{ij}-\ln C_{ij})/\sigma_{a_{ij}})\). Here \(C_{ij}\) is the median of the lognormal distribution. The appropriate model and its parameters must be specified before using either formula.

Worked Example: Same Prediction, Two LLOQs

Suppose a hypothetical sample has predicted concentration \(C=0.080\) mg/L. Under an additive normal residual model, let the measurement standard deviation be \(0.020\) mg/L for either assay. Calculate the BQL contribution for an assay with LLOQ 0.100 mg/L and for one with LLOQ 0.020 mg/L.

The model is \(Y\sim N(0.080,0.020^2)\), and the report is BQL. The required contribution is \(P(Y<L)=\Phi((L-C)/0.020)\). For \(L=0.100\) mg/L:

z=\frac{0.100-0.080}{0.020}=1.000, \qquad P(Y<0.100)=\Phi(1.000)=0.8413.

For \(L=0.020\) mg/L:

z=\frac{0.020-0.080}{0.020}=-3.000, \qquad P(Y<0.020)=\Phi(-3.000)=0.001350.

Thus, the same predicted concentration and residual model give very different probabilities for a BQL report at the two thresholds. The calculation assumes that the stated standard deviation applies to both assays and that the additive normal model is appropriate for these concentration measurements. If either assumption is wrong, these numerical contributions are not justified.

Worked Example: A Time-Varying Threshold Under a Lognormal Model

Consider two hypothetical BQL records from one subject. At each time, the predicted concentration is \(0.080\) mg/L and the log-scale standard deviation is \(0.250\). The first record has LLOQ \(0.100\) mg/L; the second has LLOQ \(0.020\) mg/L. Find both contributions under \(\ln(Y)\sim N(\ln(0.080),0.250^2)\).

For a positive threshold, the BQL probability is \(\Phi((\ln L-\ln C)/0.250)\). At the first time:

z_1=\frac{\ln(0.100/0.080)}{0.250} =\frac{\ln(1.25)}{0.250} =0.8926, \qquad \mathcal{L}_1=\Phi(0.8926)\approx 0.8139.

At the second time:

z_2=\frac{\ln(0.020/0.080)}{0.250} =\frac{\ln(0.25)}{0.250} =-5.545, \qquad \mathcal{L}_2=\Phi(-5.545)\approx 1.47\times 10^{-8}.

The product of these contributions is approximately \(0.8139(1.47\times10^{-8})=1.20\times10^{-8}\), under conditional independence. This small product is not, by itself, evidence of a coding error: one BQL report is extremely unlikely under the stated model and low threshold. The example assumes that both records genuinely use the stated lognormal residual model and that the listed thresholds are the thresholds applied by the laboratory.

Working Through an Assay Switch

When a subject switches assays, retain the PK prediction for each record and use the observation model and LLOQ that apply to that record. A switch does not reset the subject’s concentration prediction. If assay calibration or bias differs, that difference must be represented in the observation model; it should not be mistaken for a change in the underlying PK trajectory.

Worked Example: Quantified and BQL Records Across Assays

A hypothetical subject has three observations. For this calculation, assume conditionally independent additive normal errors. Assay A has residual standard deviation \(0.050\) mg/L and LLOQ \(0.050\) mg/L. Assay B has residual standard deviation \(0.020\) mg/L and LLOQ \(0.100\) mg/L. The first record uses A, is quantified at \(y_1=0.320\) mg/L, and has prediction \(C_1=0.300\) mg/L. The second uses B, is BQL, and has prediction \(C_2=0.080\) mg/L. The third returns to A, is BQL, and has prediction \(C_3=0.040\) mg/L. Calculate the three contributions and their product.

For the quantified first record, the normal density contribution is:

\mathcal{L}_1 =\frac{1}{0.050\sqrt{2\pi}} \exp\left[-\frac{(0.320-0.300)^2}{2(0.050)^2}\right] \approx 7.365\;(\mathrm{mg/L})^{-1}.

The units are inverse concentration because this is a density. For the BQL record on assay B, use assay B’s LLOQ and standard deviation:

\mathcal{L}_2 =\Phi\left(\frac{0.100-0.080}{0.020}\right) =\Phi(1.000) =0.8413.

For the third record, use assay A’s LLOQ and standard deviation, not assay B’s values:

\mathcal{L}_3 =\Phi\left(\frac{0.050-0.040}{0.050}\right) =\Phi(0.200) =0.5793.

Under the stated conditional-independence assumption, the subject’s conditional likelihood contribution is:

\mathcal{L}_{i\mid\eta_i} =\mathcal{L}_1\mathcal{L}_2\mathcal{L}_3 \approx 7.365(0.8413)(0.5793) \approx 3.589\;(\mathrm{mg/L})^{-1}.

The product carries inverse-concentration units because it contains one density and two probabilities. Its magnitude should not be interpreted as a probability. This calculation assumes the supplied predictions, assay-specific standard deviations, thresholds, and conditional independence are correct. If assay A and assay B have different calibration biases, those biases would need to be included before evaluating the corresponding density or probability.

Why the Censored Contribution Has This Form

A BQL record supplies an event, not an exact measured concentration. Every possible measurement below the threshold is consistent with the report, so the likelihood contribution must add their density over the compatible region. For a continuous observation model, this is the cumulative distribution function evaluated at the record’s LLOQ:

Identify the record’s status and metadata.
Determine whether it is quantified or BQL, which assay was used, and the threshold applied to that sample.

Choose the observation distribution.
Use the model appropriate to the assay and record, including any specified calibration or residual-variance terms.

Evaluate the matching contribution.
For a quantified value, evaluate the density at that value. For a BQL report, evaluate the cumulative probability below that record’s LLOQ.

Combine records under the dependence model.
Multiply the contributions only when the conditional-independence assumption is appropriate; otherwise use the specified joint distribution.

This construction is the record-specific version of censored likelihood contributions: the event probability is evaluated at the observed record’s censoring boundary. It also explains why replacing a BQL value with zero, half the LLOQ, or another point value is not equivalent. A point substitution assigns a density to a fabricated measurement; the censored likelihood assigns probability to the range actually reported.

Diagnostics, Alternatives, and Limitations

The most consequential implementation errors are often in the data mapping rather than in the probability formula. Verify that every record has the correct assay identifier, threshold, units, and BQL status. In particular, check that the threshold is on the same concentration scale as the observation model. A threshold recorded in different units must be converted before it enters the likelihood. For a lognormal model, the threshold must be positive so that \(\ln L_{ij}\) is defined.

Useful checks include calculating selected contributions by hand, confirming that a higher LLOQ gives a larger BQL probability when the distribution is fixed, and reviewing predicted versus observed BQL patterns within assay and time strata. A BQL probability that is numerically zero or one across many records may indicate a genuine mismatch between predictions and reports, but can also result from an incorrect threshold, unit conversion, assay assignment, or residual model. Compare the implemented calculation with the formulas for the chosen error distribution rather than assuming that a surprising result is automatically a numerical problem.

The main alternative is to simplify the data to a common threshold or to substitute numerical values for BQL reports. A common-threshold approach can discard information when actual thresholds differ, while substitution fails to represent the interval information in a censored record. A censored-likelihood approach retains the actual threshold for each observation, but it depends on accurate assay metadata and a suitable observation model. The likelihood does not correct for an unmodeled assay bias, a threshold that was recorded incorrectly, or informative decisions about which assay to use.

A further limitation is that the simple product of record-level terms assumes conditional independence. If the same specimen is tested by two assays, or repeated results share errors, multiplying their separate contributions may count dependent information as if it were independent. The joint measurement process must then be modeled. Likewise, the calculations above are conditional on the PK prediction and any individual effects used to form it. They do not replace whatever integration over individual effects is required for population-level estimation.

Common mistake: “This subject used assay A first, so I will use assay A’s LLOQ for all of the subject’s later samples.” This is wrong when the later records used another assay or a different threshold. The correction is to link each observation to its own assay and LLOQ, then evaluate its contribution using that record’s observation model. The subject’s PK prediction can be shared across records, but the measurement contribution is record-specific.

Key takeaway: For every BQL record, use the probability that its own observation model produces a value below its own LLOQ. For every quantified record, use the corresponding density at the reported value. Assay switches require accurate record-level mapping and any necessary assay-specific calibration or residual model.

Check Your Understanding

Work each item using the record-specific threshold and model. Solutions follow the questions.

Under \(Y\sim N(0.060,0.010^2)\), find the BQL contribution when \(L=0.070\) mg/L.
Under \(\ln(Y)\sim N(\ln(0.050),0.200^2)\), find the BQL contribution when \(L=0.050\) mg/L.
A quantified record has density \(f(y)=4.0\;(\mathrm{mg/L})^{-1}\) at its observed value. A separate BQL record has contribution 0.30. Under conditional independence, what is their joint contribution and its units?
Explain why a BQL result at a lower LLOQ can provide different evidence than a BQL result at a higher LLOQ, even when both have the same prediction.

Solutions. (1) \(z=(0.070-0.060)/0.010=1.000\), so the contribution is \(\Phi(1.000)=0.8413\). (2) \(z=(\ln(0.050)-\ln(0.050))/0.200=0\), so the contribution is \(\Phi(0)=0.5000\). (3) The product is \(4.0(0.30)=1.2\;(\mathrm{mg/L})^{-1}\); it is a likelihood contribution, not a probability. (4) A lower threshold defines a smaller event \(Y<L\), whose probability under the same distribution is no greater than that for a higher threshold. Therefore, a BQL report at the lower threshold can be much less likely, and more discordant with the prediction, than a BQL report at the higher threshold.