Likelihoods With Multiple Assay Limits

Why Multiple Assay Limits Change the Likelihood

A below-quantification-limit (BQL) result is not a measured concentration at the limit. It reports an event: the underlying assay result fell below a specified threshold. If different laboratories, assay methods, or study periods use different known lower limits of quantification (LLOQs), the event itself differs from record to record. A likelihood that uses the wrong threshold assigns the wrong probability to that record and can therefore affect estimates of both typical concentration profiles and between-subject variability.

The earlier material on applying the LLOQ on original and log scales establishes that the threshold must be applied on the scale of the observed assay result, with the corresponding probability calculated under the residual-error model. Here the key extension is to attach the correct threshold to each observation, rather than assuming that one common LLOQ applies throughout the data set.

Definition: Let \(Y_{ij}\) be the latent assay result for subject \(i\) at observation \(j\), and let \(L_{ij}\) be the known LLOQ that applies to that record. A quantified record reports a value \(y_{ij}\) and contributes its observation density. A BQL record reports \(Y_{ij}<L_{ij}\) and contributes the probability \(P(Y_{ij}<L_{ij}\mid\eta_i)\), conditional on subject \(i\)'s random effects \(\eta_i\) and the model covariates.

Record-Specific Contributions

Suppose the model specifies the conditional density \(f_{ij}(y\mid\eta_i)\) and cumulative distribution function \(F_{ij}(y\mid\eta_i)\) for the assay result. For a continuous observation, the likelihood contribution for a quantified result is its density evaluated at the reported value. The contribution for a BQL result is the probability of falling below its own threshold. These are different kinds of contributions: a density for an exact value and a probability for an interval event.

\ell_{ij}(\eta_i)= \begin{cases} f_{ij}(y_{ij}\mid\eta_i), & \text{if record }(i,j)\text{ is quantified},\\[4pt] F_{ij}(L_{ij}\mid\eta_i), & \text{if record }(i,j)\text{ is BQL}. \end{cases}

For example, if the model is normal on the original concentration scale, \(Y_{ij}\mid\eta_i\sim N(m_{ij}(\eta_i),\sigma^2)\), then a BQL record contributes \(\Phi((L_{ij}-m_{ij}(\eta_i))/\sigma)\). If instead \(\log Y_{ij}\mid\eta_i\sim N(\mu_{ij}(\eta_i),\sigma^2)\), with \(L_{ij}>0\), its contribution is \(\Phi((\log L_{ij}-\mu_{ij}(\eta_i))/\sigma)\). Here \(\Phi\) is the standard normal cumulative distribution function. In either case the threshold is the record-specific \(L_{ij}\), expressed on the scale required by the model.

Assumptions: The applicable LLOQ is known for each record, and BQL status means that the assay result was below that threshold. Conditional on the subject’s random effects and model covariates, the observation process follows the stated residual distribution. If observations are conditionally independent, their contributions multiply within a subject. These assumptions do not require all assays to have the same LLOQ; they do require any material assay-related differences in the observation distribution to be represented appropriately.

Write \(\delta_{ij}=1\) for a quantified record and \(\delta_{ij}=0\) for a BQL record. Under conditional independence, the subject-specific conditional likelihood is the product of the contributions across that subject’s records. The marginal likelihood integrates this conditional likelihood over the random-effects distribution. For independent subjects, the full likelihood is the product of their marginal likelihoods.

L_i(\eta_i)=\prod_j \ell_{ij}(\eta_i), \qquad L_i=\int L_i(\eta_i)\,g(\eta_i)\,d\eta_i, \qquad L=\prod_i L_i,

The order matters: first multiply the records’ contributions conditional on the same subject-level random effects, then integrate over those effects. Replacing the subject-level integral by a product of separately integrated record contributions would generally discard the dependence induced by the shared random effects.

Worked Examples

Worked Example: Different LLOQs Give Different BQL Probabilities

Consider a hypothetical log-scale residual model for one record at a fixed value of the subject’s random effect. Let the predicted median concentration be \(0.40\) mg/L, so \(\mu=\log(0.40)\), and let the residual standard deviation on the log scale be \(\sigma=0.50\). Calculate the BQL probability at two assays’ limits, \(0.20\) and \(0.30\) mg/L.

For \(L=0.20\) mg/L, the standardized threshold is \(\{\log(0.20)-\log(0.40)\}/0.50=-1.3863\). Thus \(P(Y<0.20)=\Phi(-1.3863)\approx0.08283\).

For \(L=0.30\) mg/L, the standardized threshold is \(\{\log(0.30)-\log(0.40)\}/0.50=-0.5754\). Thus \(P(Y<0.30)=\Phi(-0.5754)\approx0.2825\).

The higher LLOQ makes a BQL result more probable under this same model. If these are two conditionally independent observations from the same subject at the stated random-effect value, and both are BQL, their conditional joint contribution is \(0.08283\times0.2825\approx0.02340\). That product is conditional on the shared random effect; it is not, in general, the marginal joint probability after random effects are integrated out.

Worked Example: Two BQL Records With Assay-Specific Limits

Suppose two hypothetical independent subjects have the same model-predicted median concentration, \(m=0.20\) mg/L, and log-scale residual standard deviation \(\sigma=0.40\). Subject A has a BQL result from an assay with LLOQ \(0.10\) mg/L; subject B has a BQL result from an assay with LLOQ \(0.25\) mg/L. For this calculation, treat the model prediction as fixed rather than integrating over random effects.

For A, the probability contribution is \[ \Phi\left(\frac{\log(0.10)-\log(0.20)}{0.40}\right) =\Phi(-1.7329)\approx0.0416. \]

For B, it is \[ \Phi\left(\frac{\log(0.25)-\log(0.20)}{0.40}\right) =\Phi(0.5579)\approx0.7115. \]

Since the subjects are independent, the likelihood contribution from these two BQL records, under the stated fixed-prediction calculation, is approximately \(0.0416\times0.7115=0.0296\). Using \(0.10\) mg/L for both records would give B the wrong probability; using \(0.25\) mg/L for both would give A the wrong probability. The LLOQ belongs to the record, not to the data set as a whole.

Worked Example: Combining a Quantified Value and a BQL Record

For a hypothetical subject, assume two conditionally independent observations at the same predicted median, \(0.40\) mg/L, and \(\sigma=0.50\) on the log scale. The first is quantified at \(y=0.50\) mg/L using an assay with LLOQ \(0.20\) mg/L. The second is BQL using an assay with LLOQ \(0.30\) mg/L. Calculate the conditional likelihood contribution, using density with respect to concentration for the quantified result.

For the quantified value, the lognormal density on the concentration scale is \[ f_Y(y)=\frac{1}{y\sigma\sqrt{2\pi}} \exp\left[-\frac{1}{2}\left\{\frac{\log y-\log(0.40)}{0.50}\right\}^2\right]. \] At \(y=0.50\) mg/L, the standardized log residual is \(\log(0.50/0.40)/0.50=0.4463\). Therefore \(f_Y(0.50)\approx0.3611/(0.50\times0.50)=1.444\) per mg/L.

The second record contributes the BQL probability at \(0.30\) mg/L, calculated as \(\Phi(-0.5754)\approx0.2825\). Multiplying the conditional contributions gives \(1.444\times0.2825\approx0.4081\) per mg/L. The density has units inverse to concentration; the BQL probability is dimensionless. This is a conditional subject contribution at the specified random-effect value, not the marginal likelihood after integrating over random effects.

Justification and Implementation

The censored contribution follows directly from what the assay reports. A BQL label does not reveal the exact \(Y_{ij}\), but it does reveal that \(Y_{ij}\) belongs to the interval below \(L_{ij}\). Consequently, conditional on \(\eta_i\), the probability of the observed event is the integral of the density over that interval:

P(Y_{ij}<L_{ij}\mid\eta_i) =\int_{-\infty}^{L_{ij}} f_{ij}(y\mid\eta_i)\,dy =F_{ij}(L_{ij}\mid\eta_i).

The derivation does not depend on the threshold being common across records. Each BQL event has its own upper endpoint, so substituting \(L_{ij}\) into the CDF gives the required probability for that event. If the model is specified for \(\log Y_{ij}\), the equivalent event is \(\log Y_{ij}<\log L_{ij}\); the limit must therefore be transformed along with the observation scale.

Attach the assay metadata.
For every record, verify the assay, laboratory, study period, units, and LLOQ that actually apply. Convert units consistently before calculating a likelihood contribution.

Classify the record.
Use the reported concentration and the recorded status to determine whether the contribution is an exact-value density or a below-threshold probability.

Evaluate on the model scale.
Use \(f_{ij}(y_{ij}\mid\eta_i)\) for quantified results and \(F_{ij}(L_{ij}\mid\eta_i)\) for BQL results, transforming the limit when the model is on a transformed scale.

Assemble the mixed-effects likelihood.
Multiply contributions within subject conditional on \(\eta_i\), then integrate over the random-effects distribution and combine subjects.

Diagnostics, Alternatives, and Limitations

The most important data checks are record-level checks. Confirm that each assay-period combination maps to the intended limit, that concentrations and limits use compatible units, and that a BQL record has not been assigned a fabricated exact concentration. Check the software’s likelihood calculations on a few hand-worked records, including records near the threshold and records from each assay. On a log scale, verify explicitly that the calculation uses \(\log L_{ij}\), not \(L_{ij}\).

A common mistake is to replace all assay-specific thresholds with the largest LLOQ, on the assumption that this is conservative. It is not the likelihood of the recorded data: for a record from a more sensitive assay, it changes the event being modeled and inflates its BQL probability. Another mistake is to use the correct numeric limit but on the wrong scale. The correction in both cases is to use the limit and units that apply to that record, then evaluate the corresponding CDF on the model’s observation scale.

A common LLOQ is appropriate only when the applicable limits truly are common, or when the data have been consistently harmonized so that the modeled censoring events share a threshold. Otherwise, the record-specific censored likelihood preserves the information in the distinct assay limits. If assays differ not only in LLOQ but also in bias, precision, or residual distribution, using separate thresholds alone may be insufficient: those differences may need to be represented in the observation model. The likelihood also relies on the recorded BQL status and threshold being accurate; incorrect assay metadata cannot be repaired by the probability calculation.

Very small CDF probabilities can cause numerical underflow if likelihood contributions are multiplied directly. Implementations typically work with log contributions, summing log densities and log probabilities before evaluating the random-effects integral. A useful diagnostic is to compare a few computed contributions with direct CDF or density calculations. Large discrepancies may indicate an incorrect threshold, unit conversion, scale transformation, or status code. The method accounts for censoring under its stated model; it does not by itself resolve model misspecification or assay-calibration problems.

Key takeaway: For every BQL record, use the probability below that record’s own known LLOQ. Combine that probability with quantified-value densities at the subject level conditional on random effects, and integrate the resulting subject likelihood over the random-effects distribution.

Check Your Understanding

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

A lognormal residual model is used, and a BQL record has LLOQ \(0.12\) mg/L. Which threshold appears in the normal CDF: \(0.12\) or \(\log(0.12)\)?
Two independent subjects have BQL probabilities \(0.10\) and \(0.40\) under their respective record-specific limits. What is their joint likelihood contribution?
Why is it generally incorrect to integrate each record’s contribution over a subject’s random-effects distribution separately and then multiply the results?
A quantified concentration is reported in ng/mL, but its LLOQ is stored in mg/L. What must be checked before evaluating the likelihood?

Solutions. 1. Use \(\log(0.12)\) in the normal CDF because the model is for \(\log Y\); the corresponding standardized value is \(\{\log(0.12)-\mu_{ij}(\eta_i)\}/\sigma\). 2. The subjects are independent, so their joint contribution is \(0.10\times0.40=0.040\). 3. Records from the same subject share \(\eta_i\), which induces dependence after the random effects are integrated out; multiply conditional contributions first, then integrate their product. 4. Convert one value so the concentration and limit use the same units, and confirm the correct assay-specific limit before calculating a density or probability.