Applying the LLOQ on Original and Log Scales

Why the Scale of the LLOQ Matters

An assay reports a below-quantification-limit result in concentration units, but an NLME model may describe the response on a log-concentration scale. The laboratory threshold does not change when the response is transformed; its numerical representation does. The censoring event must therefore be translated using the same logarithm and units as the modeled response. Otherwise, a correctly implemented model can assign the observation the wrong probability.

The essential operation is simple: because the logarithm is strictly increasing, a concentration below the LLOQ is also a log concentration below the logarithm of the LLOQ. The important details are keeping the logarithm base, concentration units, and residual-error model consistent.

Writing the ratio to a reference concentration makes the logarithm’s argument dimensionless. In practice, a model may write \(\ln C\) as shorthand for the logarithm of the numerical concentration in a specified unit. That convention is valid only if the same unit is used for the response, model prediction, and transformed LLOQ. Changing units shifts the numerical log values, so a model and its threshold must be shifted together.

Translating the Censoring Event

For a continuous assay response, a report of “below LLOQ” means the underlying concentration falls below the assay threshold. On the original scale, the event is \(C<L\). On a log scale, it is \(Z<z_L\), where \(z_L\) is calculated with the model’s exact logarithm convention. If the reporting rule instead includes equality, it may be written \(C\leq L\). Under a continuous residual distribution, the probability of exactly \(C=L\) is zero, so these versions have the same probability. Rounding or a discrete reporting mechanism may require more careful treatment.

This event translation does not itself specify a probability model. That comes from the assumed residual error. For example, if the modeled log response conditional on subject-level random effects is normal, the probability of a BQL record is a normal cumulative probability evaluated at the transformed threshold. If the model instead uses additive normal residual error on the original concentration scale, the probability is evaluated using \(L\), not \(\log L\). The threshold is translated to the scale of the modeled response; the error distribution is not changed by that translation.

Below-LLOQ Probabilities Under a Log-Scale Model

Suppose the response is modeled as log concentration and, conditional on the subject’s random effects, the observation follows a normal residual model:

Here \(Z_{ij}\) is the log concentration for subject \(i\) at observation \(j\), \(m_{ij}(\eta_i,\theta)\) is its conditional model mean on the log scale, \(\eta_i\) denotes the subject-level random effects, \(\theta\) denotes the population model parameters, and \(\sigma\) is the residual standard deviation on the log scale. Under these assumptions, the probability of a BQL observation conditional on \(\eta_i\) is

where \(\Phi\) is the standard normal cumulative distribution function. This is a conditional probability: it refers to the model prediction for the specified subject-level random effects. A population-average probability generally requires averaging these conditional probabilities over the random-effects distribution. It is not, in general, obtained by setting every random effect to zero and treating that single conditional probability as the population average.

The Residual Model Determines the Probability

Applying the right threshold does not make different residual-error models equivalent. Consider an additive normal model on the original concentration scale:

Here \(F_{ij}\) is the conditional mean concentration and \(\sigma_C\) is the residual standard deviation in concentration units. Its below-LLOQ probability is

This formula uses the original-scale LLOQ \(L\). It is not the log-scale formula with \(L\) substituted for \(z_L\). Conversely, in a log-scale model, the threshold used in the normal probability calculation is \(z_L\), not \(L\). The two models encode different distributions and can produce different BQL probabilities.

Logarithm Bases, Likelihoods, and Interpretation

A model using base-10 logs must use a base-10 threshold. For example, if \(L=0.20\) mg/L and \(C_{\mathrm{ref}}=1\) mg/L, then \(z_L=\log_{10}(0.20)=-0.6990\). Pairing that threshold with a natural-log response would be a scale error. If the same response is represented using natural logs instead, then \(\ln(0.20)=-1.609\), and the model’s mean and residual standard deviation must also be expressed on that natural-log scale.

For a natural-log response \(Z=\ln(C/C_{\mathrm{ref}})\), changing to base-10 logs gives \(Z_{10}=Z/\ln(10)\). Thus the model mean and residual standard deviation also divide by \(\ln(10)\). A threshold cannot be converted in isolation while leaving the model’s parameters on their original scale.

There is also a distinction between the mean of log concentration and the log of mean concentration. Under the conditional normal model above, the conditional median concentration is \(C_{\mathrm{ref}}\exp(m_{ij})\), while the conditional arithmetic mean concentration is \(C_{\mathrm{ref}}\exp(m_{ij}+\sigma^2/2)\). In general, the model’s log-scale mean \(m_{ij}\) is not the logarithm of its arithmetic mean concentration. The BQL probability formula uses the log-scale mean and variance specified by the residual model; it should not silently replace \(m_{ij}\) with the log of an arithmetic mean.

For a quantified observation, a lognormal model can be expressed as a density on either the log or concentration scale. If \(Z=\ln(C/C_{\mathrm{ref}})\), the change of variables gives \(f_C(c)=f_Z(\ln(c/C_{\mathrm{ref}}))/c\). The factor \(1/c\) is the density Jacobian. For a BQL observation, the contribution is a cumulative probability, \(P(C<L)=P(Z<z_L)\), so the threshold is translated and no density Jacobian is applied to that probability.

Checks, Alternatives, and Common Pitfalls

Before using a transformed threshold, verify that the recorded LLOQ and model response share the same concentration units and logarithm base. Then check that the residual standard deviation is on the response scale used in the probability calculation. A log-scale standard deviation is not in mg/L, and an original-scale standard deviation is not directly usable with a log threshold.

A useful diagnostic is to compare observed BQL frequencies with probabilities predicted under the fitted model, at comparable times or prediction ranges. A marked discrepancy can indicate a scale mismatch, an incorrect residual-error assumption, or a model that poorly describes concentrations near the assay limit. Such a comparison is a diagnostic, not proof that one specific component is at fault. Also verify that the model’s implied response and the assay’s reporting rule correspond to the same measurement process.

The principal alternatives differ in the assumed response scale and error structure, not in the physical meaning of “below the LLOQ.” An original-scale additive-error model uses the threshold \(L\) and an original-scale residual distribution. A log-scale normal-error model uses \(z_L\) and implies a lognormal concentration distribution. A proportional-error model on the original scale requires its own probability calculation under its specified error distribution. None should be selected merely to make a threshold calculation convenient.

• Scale mixing: Using \(L\) in a log-scale normal CDF mixes concentration units with log units. Use \(z_L\).
• Base mixing: Using \(\ln L\) for a base-10 model, or vice versa, changes the censoring event numerically. Match the model’s base.
• Unit mixing: Taking a log of a threshold in one unit while predictions use another introduces an additive shift. Convert both the threshold and model representation consistently.
• Wrong error model: Transforming \(L\) does not turn an original-scale additive-error model into a log-scale model. Use the probability formula implied by the stated residual distribution.
• Typical-subject shortcut: A conditional probability at \(\eta_i=0\) is not generally the population-average BQL probability. The latter averages over subject-level random effects.