Censored Likelihoods Under Common Residual Error Distributions

Why the Residual Model Determines a BQL Probability

A below-quantification (BQL) record does not provide an exact concentration value. It tells us that the concentration measurement fell below the assay’s lower limit of quantification (LLOQ). The likelihood contribution for that record is therefore the probability of observing a value below the LLOQ under the model—not a density evaluated at a substituted concentration.

That probability depends on the residual error distribution. In particular, the same model-predicted concentration and LLOQ can imply different BQL probabilities under additive, proportional, combined, and lognormal error. The scale also matters: with additive, proportional, or combined normal errors, compare concentrations on the original scale; with lognormal error, take logarithms of both the prediction and the LLOQ before standardizing.

From the Measurement Model to the Tail Probability

The calculation follows the same principle for each error model. First state how the observation is distributed given \(C_{\mathrm{pred}}\). Then express the BQL event as an inequality involving the residual error, standardize that error, and evaluate the standard normal cumulative distribution function \(\Phi\). Here, \(\Phi(z)=P(Z\leq z)\) for \(Z\sim N(0,1)\).

These probabilities are conditional on the prediction and the residual-error parameters used in the model. This tutorial uses \(C_{\mathrm{pred}}\) as the conditional central prediction. In the lognormal model below, it is the median of \(Y\), not its arithmetic mean.

Additive Residual Error

Under additive normal error, residual variability has a constant standard deviation in concentration units. The model is \(Y=C_{\mathrm{pred}}+\epsilon_a\), where \(\epsilon_a\sim N(0,\sigma_a^2)\). Thus \(Y\mid C_{\mathrm{pred}}\sim N(C_{\mathrm{pred}},\sigma_a^2)\), and the LLOQ and prediction are compared directly on the concentration scale.

The model assumes normally distributed, mean-zero additive residual error with the same standard deviation \(\sigma_a\) across predictions. It can assign positive probability to negative measured concentrations. That may be a reasonable approximation in some ranges, but it can be problematic when predictions are low relative to \(\sigma_a\).

Proportional Residual Error

Under proportional normal error, the residual standard deviation grows in proportion to the prediction. Write \(Y=C_{\mathrm{pred}}(1+\epsilon_p)\), where \(\epsilon_p\sim N(0,\sigma_p^2)\). Conditional on \(C_{\mathrm{pred}}\), the observation is normal with mean \(C_{\mathrm{pred}}\) and standard deviation \(C_{\mathrm{pred}}\sigma_p\), provided \(C_{\mathrm{pred}}>0\). The parameter \(\sigma_p\) is dimensionless: it is a proportional residual standard deviation, not a standard deviation in concentration units.

The subtraction \(L-C_{\mathrm{pred}}\) and the standard deviation \(C_{\mathrm{pred}}\sigma_p\) are both on the concentration scale. This model also permits negative observations because its conditional distribution is normal; proportional error does not, by itself, mean the observed concentration is constrained to be positive.

Combined Additive and Proportional Error

A common combined model adds an additive component and a prediction-scaled component. Let

where \(\epsilon_a\sim N(0,\sigma_a^2)\) and \(\epsilon_p\sim N(0,\sigma_p^2)\) are independent. Conditional on \(C_{\mathrm{pred}}\), the observation is normal with mean \(C_{\mathrm{pred}}\). Independence matters: it makes the variance of the sum equal to the sum of the component variances.

Both variance components have concentration-squared units: \(\sigma_a\) has concentration units, while \(C_{\mathrm{pred}}\sigma_p\) has concentration units. The square root therefore returns a standard deviation on the original concentration scale.

Lognormal Residual Error

A lognormal measurement model keeps \(Y\) positive. Define it by \(\log Y=\log C_{\mathrm{pred}}+\epsilon_l\), where \(\epsilon_l\sim N(0,\sigma_l^2)\). Equivalently, \(Y=C_{\mathrm{pred}}\exp(\epsilon_l)\). Under this parameterization \(C_{\mathrm{pred}}\) is the conditional median, since the median of \(\exp(\epsilon_l)\) is 1. The conditional arithmetic mean is \(C_{\mathrm{pred}}\exp(\sigma_l^2/2)\).

For a positive LLOQ, \(Y<L\) is equivalent to \(\log Y<\log L\). The comparison is made on the log-concentration scale, and the standard deviation \(\sigma_l\) is also on that scale.

Justification and Model Checks

Each formula follows by rewriting the BQL event in terms of a standard normal variable. For example, under the combined model, independence and normality give a normal residual with variance \(\sigma_a^2+(C_{\mathrm{pred}}\sigma_p)^2\). Subtracting \(C_{\mathrm{pred}}\) from both sides of the event and dividing by this positive standard deviation yields the standard normal event:

The standardized residual on the left has a standard normal distribution, so the probability of the event is \(\Phi\) evaluated at the standardized LLOQ on the right. The same argument gives the additive and proportional formulas. For lognormal error, first take logarithms, then standardize \(\log Y-\log C_{\mathrm{pred}}\).

Before using a formula, check that its assumptions and units match the fitted residual model:

• For additive error, \(\sigma_a\) must be in concentration units and the model assumes a constant residual standard deviation.
• For proportional error, \(C_{\mathrm{pred}}\sigma_p\) must be used as the concentration-scale standard deviation; the model assumes variability scales with prediction.
• For combined error, both components must be on the stated scales, and the variance-sum formula assumes independent components.
• For lognormal error, both \(L\) and \(C_{\mathrm{pred}}\) must be positive, and the prediction must use the stated log-scale location parameterization.

A useful diagnostic is to compare the model’s implied BQL probabilities with the observed pattern across records having different predictions and LLOQs. Systematic disagreement can indicate a residual distribution or variance pattern that does not describe the data well. This check does not by itself identify the cause; data handling, assay characteristics, and other model assumptions also warrant review.

Comparing the Models and Avoiding Scale Errors

There is no universally best choice among these models. Additive error can be useful when absolute variability is roughly constant; proportional error when variability is closer to constant in relative terms; and combined error when both patterns are needed. A lognormal model is appropriate when positive measurements and multiplicative variation are central to the model. The choice should reflect the measurement process and observed residual behavior, rather than being made solely because one formula is convenient.

Common mistake: “The residual error is proportional, so I should compare \(L\) with \(C_{\mathrm{pred}}\) after taking logarithms.” That reasoning confuses proportional normal error with lognormal error. In the proportional normal model, \(Y=C_{\mathrm{pred}}(1+\epsilon_p)\), so the BQL probability uses \((L-C_{\mathrm{pred}})/(C_{\mathrm{pred}}\sigma_p)\) on the concentration scale. The lognormal model instead specifies a normal error for \(\log Y\), so it uses \(\log(L/C_{\mathrm{pred}})/\sigma_l\). These are different distributions and generally yield different probabilities.