Why a Below-Quantification Result Is Still Data
A concentration reported as below the lower limit of quantification (LLOQ) is not a measured value of zero, nor is it an exact measurement at the LLOQ. It tells us that the concentration fell in a known region of the measurement scale. That event can carry information about an individual’s pharmacokinetic (PK) parameters, especially when plausible parameter values predict substantially different probabilities of falling below the assay limit.
The update must respect what was observed. Replacing a below-quantification result with zero, half the LLOQ, or the LLOQ treats an unobserved concentration as if it had been measured at a particular value. A censored likelihood instead assigns the observation the probability of the event the assay actually reported. The residual error model determines that probability, so the calculation follows the residual error model and posterior updating methods introduced in the earlier material on Residual Error Models and Posterior Updating.
Constructing the Censored Likelihood
For an additive normal residual model, suppose the latent assay-scale concentration satisfies \(C_i\mid\theta\sim N(f_i(\theta),\sigma^2)\), where \(\sigma\) is the residual standard deviation in concentration units. The probability of a below-limit result is then the normal cumulative distribution function at the standardized limit:
Here \(\Phi\) is the standard normal cumulative distribution function. If the residual model is lognormal, with \(\log C_i\mid\theta\sim N(\log f_i(\theta),\sigma_{\log}^2)\) for positive \(f_i(\theta)\), then the corresponding contribution is:
The choice between these expressions is not a matter of convenience: it must match the residual model for the assay-scale measurement. For either model, a quantified result contributes its usual density at the observed concentration, while a below-limit result contributes a probability. Under conditional independence of observations given \(\theta\), these contributions multiply. With prior density \(p(\theta)\), the posterior is proportional to prior times all quantified densities times all censored probabilities.
where \(Q\) indexes quantified observations with measured concentrations \(c_i\), and \(B\) indexes below-limit observations. The expression assumes the thresholds and reporting process are known and that, conditional on the modeled parameters, the observation errors have the dependence structure specified by the likelihood. If residual errors are correlated, the joint probability of the censored events must reflect that dependence; multiplying marginal probabilities would not generally be valid.
Worked Example: Probability of a Below-Limit Result
Suppose a hypothetical sample has an additive normal residual model with predicted concentration \(f(\theta)=1.50\) mg/L and residual standard deviation \(\sigma=0.50\) mg/L. The LLOQ is \(L=1.00\) mg/L. Calculate the likelihood contribution for a report of below the LLOQ.
The reported event is \(C<1.00\) mg/L, not a measurement at zero or 1.00 mg/L. Standardizing the limit gives:
Therefore, the likelihood contribution is \(\Phi(-1.00)=0.1587\). Under this parameter value and residual model, the probability of a below-LLOQ result is about 15.9%. This is the contribution of this observation to the likelihood; it is not the posterior probability that the parameter value is correct.
How the Observation Changes the Posterior
For a fixed threshold and residual standard deviation in the additive normal model, the censored likelihood decreases as the predicted concentration increases. This can be seen directly by differentiating its log with respect to the predicted concentration \(m=f_i(\theta)\). Write \(z=(L_i-m)/\sigma\), and let \(\phi\) denote the standard normal density. Since \(dz/dm=-1/\sigma\) and \(\Phi'(z)=\phi(z)\), the derivative is:
Thus, if increasing a parameter raises the predicted concentration at that sample time, a below-limit result tends to shift posterior support away from larger values of that parameter. If increasing the parameter lowers predicted concentration, the direction of the shift reverses. This is a likelihood statement, not a universal claim that BLQ observations always increase a particular PK parameter: the direction depends on the model’s relationship between the parameter and concentration at the sampling time.
The observation can be weak or strong evidence. If nearly all plausible parameter values predict a concentration far below the limit, the event is unsurprising across the posterior and may change it little. If plausible values predict very different probabilities of falling below the limit, the event can be informative. Its information is bounded in a practical sense by the fact that the exact concentration below the limit was not observed.
Worked Example: Updating a Discrete Individual Parameter Prior
Consider a hypothetical individual with three candidate parameter values. At a particular time, they predict concentrations of 0.80, 1.20, and 1.60 mg/L, respectively. The prior probabilities are 0.20, 0.50, and 0.30. Assume an additive normal residual standard deviation of 0.40 mg/L, an LLOQ of 1.00 mg/L, and one below-LLOQ report. The predictions and residual model are assumed to apply at this sampling time.
For each candidate, calculate \(P(C<1.00\mid\theta)=\Phi((1.00-f(\theta))/0.40)\). The standardized limits are 0.50, \(-0.50\), and \(-1.50\), giving probabilities 0.6915, 0.3085, and 0.06681. Multiply each by its prior probability:
The normalizing constant is \(0.1383+0.1543+0.02004=0.3126\). Dividing each product by this total gives posterior probabilities of approximately 0.442, 0.494, and 0.0641. The below-limit result reduces support for the candidate predicting 1.60 mg/L and increases the relative support for candidates predicting lower concentrations. If these candidates correspond to increasing clearance at the specified time, the update favors higher clearance; that conclusion depends on that stated parameter-prediction relationship.
Substitution Is a Different Observation Model
Substitution changes the data being modeled. For example, substituting \(L/2\) and using an ordinary normal density treats the concentration as an exact measurement at \(L/2\). Deleting the result treats it as providing no likelihood information. Neither represents the event \(C<L\). Substitution can sometimes be used as an approximation for a particular analysis, but it is not the censored likelihood and may distort the posterior, particularly when there are many below-limit observations or when the probability of censoring varies substantially across plausible parameter values.
Worked Example: Comparing Censoring With Half-LLOQ Substitution
Use the same three candidate predictions and prior probabilities as in the preceding example. For comparison only, suppose someone substitutes \(L/2=0.50\) mg/L and treats it as an exact observation under the same additive normal model with \(\sigma=0.40\) mg/L. The normal density has a common factor \(1/(\sqrt{2\pi}\sigma)\) for all three candidates, so posterior weights can be calculated using only \(\exp(-(0.50-f)^2/(2\sigma^2))\).
The likelihood factors, omitting that common constant, are approximately 0.7548, 0.2163, and 0.0228. After multiplication by prior probabilities, the unnormalized weights are \(0.15096\), \(0.10815\), and \(0.00684\). Their sum is 0.26595, so the substituted-data posterior probabilities are approximately 0.568, 0.407, and 0.0257.
These differ from the censored-likelihood results of approximately 0.442, 0.494, and 0.0641. The difference arises because the two analyses encode different information: one observes the event \(C<1.00\) mg/L, while the other pretends to observe \(C=0.50\) mg/L. The numerical comparison does not establish that substitution always shifts a posterior in this direction; the effect depends on the prior, residual model, threshold, and candidate predictions.
Assumptions, Diagnostics, and Limits
Before using a censored contribution, verify that the reported status and threshold correspond to the modeled sample. The threshold may differ by assay, batch, dilution, or sample, so use the applicable \(L_i\) rather than assuming one universal LLOQ. Confirm that the residual error model is appropriate on the scale used and that the prediction and residual standard deviation use compatible units. If the model is lognormal, predictions and thresholds must be positive. The likelihood also assumes the BLQ status was generated by the stated quantification rule; additional sample-processing or reporting mechanisms may need to be represented if they affect whether a result is reported.
Useful checks include comparing predicted probabilities of BLQ results with the observed pattern across sampling times, examining whether the posterior changes sharply when a threshold or residual-error assumption is varied, and checking whether the posterior is driven almost entirely by repeated censored records. Such checks are especially important when predictions lie far from the threshold: the normal CDF can be extremely small, and numerical calculations on the probability scale can underflow. Computing log-CDF contributions with stable numerical methods avoids taking the logarithm of a rounded zero.
If a result is known to lie between two reporting limits, it is interval-censored rather than simply left-censored. For a continuous measurement with cumulative distribution function \(F\), the likelihood contribution is \(P(L_{\mathrm{low}}\le C<L_{\mathrm{high}}\mid\theta)=F(L_{\mathrm{high}}\mid\theta)-F(L_{\mathrm{low}}\mid\theta)\). This is distinct from a left-censored contribution \(F(L\mid\theta)\). The observation’s reporting category determines which probability belongs in the likelihood.
The main practical alternatives are therefore to use the censoring event directly, substitute a value, or omit the result. The first uses the available status information under an explicit measurement model. Substitution imposes an artificial exact value, and omission discards the event information. The censored approach is not automatically reliable: misspecified residual error, incorrect limits, dependence ignored in the likelihood, or a poorly identified individual parameter can still produce misleading posteriors. It also cannot recover the exact concentration that the assay did not quantify.
Record whether the sample was quantified, left-censored, or reported within an interval, and identify its applicable threshold or thresholds.
For a below-limit record, calculate the probability of that event under the model rather than evaluating a density at an invented concentration.
Multiply the relevant contributions by the individual prior, accounting for any dependence specified by the model, and normalize or sample from the resulting posterior.
Check units, thresholds, residual assumptions, predicted BLQ probabilities, and sensitivity of the posterior to plausible modeling choices.
Worked Example: Information From Two Below-Limit Results
Suppose, hypothetically, that two conditionally independent samples share the same prediction and residual standard deviation for each of three candidate parameter values. The predicted concentrations are 0.80, 1.20, and 1.60 mg/L; the prior probabilities are equal; \(\sigma=0.40\) mg/L; and both samples are reported below \(L=1.00\) mg/L. Assume independence conditional on the candidate value and the stated residual model.
The single-sample event probabilities are 0.6915, 0.3085, and 0.06681, as calculated earlier. Conditional independence means that the likelihood for the two reports is the square of each probability. With equal prior probabilities, the unnormalized weights are \(0.6915^2=0.4782\), \(0.3085^2=0.09517\), and \(0.06681^2=0.004463\). Their sum is 0.5778, giving posterior probabilities of approximately 0.828, 0.165, and 0.00772.
Two repeated below-limit results produce a sharper update than one in this example because their likelihood contributions multiply. This conclusion relies on conditional independence. If the two measurements share correlated errors or another modeled dependence, simply squaring the marginal probability would be incorrect; the joint probability of both events is required.
Check Your Understanding
Answer each question, then check the solutions below.
- Under an additive normal model, \(f(\theta)=1.30\) mg/L, \(\sigma=0.30\) mg/L, and \(L=1.00\) mg/L. What likelihood contribution corresponds to a below-limit report?
- Why is a below-LLOQ result not equivalent to an exact observation at half the LLOQ?
- If increasing a parameter decreases the predicted concentration at a sample time, how does a below-limit result tend to affect support for larger parameter values?
- Under what condition may likelihood contributions for two below-limit observations be multiplied as separate marginal probabilities?
Solutions. (1) The standardized limit is \(z=(1.00-1.30)/0.30=-1.00\), so the contribution is \(\Phi(-1.00)=0.1587\). (2) The assay reports the event that the concentration is below a threshold, not the concentration at a chosen point in that range. Substitution instead uses a density evaluated at an invented exact value. (3) Since larger parameter values then predict lower concentrations, they tend to make the below-limit event more probable and receive greater relative likelihood support, subject to the prior and other data. (4) The observations must be conditionally independent given the modeled parameters and other relevant quantities; otherwise use their joint event probability under the dependence model.