Why Below-Quantification Data Need Their Own Check
A concentration reported as below the lower limit of quantification (LLOQ) is not a measured concentration equal to zero, nor is it a missing result. It tells us that the latent concentration lies below a known threshold. A visual predictive check (VPC) should preserve that information: it can compare how often the observed data fall below quantification with how often replicated data from the model do so.
This check is particularly useful when many observations are near the assay limit, such as during a low-concentration part of a dosing interval. A model can give plausible summaries for quantified concentrations while predicting too few or too many below-quantification observations. Looking only at concentration quantiles may fail to reveal that discrepancy, especially when a substantial part of the distribution is censored.
For a VPC, divide observations into time bins using the chosen binning scheme. In bin \(b\), let \(n_b\) be the number of observations with a known quantification status and let \(B_b\) be the number of those observations below quantification. The observed proportion below quantification is \(B_b/n_b\). An observation with a known below-quantification result belongs in both the numerator and denominator. A genuinely missing observation, with no usable assay result or censoring status, does not.
Constructing the Proportion Check
For each simulated replicate, generate data under the fitted population PK model using the relevant design, including subject-level structure, observation times, and residual variability. Then apply the same censoring rule used for the observed records: a simulated concentration below that record's LLOQ is classified as below quantification. Calculate the proportion for each time bin in each replicate. The distribution of those replicate proportions gives a model-based predictive reference for the observed proportion.
Here \(m\) indexes a simulated replicate, and the simulated indicator \(R_{ij}^{(m)}\) is formed by comparing the simulated concentration with the LLOQ applicable to that observation. If LLOQs vary by record, the simulation must use those record-specific limits; applying one common threshold would evaluate a different observation process. Likewise, the bin definitions and the records included in each bin should match between observed and simulated data.
A practical plot shows the observed bin proportions against time, together with a central range of simulated proportions in each bin. For example, the 5th and 95th percentiles across replicates form a pointwise 90% predictive reference range. This is a range of outcomes that the model can generate under the specified design; it is not a confidence interval for the true proportion and it is not a simultaneous guarantee across all bins. The pattern across bins matters: repeated departures in a time region may indicate that the model predicts the timing or extent of low concentrations poorly.
Why the Simulation Must Preserve the Observation Structure
The bin proportion is an average of indicator values. Its expectation is the average probability of being below the relevant limit across the records in the bin. But its sampling variability need not follow a simple binomial formula. Repeated observations from the same subject can be correlated because they share subject-specific parameters and other sources of dependence. In general, the variance of the average includes covariance terms:
A binomial variance calculation treats the indicators as independent with a common probability. That may be a poor approximation when subjects contribute multiple records or when probabilities differ across records. Simulating replicate data with the subject structure and design represented appropriately allows the predictive distribution to reflect the dependence and heterogeneity encoded by the model. The simulation procedure still depends on the model being an adequate representation for the intended check; a reference range cannot correct a misspecified model or an incorrectly represented assay process.
Worked Example: Comparing Two Time-Bin Proportions
Suppose a hypothetical VPC has 60 evaluable observations in each of two time bins. In the earlier bin, 18 results are below quantification; in the later bin, 6 are. Across 1,000 hypothetical simulated replicates, the central 90% reference range for the earlier-bin proportion is 0.18 to 0.38, and for the later-bin proportion it is 0.04 to 0.15. Calculate the observed proportions and interpret them.
For the earlier bin, \(\widehat{p}_{1,\mathrm{obs}}=18/60=0.300\), or 30.0%. For the later bin, \(\widehat{p}_{2,\mathrm{obs}}=6/60=0.100\), or 10.0%.
The earlier-bin value, 0.300, lies inside its simulated range, 0.18 to 0.38. The later-bin value, 0.100, also lies inside its range, 0.04 to 0.15. In these bins, the observed proportions are not outside the stated pointwise predictive ranges. This does not prove that the model is correct; the ranges show how variable these summaries can be under the simulated model and design.
The interpretation would change if, for example, the observed later-bin proportion were 0.250: that value would exceed the simulated upper reference value of 0.15 and would flag a discrepancy for investigation. It would not, by itself, identify the cause of that discrepancy.
Worked Example: Applying the Same LLOQ to a Simulated Replicate
Consider ten hypothetical simulated concentrations in a time bin, all with an LLOQ of 0.50 mg/L: 0.20, 0.30, 0.45, 0.55, 0.70, 0.80, 1.00, 1.20, 1.50, and 2.00 mg/L. Classify the results and calculate the simulated proportion below quantification.
The first three concentrations are less than 0.50 mg/L, so their indicators are 1. Each of the remaining seven is at or above 0.50 mg/L, so its indicator is 0. Thus the replicate has \(B_b^{(m)}=3\) below-quantification results out of \(n_b=10\) records.
The replicate proportion is \(3/10=0.300\), or 30.0%. The value 0.50 mg/L itself would not be classified as below quantification under the stated strict rule \(Y<L\). If these were observed records, the first three would be recorded as below quantification rather than as exact concentrations of 0.20, 0.30, and 0.45 mg/L. Those numerical values here are available only because this example lists the simulated latent concentrations for illustrating classification.
What Quantile Summaries Can and Cannot Show
A conventional VPC may compare observed concentration quantiles with quantiles from simulated data. Censoring limits what can be learned from the observed concentration distribution. If the lowest 30% of results are below the LLOQ, their exact ordering and values are unknown. The below-quantification proportion is identifiable, but the detailed shape of the concentration distribution below the limit is not determined by those records.
In particular, a quantile whose rank falls among the censored observations cannot be calculated as an exact concentration from the observed data. Quantiles above that part of the distribution may be estimable from quantified values, but the precise result can depend on the sample-quantile convention, especially when a calculation interpolates between ranks that include censored records. Any displayed quantile should therefore be checked against the censored fraction and the method used to calculate it.
Worked Example: Identifying an Unavailable Quantile
A hypothetical bin contains 20 observations, of which 7 are below an LLOQ of 0.20 mg/L. The other 13 quantified concentrations, in increasing order, are 0.24, 0.28, 0.31, 0.35, 0.40, 0.46, 0.55, 0.63, 0.80, 1.10, 1.50, 2.00, and 2.80 mg/L. Which order-statistic quantiles can be read directly from the observed data?
The censored fraction is \(7/20=0.350\), or 35.0%. The first seven ordered concentrations are known only to be below 0.20 mg/L. Under the inverse empirical-distribution convention, the 10th percentile uses rank \(\lceil 0.10(20)\rceil=2\), and the 25th percentile uses rank \(\lceil 0.25(20)\rceil=5\). Both ranks are censored, so neither percentile has an identified concentration value.
The median under that convention uses rank \(\lceil 0.50(20)\rceil=10\). Since seven records are censored, this is the third quantified value: 0.31 mg/L. The median is identifiable under this convention, although another quantile convention may interpolate differently. The key conclusion is not that every percentile below or above 35% automatically has the same calculation, but that the ranks needed for the stated quantile must be supported by uncensored values.
Diagnostics, Alternatives, and Common Pitfalls
When a proportion plot looks surprising, check the data handling before attributing the pattern to the PK model. Confirm that censored observations are included in the denominator, that truly missing records are handled consistently, that the applicable LLOQ is correctly assigned, and that simulated values are censored using the same rule. Also inspect the number of observations per bin: a proportion based on few records changes in relatively large steps and may vary substantially between replicates. A pattern that changes sharply at an assay-limit change may reflect the assay or its coding and should be investigated in context.
The proportion check complements, rather than replaces, concentration summaries. Quantified-concentration quantiles describe the distribution where measurements are available; the below-quantification proportion summarizes how much probability lies below the limit. If censoring is slight, quantile comparisons may remain informative for much of the distribution. If censoring is extensive, the proportion check is often the clearer observed-data comparison, while quantiles within the censored region should not be reported as observed concentrations.
| Approach | What it summarizes | Main limitation |
|---|---|---|
| Bin-wise below-quantification proportion | Frequency of results below the applicable LLOQ | Does not reveal the distribution of concentrations below that limit |
| Quantile VPC for quantified concentrations | Selected parts of the observed concentration distribution | Quantiles relying on censored ranks are not identified as concentrations |
| Replacing censored values with a fixed number | A constructed distribution after imputation | Creates unsupported values and can distort proportions and quantiles |
A common mistake is to substitute zero or LLOQ/2 for every below-quantification observation and then treat the substituted values as measured concentrations. The reasoning is that a numeric data set is needed to calculate quantiles. The correction is that the observed information is an inequality, not an exact concentration: use the censoring indicator for the proportion check, and do not present imputed values as observed measurements. A specified imputation method may be used for a separate analysis when justified, but it does not turn the original censored values into exact observations.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A bin has 50 records with known assay status, including 12 below-quantification results and 3 genuinely missing results excluded from those 50. What is the observed proportion below quantification?
- Why should a simulated replicate use the record-specific LLOQ when limits differ across records?
- In a bin, 40% of observations are below quantification. What can be concluded about the observed 20th percentile as an exact concentration?
- Why can repeated observations from the same subject make a binomial variance approximation inadequate?
- An observed below-quantification proportion falls just outside a pointwise 90% simulated range in one time bin. What does this show, and what does it not show?
Solutions. (1) The proportion is \(12/50=0.240\), or 24.0%; the three genuinely missing records are not in the denominator because their assay status is unknown. (2) Each simulated result must be classified by the threshold that would have applied to its corresponding observed record; otherwise the observed and simulated indicators represent different censoring rules. (3) The 20th-percentile rank lies within the censored portion, so its exact concentration is not identified; the data establish only that the relevant values are below the applicable limit. (4) Shared subject-level effects can induce covariance between indicators, so the variance of the average includes covariance terms and may differ from the independent, common-probability binomial variance. (5) The observation is outside that bin's stated pointwise reference range and flags a possible predictive discrepancy. It does not establish the cause, prove model failure, or imply a simultaneous error rate across all bins.