VPCs with Below-Quantification-Limit Data

Why Censoring Changes a VPC

When a concentration is reported as below the limit of quantification, its exact value is not observed. The record still contains information: the concentration fell below a known threshold. A visual predictive check (VPC) that treats the record as an ordinary missing value, or replaces it with an arbitrary number without applying the same rule to simulations, can therefore misrepresent what the data say.

The key distinction is between the latent concentration generated by the model and the value and status recorded after applying the assay's reporting rule. A useful VPC must reproduce that reporting rule in every simulated replicate. It can then assess two features separately: how often simulated observations are below the limit, and how the observed and simulated concentrations compare among observations that are quantifiable.

For a standard VPC, observations are grouped into time bins, and summaries such as medians or other quantiles are compared with the corresponding summaries from simulated data. With censored data, the summaries need an explicit rule. In particular, a quantile of quantified concentrations describes the distribution conditional on being quantifiable; it is not a quantile of all latent concentrations.

Match the Observation Process in Simulations

For each simulated dataset, generate concentrations under the fitted model and apply the same observation and censoring rules used for the data. This includes the applicable LLOQ for each record, any relevant assay-specific rules, and the same definition of the time bins. If the model produces a latent simulated concentration \(Y^{\mathrm{rep}}_{ij}\), classify it as BLQ when \(Y^{\mathrm{rep}}_{ij}<L_{ij}\), and as quantifiable otherwise. Calculate VPC summaries from the resulting simulated records using the same summary rule used on the observed records.

This matching requirement is important whether the plotted quantity is a BLQ fraction or a concentration quantile. For example, when the observed median is calculated only among quantified records, each simulated median must also be calculated only among simulated records that pass the same LLOQ rule. Otherwise, the two medians describe different populations. Conversely, if censored records are represented at the LLOQ in a display, simulated BLQ records must be represented that way too.

Two Useful Views of the Data

A quantified-only VPC omits BLQ records from concentration quantiles. In each bin, it summarizes only concentrations at or above their LLOQs, and simulations use the same restriction. The resulting curves answer a conditional question: among records that are quantifiable, do the observed concentrations follow the simulated pattern? This view can be useful for examining the shape and trend of the quantified concentrations, but by itself it does not show whether the model predicts the correct number of BLQ records.

A complementary display retains the censored records as statuses. Plot the proportion BLQ in each bin, with simulation-based prediction intervals, alongside the quantified-only concentration summaries. A separate BLQ-fraction panel is usually easier to interpret than encoding a censored concentration as if its exact value were known. A combined display may instead mark BLQ observations at the LLOQ, but the markers must be identified as censored, not measured values.

What Censoring Does to Quantiles

The effect of threshold substitution can be seen directly from the distribution. Assume, for this justification, one common LLOQ \(L\) and a continuous latent concentration \(Y\). If every value below \(L\) is displayed as \(L\), the displayed variable is \(C=\max(Y,L)\). For \(y\geq L\), its cumulative distribution is \(P(C\leq y)=P(Y\leq y)\). At \(L\), the display creates a point mass of size \(P(Y<L)\): distinct latent concentrations all appear at the same plotted value.

Thus, if at least half the latent values are below \(L\), a median of the threshold-substituted display is \(L\), even though the median latent concentration need not equal \(L\). More generally, the substituted quantiles can be pulled toward the threshold. This is why a plot that retains BLQ records should distinguish their status, and why a standard quantile calculated from substituted values needs cautious interpretation.

By contrast, the distribution of quantified values is conditional on \(Y\geq L\). For \(y\geq L\), its cumulative distribution is:

where \(F_Y\) is the latent concentration distribution function. The expression assumes a continuous distribution, so that the probability of a value exactly equal to \(L\) is zero. It makes clear that quantified-only quantiles characterize the upper, selected portion of the distribution. Changing the LLOQ changes that conditional distribution, even if the latent concentration distribution stays the same.

Reading the Joint Pattern

The BLQ fraction and the quantified-only concentration quantiles should be read together. A model can reproduce the median and upper quantiles among quantified records while predicting too few BLQ records. In that situation, its conditional fit above the threshold may look adequate even though its predictions for the full set of records do not. The reverse can also occur: the simulated BLQ fraction can agree with the observed fraction while the quantified concentrations show a systematic trend or level mismatch.

A useful diagnostic sequence is to check the data-processing assumptions first, then the plot:

• Confirm that observed and simulated records use the same LLOQ for each measurement. If limits vary, apply each record's own limit rather than silently treating all limits as identical.
• Check the bin counts and the number of quantified observations in each bin. Quantiles based on very few quantified records are unstable or may be undefined.
• Inspect the observed BLQ fraction against its simulation-based range across bins. A sustained departure can indicate that the model and observation process together predict the wrong frequency of values below the threshold.
• Inspect quantified-only quantiles separately. A trend in the median or other quantiles can reveal a mismatch among quantifiable concentrations even when BLQ fractions agree.
• Verify the actual recording convention. If BLQ values were stored using a placeholder such as zero or half the LLOQ, recover or preserve the BLQ status rather than treating the placeholder as an exact concentration.

When a bin has no quantified records, its quantified-only median is undefined; it is not zero. When it has only one or two quantified records, its quantiles provide little information about the distribution. Show the bin's BLQ count or fraction and indicate the limited quantified sample rather than drawing an apparently precise concentration trend.

Alternatives, Limitations, and a Common Pitfall

The clearest general approach is to show quantified-only concentration summaries together with a BLQ-fraction panel. It keeps the conditional concentration question separate from the censoring-frequency question. Omitting BLQ records from every displayed result is simpler, but it discards information about how often the assay reports values below its limit. Marking BLQ observations at the LLOQ can retain their positions in a combined plot, provided the symbols and caption make their censored status unmistakable. None of these displays recovers the unobserved exact values below the threshold.

A common mistake is to replace every BLQ observation with zero (or half the LLOQ), calculate ordinary quantiles, and compare them with simulations that use a different rule or no censoring rule at all. The reasoning is that a numerical placeholder lets all observations enter the same calculation. The correction is that the placeholder is not the measured concentration: it changes the distribution of plotted values and may create artificial ties or shift quantiles. Preserve BLQ status, apply the same reporting rule to simulations, and state whether each concentration summary is conditional on quantification.

A VPC also cannot separate every possible source of disagreement. A BLQ-frequency mismatch may reflect model predictions, an incorrectly specified or applied LLOQ, or errors in the data-processing and simulation workflow. The check evaluates the predictions after the stated observation process; it does not independently validate that process. When limits vary substantially across records, the pooled BLQ fraction may be hard to interpret, so inspect the limits and consider summaries that do not mix incomparable threshold settings.