Why Deletion and Substitution Change the Data
A below-quantification (BQL) result does not report an exact concentration. It reports that the concentration was below the assay’s lower limit of quantification (LLOQ). That distinction matters: a BQL record contains information about the concentration, but not the same information as a measured zero or a precise value chosen by an analyst.
Two shortcuts are common. Deletion removes BQL records from the analysis. Substitution assigns each BQL record a fixed value, such as zero, LLOQ/2, or the LLOQ, and analyzes that value as if it had been measured exactly. Either choice changes what the data say. Deletion discards the threshold information; substitution invents an exact concentration and can distort the likelihood and resulting parameter estimates.
This tutorial compares these shortcuts with retaining the censoring information in a likelihood analysis. The M3 method and the M4 method address BQL records within model-based approaches; here the emphasis is on understanding what deletion and substitution do to the information. The exact consequences depend on the pharmacokinetic model, the residual error model, the sampling design, and how many observations are BQL.
What Each Choice Does to the Likelihood
Let \(\theta\) denote the model parameters, and suppose the model specifies a distribution for a concentration \(Y\) conditional on the covariates and, where relevant, the individual random effects. For a quantified result \(y\), the likelihood contribution is the density \(f(y\mid\theta)\). For a BQL result with LLOQ \(L\), the observation is the event \(Y<L\), whose likelihood contribution is the probability \(P(Y<L\mid\theta)\). These are different kinds of evidence: an exact value and an interval event.
The contribution \(1\) for a deleted record means that it contributes no parameter-dependent information. Substitution instead treats the chosen value \(y^*\) as an exact measurement. Neither shortcut represents the event \(Y<L\). In a population model, the full likelihood may also need to account for individual random effects; the comparison above describes the observation-level contribution conditional on those effects.
The difference can also be seen from the parameter score, the derivative of the log-likelihood with respect to a parameter. A BQL event contributes the derivative of \(\log P(Y<L\mid\theta)\); deletion contributes zero, and substitution contributes the derivative of \(\log f(y^*\mid\theta)\). Thus these choices can pull estimates in different directions and give different information about precision.
Worked Example: Comparing Likelihood Information for One BQL Record
Consider a hypothetical additive normal residual model with known standard deviation \(0.5\) concentration units. Compare two candidate predictions, \(\mu=1\) and \(\mu=3\), for one observation with LLOQ \(L=2\). Under this model, \(Y\sim N(\mu,0.5^2)\). The record is BQL, so the observed information is \(Y<2\), not a measured concentration.
Retaining the BQL event. Standardizing the threshold gives \(z=(2-\mu)/0.5\). At \(\mu=1\), \(z=2\), so \(P(Y<2\mid\mu=1)=0.9772\). At \(\mu=3\), \(z=-2\), so \(P(Y<2\mid\mu=3)=0.02275\). The BQL event therefore favors the lower prediction: its likelihood contribution is about \(0.9772/0.02275=42.96\) times as large at \(\mu=1\).
Deleting the record. Both candidate predictions receive contribution \(1\). The record no longer distinguishes them.
Substituting \(L/2=1\). The analysis now treats \(y^*=1\) as an exact value. The normal density is \(f(1\mid\mu)=\{0.5\sqrt{2\pi}\}^{-1}\exp[-(1-\mu)^2/(2(0.5)^2)]\). It is \(0.7979\) at \(\mu=1\) and \(0.0002677\) at \(\mu=3\). This is not the BQL probability: it is the density at a fabricated exact value. It favors \(\mu=1\) much more strongly than the observed threshold event does.
The numerical comparison depends on the specified residual model, but the conceptual difference does not: an interval event, no observation, and an exact substituted value are distinct likelihood statements.
How Shortcuts Distort Summaries and Estimates
Deletion is particularly risky because BQL status is generally related to the unobserved concentration: low concentrations are more likely to fall below the LLOQ. The retained quantified observations are therefore selected by the assay threshold. A mean calculated after deletion describes the quantified subset, not necessarily the full set of concentrations that would have been observed without the threshold.
Substitution also changes the data, but its effect depends on the chosen value and the analysis. Replacing with zero or LLOQ/2 often pulls summaries downward when the true BQL concentrations are positive. Replacing with the LLOQ can pull them upward. There is no universal direction for bias in a fitted pharmacokinetic parameter: model structure, time of the BQL observations, and parameter correlations all matter.
Worked Example: A Pooled Mean Under Different Handling Choices
In a hypothetical set of six concentrations measured under comparable conditions, two quantified values are \(4\) and \(6\) concentration units, and four results are BQL at \(L=2\) concentration units. Calculate the naïve arithmetic mean under deletion and under three fixed-value substitutions. This simple calculation illustrates the data changes; it is not a recommendation to estimate a PK parameter by pooling observations.
Deletion. The mean of the two retained values is \((4+6)/2=5.000\) concentration units. It is the mean of the quantified subset, not the mean of all six underlying concentrations.
Substitution by zero. The resulting mean is \((4+6+4(0))/6=10/6=1.667\) concentration units.
Substitution by \(L/2=1\). The resulting mean is \((4+6+4(1))/6=14/6=2.333\) concentration units.
Substitution by \(L=2\). The resulting mean is \((4+6+4(2))/6=18/6=3.000\) concentration units. Each calculation answers a different, artificially constructed question. None recovers the unknown exact concentrations merely by arithmetic; the BQL records establish only that each concentration was below \(2\).
Worked Example: Substitution Can Flatten an Estimated Terminal Slope
Suppose a hypothetical noiseless concentration profile follows \(C(t)=8\exp(-kt)\), with time \(t\) in hours and concentration in concentration units. The four concentrations at \(t=0,1,2,3\) hours are \(8,4,2,1\), corresponding to \(k=\ln(2)=0.6931\) per hour. Let the LLOQ be \(3\) concentration units. The first two observations are quantified; the last two are BQL.
Deletion. Fitting a straight line to \(\ln C\) against time using only the quantified values gives slope \((\ln4-\ln8)/(1-0)=-\ln2=-0.6931\) per hour. The estimated elimination rate, the negative of that slope, is \(0.6931\) per hour. In this deliberately noiseless example, deletion happens to recover the rate from the two quantified points, but it still discards both late-time threshold constraints. That favorable numerical result is not a general property of deletion.
Substitution by \(L/2=1.5\). The analyzed values become \(8,4,1.5,1.5\). In the regression of \(\ln C\) on \(t\), the mean time is \(1.5\) hours, and the mean log concentration is \((\ln8+\ln4+2\ln1.5)/4=1.0692\). The slope is
The implied rate is \(0.6003\) per hour, about \(13.4\%\) below the true \(0.6931\) per hour. Here substitution flattens the estimated terminal slope because it turns two distinct late concentrations into the same invented value. Different profiles, thresholds, and substitutions can produce different effects.
Assumptions, Diagnostics, and Alternatives
Neither deletion nor fixed-value substitution is generally justified by the fact that a software package accepts the resulting data. Deletion may be less consequential when BQL observations are rare and contribute little to the parameter information of interest, but that must be assessed rather than assumed. It is not enough to say that the remaining observations are “complete”: the chance of being retained depends on whether a concentration crosses the quantification threshold.
Substitution requires treating the assigned value as if it were an exact measurement. That assumption is not supported by the BQL record. A chosen value can also be incompatible with the model’s scale. For example, taking logarithms of a substituted zero is undefined, while replacing zero with an arbitrary small positive value makes results depend on that extra choice. Substituting the LLOQ or a fraction of it does not solve the information problem; it changes the fabricated exact value.
Useful checks include tabulating BQL counts and proportions by time after dose, dose group, and other relevant covariates; checking whether BQLs cluster in a region that informs a parameter of interest; and refitting with alternative handling choices as a sensitivity analysis. Large changes in estimates or uncertainty across deletion, zero, and LLOQ-based substitutions are a warning that the shortcut materially affects conclusions. Similar results do not prove that the shortcuts are valid; they indicate only that those choices had limited impact on the estimates in that analysis.
| Approach | Information used for a BQL record | Main concern |
|---|---|---|
| Deletion | No parameter-dependent contribution | Removes threshold information and selects the quantified subset |
| Fixed substitution | Density as if the chosen value were measured exactly | Creates an artificial exact value; results depend on the choice |
| Censored likelihood | Probability of falling below the LLOQ under the model | Requires an appropriate model for the observation process and residual error |
A censored-likelihood approach retains the event information rather than discarding it or converting it to a point measurement. Its validity still depends on a suitable model and correct implementation; it is not immune to model misspecification or sparse information. The residual error distribution must be appropriate for the concentration data and thresholding process. The comparison of common residual error distributions is a separate modeling question.
Common Mistake and Key Takeaway
Wrong reasoning: “A BQL result is probably close to zero, so replacing it with zero or LLOQ/2 is equivalent to using the observation.” The correction is that the observation states an inequality, \(Y<L\), not a point value. Even if a selected substitute seems plausible, its exactness is an added assumption that changes the likelihood. Likewise, deleting the record is not neutral: it removes information and can leave a selected subset of observations.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- An assay has LLOQ \(L=1.2\) concentration units. What does a BQL result establish, and what does it not establish?
- Why does deletion contribute no information about \(\theta\) from the BQL record in a likelihood analysis?
- For a BQL record, why is substituting \(L/2\) not equivalent to using the probability of being below \(L\)?
- If BQLs occur mostly at late times, name one PK quantity whose estimate could be affected and explain why the direction of change is not guaranteed.
Solutions. (1) It establishes \(Y<1.2\) concentration units; it does not establish \(Y=0\), \(Y=0.6\), or any other exact value. (2) Once the record is deleted, its likelihood contribution is \(1\), which is constant in \(\theta\), so its log-likelihood contribution and score are zero. (3) The probability uses all model values below the threshold, \(P(Y<L\mid\theta)\); substitution instead uses the density at the single value \(L/2\), \(f(L/2\mid\theta)\). (4) The terminal elimination rate could be affected because late concentrations help constrain the terminal decline. The direction depends on the model, the observed profile, and the handling choice, so it cannot be inferred from the timing alone.