Modeling Concentrations Below Quantification Limits

Why Below-Quantification Results Need a Model

A concentration reported as below the lower limit of quantification (LLOQ) is not an exact measurement of zero, nor is it an ordinary missing value. It conveys an inequality: the concentration was not quantified, but it is understood to be below a known assay limit. Replacing that inequality with a single number discards information and can distort the fitted concentration-time curve. The distortion can matter particularly when below-quantification observations occur in a terminal profile and therefore affect estimated exposure.

The central modeling choice is whether to use the information that a result lies below the limit in the likelihood, or to assign a numerical value to it. A censored likelihood retains the inequality. Substitution methods instead treat an assigned value, such as zero or half the limit, as if it had been observed exactly. Neither a likelihood model nor a substitution can correct a wrongly recorded assay limit, a misspecified concentration model, or an inappropriate treatment of observations that were never collected.

Definition: Let $L$ be a known quantification limit. A below-quantification observation reports that the true concentration $C$ satisfies $0 \leq C < L$; it does not report an exact value of $C$. If concentrations are modeled on the log scale, the observation reports $Y=\log C < \log L$, provided $C>0$.

For a simple illustration, suppose the log concentration at a sampling time follows a normal distribution with model-predicted mean $\eta$ and residual standard deviation $\sigma$. Thus $Y \sim N(\eta,\sigma^2)$. An exact quantified concentration $c$ contributes its density at $y=\log c$; a below-quantification result contributes the probability that $Y$ falls below the log limit. This illustration assumes the assay limit is known, censoring is determined by that limit, and the stated normal distribution adequately describes residual variation at the sampling time.

Likelihood contributions: For an exact quantified concentration $c \geq L$, the lognormal density contribution is $$ f_C(c\mid\eta,\sigma)=\frac{1}{c\sigma}\phi\!\left(\frac{\log c-\eta}{\sigma}\right). $$ For a below-quantification observation, the contribution is $$ P(C<L\mid\eta,\sigma)=\Phi\!\left(\frac{\log L-\eta}{\sigma}\right). $$ Here $\phi$ and $\Phi$ are the standard normal density and cumulative distribution functions. The exact-observation density includes the $1/c$ Jacobian because the density is expressed on the concentration scale.

The contribution for a censored result is a probability, not the density evaluated at $L$, at zero, or at a substituted concentration. With several conditionally independent observations, the likelihood is the product of their exact-density and censoring-probability contributions. In a mixed-effects model, those contributions are combined conditional on an individual's random effects and then integrated over their distribution. That integration matters: treating observations from the same individual as independent when the model says they share random effects changes the likelihood.

How the Censored Contribution Works

The censored likelihood contribution changes with the predicted concentration. If the model predicts a concentration far below $L$, a below-limit observation is likely and contributes relatively little evidence against that prediction. If the model predicts a concentration far above $L$, the same observation is surprising and reduces the likelihood. It does not, by itself, establish the exact concentration or the direction of an individual's measurement error.

For a lognormal model, $\eta$ is the mean of log concentration and $\exp(\eta)$ is the median concentration at the sampling time. If the residual standard deviation $\sigma$ is constant, the censored contribution is $\Phi(z)$, where $z=(\log L-\eta)/\sigma$. Differentiating its log with respect to $\eta$ gives the censored observation's contribution to the score:

\frac{\partial}{\partial\eta}\log\Phi(z) = -\frac{\phi(z)}{\sigma\Phi(z)}.

The score is negative: a below-limit observation tends to pull the fitted log concentration downward. Its size depends on how incompatible the predicted value is with being below the limit. This is a useful way to see why censoring is not equivalent to inserting a fixed value. The likelihood contribution changes as the model prediction changes; a substituted value contributes as though that value itself had been measured.

Worked Example: How Informative Is One Below-Limit Result?

Suppose the quantification limit is $L=2.00$ concentration units and the residual standard deviation on the log scale is $\sigma=0.400$. Compare two hypothetical model predictions: a median concentration of $1.50$ units and a median of $4.00$ units. Assume the lognormal residual model stated above.

For median $1.50$, $\eta=\log(1.50)$. The probability of a below-limit result is $$ \Phi\!\left(\frac{\log(2.00)-\log(1.50)}{0.400}\right) = \Phi(0.7192) \approx 0.764. $$

For median $4.00$, $\eta=\log(4.00)$, so the probability is $$ \Phi\!\left(\frac{\log(2.00)-\log(4.00)}{0.400}\right) = \Phi(-1.733) \approx 0.0416. $$

Thus this one below-limit result is much more compatible with the first prediction than the second: its likelihood contribution is about $0.764/0.0416=18.4$ times as large under the first prediction. The result does not prove that the concentration was $1.50$ units. It indicates how the observation weighs the two predictions under the stated model.

Substitution and Its Consequences

Common substitution rules replace every below-quantification result with zero, $L/2$, or $L$. These choices are easy to implement, but none is a general consequence of the observation. Substituting $L$ treats a censored concentration as exactly at the limit; substituting zero or $L/2$ also assigns an exact value that the assay did not provide. On the log scale, substituting zero is not even directly defined without an additional convention.

Substitution can affect a fitted model in more than one way. It can alter residuals and parameter estimates, affect the estimated decline in the terminal phase, and thereby alter model-derived AUC or other exposure measures. It can also affect uncertainty because substituted values are treated as exact, although the actual observation only supplies an inequality. The size and direction of these effects depend on the sampling times, the number and pattern of censored results, and the model.

Worked Example: A Small Censored-Normal Estimate

Consider five hypothetical log concentrations from one sampling-time model. Three quantified observations are $0.200$, $0.400$, and $0.600$. Two further observations are below the limit, whose log value is $0$. Assume independent normal residuals with known $\sigma=0.500$, a common unknown log-scale mean $\eta$, and no random effects in this illustrative calculation. These assumptions make the likelihood calculation transparent; they are not a claim that such a model is adequate for every study.

Ignoring constants that do not depend on $\eta$, the log likelihood is $$ \ell(\eta) = -\frac{1}{2(0.500)^2}\sum_{i=1}^{3}(y_i-\eta)^2 + 2\log\Phi\!\left(\frac{0-\eta}{0.500}\right). $$

The score equation is $$ \frac{1.200-3\eta}{0.250} - \frac{2\phi(-\eta/0.500)}{0.500\Phi(-\eta/0.500)} =0. $$ The first term comes from the three exact log concentrations; the second is the contribution from the two censored observations. At $\eta=0.093$, the exact-data term is $(1.200-3(0.093))/0.250=3.684$. The censored term is approximately $-2(0.920/0.500)=-3.680$, making the score approximately $0.004$. Numerical maximization therefore gives $\hat{\eta}\approx 0.093$.

For comparison, replacing each censored concentration by half the limit gives log values $\log(0.500)=-0.6931$. The mean of the five substituted log values is $$ \frac{0.200+0.400+0.600+2(-0.6931)}{5} = -0.0373. $$ This differs from the censored-likelihood estimate because the substitution procedure treats both censored results as exact log concentrations of $-0.6931$, whereas the likelihood uses only the information that each is below zero on the log scale. The comparison illustrates the methods; it does not establish that the censored estimate is reliable if the normal model or its independence assumption is wrong.

From Concentration Fits to Bioequivalence

In model-based bioequivalence, fitted concentration-time profiles can be used to derive exposure metrics such as AUC and $C_{\max}$, and formulation contrasts can then be summarized on the appropriate scale. A censored result can affect these outputs through the fitted curve even though it is not itself an exposure metric. The impact is often most consequential when many below-limit observations constrain the late profile or when few quantified samples anchor that portion of the curve.

A bioequivalence decision is not determined by the censored likelihood contribution alone. It depends on the estimated formulation contrast, its uncertainty, the chosen exposure metric, and the prespecified equivalence limits. A change in censoring treatment can change the estimate, its standard error, or both. Consequently, conclusions should be reported in context and tied to the method and assumptions used.

Worked Example: A Censoring Choice Changes an Equivalence Decision

Suppose two hypothetical analyses use the same concentration-time model and dataset but handle below-limit observations differently. The censored-likelihood analysis estimates a model-derived AUC geometric mean ratio of $0.940$, with standard error $0.0800$ for its log ratio. A half-limit substitution analysis estimates a ratio of $0.910$, with log-ratio standard error $0.100$. These are hypothetical fitted outputs, supplied here to illustrate the decision calculation; they are not raw data from which the fits can be reproduced.

For illustration, use a 90% confidence interval with the normal critical value $1.645$, and equivalence limits of $0.800$ and $1.250$. For the censored-likelihood analysis, the log ratio is $\log(0.940)=-0.06188$, so the interval on the log scale is $$ -0.06188 \mathbin{\pm} 1.645(0.0800) = (-0.1935,\ 0.06972). $$ Exponentiating the endpoints gives a ratio interval of approximately $(0.824,\ 1.072)$. This interval lies within $0.800$ to $1.250$, so it meets the illustrative interval criterion.

For the substitution analysis, $\log(0.910)=-0.09431$, and $$ -0.09431 \mathbin{\pm} 1.645(0.100) = (-0.2588,\ 0.07019). $$ Exponentiating gives approximately $(0.772,\ 1.073)$. Its lower endpoint is below $0.800$, so this interval does not meet the same criterion. The two illustrative analyses therefore lead to different interval-based conclusions. This difference warns that censoring treatment can matter; it does not show which result is correct without assessing the model and assumptions.

Assumptions, Diagnostics, and Alternatives

A censored likelihood is appropriate only when its assumptions describe how the data arose. Check that the assay limit is correctly recorded for the relevant samples, that a below-limit flag represents censoring rather than an uncollected or otherwise missing measurement, and that the concentration model is plausible over the relevant times and formulations. The assumed residual distribution and, where applicable, the random-effects structure must also be suitable. A likelihood calculation can be mathematically correct while its assumptions are scientifically implausible.

Useful diagnostics include plotting quantified observations and below-limit indicators against time and formulation, comparing the observed frequency of below-limit results with model-predicted probabilities, and checking whether model predictions place censored observations in plausible regions. Examine whether censoring is concentrated in a particular formulation, period, or phase of the profile. If nearly all observations in a region are censored, the data may provide limited information about the precise concentrations there: a model can predict below the limit without the observations identifying how far below it.

The principal alternatives have different information costs. Censored likelihood uses the inequality directly, conditional on the specified model. Substitution is simple but turns inequalities into assumed exact values. Omitting below-limit observations discards their inequality information and may leave an unrepresentative set of quantified samples. Treating them as zero imposes a particularly strong value assumption and is not the same as modeling censoring. These alternatives should not be described as interchangeable implementations of one analysis.

Approach	What it uses	Main concern
Censored likelihood	The probability that the modeled concentration is below the known limit	Depends on the assumed concentration and residual model
Substitution	An assigned value such as zero, $L/2$, or $L$	Treats an unobserved exact value as if it were measured
Omission	Only quantified observations	Discards the inequality information and can change which observations inform the fit

A frequent mistake is to reason, “Below the limit means zero, so I will enter zero and fit the model.” The correction is that the observation establishes $C<L$, not $C=0$; use the corresponding probability contribution when the model and censoring mechanism support it. Another mistake is to assume that using a censored likelihood guarantees unbiased exposure estimates. It does not: the likelihood still relies on correct limits, an adequate model, and appropriate handling of dependence and missingness.

The method also has limits. If the assay limit is uncertain or varies by sample, that variation must be represented rather than silently replaced by one common threshold. If the recorded category denotes an interval other than $0\leq C<L$, its actual bounds should determine the probability contribution. If observations are missing for reasons other than the assay limit, they are not automatically censored observations; the distinction addressed in the earlier material on missing concentrations and dropout remains important. When many results are censored, the data may not support precise estimates of the lower profile, and conclusions can become sensitive to modeling assumptions.

Key takeaway: A below-quantification result is an inequality. A censored likelihood uses the probability of that inequality under the model; substitution replaces it with an assumed exact value. Because these choices can affect model-derived exposure and bioequivalence intervals, report the censoring rule, assess its assumptions, and interpret conclusions accordingly.

Check Your Understanding

Answer each question, then compare with the solutions below.

A sample has a known limit $L$, and its result is below that limit. What information does this contribute, and why is it not an exact zero?
Under a lognormal model, write the likelihood contribution for a below-limit result in terms of $\eta$, $\sigma$, and $L$.
Why can replacing a below-limit result by $L/2$ change a fitted concentration-time profile?
An analysis reports a 90% ratio interval of $(0.79,1.10)$ against equivalence limits of $0.80$ and $1.25$. Does the interval meet the criterion?
Name one diagnostic for a censored-likelihood analysis and one reason its conclusions could still be unreliable.

Solutions. (1) It contributes the information $0\leq C<L$; it does not establish $C=0$. (2) The contribution is $\Phi((\log L-\eta)/\sigma)$. (3) Half-limit substitution treats the assigned concentration as exact, changing residuals and potentially model parameters and derived exposure. (4) No. Although $1.10<1.25$, the lower endpoint $0.79$ is below $0.80$, so the interval is not entirely within the limits. (5) One diagnostic is to compare observed below-limit frequencies with model-predicted probabilities across time or formulation. The result may still be unreliable if, for example, the residual distribution or concentration-time model is inadequate.