Likelihoods for Censored Biomarker Measurements

Why a Below-Limit Result Is Not a Measured Value

A biomarker assay may report a value only when the signal is sufficiently large to quantify reliably. If the underlying assay result falls below the lower limit of quantification (LLOQ), the record may say “< 5 ng/mL” rather than report a number. This is informative: the result lies in a known range. But it is not the same as observing exactly 5 ng/mL, nor is it equivalent to having no information.

In QSP calibration, the mechanistic model predicts a biomarker level at each sampling time and under each parameter setting. An observation error model describes how the assay result varies around that prediction. For a censored record, the likelihood contribution is the probability that this error model would produce a value below the assay limit. The prediction, error scale, limit, and error distribution all affect that probability.

Definition: A measurement is left-censored at limit \(L\) when the recorded information is that its underlying assay value \(Y\) lies below \(L\), rather than that \(Y\) equals a reported numerical value. If \(Y\) has density \(p(y\mid\theta)\) under model parameters \(\theta\), its likelihood contribution is \(P(Y<L\mid\theta)\), the cumulative probability below \(L\).

Likelihood Contribution From the Error Distribution

Let \(f_i(\theta)\) denote the QSP model prediction for the biomarker concentration at observation \(i\), given parameters \(\theta\). Let \(Y_i\) be the assay result that would be obtained before censoring. For an exact observation \(y_i\), the likelihood contribution is its probability density \(p(y_i\mid\theta)\). For a result recorded only as \(Y_i<L_i\), the contribution is instead the probability of the entire compatible range:

L_i(\theta)=P(Y_i<L_i\mid\theta) =\int_{-\infty}^{L_i}p(y\mid\theta)\,dy =F(L_i\mid\theta),

where \(F\) is the cumulative distribution function (CDF) implied by the observation error model. For a continuous distribution, using \(Y_i\leq L_i\) instead gives the same probability. The density at \(L_i\) is not the relevant quantity: a censored result identifies a region of possible assay values, not a point.

For example, with additive normal error, suppose \(Y_i\mid\theta\sim N(f_i(\theta),\sigma^2)\). Here \(\sigma\) is the residual standard deviation in concentration units. Standardizing the censoring boundary gives:

P(Y_i<L_i\mid\theta) =\Phi\!\left(\frac{L_i-f_i(\theta)}{\sigma}\right),

where \(\Phi\) is the standard normal CDF. With a lognormal observation model, suppose \(\log Y_i\mid\theta\sim N(\log f_i(\theta),\sigma^2)\), for positive \(Y_i\) and \(f_i(\theta)\). In this parameterization, \(f_i(\theta)\) is the median of \(Y_i\), not its arithmetic mean. The contribution is:

P(Y_i<L_i\mid\theta) =\Phi\!\left(\frac{\log L_i-\log f_i(\theta)}{\sigma}\right).

These formulas are examples, not universal choices. For another error distribution, use its own CDF. A symmetric additive error model can assign probability to negative concentrations; if that is scientifically implausible, an error model with positive support may be more suitable. The error model should be selected for the measurement process, not just because a formula is convenient.

Assumptions and the Combined Likelihood

The formulas require a clearly defined event and a model for the underlying assay result. The reported censoring threshold must be known for each sample, and “below the limit” must genuinely mean the assay result fell below that threshold. The observation error distribution and its dependence on the model prediction must be specified. If measurements are conditionally independent given the model parameters and any modeled subject effects, their likelihood contributions multiply. Exact and censored measurements can therefore appear in the same product:

L(\theta)= \prod_{i\in E}p(y_i\mid\theta) \prod_{i\in C}F(L_i\mid\theta),

Here \(E\) indexes exact observations and \(C\) indexes left-censored observations. This product is a likelihood, not necessarily a probability density for the full data on one common scale: exact observations contribute densities, while censored observations contribute probabilities. A density carries units, such as inverse concentration; a probability is dimensionless. The product is nevertheless the correct likelihood for the mixed recorded data.

Conditional independence is an assumption, not an automatic property of repeated samples. If observations from the same subject share an error component or subject-specific effect, that dependence must be represented in the joint observation model. Similarly, if the chance of a sample being censored depends on an unmodeled process, the simple CDF contribution may not describe the full observation mechanism.

Worked Example: A Censored Result Under Additive Normal Error

Suppose a hypothetical assay result follows \(Y\mid\theta\sim N(f(\theta),\sigma^2)\), with predicted concentration \(f(\theta)=8.0\) ng/mL and residual standard deviation \(\sigma=2.0\) ng/mL. The assay reports only “< 5.0 ng/mL” when the result is below \(L=5.0\) ng/mL. Find the likelihood contribution at these parameter values.

The appropriate contribution is the probability of a result below 5.0, not the density at 5.0:

L(\theta)=\Phi\!\left(\frac{L-f(\theta)}{\sigma}\right) =\Phi\!\left(\frac{5.0-8.0}{2.0}\right) =\Phi(-1.50) \approx 0.0668.

Thus, under this hypothetical model and parameter setting, there is about a 6.68% probability of an assay result below 5.0 ng/mL. That probability is the likelihood contribution for the recorded censored result. It is not a statement that the true concentration is 5.0 ng/mL.

Worked Example: A Censored Result Under Lognormal Error

For a second hypothetical assay, suppose \(\log Y\mid\theta\sim N(\log f(\theta),\sigma^2)\), with \(f(\theta)=10.0\) ng/mL, \(\sigma=0.300\), and a lower quantification limit \(L=6.0\) ng/mL. The recorded value is “< 6.0 ng/mL.” Calculate its likelihood contribution.

On the log scale, the standardized limit is:

z=\frac{\log L-\log f(\theta)}{\sigma} =\frac{\log(6.0)-\log(10.0)}{0.300} =\frac{\log(0.6)}{0.300} \approx -1.703.

Therefore:

L(\theta)=\Phi(-1.703)\approx 0.0443.

The censored result has a probability of about 4.43% under this hypothetical prediction and error model. The lognormal standard deviation is on the log scale and is dimensionless; \(f(\theta)\) and \(L\) are concentrations in ng/mL. Using an additive normal formula here would impose a different error model and generally produce a different contribution.

Why the CDF Appears: A Short Derivation

The derivation follows directly from what was recorded. Let \(Y\) be the underlying assay result with density \(p(y\mid\theta)\). The observed record is not \(Y=y\) for some known \(y\); it is the event \(Y<L\). The probability of that event is the integral of the density over all compatible values. For a continuous distribution, integrating the density up to the limit is precisely the CDF at that limit:

P(Y<L\mid\theta) =\int_{-\infty}^{L}p(y\mid\theta)\,dy =F(L\mid\theta).

This also explains why replacing the censored value with the limit is not a likelihood derivation. A substitution treats \(Y=L\) as an exact observation and uses \(p(L\mid\theta)\); the actual record gives an event and uses \(P(Y<L\mid\theta)\). The two quantities have different meanings and, in general, different units.

Worked Example: Combining an Exact and a Censored Result

Consider two hypothetical independent assay observations under the same additive normal model with prediction \(f(\theta)=8.0\) ng/mL and \(\sigma=2.0\) ng/mL. The first is exactly measured as \(y_1=9.0\) ng/mL. The second is reported as \(Y_2<5.0\) ng/mL. Calculate their joint likelihood contribution at these parameter values.

For the exact observation, the normal density is:

p(9.0\mid\theta) =\frac{1}{2.0\sqrt{2\pi}} \exp\!\left[-\frac{(9.0-8.0)^2}{2(2.0)^2}\right] \approx 0.1760\;(\text{ng/mL})^{-1}.

For the censored observation, the CDF contribution is:

P(Y_2<5.0\mid\theta) =\Phi\!\left(\frac{5.0-8.0}{2.0}\right) =\Phi(-1.50) \approx 0.0668.

By conditional independence, multiply the two contributions:

L(\theta)=0.1760\times0.0668 \approx 0.0118\;(\text{ng/mL})^{-1}.

The result is a joint likelihood contribution for one exact and one censored record. Its units come from the density for the exact observation; the censored observation contributes a dimensionless probability.

Diagnostics, Alternatives, and Limitations

A likelihood can be calculated correctly while the measurement model is still unsuitable. Check that the limit is the assay’s relevant threshold for each sample, that units and log transformations match, and that the assumed error distribution is defensible for the biomarker and assay. For fitted parameter values, compare the number and pattern of predicted censored results with what was observed. A model that predicts almost no results below the limits when many were censored, or predicts many when few were censored, may be inconsistent with the data or the measurement assumptions. Examine exact observations on the appropriate scale as well; a poor fit among uncensored values is not repaired by the censored likelihood.

For several possible ways to handle the same records, the differences are consequential:

Approach	What it uses	Main concern
CDF likelihood	The probability that the assay result lies below its limit	Requires a suitable error model and a correctly specified limit
Substitute the limit	An artificial exact value at the threshold	Confuses an event with a point observation and can distort calibration
Discard the record	No contribution from the censored sample	Throws away the information that the result was below the limit

Other censoring patterns require corresponding event probabilities. For an interval-censored result known only to lie between \(A\) and \(B\), the contribution is \(F(B\mid\theta)-F(A\mid\theta)\). For a right-censored result known to exceed \(R\), it is \(1-F(R\mid\theta)\). These forms follow the same principle: use the probability of the recorded range. If censoring depends on other unmodeled variables, or the threshold itself is uncertain, the simple contributions above may need to be extended to model those features.

Common Mistake

Wrong reasoning: “The assay says the biomarker is below 5 ng/mL, so I will enter 5 as the observed concentration.” This treats the limit as an exact measurement even though the actual assay result could be anywhere below it. It also replaces a probability with a density, which is not the same likelihood contribution.

Correction: Write down the event actually reported, such as \(Y<5\) ng/mL, then calculate its probability under the specified observation model. For additive normal error, use \(\Phi((5-f(\theta))/\sigma)\); for another error model, use its CDF at the limit. Use the limit as an exact value only if the assay genuinely measured that exact value.

Key takeaway: A below-quantification-limit record contributes the probability of the censored region under the observation error model. In a left-censored record this is the CDF evaluated at the limit, not the density evaluated there and not an exact observation at the limit.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Under additive normal error, \(Y\mid\theta\sim N(7, 2^2)\). What is the likelihood contribution for a record \(Y<5\)?
Under the lognormal parameterization \(\log Y\mid\theta\sim N(\log f, \sigma^2)\), write the contribution for \(Y<L\).
Why is substituting the LLOQ for a censored value not equivalent to using its censoring likelihood?
For a record known only to fall between \(A\) and \(B\), state its likelihood contribution in terms of the CDF.

Solutions. (1) The standardized boundary is \((5-7)/2=-1\), so the contribution is \(\Phi(-1)\approx0.1587\). (2) The contribution is \(\Phi((\log L-\log f)/\sigma)\), since the event is \(Y<L\), equivalently \(\log Y<\log L\). (3) Substitution treats the limit as an exact value and uses a density there; censoring records an event over a range and contributes its probability, obtained by integrating the density. (4) The contribution is \(P(A<Y<B\mid\theta)=F(B\mid\theta)-F(A\mid\theta)\) for a continuous distribution.