Deriving the Censored Likelihood Contribution

Why the Observation Type Determines the Likelihood

A concentration reported as an exact measured value and a concentration reported as below the lower limit of quantification (LLOQ) do not provide the same information. An exact value identifies a location on the concentration scale. A BQL record identifies an event: the unobserved concentration lies at or below the assay limit. The likelihood must represent the information actually observed, rather than treating the BQL record as though the concentration had been measured at the limit.

This tutorial derives the two contributions using a continuous observation distribution. The derivation is conditional on the model parameters and any quantities treated as fixed when specifying that distribution. It does not integrate over random effects or derive a hierarchical likelihood.

Definition: Let \(Y\) be a continuous concentration with density \(f(y\mid\theta)\) and cumulative distribution function \(F(y\mid\theta)=P(Y\leq y\mid\theta)\), where \(\theta\) denotes the parameters of the observation distribution. An exact observation \(Y=y\) contributes \(f(y\mid\theta)\) to the likelihood. A left-censored observation known only to satisfy \(Y\leq L\), for a known LLOQ \(L\), contributes \(F(L\mid\theta)=P(Y\leq L\mid\theta)\).

A density is not the probability of observing one exact value: for a continuous random variable, \(P(Y=y\mid\theta)=0\). Rather, a density describes probability per unit of the measurement scale near \(y\). The BQL event, in contrast, has a probability that can be positive, so its contribution is a cumulative probability.

Deriving the Exact-Value Contribution

For a small positive interval width \(\Delta y\), the probability that a continuous observation falls in a narrow interval starting at \(y\) is

P(y\leq Y<y+\Delta y\mid\theta) =\int_y^{y+\Delta y} f(u\mid\theta)\,du.

Dividing by the interval width gives the average density over that interval. If \(f(\cdot\mid\theta)\) is continuous at \(y\), this average converges to \(f(y\mid\theta)\) as the interval shrinks. This gives the density interpretation of an exact observation. The standard likelihood contribution for a continuous exact observation is therefore \(f(y\mid\theta)\); the continuity condition justifies this small-interval argument and should not be silently assumed for an arbitrary density at a discontinuity.

\lim_{\Delta y\to 0^+} \frac{P(y\leq Y<y+\Delta y\mid\theta)}{\Delta y} = \lim_{\Delta y\to 0^+} \frac{1}{\Delta y}\int_y^{y+\Delta y} f(u\mid\theta)\,du =f(y\mid\theta), \quad\text{if } f(\cdot\mid\theta)\text{ is continuous at }y.

For example, if the observation model is normal with mean \(\mu\) and standard deviation \(\sigma>0\), then

f(y\mid\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left[-\frac{1}{2}\left(\frac{y-\mu}{\sigma}\right)^2\right].

The density is expressed in inverse units of concentration. It can be greater than 1; that is not a probability error. Its numerical value changes with the measurement units, while probabilities for corresponding intervals do not.

Worked Example: An Exact Concentration Under a Normal Model

Suppose a hypothetical exact concentration is \(y=1.20\) mg/L. For this calculation, take the specified observation distribution to be normal with \(\mu=1.00\) mg/L and \(\sigma=0.250\) mg/L. The normal density is continuous, so the small-interval justification applies.

Standardize the observation: \(z=(1.20-1.00)/0.250=0.800\). The exact-value contribution is

f(1.20\mid 1.00,0.250) = \frac{1}{0.250\sqrt{2\pi}}\exp\!\left(-\frac{0.800^2}{2}\right) = \frac{0.289692}{0.250} \approx 1.159\;(\mathrm{mg/L})^{-1}.

Thus the likelihood factor for this exact record is approximately \(1.159\) per mg/L. It is a density, not a probability of 1.159. For instance, the probability of falling in a sufficiently narrow interval around 1.20 mg/L is approximately this density multiplied by the interval width in mg/L.

Deriving the Left-Censored Contribution

A BQL result at LLOQ \(L\) tells us that the latent concentration fell in the set of values at or below \(L\). The probability of that observed event under the model is the integral of the density over the entire compatible range. By the definition of the cumulative distribution function, this integral is \(F(L\mid\theta)\).

P(Y\leq L\mid\theta) = \int_{-\infty}^{L} f(u\mid\theta)\,du = F(L\mid\theta).

This is the likelihood contribution because the observed data are the event \(Y\leq L\), not a specific value of \(Y\). For a continuous distribution, \(P(Y=L\mid\theta)=0\), so the conventions \(Y<L\) and \(Y\leq L\) give the same probability. The cumulative probability includes all possible concentrations consistent with the BQL record; evaluating the density at \(L\) would instead describe local density near that single boundary value.

Worked Example: A BQL Record at 0.50 mg/L

Consider a hypothetical BQL record with \(L=0.500\) mg/L. Suppose its specified concentration distribution is normal with \(\mu=0.800\) mg/L and \(\sigma=0.400\) mg/L. These parameters define a continuous distribution, and the record is assumed to mean that the concentration is at or below the stated LLOQ.

Standardizing the LLOQ gives \(z=(0.500-0.800)/0.400=-0.750\). Therefore, the appropriate likelihood contribution is

P(Y\leq 0.500\mid\mu=0.800,\sigma=0.400) = \Phi(-0.750) \approx 0.2266,

where \(\Phi\) is the standard normal cumulative distribution function. The BQL record contributes a probability of approximately 0.2266. For comparison, the density at the LLOQ under this same distribution is

f(0.500\mid 0.800,0.400) = \frac{\phi(-0.750)}{0.400} = \frac{0.301137}{0.400} \approx 0.7528\;(\mathrm{mg/L})^{-1}.

The values differ because they answer different questions. \(0.2266\) is the probability of the full event \(Y\leq0.500\) mg/L; \(0.7528\) per mg/L is the density at the boundary. The latter is not the contribution for the BQL record.

Combining Records in a Likelihood

If observations are conditionally independent under the specified model, their likelihood contributions multiply. An exact observation contributes a density factor, while a BQL observation contributes a cumulative-probability factor. Taking logarithms converts that product into a sum, which is often more convenient for estimation. Conditional independence is an assumption here: if observations are dependent, the product of marginal contributions is not generally the joint likelihood.

Worked Example: Two Exact Values and One BQL Record

Suppose three hypothetical records are conditionally independent under their specified distributions. Two are exact concentrations, \(y_1=1.00\) mg/L and \(y_2=1.20\) mg/L, each modeled as normal with \(\mu=1.00\) mg/L and \(\sigma=0.400\) mg/L. The third record is BQL at \(L=0.500\) mg/L, modeled as normal with \(\mu=0.800\) mg/L and \(\sigma=0.400\) mg/L. All exact-value densities are continuous at the observed values, and the BQL event is defined as \(Y_3\leq L\).

For the first exact value, \(z_1=(1.00-1.00)/0.400=0\), so its density factor is \(\phi(0)/0.400=0.398942/0.400\approx0.9974\;(\mathrm{mg/L})^{-1}\). For the second, \(z_2=(1.20-1.00)/0.400=0.500\), giving \(\phi(0.500)/0.400=0.352065/0.400\approx0.8802\;(\mathrm{mg/L})^{-1}\).

For the BQL record, \(z_L=(0.500-0.800)/0.400=-0.750\), so its contribution is \(\Phi(-0.750)\approx0.2266\), a probability. Multiplying the factors gives the likelihood for these three records:

L = \left(\frac{0.398942}{0.400}\right) \left(\frac{0.352065}{0.400}\right) \Phi(-0.750) \approx (0.9974)(0.8802)(0.2266) \approx 0.1989\;(\mathrm{mg/L})^{-2}.

The units are inverse concentration squared because there are two density factors and one dimensionless probability factor. Using the unrounded factors, the log-likelihood is approximately \(\log(0.997356)+\log(0.880163)+\log(0.226627)=-0.00265-0.1276-1.4844=-1.615\). The product and log-likelihood describe the same likelihood on different scales.

If the BQL record were incorrectly treated as an exact observation at 0.500 mg/L, its factor would be the density \(0.7528\;(\mathrm{mg/L})^{-1}\), not the event probability \(0.2266\). This would encode a different observation and produce a different likelihood, with the wrong dimensional structure for the recorded data.

Assumptions, Checks, and Common Failure Modes

The contributions above rely on a clear observation model and an accurate description of what was recorded. The exact-value formula assumes a continuous distribution with a density for the measured concentration. The censored formula assumes the record means precisely that the concentration is at or below the known LLOQ, and uses the corresponding CDF of that same modeled variable. For a product likelihood across records, conditional independence must also be defensible.

Several checks help catch implementation errors before interpreting estimates:

Check the event and boundary. A left-censored record at \(L\) must use \(P(Y\leq L)\). Using an upper-tail probability reverses the event. Confirm that each record's LLOQ is in the same concentration units and scale as the modeled observation.
Check contribution types. Exact continuous values use densities; censoring events use probabilities. A density can exceed 1, but a probability must lie between 0 and 1.
Check the data encoding. A BQL flag must not be converted into an invented exact concentration at the LLOQ or another substituted value when calculating this censored likelihood. The record provides an inequality, not that substituted measurement.
Check numerical calculations. Compare a computed CDF with the expected direction: increasing \(L\) cannot decrease \(F(L\mid\theta)\). Extremely small probabilities can underflow if computed directly, so implementations commonly evaluate log-CDF values using numerically stable routines.

A useful derivation check is to consider what information the record conveys. Integrating the density over the full event \(Y\leq L\) retains all concentrations compatible with the assay report. Substituting the density at \(L\) retains only local information at the boundary, while deleting the record discards the event altogether. These alternatives therefore do not represent the same likelihood. They may be used as different data-handling approaches, but they should not be described as the likelihood contribution derived here.

Key takeaway: An exact continuous concentration contributes a density evaluated at the observed value. A BQL observation contributes the cumulative probability up to the LLOQ. The distinction follows from the observed information: a point measurement versus a censored event.

Check Your Understanding

Answer each question, then compare with the solutions below.

An exact concentration \(y=0.70\) mg/L is observed under a continuous model. What quantity enters the likelihood, and is it necessarily a probability?
A record is BQL at \(L=0.25\) mg/L. Write its likelihood contribution using the model CDF.
For a continuous distribution, why does a BQL record not contribute \(f(L\mid\theta)\)?
Two exact records and one BQL record are conditionally independent. How do their contributions combine, and how many density factors are present?

Solutions. (1) The contribution is \(f(0.70\mid\theta)\), a density, not necessarily a probability; it has inverse-concentration units. (2) The contribution is \(F(0.25\mid\theta)=P(Y\leq0.25\mid\theta)\). (3) The record identifies the event that \(Y\) lies at or below the limit, whose probability is the integral of the density over that range; the density at \(L\) describes only the local concentration of probability near the boundary. (4) Multiply the two exact-value densities by the BQL event probability, \(f(y_1\mid\theta)f(y_2\mid\theta)F(L\mid\theta)\). There are two density factors and one dimensionless probability factor.