Lognormal Observation Models With BQL Data

Why the Lognormal Model Changes the BQL Contribution

A lognormal observation model is useful when concentrations are positive and residual variation is naturally described as multiplicative. With below-quantification-limit (BQL) data, however, the likelihood contribution for a censored record is not a density evaluated at the LLOQ. It is the probability that the unobserved concentration lies below that limit. The lognormal model makes this probability a normal tail area after the LLOQ is transformed to the log scale.

The calculations below are conditional on an individual's model prediction. In an NLME model, that prediction depends on the population parameters and the individual's random effects. The residual observation model specifies how a measured concentration varies around that prediction.

Definition: Let $Y_{ij}$ be the concentration for individual $i$ at record $j$, and let $f_{ij}=f_{ij}(\theta,\eta_i)>0$ be its model-predicted concentration. A lognormal observation model is $$ \log Y_{ij}=\log f_{ij}+\epsilon_{ij}, \qquad \epsilon_{ij}\sim N(0,\sigma^2), \qquad \sigma>0. $$ Here $\log$ is the natural logarithm. Under this parameterization, $f_{ij}$ is the conditional median of $Y_{ij}$, not its conditional arithmetic mean. The conditional median absolute deviation is not specified by $\sigma$; rather, $\sigma$ is the standard deviation of log concentration.

Assume residual errors are conditionally independent across records given the individual's random effects and model parameters. Assume each BQL record means the true measured concentration is below a known, positive LLOQ $L_{ij}$, and that the censoring threshold is determined by the assay rather than by the unobserved concentration. A lognormal model also requires positive concentrations: it does not assign probability to zero or negative values. Since this distribution is continuous, the distinction between $Y_{ij}<L_{ij}$ and $Y_{ij}\leq L_{ij}$ has probability zero.

Exact Concentrations: The Density and Its Jacobian

For a quantified record with observed concentration $y_{ij}>0$, the likelihood contribution is a density on the concentration scale. To obtain it, set $Z_{ij}=\log Y_{ij}$. The normal density of $Z_{ij}$ is evaluated at $\log y_{ij}$, and converting from log concentration back to concentration requires the change-of-variable factor $\left|d\log y_{ij}/dy_{ij}\right|=1/y_{ij}$.

f_Y(y_{ij}\mid f_{ij},\sigma) = \frac{1}{y_{ij}\sigma\sqrt{2\pi}} \exp\left\{-\frac{1}{2} \left(\frac{\log y_{ij}-\log f_{ij}}{\sigma}\right)^2\right\}, \qquad y_{ij}>0.

The factor $1/y_{ij}$ is essential when the likelihood is written for the observed concentration. Equivalently, one may calculate a normal density for $\log y_{ij}$, but the likelihood for the original concentration then includes the Jacobian. Omitting it changes the likelihood being evaluated.

Worked Example: Likelihood Contribution of a Quantified Concentration

Suppose a hypothetical record has observed concentration $y=0.900$ mg/L, model prediction $f=0.800$ mg/L, and log-scale residual standard deviation $\sigma=0.250$. The assumptions are satisfied for this calculation because both concentrations are positive and the stated model treats $f$ as the conditional median.

First standardize the log concentration:

z_y=\frac{\log(0.900)-\log(0.800)}{0.250} =\frac{\log(1.125)}{0.250} =0.4711.

Then substitute into the concentration-scale density:

f_Y(0.900\mid0.800,0.250) = \frac{1}{(0.900)(0.250)\sqrt{2\pi}} \exp\left(-\frac{0.4711^2}{2}\right) \approx 1.587\;(\mathrm{mg/L})^{-1}.

This is a density contribution, so it has units inverse to concentration. It is not the probability of observing exactly 0.900 mg/L; for a continuous observation, the probability of any single exact value is zero.

BQL Records: Transforming the LLOQ

For a BQL record, the observed information is the event $Y_{ij}<L_{ij}$, not an exact concentration. Taking logarithms preserves the inequality because the logarithm is increasing. Standardizing the resulting normal variable gives the censored likelihood contribution:

Result: Under the lognormal observation model, a BQL record with LLOQ $L_{ij}>0$ contributes the conditional probability $$ P(Y_{ij}<L_{ij}\mid f_{ij},\sigma) = P(\log Y_{ij}<\log L_{ij}\mid f_{ij},\sigma) = \Phi\left(\frac{\log L_{ij}-\log f_{ij}}{\sigma}\right), $$ where $\Phi$ is the standard normal cumulative distribution function. The standardized threshold is the log ratio of the LLOQ to the prediction, divided by the log-scale standard deviation.

This is a lower-tail probability. If the LLOQ is below the prediction, the standardized threshold is negative and the BQL probability is below one-half; if the LLOQ is above the prediction, the probability is above one-half. These are conditional probabilities given the model prediction and residual variability.

Worked Example: Probability of a BQL Result

For a hypothetical record, let $f=0.800$ mg/L, $\sigma=0.250$, and the LLOQ be $L=0.500$ mg/L. The prediction and LLOQ are positive, the LLOQ is known, and the BQL status is treated as censoring at that threshold.

Transform and standardize the LLOQ:

z_L=\frac{\log(0.500)-\log(0.800)}{0.250} =\frac{\log(0.625)}{0.250} =\frac{-0.4700}{0.250} =-1.880.

Thus the probability contribution is

P(Y<0.500\mid f=0.800,\sigma=0.250) =\Phi(-1.880) \approx 0.03005.

Under this model, the conditional probability of a BQL result at this record is about 3.005%. The contribution is a probability and therefore has no concentration units. Using the density at $0.500$ mg/L instead would not represent the observed event.

Combining Quantified and Censored Records

Conditional independence means that the likelihood for a collection of records, conditional on the individual's random effects, is the product of their contributions. A quantified record contributes a density; a BQL record contributes a probability. In log-likelihood calculations, these factors become a sum of log densities and log probabilities. For an individual in an NLME model, the marginal likelihood additionally averages the conditional likelihood over the distribution of that individual's random effects.

L_i(\theta,\eta_i) = \prod_{j\in Q_i} f_Y(y_{ij}\mid f_{ij}(\theta,\eta_i),\sigma) \prod_{j\in B_i} \Phi\left( \frac{\log L_{ij}-\log f_{ij}(\theta,\eta_i)}{\sigma} \right), \qquad L_i(\theta)=\int L_i(\theta,\eta)\,g(\eta;\Omega)\,d\eta.

Here $Q_i$ and $B_i$ denote the quantified and BQL records for individual $i$, respectively, and $g(\eta;\Omega)$ is the random-effects density. The integral is the usual marginalization over random effects; the distinctive contribution for this topic is the lognormal density or normal tail probability inside the conditional likelihood.

Worked Example: One Quantified and One BQL Record

Consider two hypothetical, conditionally independent records for one individual. At the first record, $y_1=0.900$ mg/L; at the second, the result is BQL with $L_2=0.700$ mg/L. Suppose the prediction at each record is $f=0.800$ mg/L and $\sigma=0.250$. Both predictions, the quantified value, and the LLOQ are positive, and both observations use the same stated residual variability.

From the exact-density calculation, the first contribution is approximately $1.5868\;(\mathrm{mg/L})^{-1}$. For the BQL record:

z_{L_2}=\frac{\log(0.700)-\log(0.800)}{0.250} =\frac{\log(0.875)}{0.250} =-0.5341, \qquad P(Y_2<0.700\mid f=0.800)=\Phi(-0.5341)\approx0.2967.

The conditional likelihood contribution for the pair is the product:

L_{\text{pair}} \approx(1.5868)(0.2967) \approx0.4708\;(\mathrm{mg/L})^{-1}.

The unit is inverse concentration because one factor is a concentration density and the other is dimensionless. In a full NLME analysis, this conditional product is evaluated at random effects and then integrated over their population distribution as appropriate.

Parameterization, Comparisons, and Checks

The median parameterization must match the meaning assigned to the structural prediction. If $f_{ij}$ is intended to be the arithmetic mean concentration instead, the lognormal model must be shifted: for log-scale standard deviation $\sigma$, use $\log Y_{ij}=\log f_{ij}-\sigma^2/2+\epsilon_{ij}$. This shift makes $E(Y_{ij})=f_{ij}$. In that parameterization, replace $\log f_{ij}$ in both the density and the BQL probability by $\log f_{ij}-\sigma^2/2$. Do not use the mean parameterization in one contribution and the median parameterization in another.

Compared with additive normal error, the lognormal model respects positive concentrations and expresses residual variability multiplicatively. It implies a concentration-scale spread that increases with the prediction. Unlike a normal model on the concentration scale, it cannot generate negative measurements. A normal model on log concentration is mathematically equivalent to a lognormal model on concentration only when the transformation's Jacobian is handled correctly for exact observations. For BQL records, both descriptions give the same event probability when the threshold is transformed consistently.

Before interpreting fitted results, check that the implementation uses the correct observation status and threshold for every record, that no zero or negative value is passed to a logarithm, and that the censored contribution is a CDF probability rather than a density. Confirm that predictions and LLOQs use the same concentration units, since their ratio must be dimensionless. If records have different LLOQs, use the threshold belonging to each record. A lognormal model may be unsuitable if the observation process allows meaningful zero or negative values, or if log-scale residual variation is a poor description of the quantified data.

Common pitfall: Replacing a BQL record by its LLOQ and evaluating $f_Y(L_{ij}\mid f_{ij},\sigma)$ treats the record as an exact concentration. The observation only establishes $Y_{ij}<L_{ij}$, so the correct contribution is $\Phi((\log L_{ij}-\log f_{ij})/\sigma)$. A density at the threshold and a probability below the threshold are different likelihood contributions.

Check Your Understanding

For each question, identify the appropriate likelihood contribution or interpretation. Solutions are provided after the questions.

With $f=1.00$ mg/L, $\sigma=0.200$, and an exact observation $y=1.00$ mg/L, write the concentration-scale density contribution.
With $f=1.00$ mg/L, $\sigma=0.200$, and $L=0.800$ mg/L, calculate the standardized BQL threshold and state whether the BQL probability is above or below one-half.
Why does an exact lognormal density contain a factor $1/y$, while a BQL contribution does not?
A record is BQL below $0.400$ mg/L. Should its contribution be the density at $0.400$ mg/L or a probability? Give the probability formula.

Solutions. (1) The contribution is $ [1/(1.00\cdot0.200\sqrt{2\pi})]\exp(0)\approx1.995\;(\mathrm{mg/L})^{-1}$. (2) $z_L=\log(0.800/1.00)/0.200=-1.116$ (approximately), so $\Phi(z_L)<0.5$. (3) The factor $1/y$ is the change-of-variable Jacobian from log concentration to concentration for a density; a BQL record uses an event probability, obtained from the normal CDF. (4) It is a probability: $\Phi((\log(0.400)-\log f)/\sigma)$, using that record's prediction $f$ and log-scale standard deviation $\sigma$.