The M4 Method for BQL Records

What M4 Does With a BQL Record

A below-quantification-limit (BQL) result is not an observed concentration of zero. It tells us that the concentration was below the assay’s lower limit of quantification (LLOQ). The M3 method uses that information as a censored observation. By contrast, M4 treats each BQL concentration record as missing from the concentration likelihood. It does not replace the result with a numerical concentration or use the fact that it fell below the LLOQ.

The omission applies to the record, not automatically to the entire subject. If a subject has quantified concentrations at some times and BQL records at others, M4 retains the quantified observations and omits only the BQL concentration contributions. This can preserve useful information from a subject’s profile, but it discards information carried by the pattern of BQL results.

Definition: Under M4, a quantified record contributes its usual concentration density to the likelihood, while a BQL record contributes a factor of 1 (equivalently, zero to the log-likelihood). Other quantified records from the same subject remain in the likelihood.

To make the likelihood explicit, suppose \(y_{ij}\) is the transformed concentration for subject \(i\) at record \(j\), and \(f(y_{ij}\mid\eta_i;\theta)\) is its density under the population model. Here \(\theta\) denotes population and residual model parameters, and \(\eta_i\) denotes that subject’s random effects. Let \(Q_i\) be the set of quantified records for subject \(i\). Assuming conditional independence of residual errors given the subject’s random effects, M4 uses the marginal likelihood

L_{\mathrm{M4}}(\theta) = \prod_i \int \left\{\prod_{j\in Q_i} f(y_{ij}\mid\eta_i;\theta)\right\} p(\eta_i;\Omega)\,d\eta_i .

The integral is important: records from one subject share the same random effects. M4 omits a BQL record’s concentration contribution inside the subject-specific product, but it does not replace the shared-random-effect model with independent effects for each retained observation.

Worked Likelihood Contributions

Worked Example: One Subject With Quantified and BQL Records

Consider a hypothetical subject with two quantified concentrations, 8 and 4 ng/mL, followed by a BQL result. For illustration, use log concentration \(y=\ln(C)\), a normal residual model with standard deviation \(\sigma=0.20\), and condition on the subject effect so the predicted log concentrations at the two quantified times are \(\ln(8)\) and \(\ln(4)\). The third predicted concentration is 1.5 ng/mL, and the LLOQ is 2 ng/mL.

For a normal density evaluated at its mean, the quantified-record density is

f(y\mid\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac{(y-\mu)^2}{2\sigma^2}\right], \qquad f(\mu\mid\mu,0.20) = \frac{1}{0.20\sqrt{2\pi}} \approx 1.995.

Both quantified observations equal their conditional predictions, so their joint conditional contribution is \(1.995\times1.995\approx3.979\). The BQL result contributes 1 under M4. Thus the conditional M4 contribution for these records is approximately 3.979, before integrating over the subject’s random effects. This is a density contribution, not a probability, so it need not be less than 1.

The subject is not discarded: both quantified observations remain. But the likelihood does not reward or penalize the model for predicting 1.5 ng/mL at the BQL time, even though that prediction is below the stated LLOQ.

Worked Example: Retaining Two Records With a Shared Random Effect

Use a simple hypothetical quadrature approximation for one subject’s random effect \(\eta_i\): it takes values \(-0.20\) and \(0.20\), each with probability 0.5. At two quantified times, suppose the population predictions on the log scale are \(\ln(8)\) and \(\ln(4)\), and the observed values equal those predictions. Let \(\sigma=0.20\). A third record is BQL.

At either quadrature point, adding \(\eta_i\) shifts both predictions by the same amount. Each quantified residual is therefore \(0.20\) or \(-0.20\), so each density is

\frac{1}{0.20\sqrt{2\pi}} \exp\left[-\frac{(0.20)^2}{2(0.20)^2}\right] = 1.995\exp(-0.5) \approx 1.210.

The two-record conditional product is \(1.210^2\approx1.464\) at either quadrature point. Averaging over the two equally weighted points gives the M4 marginal contribution \(0.5(1.464)+0.5(1.464)=1.464\). The BQL record adds a factor of 1 at both points and does not change that result.

This calculation retains both quantified observations jointly under the same \(\eta_i\), then averages over that shared effect. Dropping the whole subject would lose the contribution 1.464. Treating each quantified record as having a separate random effect would instead change the model and its likelihood.

Why Omitting BQL Information Is an Approximation

A BQL result carries an inequality: the latent measured concentration was below the LLOQ. In a censored likelihood, that record contributes the probability of falling below the limit. For example, with a normal model on log concentration, conditional mean \(\mu\), and standard deviation \(\sigma\), the contribution is

P(Y<\ln(\mathrm{LLOQ})\mid\mu,\sigma) = \Phi\left(\frac{\ln(\mathrm{LLOQ})-\mu}{\sigma}\right),

where \(\Phi\) is the standard normal cumulative distribution function. M4 replaces this parameter-dependent probability by 1. The M3 method uses the censored probability contribution; M4 does not. That difference matters whenever the probability of being BQL changes with the model parameters.

Justification: For a quantified record, the observed-data likelihood includes its density \(f(y\mid\theta)\). For a BQL record, the data do not give an exact \(y\); they give the event \(Y<\ln(\mathrm{LLOQ})\), whose likelihood contribution is \(F(\ln(\mathrm{LLOQ})\mid\theta)\). M4 substitutes 1 for this latter contribution. Since \(\log(1)=0\), it removes both that record’s log-likelihood contribution and its parameter-dependent score.

The omitted score is not generally zero. If \(F\) depends on a parameter \(\theta_k\), its contribution to the score is \(\partial\log F/\partial\theta_k=(1/F)(\partial F/\partial\theta_k)\). M4 drops this term. The approximation is least consequential when BQL records are rare or their probabilities change very little over the plausible parameter range. It can be consequential when BQL status strongly depends on predicted concentrations.

Worked Example: Size and Direction of an Omitted Score

Suppose, hypothetically, that log concentration is normal with \(\mu=\ln(1.5)\), \(\sigma=0.25\), and LLOQ \(=2\) ng/mL. The standardized LLOQ is

z=\frac{\ln(2)-\ln(1.5)}{0.25} =\frac{0.2877}{0.25} \approx1.151.

The censored probability is approximately \(\Phi(1.151)=0.875\), so its log-likelihood contribution is \(\ln(0.875)\approx-0.1335\). M4 uses a contribution of 1 and log contribution 0, omitting this information.

For the censored contribution, the derivative with respect to \(\mu\) is \(-\phi(z)/[\sigma\Phi(z)]\), where \(\phi\) is the standard normal density. Using \(\phi(1.151)\approx0.206\), the derivative is approximately \(-0.206/[0.25(0.875)]=-0.942\). M4’s derivative for that record is zero. The negative censored score indicates that, at these values, this BQL event provides likelihood support for a lower \(\mu\); M4 omits that support. This is a local illustration, not a universal direction of bias in a fitted PK model.

Assumptions, Diagnostics, and Failure Modes

The quantified-record part of an M4 analysis still relies on the specified structural PK model, residual error model, random-effect distribution, and conditional dependence assumptions being appropriate for the retained data. M4 does not make those assumptions more or less valid. Its additional approximation is that the BQL status can be ignored for estimating the parameters of interest.

That approximation is most defensible when the BQL pattern contains little parameter information—for example, when very few records are BQL, or when their chance of being BQL is nearly unchanged across plausible model parameters. It is more concerning when BQL records cluster in a parameter-informative region, such as later times where concentrations are declining, or when many subjects have profiles with only a few quantified observations. The measured values that remain quantified are selected by being above the limit; ignoring that selection can make their distribution unrepresentative of the full concentration process.

Inspect where BQL records occur. A concentration-time plot that shows BQL records concentrated in the terminal portion of profiles warns that M4 is omitting information about the low-concentration trajectory.
Check fit to the retained observations. Examine whether quantified residuals and fitted profiles show systematic patterns. A good fit to quantified points alone does not establish that ignoring BQL status is harmless.
Assess sensitivity to the censoring treatment. Compare M4 estimates and uncertainty with estimates from a method that includes the censored likelihood contribution, when feasible. Material changes indicate that the omitted BQL information affects inference.
Watch for weak information and unstable estimates. If few quantified observations remain in informative portions of profiles, parameter estimates may be sensitive to model assumptions or data selection. Convergence alone does not show that the approximation is adequate.

There is no universal direction for M4 bias. Omitting low-concentration evidence may alter estimates of elimination, exposure, residual variability, or random-effect variability, but the direction and magnitude depend on the sampling design, model, and distribution of BQL records. In particular, it is not safe to assert that M4 must always estimate a slower or faster terminal decline.

How M4 Differs From the Main Alternative

Feature	M4	M3
Quantified record	Uses its concentration density	Uses its concentration density
BQL record	Omits its concentration contribution; factor 1	Uses the probability of being below the LLOQ
Other records from the same subject	Retains quantified records	Retains quantified records
Information from BQL status	Discarded	Included through the censoring probability
Main practical trade-off	Simpler likelihood, but potentially informative BQL patterns are ignored	Uses the censoring information, but requires evaluating the censored likelihood

M4 can be attractive as a simpler approximation, particularly when BQL observations are sparse and contribute little information. It is not equivalent to M3, and it should not be described as “using the BQL results as zero” or as “discarding every subject with any BQL record.” Its defining operation is narrower: omit the likelihood term for each BQL concentration while retaining the other usable records.

Common Mistake

Wrong reasoning: “A subject has a BQL result, so none of that subject’s data can be used under M4.”

Correction: M4 omits the BQL record’s concentration contribution, not the subject’s entire profile. If the same subject has quantified observations, their densities remain in the subject’s joint likelihood and are integrated over that subject’s shared random effects. A different mistake is to treat the omitted BQL contribution as evidence that the concentration was exactly zero; M4 makes no such substitution.

Key takeaway: M4 keeps quantified concentration information, including quantified records from subjects who also have BQL results, but it assigns no likelihood information to the BQL status itself. Its simplicity is useful only when that discarded information is not important for the inference being made.

Check Your Understanding

Answer each question, then check the solutions below.

A subject has two quantified concentrations and three BQL records. Which records contribute concentration likelihood terms under M4?
In a log-normal concentration model, why can a BQL record provide information about a model parameter even though its exact concentration is unknown?
For the hypothetical normal model with \(\mu=\ln(1.5)\), \(\sigma=0.25\), and LLOQ \(=2\) ng/mL, what is the approximate M3 probability contribution, and what does M4 use instead?
Why must quantified observations from the same subject be handled jointly when the model includes a shared subject random effect?

Solutions. (1) The two quantified records contribute their densities; each BQL record contributes a factor of 1, so its concentration term is omitted. (2) The BQL status says the concentration fell below the LLOQ, an event whose probability depends on parameters such as the predicted mean and residual variability. (3) The standardized limit is \(z\approx1.151\), giving \(\Phi(z)\approx0.875\) under the censored likelihood; M4 uses 1. (4) The records share the same random effect, so their conditional densities are multiplied for each value of that effect and the joint product is then integrated over its distribution.

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