From a Censored Record to a Subject-Level Likelihood
A below-quantification-limit (BQL) record does not provide an exact concentration. It tells us that the unobserved measurement fell below a known assay limit. In a nonlinear mixed-effects model, that event is evaluated conditional on the subject’s random effects, just as an exact measurement’s density is evaluated conditional on those effects. The subject-level likelihood combines all of that subject’s exact values and BQL events before averaging over the random-effects distribution.
This order matters. The random effects are shared across a subject’s records. Conditional on a particular random-effects value, record-level contributions may multiply under the model’s conditional-independence assumption. But the resulting product must then be integrated over the subject-level random-effects distribution. Separately averaging each record’s contribution and multiplying those averages generally gives a different answer.
Let \(E_i\) be the set of subject \(i\)’s records reported as exact and \(C_i\) the set reported as BQL. Under conditional independence of records given the random effects and model parameters, the conditional subject-level likelihood is
An empty product is defined to equal 1. Thus, a subject with only exact records has a product of densities, while a subject with only BQL records has a product of event probabilities. The likelihood combines density and probability contributions: exact continuous measurements contribute densities, whereas censored records contribute probabilities. A density is not itself a probability, and its numerical value can depend on the measurement units.
Assumptions and Marginalization
The conditional expression relies on several assumptions. First, each record’s status is correctly represented: an exact record supplies its measured value, and a BQL record supplies the event that the measurement is at or below its applicable limit. Second, the measurement distribution and the limit are specified on the same scale. Third, conditional on \(\eta_i\) and \(\theta\), the records are independent, so their contributions multiply. If the model includes residual dependence across records, this product must be replaced by the appropriate joint conditional density or event probability.
Finally, the random effects are not observed. Suppose their population distribution has density \(p(\eta_i\mid\Omega)\), where \(\Omega\) denotes its distributional parameters. The marginal likelihood for the observed records of subject \(i\) is obtained by integrating the conditional likelihood over that distribution:
This is an average of the subject’s conditional likelihood over plausible random-effects values, weighted by their population distribution. It is not generally the conditional likelihood evaluated at an estimated or “typical” random effect. If subjects are independent draws from the population, their marginal likelihood contributions multiply:
The integral is often not available in closed form and must be approximated numerically. The particular integration method is a computational choice; it does not change the definition of the marginal likelihood. In computation, likelihoods are commonly handled on a log scale to reduce numerical underflow. The log of a product of record-level contributions is a sum conditional on \(\eta_i\), but the integration over \(\eta_i\) still applies to the likelihood, not to that conditional log-likelihood as though it were an ordinary probability density.
Worked Examples
Worked Example: Combining Exact and BQL Contributions
Hypothetical data. A subject has two exact concentrations, 8 and 4 mg/L, and one BQL record with limit 5 mg/L. At a fixed random-effects value \(\eta_i\), suppose the model gives conditional densities \(f(8\mid\eta_i,\theta)=0.12\) per mg/L and \(f(4\mid\eta_i,\theta)=0.08\) per mg/L, and a conditional probability \(F(5\mid\eta_i,\theta)=0.30\). Assume the records are conditionally independent.
The exact records contribute their densities, and the BQL record contributes the probability of being at or below 5 mg/L. Therefore,
The units reflect two density contributions; the BQL probability is dimensionless. This is the conditional likelihood contribution at the specified \(\eta_i\), not yet the subject’s marginal likelihood.
Worked Example: Approximating the Marginal Likelihood
Hypothetical quadrature calculation. For the subject above, suppose an integration rule approximates the random-effects distribution using three nodes, \(\eta=-1,0,1\), with weights \(0.25,0.50,0.25\). Suppose the full conditional likelihoods at those nodes, including both exact densities and the BQL probability, are respectively \(0.00012\), \(0.000288\), and \(0.00020\;(\mathrm{mg/L})^{-2}\). The weights sum to 1, so the weighted sum approximates the integral.
The result averages the conditional subject-level likelihood across the random-effects distribution. The quadrature nodes and weights are stipulated for this example; in an actual fit, their adequacy and the numerical accuracy of the integration must be assessed for the model and data.
Worked Example: Why Contributions Must Be Combined Before Integration
Hypothetical two-record example. Consider two BQL events from one subject. For illustration, suppose the random effect has two possible values, each with probability \(0.5\). At the first value, the two conditional BQL probabilities are \(0.10\) and \(0.80\); at the second, they are \(0.80\) and \(0.10\). Assume conditional independence.
At either random-effects value, the conditional probability of both events is the product \(0.10\times0.80=0.08\). Integrating that joint contribution gives
If instead each record is integrated separately, each marginal probability is \((0.5)(0.10)+(0.5)(0.80)=0.45\). Multiplying those gives \(0.45\times0.45=0.2025\), which is not the joint subject likelihood. The separate calculation treats the two records as if their random effects were independently drawn. In the hierarchical model, both records share the same \(\eta_i\), so the correct calculation integrates their conditional product.
Why the Order of Operations Is Necessary
The marginalization formula follows from the law of total probability (or, for exact measurements, the corresponding density version). Conditional on \(\eta_i\), the joint contribution of the subject’s observed values and censoring events is \(L_i(\theta\mid\eta_i)\). The joint distribution of the observations and the random effect factors into this conditional contribution times the random-effects density. Integrating out the unobserved random effect gives the marginal contribution:
Use the model’s conditional density for each exact value and conditional event probability for each BQL record.
Multiply record contributions only when conditional independence given \(\eta_i\) and \(\theta\) is part of the model.
Integrate the complete conditional subject likelihood against \(p(\eta_i\mid\Omega)\) to obtain \(L_i(\theta,\Omega)\).
The key distinction is that conditional independence does not usually imply marginal independence. A shared random effect can induce dependence among a subject’s measurements after that effect is integrated out. This is why the random-effects integral encloses the full product of the subject’s exact and censored contributions.
Diagnostics, Alternatives, and Limitations
Likelihood construction should be checked before interpreting parameter estimates. Useful checks include confirming that each record is classified once as exact or BQL, matching each BQL record to its correct limit, and verifying that exact-value densities and censoring probabilities use the same measurement scale and conditional model. A BQL contribution must be a probability between 0 and 1. An exact contribution is a density and need not be below 1. The units of a subject’s likelihood depend on how many continuous exact observations it contains.
A useful implementation check is to compare the marginal likelihood from increasingly accurate integration settings or independent numerical methods, where feasible. Material changes can indicate inadequate integration. Very small likelihood products can also underflow in direct arithmetic; summing conditional log contributions and using stable log-scale integration helps address this numerical issue. These checks diagnose computation and construction, but they do not establish that the chosen structural, residual, or random-effects model is scientifically adequate.
| Approach | What it represents | Main distinction |
|---|---|---|
| Conditional subject likelihood | Likelihood given a fixed \(\eta_i\) | Useful for constructing the record-level contributions, but it has not averaged over the unobserved subject effect. |
| Marginal subject likelihood | Conditional subject likelihood integrated over \(p(\eta_i\mid\Omega)\) | Accounts for population variation and is the subject contribution used in the marginal likelihood. |
| Product of separately marginalized records | Each record is averaged over a random effect before record contributions are multiplied | Generally incorrect for a shared subject effect because it removes the dependence induced by that effect. |
The marginal likelihood is only as appropriate as its underlying assumptions. Conditional dependence, an incorrectly specified random-effects distribution, or an incorrectly handled censoring limit can make the stated likelihood unsuitable. When all records for a subject are BQL, the likelihood still has a defined form, but the data may provide limited information about that subject’s exact concentration trajectory. Numerical integration can also be costly in models with many random effects. None of these limitations justifies replacing a BQL event with an invented exact measurement in this likelihood.
Common Mistakes
- Using a density for a BQL record. The record reports an event, not an exact value at the limit. Use the conditional probability of falling at or below that limit.
- Multiplying after integrating each record separately. This generally ignores the shared subject effect. Form the conditional product first, then integrate it.
- Calling the conditional likelihood the marginal likelihood. A likelihood evaluated at a fixed \(\eta_i\) has not accounted for uncertainty and population variation in that unobserved effect.
- Assuming any product is justified. The product over records requires conditional independence. If the model specifies residual dependence, use its joint contribution instead.
- Mixing scales or units. A limit on one scale cannot be used with a distribution defined on another without the correct transformation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A subject has one exact record and two BQL records. What contribution does each type of record make to the conditional subject likelihood?
- Why is the random-effects integral taken around the complete conditional product rather than separately around each record’s contribution?
- Suppose a subject has only BQL records. Is the conditional likelihood necessarily zero? Explain.
- What assumption justifies multiplying record contributions conditional on the random effects?
Solutions. (1) The exact record contributes its conditional density at the observed value; each BQL record contributes its conditional probability of being at or below its limit. Their product is the conditional subject likelihood if records are conditionally independent. (2) The subject’s records share the same random effect. Integrating the product preserves that shared effect and the dependence it induces; separate integration generally corresponds to different effects for different records. (3) No. It is the product of the conditional probabilities of the observed BQL events, which can be positive. It can be small, but BQL status alone does not make the likelihood zero. (4) Conditional independence of records given the subject random effects and model parameters. If that assumption does not hold, the joint conditional distribution must replace the product.