Why the distinction matters
A fast–slow pattern in pharmacokinetic (PK) data can be represented in two fundamentally different ways. A latent class is unobserved: each subject’s class is inferred probabilistically from that subject’s PK observations. A measured genotype or phenotype is recorded independently as a known grouping variable. Although both approaches may produce fast and slow groups in a model, they do not use the same likelihood or make the same claim.
This distinction matters because uncertainty about a latent class belongs in the likelihood. A measured grouping, by contrast, is conditioned on as observed. Treating a measured genotype as proof of a latent PK class can be biologically unjustified; treating a genuinely unobserved class as known discards uncertainty. The choice should follow what was measured and what the model says about the relationship between the measurement and PK behavior.
Likelihood when the class is latent
Suppose there are \(K\) possible classes. Let \(\pi_{ic}=P(c_i=c\mid x_i)\) be subject \(i\)’s probability of class \(c\), potentially depending on observed covariates \(x_i\). For class \(c\), let \(p(y_i\mid b_i,c,\theta)\) be the conditional density of the subject’s observations under the PK and residual-error model, and let \(p(b_i\mid c,\theta)\) be the random-effects density. The class-specific subject likelihood after integrating over random effects is
Because the class is unknown, the observed-data contribution for subject \(i\) is the probability of the data under all possible classes, weighted by their class probabilities:
Under the usual assumption that subjects are independent conditional on the model parameters and their covariates, the likelihood for all subjects is \(L(\theta)=\prod_i L_i(\theta)\). The sum over classes occurs within each subject’s contribution; subjects do not share one common class. This is the two-class NLME likelihood expressed with class-specific integrated likelihoods.
The sum cannot generally be moved outside the product. In general, \(\prod_i\sum_c\pi_{ic}L_{ic}\) is not equal to \(\sum_c\pi_c\prod_iL_{ic}\). The latter would describe a different structure in which one shared class applies to the entire data set. In a subject-level metabolizer mixture, each subject has their own class.
Worked Example: Marginalizing One Subject’s Latent Class
For a simplified discrete PK outcome, suppose the probability of subject \(i\)’s observed outcome is \(0.08\) under the fast class and \(0.02\) under the slow class. The model assigns probabilities \(0.30\) and \(0.70\) to those classes. Find the subject’s likelihood contribution when class is latent.
The assumptions are that the two class probabilities sum to one and that the stated outcome probabilities are conditional on class. The subject’s class is not observed, so
The contribution is \(0.038\). It is not \(0.08\), because fast status has not been established, and it is not \(0.02\), because slow status has not been established. Nor is it the unweighted average of the two probabilities: the class weights are part of the model.
Likelihood when a grouping variable is observed
Now suppose a recorded variable \(g_i\) identifies the group used in the PK model. If \(g_i\) is known and is modeled as corresponding directly to group-specific PK parameters, the likelihood conditions on that value. For example, if the observed group is \(c_i\), its subject-level contribution is
There is no sum over classes for this subject: the group is observed. The likelihood across subjects is the product of their contributions conditional on their recorded groups. A genotype may instead be a covariate that predicts PK parameters without defining a fast or slow class; in that case the model uses genotype-dependent parameters, and it still does not assign an unobserved class unless one is explicitly included.
A directly measured phenotype can be used as a known grouping variable when the scientific model defines the groups by that measurement. That does not prove the phenotype category is identical to a latent class that would be inferred from PK. For example, a laboratory phenotype category may be associated with clearance but may not perfectly separate all subjects into two PK distributions. The meaning of the grouping must therefore be stated, rather than inferred from the fact that both descriptions use words such as “fast” and “slow.”
Worked Example: Using a Recorded Phenotype Group
Suppose a subject’s measured phenotype is recorded as fast. For the subject’s observed PK outcome, the model gives conditional probability \(0.08\) in the fast group. The slow-group probability would be \(0.02\), but the subject’s phenotype is known. What is the likelihood contribution?
Under the stated model, the observed phenotype defines the group, and the likelihood is conditional on that recorded group. Therefore,
The value is not \(0.30(0.08)+0.70(0.02)=0.038\), because that calculation would treat the class as unobserved and apply latent-class probabilities. Here the measurement has already specified the grouping used by the model. This calculation is appropriate only if the measured phenotype is intended to define the modeled group; if it merely predicts class membership, a latent-class formulation may still be needed.
Observed covariate versus observed class
An important middle case is an observed genotype that predicts, but does not determine, a latent PK class. Let \(g_i\) be the measured genotype. If the model states that genotype changes class probabilities, then \(\pi_{ic}=P(c_i=c\mid g_i)\), but \(c_i\) remains unobserved. The contribution is still
Thus, observing a covariate does not remove a latent-class sum merely because that covariate is informative about class. The sum disappears only when the grouping itself is observed and the model treats that observed group as the relevant class. This distinction separates “genotype is associated with class membership” from “genotype is the class label.”
Worked Example: A Measured Genotype Predicts a Latent Class
A subject has a measured genotype \(g_i\). Under a model for this genotype, the probability of the latent fast class is \(0.70\), and the probability of the latent slow class is \(0.30\). For the subject’s PK data, the integrated likelihoods are \(0.04\) under fast and \(0.01\) under slow. Calculate the contribution conditional on genotype.
The genotype is observed, but class remains latent. The genotype-specific class probabilities weight the two class-specific likelihoods:
The contribution is \(0.031\). Using \(0.04\) alone would incorrectly treat the genotype as a certain fast-class assignment. Using \(0.70(0.04)+0.30(0.01)\) is appropriate because the stated model makes genotype predictive, not determinative, of the latent class.
Missing measurements and model assumptions
If a grouping measurement is missing, its handling depends on the data and missingness assumptions. If genotype is missing and the model specifies genotype probabilities \(P(g)\), a joint model can marginalize over the unobserved genotype. If the genotype predicts a latent class, both uncertainties may be relevant. A simple contribution could be written as
This expression is not automatic: it requires a model for genotype frequencies and class probabilities, and its interpretation depends on how the subject entered the analysis. If missingness depends on unobserved genotype or PK data, the missingness mechanism may also need to be modeled. Simply assigning a missing genotype to the most common category is not equivalent to marginalizing over uncertainty.
The usual likelihood comparisons also rely on a correctly specified observation model, random-effects distribution, class-specific PK model, and class-probability model. With an observed grouping, the recorded group must be measured and coded reliably for the intended analysis. With a latent class, the mixture must be identifiable enough for its class-specific parameters and weights to be estimated from the PK data. Weak separation or excessive overlap can make individual class probabilities uncertain even when the population model can be fit.
Worked Example: Marginalizing a Missing Group Measurement
For a simplified discrete outcome, suppose a missing genotype is fast-associated with probability \(0.40\) and slow-associated with probability \(0.60\). Conditional on those genotype groups, the outcome probabilities are \(0.06\) and \(0.02\). Assume these are exhaustive categories and that the genotype distribution applies to this subject. Find the likelihood contribution after accounting for the missing genotype.
Since the genotype is not observed, sum over its possible values:
The contribution is \(0.036\). This is a valid marginal contribution under the stated genotype-frequency assumption; it is not the contribution conditional on either genotype. If genotype also predicts a separate latent PK class, the model would need to sum over both genotype and class, with the appropriate conditional probabilities.
Comparing the formulations and checking the analysis
| Question | Latent PK class | Observed grouping variable |
|---|---|---|
| What is recorded? | PK data; class is unobserved | PK data and the group value |
| Subject-level likelihood | Sum over classes, after integrating random effects | Use the contribution for the recorded group |
| Interpretation | Classes represent unobserved subpopulations inferred from PK | PK distributions are compared conditional on measured groups |
| Uncertainty in group assignment | Included through the mixture likelihood | Not included unless measurement error or uncertain classification is modeled |
A useful check is to trace each subject’s data through the model: what was actually observed, which quantities remain unknown, and which sums or integrals are therefore required? If the grouping is observed, verify that the analysis uses the recorded value and that the scientific interpretation matches the measurement. If class is latent, verify that the sum over classes is inside the product over subjects. If a measured covariate only predicts class, retain the latent-class sum and use the covariate-dependent class probabilities.
Warning signs include a “known class” assigned solely from a genotype that is only probabilistically associated with metabolism, or latent-class probabilities reported as though they were observed labels. Another warning sign is a likelihood that multiplies class-specific terms for a subject without weighting or summing over the unobserved class. Conversely, applying mixture weights to subjects whose modeled group is directly observed changes the likelihood and fails to condition on the available information.
The approaches answer different questions. A latent mixture asks whether PK observations support unobserved subpopulations and estimates their class structure. A measured-group model asks whether PK behavior differs across recorded groups. An observed genotype can serve as a predictor, define groups, or inform latent class probabilities; these are distinct model specifications. None is universally preferable: the right choice depends on the measurement process and the scientific target.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A PK model gives class-specific integrated likelihoods \(0.05\) and \(0.01\), with latent class probabilities \(0.20\) and \(0.80\). What is the subject’s marginal likelihood contribution?
- A phenotype measurement directly defines the subject’s modeled group as slow. The slow-group contribution is \(0.03\). Should the likelihood contribution include a mixture-weighted sum over fast and slow? Explain.
- A genotype is observed and predicts a latent fast class with probability \(0.60\). What likelihood structure should be used if the fast and slow integrated likelihoods are \(L_{iF}\) and \(L_{iS}\)?
- Why is multiplying each subject’s fast- and slow-class likelihoods and then summing over one class not generally the subject-level mixture likelihood?
Solutions. (1) \(0.20(0.05)+0.80(0.01)=0.010+0.008=0.018\). (2) No. If the recorded phenotype defines the modeled group, condition on slow and use \(0.03\); a mixture sum would treat the class as unobserved. (3) Retain the latent-class sum: \(0.60L_{iF}+0.40L_{iS}\). (4) Each subject has their own latent class. The correct data likelihood is the product of subject-level sums, \(\prod_i\sum_c\pi_{ic}L_{ic}\); summing after multiplying instead imposes a shared class across subjects.