Worked Analysis of Fast and Slow Clearance Subgroups

From a Two-Class Model to an Analysis

A latent fast/slow clearance model does not assign each subject to a class before fitting. Instead, it represents each subject’s concentration-time data as arising from one of two unobserved classes, with class-specific clearance distributions and an estimated population prevalence. The analysis must therefore estimate the kinetic parameters and the mixture proportions together, then use each subject’s observed data to update the class probabilities.

The worked analysis below uses a one-compartment, intravenous bolus model to keep the calculations transparent. The same likelihood logic applies when the concentration prediction comes from a more complex structural model. The main targets here are class-specific clearance, fast-class prevalence, and subject-level posterior probabilities—not a hard assignment of every subject to one group.

Model and assumptions: For subject $i$, let $Z_i$ be an unobserved class, either $F$ (fast) or $S$ (slow), with $P(Z_i=F)=\pi_F$ and $P(Z_i=S)=1-\pi_F$. For dose $D_i$, the one-compartment IV-bolus concentration prediction at time $t_{ij}$ is $$ f_{ij}=\frac{D_i}{V}\exp\left(-\frac{CL_i t_{ij}}{V}\right). $$ Within class $k$, use $\log(CL_i)=\log(CL_k)+\eta_i$, where $\eta_i\sim N(0,\omega_{CL}^2)$. Thus $CL_k$ is the class-specific median clearance, and $\omega_{CL}$ is the shared between-subject standard deviation on the log-clearance scale. Assume $CL_F>CL_S>0$, a common positive volume $V$, and observations satisfying $\log(Y_{ij})=\log(f_{ij})+\epsilon_{ij}$, with independent $\epsilon_{ij}\sim N(0,\sigma^2)$. Subjects are independent; class membership is independent across subjects; and, conditional on class and random effect, observations follow the stated error model. This example assumes no concentration censoring, no class-membership covariates, and a common residual and clearance-variability structure across classes.

These assumptions define a particular analysis, not universal properties of mixture models. In an application, the structural model, residual error, random-effects structure, and class-membership model must be scientifically defensible. Here, sharing $V$, $\omega_{CL}$, and $\sigma$ focuses the distinction between classes on clearance and avoids asking the data to estimate many class-specific parameters at once.

The Subject-Level Mixture Likelihood

Write $x_{ij}=\log(Y_{ij})$. For a given subject, the likelihood conditional on class $k$ and random effect value $\eta$ is the product of the normal densities of the log observations. Because $\eta_i$ is unobserved, the class-specific likelihood $L_{ik}$ integrates that conditional likelihood over the distribution of $\eta_i$. The subject’s likelihood then averages the two class-specific likelihoods using their population prevalences.

L_{ik}=\int \left\{\prod_j \frac{1}{\sigma}\phi\left( \frac{x_{ij}-\log f_{ij}(CL_k e^\eta)}{\sigma} \right)\right\} \frac{1}{\omega_{CL}}\phi\left(\frac{\eta}{\omega_{CL}}\right)\,d\eta, \qquad L_i=\pi_F L_{iF}+(1-\pi_F)L_{iS}.

Here $\phi$ is the standard normal density, and $f_{ij}(CL_k e^\eta)$ means that the concentration prediction uses clearance $CL_k e^\eta$. The integral accounts for the fact that a subject’s clearance need not equal the class median. In practice it is evaluated numerically, for example by quadrature or an appropriate approximation. The population parameters are estimated by maximizing the marginal log likelihood $\ell=\sum_i\log(L_i)$. The log of the sum is essential: summing class log likelihoods, or choosing a class first and fitting as if that choice were certain, does not give this mixture likelihood.

Worked Examples

Worked Example: Translating Class Clearances Into Concentrations

Suppose a hypothetical subject receives $D=100$ mg intravenously, and a proposed model has $V=20$ L, $CL_S=2.0$ L/h, and $CL_F=5.0$ L/h. Ignore the individual random effect for this prediction and compare the class-typical concentrations at $t=4$ h.

The common starting concentration is $D/V=100/20=5.00$ mg/L. For the slow-clearance class, $f_S=5.00\exp[-(2.0\text{ L/h})(4\text{ h})/(20\text{ L})] =5.00e^{-0.4}=3.352$ mg/L.

For the fast-clearance class, $f_F=5.00\exp[-(5.0\text{ L/h})(4\text{ h})/(20\text{ L})] =5.00e^{-1}=1.839$ mg/L. The fast class has the higher clearance and therefore the lower class-typical concentration at this post-dose time. This calculation checks the direction and units implied by the model; it does not classify an actual subject.

Worked Example: Updating a Subject’s Class Probability

Suppose a hypothetical fitted model estimates $\pi_F=0.30$. For one subject’s complete concentration record, numerical integration of the assumed random-effects model gives $L_{iF}=0.012$ and $L_{iS}=0.0030$, in the same likelihood scale. These are class-conditional marginal likelihoods: each already integrates over the subject’s clearance random effect.

The posterior fast-class probability is the fast-class contribution to the mixture likelihood divided by the total contribution:

P(Z_i=F\mid \mathbf{x}_i) =\frac{\pi_F L_{iF}}{\pi_F L_{iF}+(1-\pi_F)L_{iS}} =\frac{(0.30)(0.012)}{(0.30)(0.012)+(0.70)(0.0030)} =\frac{0.0036}{0.0057} =0.6316.

Given this model and the subject’s observed record, the posterior probability of fast-class membership is about 0.632. The prior prevalence was 0.30; the data shift the probability upward because the fast-class likelihood is larger. The result is not a certainty that the subject belongs to the fast class, nor is it a probability that the model’s assumptions are correct.

Worked Example: Interpreting a Hypothetical Fitted Analysis

Consider a hypothetical data set of $120$ subjects analyzed using the stated shared-parameter model. Suppose numerical maximum likelihood gives $\hat{\pi}_F=0.25$, $\widehat{CL}_S=2.4$ L/h, $\widehat{CL}_F=6.0$ L/h, $\hat V=18$ L, $\hat\omega_{CL}=0.25$, and $\hat\sigma=0.12$. These values are illustrative fitted results, not data from a real study.

The ordering check is satisfied because $6.0>2.4$ L/h. The fitted prevalence means that the model assigns 25% of the population to the fast class before conditioning on an individual’s concentration data. It does not mean that exactly 30 of these 120 subjects must have high posterior probabilities, or that a hard classification would produce 30 fast subjects.

For a 90 mg dose, the class-typical predictions at 6 h are $90/18=5.00$ mg/L initially. The slow-class value is $5.00\exp[-(2.4)(6)/18]=5.00e^{-0.8}=2.247$ mg/L; the fast-class value is $5.00\exp[-(6.0)(6)/18]=5.00e^{-2}=0.6767$ mg/L. Clearance is in L/h, time in h, and volume in L, so each exponent is dimensionless and the predicted concentrations are in mg/L.

For a hypothetical subject whose integrated class likelihoods are $L_{iF}=0.0040$ and $L_{iS}=0.0010$, the posterior is $(0.25)(0.0040)/[(0.25)(0.0040)+(0.75)(0.0010)] =0.0010/0.00175=0.5714$. Although the fast-class likelihood is four times the slow-class likelihood, the posterior probability is about 0.571 rather than 0.80 because the prior prevalence favors the slow class.

Why the Posterior Formula Holds

The posterior probability follows directly from conditioning on the subject’s data. The joint contribution of fast-class membership and the data is $P(Z_i=F,\mathbf{x}_i)=\pi_F L_{iF}$. The data can arise from either class, so their total likelihood is $\pi_F L_{iF}+(1-\pi_F)L_{iS}$. Dividing the joint contribution by this total gives the posterior expression. This also shows why prevalence and subject-level evidence must both be used.

P(Z_i=F\mid\mathbf{x}_i) =\frac{P(Z_i=F,\mathbf{x}_i)}{P(\mathbf{x}_i)} =\frac{\pi_F L_{iF}}{\pi_F L_{iF}+(1-\pi_F)L_{iS}}.

The derivation depends on the two classes being mutually exclusive and exhaustive in the specified model, and on $L_{iF}$ and $L_{iS}$ being computed under the same observation model and on the same data scale. A conditional likelihood evaluated at a selected random effect cannot be substituted for an integrated class likelihood without changing the calculation.

Fitting, Checks, and Alternatives

A practical fit maximizes the marginal likelihood over the class medians, mixture prevalence, and the shared parameters. Constrain $0<\pi_F<1$, positive variances, and $CL_F>CL_S>0$; for example, optimize positive clearance parameters through log scales and represent the clearance separation by a positive parameter. The class-order constraint prevents the labels from swapping during optimization. The earlier material on stable initialization and class ordering is useful when selecting starting values and checking that different starts reach comparable solutions.

After fitting, inspect whether the likelihood has a stable optimum, whether parameter estimates are finite and away from constraints without scientific reason, and whether class-specific clearances and prevalence are reasonably determined. The earlier material on mixture diagnostics and identifiability addresses broader diagnostic tools; for this worked analysis, the central warning is that a nominally successful optimizer does not establish that the two classes are distinguishable. If the clearances nearly coincide, prevalence is near a boundary, or multiple parameter combinations give similar likelihoods, subject-level probabilities may be highly sensitive to the fitted parameters.

A two-class model is not the only explanation for heterogeneous clearance. A one-class model with a continuous clearance distribution may represent the population better when variability is gradual rather than subgroup-like. If metabolizer phenotype is measured, an observed-phenotype model may be preferable to treating membership as latent. These alternatives answer different scientific questions; a better numerical fit alone does not establish that discrete biological subgroups exist.

The model is also limited by its structural and distributional assumptions. A misspecified concentration-time model, residual error pattern, or random-effects distribution can distort both class separation and posterior probabilities. Sharing $V$ and variability across classes may be a useful parsimonious starting point, but it could be inadequate if there is strong evidence that those features differ. Conversely, allowing many class-specific parameters can make the mixture weakly identified. Posterior class probabilities are conditional on the fitted model and its parameters; they should not be read as error-free labels.

Common Pitfall

Common mistake: “The fast-class likelihood is four times larger, so the subject has an 80% probability of being fast.” This ignores prevalence. The correct posterior is $\pi_F L_{iF}/[\pi_F L_{iF}+(1-\pi_F)L_{iS}]$. For example, with $\pi_F=0.25$, $L_{iF}=4L_{iS}$, the posterior is $0.25(4)/[0.25(4)+0.75(1)]=0.571$, not 0.80. Also, a posterior probability is not a hard class assignment.

Check Your Understanding

Answer each question, then compare with the worked solutions below.

If $\pi_F=0.20$, $L_{iF}=0.006$, and $L_{iS}=0.002$, what is the posterior fast-class probability?
Why must the random-effect distribution be integrated in $L_{ik}$ under the stated model?
For $D=80$ mg, $V=16$ L, $CL_F=4$ L/h, and $t=4$ h, calculate the fast-class typical concentration.
What does a fitted $\pi_F=0.35$ say, and what does it not say about a particular subject?

Solutions. 1. The numerator is $0.20(0.006)=0.0012$; the denominator is $0.0012+0.80(0.002)=0.0028$. Thus the posterior is $0.0012/0.0028=0.4286$. 2. The subject’s random clearance is unobserved and distributed within the class, so the class likelihood must average the conditional data likelihood over that distribution rather than pretend the random effect is known. 3. The starting concentration is $80/16=5.00$ mg/L and the exponent is $-(4)(4)/16=-1$, giving $5.00e^{-1}=1.839$ mg/L. 4. It is the model-estimated population prevalence of the fast class before conditioning on a particular subject’s data; that subject’s posterior probability depends on their class likelihoods as well.