Two Explanations for Clearance Differences
A population may show substantial variation in clearance for at least two different reasons. Individuals might belong to distinct latent subpopulations, each with its own clearance distribution. Alternatively, clearance might vary continuously across individuals without a discrete class structure. Both explanations can produce a wide range of observed concentrations, and sparse sampling or measurement noise can make their predictions look similar.
The comparison is therefore not simply a matter of looking at individual clearance estimates and deciding whether they form two groups. It asks whether a class-based population model predicts the observed concentration data better, and more plausibly, than a continuous random-effects model under comparable assumptions about the pharmacokinetic structure, covariates, and residual error.
Specify Both Population Models
Let \(y_i\) denote all concentration observations for individual \(i\), and let \(\eta_i\) be that individual’s random effect on log clearance. Write clearance as \(CL_i=CL_{\mathrm{typ}}\exp(\eta_i)\), where \(CL_{\mathrm{typ}}\) is a typical clearance value. The common structural model and observation model determine the conditional likelihood \(p(y_i\mid\eta_i,\theta)\), where \(\theta\) collects the remaining model parameters.
In a two-class mixture, a latent class indicator \(Z_i\) is 1 or 2, with class probabilities \(\pi_1\) and \(\pi_2=1-\pi_1\). Within each class, log-clearance effects follow a class-specific distribution. A common specification is normal within class. In a continuous comparator, the same log-clearance effect follows one continuous population distribution, commonly a normal distribution. Other continuous distributions are possible; the chosen distribution must be stated because it affects the comparison.
For a two-class model with \(\eta_i\mid Z_i=k\sim N(m_k,\omega_k^2)\), the marginal likelihood contribution for one individual is
where \(\phi(\cdot;m,v)\) is a normal density with mean \(m\) and variance \(v\). For the continuous normal comparator, \(\eta_i\sim N(m,\omega^2)\), and
These are marginal likelihoods: they average each individual’s conditional data likelihood over the relevant population distribution. This distinction matters. A mixture of two latent distributions is not the same as splitting observations into two visibly separate groups, and the continuous model does not imply that all individuals have similar clearance. Both models should use the same structural and residual-error assumptions when the purpose is to compare their clearance distributions. Covariate effects should also be handled comparably.
The main assumptions are that individuals are independent conditional on the population model, the specified likelihood adequately represents each individual’s concentration data, and the random-effects distribution or mixture is an appropriate description of between-individual variation. A mixture also assumes a meaningful class probability and class-specific distributions. A normal continuous comparator assumes one normal distribution on the log-clearance scale; it should not be described as representing every conceivable continuous distribution.
Why Latent Distributions and Observed Profiles Can Differ
A fitted distribution of \(\eta_i\) describes population variation before measurement error is added. The observed concentration data are less direct: they arise from the structural model, the individual’s random effect, sampling times, and residual error. Consequently, a clear difference between two fitted latent distributions may produce only a small difference in observed-data predictions when the design is sparse or noisy.
The reverse caution also applies. A few extreme concentrations do not by themselves establish a latent class. They could reflect continuous tail variation, residual error, an inadequate structural model, or covariate misspecification. A useful comparison considers both the implied population distribution and the predictions for complete concentration profiles under the actual design.
Worked Example: Measurement Error Blurs a Mixture
Suppose, for illustration, that a well-informed summary \(W_i\) of an individual’s log clearance is measured with independent normal error: \(W_i=\eta_i+\epsilon_i\), where \(\epsilon_i\sim N(0,0.25^2)\). Compare an equal-probability mixture with class means \(-0.35\) and \(0.35\), each with within-class standard deviation \(0.10\), to a single normal distribution matched to the mixture’s mean and variance.
The mixture mean is \(0\), and its variance is the within-class variance plus the variance of the two class means:
Thus, the matched continuous comparator has \(\eta_i\sim N(0,0.1325)\). For the observed summary \(W_i\), the mixture components each have variance \(0.10^2+0.25^2=0.0725\), so their standard deviation is \(0.2693\). The continuous model gives \(W_i\sim N(0,0.1325+0.25^2)\), with standard deviation \(\sqrt{0.195}=0.4416\).
Consider the predictive probability that \(W_i\) lies between \(-0.15\) and \(0.15\). Under the mixture, the probability in either component is the same by symmetry. For the component with mean \(0.35\), it is
The mixture therefore predicts about 19.7% of summaries in this central interval. The matched normal predicts \(\Phi(0.15/0.4416)-\Phi(-0.15/0.4416)\approx 0.266\), or 26.6%. Even after measurement error smooths the distributions, their predictions differ. In real NLME data, the summary \(W_i\) is only an illustration; the comparison should use the likelihood for the complete concentration records.
Compare Predictions for Concentration Profiles
A mixture with separated clearance classes can predict two concentrations of population probability around distinct trajectory patterns. A continuous distribution instead spreads probability across a continuum of clearances. Whether this difference is visible depends on sampling times, residual variability, and within-class variability. The two models need not predict clearly separated observed concentrations even when their latent distributions differ.
Worked Example: Clearance Classes and Predicted Concentrations
Consider a hypothetical one-compartment intravenous bolus model with dose \(100\) mg and volume \(V=10\) L. For this illustration only, assume each class has fixed clearance, with no within-class variation or residual error: class 1 has \(CL=2\) L/h and class 2 has \(CL=5\) L/h. The concentration prediction is \(C(t)=(\mathrm{Dose}/V)\exp(-CL\,t/V)\).
At \(t=2\) h, the class-specific predictions are
and
At \(t=4\) h, the corresponding predictions are \(10\exp(-0.8)\approx 4.493\) mg/L and \(10\exp(-2)\approx 1.353\) mg/L. The two class trajectories start at the same concentration but diverge over time because clearance changes the elimination rate. A continuous clearance model assigns probability to intermediate trajectories as well as to values near these two. With real residual and within-class variability, the class patterns broaden, so repeated concentration-time observations are more informative than a single concentration. This calculation illustrates predictions; it does not establish that the hypothetical population truly has two classes.
Distributional Assumptions Can Change the Story
A two-class normal mixture and a single normal random effect make specific, different shape assumptions. A normal distribution on log clearance is unimodal. A two-class mixture can be unimodal or bimodal, depending on how far apart its component means are relative to their within-class standard deviations. But observing a poor fit from the single normal comparator does not by itself prove discrete classes: another continuous distribution could fit better.
For an equal mixture with component means \(-a\) and \(a\) and common standard deviation \(s\), symmetry makes the mixture density’s derivative zero at the center. Differentiating the normal density twice shows that the sign of the mixture’s second derivative at zero is the sign of \(a^2/s^4-1/s^2\). Therefore, if \(a>s\), the center is a local minimum, which indicates two modes for this symmetric equal mixture. If \(a<s\), the center is a local maximum. This condition describes the idealized latent density; observation error can blur or eliminate visible separation in the data.
Worked Example: Checking Whether a Mixture Is Bimodal
Take an equal mixture of \(N(-0.35,0.10^2)\) and \(N(0.35,0.10^2)\) for log clearance. Here \(a=0.35\) and \(s=0.10\), so \(a>s\). The second derivative at the center is positive because
Thus zero is a local minimum of the mixture density, with two modes on either side. This supports describing this particular latent distribution as bimodal. It does not show that a data set came from such a mixture: likelihood, measurement error, and the sampling design still determine whether the two-component model is supported by observations.
Sensitivity analysis should ask whether the mixture-versus-continuous conclusion persists under reasonable alternatives for the continuous distribution and other influential assumptions. For example, a heavy-tailed continuous distribution may accommodate extreme values better than a normal distribution, while an asymmetric continuous distribution may accommodate skewness. These alternatives do not automatically reproduce two separated modes, but they may reduce the apparent advantage of a mixture. A highly flexible continuous distribution can even be multimodal, in which case the distinction may be more about how the distribution is parameterized and interpreted than about the shape it can represent.
Practical Comparison and Failure Modes
A disciplined comparison keeps the structural model, covariates, observation model, and data inclusion rules aligned. It then examines what each fitted model predicts for the observed concentrations and for new individuals under the same sampling design. Predictive checks can target the distribution of concentrations at informative times, the prevalence of unusually high or low profiles, and the frequency of trajectories resembling the putative class-specific patterns.
The fitted latent clearance distribution is useful, but it is not observed directly. Empirical Bayes estimates are individual estimates informed by both the individual’s data and the population distribution. As discussed in the earlier material on shrinkage, sparse or noisy records can pull estimates toward population values and obscure individual differences. A histogram of those estimates is therefore not a direct histogram of the true population random effects, and apparent clusters in it are not conclusive evidence of classes.
The comparison can fail when the design contains little information about clearance, when class-specific clearance values are weakly identified, or when there are too few individuals to estimate a flexible population distribution. Model misspecification can also create false separation: a class mixture might absorb an omitted covariate effect or a structural-model error. Conversely, a single normal distribution can hide meaningful subpopulation patterns when the data are informative. Interpret the result as conditional on the candidate models and assumptions, not as a model-free discovery of biological groups.
Information criteria or other summaries of relative fit can contribute to the assessment, but they do not replace predictive checks or sensitivity analysis. In particular, do not treat a better fit under one assumed continuous distribution as proof against every continuous explanation, or a better fit under a mixture as proof of discrete biological classes. The goal here is to decide whether the mixture provides a robust and useful description of the data compared with specified continuous alternatives.
Check Your Understanding
Answer each question using the model comparison principles above. Solutions follow the questions.
- In a two-class model, what is integrated over when calculating an individual’s marginal likelihood?
- A normal continuous model and a two-class mixture have the same mean and variance on log clearance. Does that imply they predict the same distribution?
- Why might a mixture’s two latent clearance modes be difficult to see in observed concentration data?
- What does the condition \(a>s\) imply for an equal mixture of normal distributions with means \(-a\) and \(a\) and common standard deviation \(s\)?
- Why is a histogram of empirical Bayes estimates not sufficient to establish two clearance classes?
Solutions. (1) Integrate the conditional likelihood over the random effect and, in the mixture, sum the two class-specific integrated likelihoods weighted by their population probabilities. (2) No. Matching the first two moments does not match distributional shape; a separated mixture can have a central valley where the matched normal has its highest density. (3) Residual error, within-class variation, sparse sampling, and the nonlinear concentration model can blur differences in latent clearance into overlapping observed profiles. (4) The center is a local minimum of the symmetric mixture density, so it is bimodal in this idealized setting. (5) Empirical Bayes estimates are informed by the population model and can be shrunken toward population values, particularly when individual records are sparse or noisy; their histogram is not the latent population distribution itself.