Why Identifiability Matters
A two-class model can converge and still provide little reliable information about its class-specific clearance values. If the classes produce similar concentration profiles, many combinations of fast clearance, slow clearance, and class prevalence may fit nearly as well. An optimizer may return a precise-looking set of numbers, but the data may not distinguish those numbers from alternatives.
Here, identifiability concerns whether the observed-data likelihood meaningfully distinguishes parameter values. Structural identifiability asks whether distinct parameter values could produce exactly the same probability distribution, even with unlimited data. Practical identifiability asks whether the available design and data distinguish the values well enough to estimate them stably. This tutorial focuses on diagnosing practical weakness in class-specific clearance and prevalence, assuming the two-class model and its class ordering have already been specified.
The Likelihood Features Behind Weak Separation
For subject \(i\), let \(L_{iF}\) and \(L_{iS}\) be that subject's likelihood contributions under the fast and slow classes. Each contribution includes the within-class NLME model, including integration over individual random effects where applicable. If subjects are independent and class membership is latent, the subject's marginal likelihood contribution is the class-weighted sum, not a product of class-specific contributions:
This likelihood assumes that the stated class-specific concentration models and residual model are appropriate, that \(\pi_F\) is a valid probability, and that the subject contributions are correctly constructed. It also assumes that the numerical likelihood calculation is accurate enough to distinguish genuine flatness from integration or optimization error. The ordering \(CL_F>CL_S\) fixes the class names, but it does not ensure that the data can distinguish the classes.
Three complementary views are especially useful. A profile likelihood shows how well a parameter is determined after allowing other parameters to adjust. A parameter-correlation or information-matrix view shows local trade-offs near the optimum. A class-specific information view asks whether the observed subjects and sampling schedule actually contain useful information for each class. None is conclusive alone.
Profile Likelihoods
To profile a parameter such as \(CL_F\), fix it at a sequence of values and re-optimize all remaining parameters at each value. Plot or tabulate the resulting maximized log likelihood, \(\ell_p(CL_F)\). A flat profile means that the data do not strongly favor one value over a range of alternatives. A sharp, well-defined maximum is more reassuring, although it does not guarantee that the model is correct.
Under regularity conditions, a rough likelihood-based 95% interval for one profiled parameter consists of values satisfying \(2\{\ell_{\max}-\ell_p\}\leq 3.84\), the 95th percentile of a chi-square distribution with one degree of freedom. This cutoff is a diagnostic reference, not an automatic guarantee of valid coverage. Mixture models can have boundary or other nonregular features, so formal interval and testing procedures may require additional justification.
Worked Example: A Broad Profile for Fast-Class Clearance
Suppose a hypothetical NLME fit estimates \(CL_F\) in L/h. At each fixed value below, all other model parameters, including \(CL_S\) and \(\pi_F\), have been re-optimized. The maximum log likelihood on this grid is \(-98.0\).
| Fixed \(CL_F\) (L/h) | Profile log likelihood | \(2\{\ell_{\max}-\ell_p\}\) |
|---|---|---|
| 6 | -104.0 | 12.0 |
| 8 | -100.0 | 4.0 |
| 10 | -98.2 | 0.4 |
| 12 | -98.1 | 0.2 |
| 14 | -99.5 | 3.0 |
| 16 | -102.0 | 8.0 |
For example, at 14 L/h, \(2\{-98.0-(-99.5)\}=3.0\). The grid values 10, 12, and 14 L/h fall within the usual one-parameter reference cutoff of 3.84, while 8 and 16 L/h do not. The profile is therefore consistent with a broad range of plausible fast-class clearances rather than a sharply determined value. The table is only a coarse grid, so it does not locate exact interval endpoints. The result is evidence of weak practical identification, not proof that \(CL_F\) is impossible to estimate.
Parameter Correlations and Local Information
Near a fitted optimum, the inverse of the observed information matrix approximates the covariance matrix of the parameter estimates when the likelihood is sufficiently regular and locally quadratic. Large absolute correlations between estimates can signal a ridge: moving one parameter in one direction can be compensated by moving another. The approximation can be unreliable when the likelihood is strongly nonquadratic, a parameter is near a boundary, or the information matrix is nearly singular. Thus, inspect the matrix alongside profiles rather than treating a correlation as a stand-alone verdict.
Worked Example: A Near-Ridge Between Log Clearances
For a hypothetical fit, suppose the estimated standard errors of \(\log(CL_F)\) and \(\log(CL_S)\) are 0.30 and 0.25, and their estimated covariance is 0.072. The corresponding variances are \(0.30^2=0.0900\) and \(0.25^2=0.0625\). Their estimated correlation is
This very high positive local correlation indicates that the two log-clearances can move together while producing similar fitted likelihoods. The covariance matrix is \(\begin{pmatrix}0.0900 & 0.072\\0.072 & 0.0625\end{pmatrix}\). Its eigenvalues are approximately 0.1495 and 0.0030, a ratio of about 50. The small eigenvalue indicates a direction of much lower local curvature than the other direction. Together, these calculations suggest a poorly conditioned estimate of the two clearances. They do not show that the class labels are wrong or establish that the model should be replaced.
A correlation can be misleading if it is calculated on a parameter scale that obscures the important trade-off. Clearance is positive, so examining log-clearance parameters is often useful: changes on that scale correspond to relative rather than absolute changes. Also check whether apparent high correlation is driven by weak information in just one parameter, an active parameter bound, or an inaccurate Hessian calculation.
Class-Specific Information in the Subjects
A fitted mixture assigns each subject a posterior probability of belonging to each class, conditional on that subject's data and the fitted parameters. For the fast class, this responsibility is
The sum \(\sum_i r_{iF}\) is a useful descriptive measure of the expected number of subjects allocated to the fast class under the fitted model. It is not a count of known class members. Further, a subject contributes strongly to estimating a class clearance only if the within-class concentration data are informative about clearance: measurements must respond to clearance under the specified model, and the sampling schedule must capture that response. Many subjects with little clearance-sensitive information do not necessarily identify a class-specific clearance.
A useful conceptual decomposition is that the marginal log-likelihood sensitivity for a fast-class parameter is weighted by the subject's fast-class responsibility. If \(\alpha_F=\log(CL_F)\), then
Thus, if most \(r_{iF}\) values are small, or if the class-specific likelihood is itself insensitive to \(\alpha_F\), information about fast clearance will be limited. This identity concerns the score; the observed information also includes changes in responsibilities and cross-parameter terms. A simple responsibility-weighted sum is therefore a diagnostic aid, not a replacement for the full marginal information matrix.
Worked Example: Few Subjects Inform the Fast-Class Clearance
Consider four hypothetical subjects with fitted fast-class responsibilities \(0.90,\ 0.70,\ 0.40,\ 0.10\). The expected fast-class count is \(0.90+0.70+0.40+0.10=2.10\) subjects, and the expected slow-class count is \(4-2.10=1.90\). This indicates that the fit does not assign a large, clearly separated group to either class.
For a rough illustration, suppose the within-fast-class information about \(\alpha_F=\log(CL_F)\), if class membership were known, is \(4.0,\ 1.0,\ 0.20,\ 0.10\) information units for these subjects. A responsibility-weighted complete-data information proxy is \(0.90(4.0)+0.70(1.0)+0.40(0.20)+0.10(0.10)=3.6+0.70+0.08+0.01=4.39\) information units. The first subject supplies most of this proxy; the last contributes almost none.
This calculation is deliberately a proxy, not the observed information from the mixture likelihood: uncertainty in class membership and parameter cross-terms affect the latter. It shows why class count alone is insufficient. If the few subjects with high fast-class responsibility also have weak concentration sensitivity to clearance, the fast-class estimate can remain unstable even when the model reports a nonzero expected class size.
Diagnosing Prevalence Instability
Prevalence can be weakly identified when the two class-specific likelihoods are similar across subjects. To see why, hold the class likelihoods fixed temporarily and differentiate the subject log likelihood with respect to \(\pi_F\):
If \(L_{iF}\) and \(L_{iS}\) are nearly equal, the numerator is near zero, so that subject's likelihood changes little as prevalence changes. When the class-specific parameters are also re-optimized, a change in \(\pi_F\) may be offset by changes in the clearances, making the profile for prevalence flatter still.
Worked Example: A Flat Profile for Fast-Class Prevalence
In a hypothetical fit, the profile log likelihood for \(\pi_F\), after re-optimizing the other parameters, is as follows. The maximum is \(-49.8\) at \(\pi_F=0.50\).
| Fixed \(\pi_F\) | Profile log likelihood | \(2\{\ell_{\max}-\ell_p\}\) |
|---|---|---|
| 0.10 | -51.8 | 4.0 |
| 0.20 | -50.7 | 1.8 |
| 0.30 | -50.1 | 0.6 |
| 0.40 | -49.9 | 0.2 |
| 0.50 | -49.8 | 0.0 |
| 0.60 | -49.9 | 0.2 |
| 0.70 | -50.2 | 0.8 |
| 0.80 | -50.8 | 2.0 |
| 0.90 | -51.7 | 3.8 |
For instance, at \(\pi_F=0.80\), the likelihood-ratio quantity is \(2\{-49.8-(-50.8)\}=2.0\), below 3.84. The profile therefore supports substantial variation in prevalence relative to the maximum; values from 0.20 through 0.80 on this grid are within the usual reference cutoff. The fitted value 0.50 should not be reported as though it were precisely determined. Because prevalence can approach a boundary and mixture likelihoods may be nonregular, this profile is best read here as a diagnostic of instability, not as a definitive confidence interval.
A Practical Diagnostic Sequence
Confirm that the fast/slow ordering is enforced, parameter bounds are understood, and likelihood evaluations are stable across reasonable starting values and numerical settings.
Fix one target parameter at a time and re-optimize the rest. Look for broad plateaus, asymmetry, multiple peaks, or endpoints determined only by imposed bounds.
Review standard errors, parameter correlations, eigenvalues or conditioning of the information matrix, and whether the local quadratic approximation agrees with the profiles.
Summarize responsibilities alongside the concentration-time information available for each class. Do not interpret soft assignments as observed class labels.
If evidence is weak, report that the data do not sharply distinguish the class-specific parameter values or prevalence. Do not turn optimizer convergence into a claim of precise identification.
Failure Modes and Limits
A flat profile can reflect true lack of information, but it can also result from incomplete re-optimization, a parameter grid that is too coarse, or numerical error. Repeat profile fits from multiple reasonable starting values, check that nuisance parameters have converged at every grid point, and refine the grid near apparent turning points. If different starts yield materially different likelihoods, resolve the optimization issue before interpreting the profile as an identifiability result.
Likewise, a nearly singular Hessian is a warning, not a diagnosis by itself. Hessians can be unstable under poor scaling or inaccurate likelihood derivatives. Compare the result with profile shapes and, where feasible, inspect derivatives or information calculations using more accurate numerical settings. Conversely, a benign-looking Hessian near one optimum cannot rule out a second optimum or a long nonquadratic ridge.
These diagnostics answer whether the fitted two-class model estimates its parameters stably; they do not determine whether a discrete mixture is preferable to a different model of clearance variability. That is a separate model-comparison question. Nor do they establish that a class has a biological interpretation. Their practical value is to prevent weakly separated estimates and unstable prevalence from being presented as certain findings.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A clearance profile has a broad plateau but the optimizer reports convergence. What does the plateau imply, and what does convergence establish?
- For \(\pi_F=0.30\), a profiled log likelihood is \(-70.4\), while the maximum is \(-69.0\). Calculate the likelihood-ratio quantity and compare it with 3.84.
- Why can a large expected number of subjects assigned to a class still fail to identify that class's clearance?
- What does a correlation of 0.97 between log-clearance estimates suggest, and why is it not conclusive on its own?
Solutions. (1) The plateau indicates that a range of clearance values fits nearly as well after nuisance parameters are re-optimized, so practical identification may be weak. Convergence establishes only that the numerical fit met its convergence criterion. (2) \(2\{-69.0-(-70.4)\}=2.8\), which is below 3.84; under the usual regularity-based reference, that value is not sharply separated from the maximum, though the cutoff is only diagnostic here. (3) Subjects may have weak clearance-sensitive sampling, or their class responsibilities may be diffuse; expected class count alone does not measure information about clearance. (4) The estimates show a strong local trade-off, potentially indicating a likelihood ridge. The correlation depends on local quadratic approximation and should be checked against profiles and numerical stability.