Identifiability of Size and Maturation Effects

Why Size and Maturation Can Be Hard to Separate

Pediatric clearance often changes with both body size and developmental age. The challenge is that size and age usually change together: older children tend to weigh more, and clearance tends to increase with both weight and maturation. If the observed data cover only a narrow age range or provide little variation in weight among children of similar ages, different combinations of size and maturation parameters may explain the observations almost equally well. The fitted model can look adequate while its individual parameter estimates remain uncertain.

This is an identifiability problem. It matters because a model may be used to predict clearance at ages or weights that were not well represented in the data. If size and maturation effects are not separately supported, those predictions can depend strongly on which parameter combination the fitting procedure selected.

A Model and Its Information

Consider a clearance model with body weight \(WT_i\), postmenstrual age \(PMA_i\), a fixed allometric size exponent of 0.75, and a sigmoidal maturation function. Here \(CL_{\mathrm{mat}}\) is the reference clearance scale, \(TM_{50}\) is the age at which the maturation function is one-half, and \(h\) controls its steepness:

For this tutorial, assume clearance is positive, the weight and age measurements are adequate, the model form is appropriate over the modeled range, and the random effect \(\eta_i\) has mean zero on the log scale. For the examples, residual errors in log clearance are independent, normally distributed, and have a common variance. The weight exponent and \(h\) are treated as fixed where stated; estimating additional parameters would generally require more information.

The central issue can be seen by taking logs and examining how the expected log clearance changes as parameters change. Write \(F_i\) for the maturation fraction. The sensitivities of the log-clearance prediction to the main parameters are:

For the second derivative, \(F_i=PMA_i^h/(PMA_i^h+TM_{50}^h)\). Differentiating its logarithm with respect to \(\log(TM_{50})\) gives \(-hTM_{50}^h/(PMA_i^h+TM_{50}^h)\), which equals \(-h(1-F_i)\). Thus, if maturation fractions are similar across the observations, changes in \(TM_{50}\) shift predictions in a similar way across those observations. A compensating change in \(CL_{\mathrm{mat}}\) can then preserve much of the fit. If weight is also strongly associated with age, the size contribution may be difficult to distinguish from the age-related contribution.

This is a local explanation of weak information, not a claim that size and maturation are always confounded. Data spanning different ages at comparable weights, or different weights at comparable ages, can help distinguish their contributions. The earlier material on body-size scaling and maturation functions supplies the model components; the task here is to assess whether the data can estimate them separately.

Parameter Correlations and Profile Likelihoods

A parameter-correlation matrix is a useful first diagnostic. For the model above, a high positive estimated correlation between \(CL_{\mathrm{mat}}\) and \(TM_{50}\) is plausible: increasing \(TM_{50}\) reduces the maturation fraction at a given age, and increasing \(CL_{\mathrm{mat}}\) can partly offset that reduction. A correlation close to either \(+1\) or \(-1\) indicates that the estimates move together strongly in the fitted uncertainty approximation. It is a warning of weak separation, not proof of structural non-identifiability. Correlations can depend on parameter scaling, the estimation method, and the local shape of the likelihood.

A profile likelihood gives a more direct view of the trade-off. To profile \(TM_{50}\), fix it at a sequence of values, re-estimate the other free parameters at each value, and record the resulting maximized likelihood. A broad, shallow profile means many values of \(TM_{50}\) fit nearly as well; a narrow, sharply curved profile around its minimum indicates greater information. A profile can reveal asymmetry or a long tail that a correlation matrix alone may hide.

In a real analysis, a profile is usually summarized using the increase in \(-2\log L\) relative to its minimum, where \(L\) is the likelihood. Under regular large-sample conditions, an increase of about 3.84 is a conventional approximate 95% likelihood-based cutoff for one profiled parameter. These conditions can fail, especially with weak identification, boundaries, or small samples; the cutoff should not be treated as an automatic guarantee of accurate coverage. Inspect the profile itself, the parameter bounds, and whether the fitted nuisance parameters remain plausible along it.

Simulation-Based Recovery Checks

Simulation-based recovery asks whether the planned design and fitting procedure can recover known parameter values when data are generated from the assumed model. Specify the parameter values, covariate distribution, sampling pattern, residual and between-subject variability, and fitting procedure. Generate repeated datasets, fit each one, and examine convergence, bias, uncertainty, parameter correlations, and interval coverage. Recovery is conditional on the simulation model: it assesses whether the design can identify parameters under those assumptions, not whether the assumptions are true.

Diagnostics, Alternatives, and Limitations

Evidence of weak identifiability can include a nearly singular parameter covariance matrix, very high parameter correlations, large standard errors, estimates that change markedly with starting values or bounds, and profile likelihoods that remain nearly flat over a wide range. Simulation recovery can reveal the same problem through biased estimates, poor coverage, frequent boundary estimates, or unstable fits. A small residual error or an apparently good prediction within the observed data does not by itself show that the parameters are separately identified.

The appropriate response depends on the scientific question. Options include collecting data that better separate age from weight, fixing a parameter only when external scientific evidence supports the value, or using a justified prior distribution and reporting how much the result depends on it. A prior can regularize an estimate, but it does not create information in the observed data. When separate estimation is not supported, report the uncertainty and consider whether a combined prediction or parameter combination is more defensible than precise-looking individual estimates.

These approaches answer different questions. Parameter correlations are quick local diagnostics; profile likelihoods explore how fit changes when one parameter is constrained; simulation-based recovery evaluates the whole design-and-analysis procedure under specified assumptions. None can establish that the chosen model is biologically correct. Likewise, fixing the allometric exponent or maturation steepness can improve estimation of remaining parameters while shifting uncertainty into the fixed assumptions. Such choices should be justified and sensitivity-tested rather than made solely to obtain convergence.