Estimating Between-Child Variability in Maturation

Population Maturation and Child-to-Child Differences

A maturation curve summarizes how a physiological capacity changes with age across a population. It does not, by itself, say whether children of the same age are nearly alike or differ substantially around that curve. That distinction matters in pediatric PBPK modeling: a curve can describe the typical age trend well while still underrepresenting the range of capacities among individual children. Conversely, apparent spread in pediatric observations does not automatically establish biological variability. Measurement error, an unsuitable mean curve, and differences in sampling can also create residual variation.

The focus here is to estimate and interpret variation between children after accounting for the population age trend. The introductory material on fitting maturation curves to sparse age-stratified data provides the context for choosing a mean curve. The additional question is whether the data contain enough information to support child-level deviations from it.

A useful model separates the age trend from child-level deviations. Let \(y_{ij}\) be a measurement for child \(i\) at occasion \(j\), expressed on a scale appropriate to the measurement. For example, \(y_{ij}\) might be the logarithm of a measured capacity relative to adult capacity. A simple random-intercept model is

Here \(a_{ij}\) is age, \(f(a_{ij};\theta)\) is the population curve on the modeled scale, \(b_i\) is the child-specific deviation, and \(\epsilon_{ij}\) is residual measurement or occasion-level error. The model assumes that child effects are independent across children, residual errors are independent across occasions and children, and \(b_i\) is independent of \(\epsilon_{ij}\). In this simple form, the child deviation is constant over the ages observed for that child; the model does not claim that every biological difference must have this form.

The parameter \(\omega^2\) is the between-child variance on the model scale, and \(\omega\) is its standard deviation. The residual variance \(\sigma^2\) describes variation not attributed to the mean curve or child effect. These components have different interpretations: repeated measurements from one child share \(b_i\), but their independent residual errors need not be shared.

The scale matters. If \(y\) is log capacity, \(f\) is the population mean of log capacity under this model; exponentiating \(f\) gives a geometric-mean-type quantity, not generally the arithmetic mean capacity. Under the stated normal random-intercept model, the arithmetic mean capacity at age \(a\), if capacity is \(\exp(y)\), is \(\exp\{f(a;\theta)+\omega^2/2+\sigma^2/2\}\) for a single observation including residual error. Therefore, an interpretation such as “the average maturation” must specify whether it refers to the modeled scale, the child-level capacity, or an observation that also includes residual error.

Estimating a Between-Child Variance

A basic variance-components calculation shows why repeated observations are valuable. Suppose, for illustration, that all children have the same number \(r\) of observations at the same modeled age, or that observations have already been adjusted for the fitted age curve. Let \(\bar{e}_i\) be child \(i\)'s mean residual from that curve. Under the model, it is the sum of the child effect and the average residual error:

The equality follows because \(b_i\) and \(\bar{\epsilon}_i\) are independent, \(\operatorname{Var}(b_i)=\omega^2\), and the mean of \(r\) independent residuals of variance \(\sigma^2\) has variance \(\sigma^2/r\). Across children, the sample variance of the child means estimates their total variance. Subtracting the expected contribution of residual error gives a simple method-of-moments estimate:

This calculation is a justification of the estimator, not a universal fitting recipe. It relies on a common \(r\), a suitable age adjustment, independent residual errors, and an estimate of residual variance. In an unbalanced data set, or when children are measured at different ages, a mixed-effects fit or a design-appropriate variance-components method is generally needed. If the subtraction is negative, the data do not provide a positive method-of-moments estimate under this calculation; setting the estimate to zero is a boundary convention, not evidence that biological variation is exactly absent.

Reading Age-Stratified Residuals

Residuals should be interpreted in relation to both age and child identity. A useful first diagnostic is to examine residuals \(e_{ij}=y_{ij}-f(a_{ij};\widehat{\theta})\) by age stratum, while retaining the information about which observations belong to the same child. A residual cloud centered near zero at each age, with repeated observations from a child tending to remain similarly above or below zero, is compatible with a child-level random intercept. A systematic shift in the average residual across age strata instead suggests that the population curve may be missing an age trend.

The distinction is important because a random effect is not a repair for a poorly fitted mean curve. If the curve systematically underpredicts values in one age range and overpredicts them in another, a random intercept may inflate the estimated spread while leaving age-related bias unresolved. Conversely, if the fitted curve is appropriate but observations from the same child cluster together, ignoring the child structure can make residuals appear to be independent when they are not.

What Sparse Data Can and Cannot Establish

Whether individual-level variability can be estimated depends on the data structure, not only on the number of observations. Repeated measurements per child help distinguish persistent child differences from occasion-level noise because observations from the same child share the random effect. Observations spread across ages help characterize the mean curve, but if each age stratum contains different children, age effects and child composition can be difficult to separate. A wide range of observations across children is not, by itself, evidence for a child random effect.

Common alternatives answer different questions. A fixed-effect model with an age curve but no child effect estimates the population trend while treating unexplained observations as residual error; it does not estimate persistent between-child variation. A separate curve or parameter estimate for every child may describe observed children but can overfit sparse records and does not automatically provide a population distribution of child effects. A random-effects model partially pools child estimates toward the population curve, but its variance estimate remains dependent on the assumed curve and error model. None of these approaches can manufacture information absent from the sampling design.

For a PBPK application, the practical implication is to avoid treating a population maturation curve as a complete description of every child unless the data support that simplification. At the same time, adding a random effect solely because pediatric observations scatter around the curve risks confusing model error with biological heterogeneity. The variance estimate is most useful when its scale, assumptions, uncertainty, and intended use in individual-level predictions are all stated.

Assumptions, Diagnostics, and Common Pitfalls

• Check residual centers by age. Persistent positive or negative means in particular age strata indicate possible mean-curve bias. Assess the age trend before attributing all unexplained spread to child effects.
• Check within-child dependence. If repeated observations from a child are more alike than observations from different children, an independent-error model may understate uncertainty or misallocate variance.
• Check whether spread changes with age. A single \(\omega^2\) assumes constant child-level variance on the modeled scale. Markedly different spreads across age strata challenge that assumption; sparse strata, however, can make apparent differences unstable.
• Check the scale and transformation. A standard deviation on a log scale is not a standard deviation in capacity units. For log-scale variation, \(\exp(\omega)\) is a multiplicative factor corresponding to one standard deviation in the child effect, not an additive change in capacity.
• Check what is identifiable. With one observation per child and no independent estimate of residual error, the two variance components cannot generally be separated by the simple model.

A frequent mistake is to calculate the spread of all age-adjusted observations and label it “between-child variability.” That spread generally combines child effects and residual error; if the age curve is biased, it can also include systematic age-related deviation. The correction is to define the variance components, account for residual error and repeated-child structure, and inspect age-stratified residuals before interpreting the estimate biologically.