Interpreting Size and Maturation Effects Together

Why Separate the Contributions?

A predicted clearance for a child may differ from another child’s predicted clearance because the children differ in both body weight and developmental maturity. A single prediction ratio describes the overall contrast, but it does not say how much of that contrast is associated with the size term and how much with the maturation term. The model equation provides a principled way to separate those factors.

The separation is useful for interpreting a model, checking calculations, and explaining pediatric dose-scaling predictions. It is a model-based decomposition: it describes how the specified equation changes when its covariates change. By itself, it does not establish that changing one covariate would cause a particular clinical outcome.

The Combined Prediction and Its Assumptions

Consider a typical clearance prediction with fixed allometric exponent $0.75$ and a sigmoidal Hill maturation function. Let $WT$ be body weight in kilograms and $a$ be age in years, measured on the age scale used to fit the model. Let $CL_{70}$ be the clearance scale at the 70-kg reference weight and at the maturation plateau. The parameters $TM$ and $h$ are the half-maturation age and Hill coefficient, respectively.

Definition: In this tutorial, the combined typical clearance prediction is $$ CL(WT,a)=CL_{70}\left(\frac{WT}{70}\right)^{0.75}M(a), \qquad M(a)=\frac{a^h}{TM^h+a^h}. $$ The size factor is $(WT/70)^{0.75}$, and the maturation factor is $M(a)$, which ranges from 0 toward 1 for positive age and positive $TM,h$. At $a=TM$, $M(a)=0.5$.

Interpreting two predictions as separate size and maturation contributions requires that both are evaluated under the same model: the same $CL_{70}$, allometric exponent, $TM$, $h$, and age definition. The comparison also assumes positive weights and ages for this Hill form. These are typical population predictions; individual random effects and residual error are not included in the equation above.

The multiplicative form matters. Size and maturation each act as a factor on the same clearance scale, so the combined ratio is the product of the two component ratios. An additive decomposition of clearance differences would not, in general, match this model.

Deriving the Ratio Decomposition

For two profiles, label the first profile 1 and the second profile 2. Dividing their predictions cancels the common clearance scale. Rearranging the resulting expression identifies the size and maturation terms separately.

\begin{aligned} \frac{CL(WT_1,a_1)}{CL(WT_2,a_2)} &= \frac{CL_{70}(WT_1/70)^{0.75}M(a_1)} {CL_{70}(WT_2/70)^{0.75}M(a_2)}\\ &= \left(\frac{WT_1}{WT_2}\right)^{0.75} \frac{M(a_1)}{M(a_2)}. \end{aligned}

The first factor is the size-only ratio: it compares the profiles’ weights while holding maturation equal. The second is the maturation-only ratio: it compares their maturation values while holding size equal. Their product is the combined prediction ratio. On the log scale, multiplication becomes addition:

\log\left(\frac{CL_1}{CL_2}\right) = 0.75\log\left(\frac{WT_1}{WT_2}\right) + \log\left(\frac{M(a_1)}{M(a_2)}\right).

This identity is a concise way to describe the model’s relative contributions. It is not a unique decomposition of an absolute clearance difference into two additive amounts: the difference $CL_1-CL_2$ depends on the scale at which the factors are applied. The ratio decomposition is exact for the specified multiplicative model.

Worked Example: Comparing Two Pediatric Profiles

Suppose a hypothetical model has $CL_{70}=4.00$ L/h, $TM=2.00$ years, and $h=2$. Compare a child weighing 20.0 kg at age 8.00 years with a child weighing 10.0 kg at age 2.00 years. These values are illustrative model inputs, not observations from a real study.

First calculate the maturation factors: $$ M(8)=\frac{8^2}{2^2+8^2}=\frac{64}{68}=0.9412, \qquad M(2)=\frac{2^2}{2^2+2^2}=\frac{4}{8}=0.5000. $$ The maturation-only ratio is $0.9412/0.5000=1.8824$. At the older age, the model assigns a larger maturation factor.

Next calculate the size-only ratio: $$ \left(\frac{20.0}{10.0}\right)^{0.75} =2^{0.75}=1.6818. $$ Thus, at the same maturation value, the 20.0-kg profile has a size-scaled prediction 1.6818 times that of the 10.0-kg profile.

The combined ratio is the product: $$ 1.6818\times1.8824\approx3.166. $$ The full predictions give the same result. For the 20.0-kg, 8.00-year profile, $$ CL=4.00\left(\frac{20.0}{70}\right)^{0.75}\frac{64}{68} \approx1.471\text{ L/h}. $$ For the 10.0-kg, 2.00-year profile, $$ CL=4.00\left(\frac{10.0}{70}\right)^{0.75}\frac{4}{8} \approx0.4647\text{ L/h}. $$ Their ratio using the displayed predictions is approximately $1.471/0.4647=3.165$, which differs slightly from the full-precision model ratio, approximately $3.166$, because of rounding the displayed predictions. The overall contrast is larger than the size-only contrast because the maturation factor is also larger in the first profile.

Counterfactual Comparisons to Isolate Each Factor

A practical interpretation strategy is to compare predictions while changing one input at a time. These intermediate predictions need not represent actual children; they are controlled model comparisons that show what each factor contributes. They should use the same parameters and reference scale as the full comparison.

Calculate the full predictions.
Evaluate the model at each profile’s weight and age, then take their ratio.

Hold maturity fixed.
Use the same maturation value in both predictions. The ratio then reduces to $(WT_1/WT_2)^{0.75}$.

Hold size fixed.
Use the same weight in both predictions. The ratio then reduces to $M(a_1)/M(a_2)$.

Check the factorization.
Multiply the two component ratios and verify that the product agrees with the full prediction ratio before rounding.

Worked Example: Isolating Maturation at a Common Weight

Use the hypothetical parameters from the previous example and compare predictions at a common weight of 10.0 kg, first at age 2.00 years and then at age 8.00 years. The common size term is $$ 4.00\left(\frac{10.0}{70}\right)^{0.75}\approx0.9295\text{ L/h}. $$

At age 2.00 years, the prediction is $0.9295\times0.5000\approx0.4647$ L/h. At age 8.00 years, it is $0.9295\times0.9412\approx0.8748$ L/h. Therefore, $$ \frac{CL(10.0,8.00)}{CL(10.0,2.00)} =\frac{0.9412}{0.5000} =1.8824. $$ The common size term cancels. This contrast isolates the maturation factor within the model; it should not be described as a clinical effect of aging without additional evidence and an appropriate study design.

Worked Example: Reporting Ratios Without Rounding Errors

A ratio calculated from unrounded model output can differ in its final reported digit from a quotient calculated using the displayed predictions. For a hypothetical size-only comparison, suppose the full-precision ratio is approximately 1.6818. If the displayed predictions are 0.8963 L/h and 0.5329 L/h, dividing those displayed values gives $$ \frac{0.8963}{0.5329}\approx1.68193, $$ which rounds to 1.6819 at four decimal places. The full-precision model ratio is approximately 1.6818; the quotient of the displayed, rounded predictions is approximately 1.6819. These are different calculations and should not be presented as though they must yield identical rounded digits.

The same issue arises for a larger contrast. Suppose the full-precision model calculation gives a ratio of approximately 11.676447, while the displayed predictions are 0.89628 L/h and 0.0767594 L/h. Dividing the displayed values gives $$ \frac{0.89628}{0.0767594}\approx11.676485, $$ which rounds to 11.6765 at four decimal places. The full-precision model ratio rounds to 11.6764. To report the model ratio, retain full precision through the division and round only the final result; if calculating from printed predictions, identify that quotient as an approximation based on rounded values.

Checks, Alternatives, and Limitations

Several checks help reveal interpretation or calculation problems. The size-only ratio should be greater than one when $WT_1>WT_2$, because the exponent is positive. The maturation-only ratio should track the relative maturation values, not the absolute ages: doubling age does not generally double maturation. The combined ratio should equal the product of the component ratios before rounding. Also check units: the factors and ratios are dimensionless, while each predicted clearance has units of volume per time.

A common mistake is to attribute the whole combined ratio to body size because weight is the more visible difference. For the first worked example, that would report 1.6818 as the total contrast, even though the maturation ratio is 1.8824 and the combined ratio is about 3.166. The correction is to evaluate both model factors and multiply their ratios. Another mistake is to add the size and maturation ratios. The model is multiplicative, so $1.6818+1.8824$ is not the combined ratio.

An alternative is to report only the two predicted clearances and their total ratio. That may be sufficient when the goal is prediction, but it conceals the model’s separate factor contributions. Reporting the component ratios alongside the total makes the interpretation more transparent. The log-scale expression is also useful because it expresses the total log-ratio as the sum of the size and maturation log-ratios; it does not change the model or imply additive effects on the clearance scale.

This interpretation is conditional on the chosen model form and parameter values. If the model does not factor size and maturation multiplicatively, the ratio need not separate into these two terms. Even when it does, the decomposition does not prove that size and age effects are independently identified by the available data, establish a causal effect, or account for uncertainty in parameter estimates. Predictions can also be sensitive to the assumed maturation function and to extrapolation beyond the ages and weights that informed the model. The decomposition explains the equation; it does not by itself validate it.

Key takeaway: Under a multiplicative size–maturation model, the combined clearance ratio is exactly the size-only ratio multiplied by the maturation-only ratio. Evaluate both factors on the same model scale, verify their product, and distinguish full-precision calculations from quotients of rounded predictions.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

For the stated allometric exponent, what is the size-only ratio for weights of 16 kg and 8 kg?
With $h=2$ and $TM=2$ years, calculate the maturation-only ratio for ages 4 and 2 years.
If the size-only ratio is 1.5 and the maturation-only ratio is 0.8, what is the combined ratio? Interpret its direction.
Why can the quotient of displayed predictions differ slightly from a ratio calculated using full-precision predictions?

Solutions. (1) The size-only ratio is $(16/8)^{0.75}=2^{0.75}\approx1.682$. (2) $M(4)=16/(16+4)=0.8000$, while $M(2)=4/(4+4)=0.5000$, so the maturation ratio is $0.8000/0.5000=1.600$. (3) The combined ratio is $1.5\times0.8=1.20$. Profile 1’s predicted clearance is 1.20 times profile 2’s, so it is 20% higher under the model despite the maturation factor reducing the contrast. (4) Rounding changes the displayed numerator and denominator before division; the quotient of those rounded values need not round to the same final digit as a ratio calculated before rounding. Retain full precision until the final reporting step.