Multiplicative Versus Additive Size-Maturation Models

Why the Way Effects Combine Matters

A size effect and a maturation effect can each be expressed as a change relative to a reference clearance. But specifying both effects does not, by itself, say how they combine. A model might multiply their relative effects, or add their deviations from the reference. These choices can give different clearance predictions for the same individual, even when both use the same size and maturation inputs.

The distinction is not merely algebraic. Multiplication says that the factors scale clearance jointly: the absolute change associated with one factor depends on the value of the other. Addition says that the two deviations contribute separately on the clearance scale. The choice therefore affects predictions and how the model’s biological assumptions should be described.

To isolate this distinction, let \(S\) denote a dimensionless size modifier and \(M\) a dimensionless maturation modifier. Each is defined relative to a reference condition, so \(S=1\) and \(M=1\) at that condition. For example, \(S\) could be supplied by a previously specified size-scaling model and \(M\) by a maturation function. The particular choices of those functions are separate from the question of how their effects combine.

Two Formulations on a Common Reference Scale

The simplest multiplicative formulation is

The predicted clearance is the reference clearance multiplied by both dimensionless modifiers. This formulation preserves the reference: setting \(S=M=1\) gives \(\mathrm{CL}_{\mathrm{mult}}=\mathrm{CL}_{\mathrm{ref}}\).

A directly comparable additive formulation is

Here, \(\beta_S\) and \(\beta_M\) are dimensionless coefficients describing the contributions of the deviations from the reference. They are often estimated from data. For a simple comparison in which each modifier contributes its full deviation, set \(\beta_S=\beta_M=1\). This gives \(\mathrm{CL}_{\mathrm{add}}=\mathrm{CL}_{\mathrm{ref}}[1+(S-1)+(M-1)]\). Like the product model, the additive model returns the reference clearance when \(S=M=1\), whatever the coefficient values.

Both models require a consistent reference and meaningful modifiers. If \(S\) or \(M\) has been defined so that its reference value is not 1, the equations must be reparameterized before making this comparison. The modifiers and coefficients should also correspond to the same population and clearance definition. In particular, the additive prediction must be positive throughout the covariate range where it is intended to be used; the equation does not guarantee positivity automatically.

What the Models Assume About Effects

The multiplicative model expresses relative scaling. Holding \(M\) fixed, increasing \(S\) from \(S_1\) to \(S_2\) changes clearance by the ratio \(S_2/S_1\). Its absolute difference is \(\mathrm{CL}_{\mathrm{ref}}M(S_2-S_1)\), so the absolute size-associated change depends on maturation. The same reasoning applies to the effect of changing \(M\): its absolute contribution depends on \(S\).

The additive model instead assigns a separate contribution to each deviation on the clearance scale. Holding \(M\) fixed, changing \(S\) by \(\Delta S\) changes predicted clearance by \(\mathrm{CL}_{\mathrm{ref}}\beta_S\Delta S\), independent of \(M\). Likewise, changing \(M\) by \(\Delta M\) changes predicted clearance by \(\mathrm{CL}_{\mathrm{ref}}\beta_M\Delta M\), independent of \(S\). The corresponding relative change still depends on the starting clearance, but the absolute difference does not.

The algebra makes the contrast explicit. Expanding the product model gives

When \(\beta_S=\beta_M=1\), the additive model contains the first three terms but omits the cross-product term \((S-1)(M-1)\). That term is not an extra parameter in the simple multiplicative model; it follows from multiplying the two modifiers. If both modifiers are below 1, their deviations are both negative and the cross-product is positive. If one is above 1 and one is below 1, the cross-product is negative. Therefore, the difference between predictions changes with the direction and magnitude of the two deviations.

This distinction concerns the scale of the model. On the original clearance scale, the product contains joint dependence through \(SM\), whereas the additive model has separate linear contributions in \(S-1\) and \(M-1\). A product can also be written as a sum on the log scale when all quantities are positive: \(\log(\mathrm{CL}_{\mathrm{mult}}/\mathrm{CL}_{\mathrm{ref}})=\log(S)+\log(M)\). That algebraic identity does not make it an additive model on the clearance scale.

Positivity, Diagnostics, and Model Comparison

A clearance prediction must be positive. If \(S>0\), \(M>0\), and \(\mathrm{CL}_{\mathrm{ref}}>0\), the simple multiplicative model produces positive clearance. In contrast, the additive model can produce zero or negative values, especially when both modifiers are substantially below 1. This is a mathematical limitation of the specified additive equation, not evidence that negative clearance is biologically meaningful.

In an analysis, compare the formulations using the same data, reference conditions, and relevant assumptions about the other model components. Examine whether predictions and residual patterns vary systematically across size and maturation. A pattern across size may indicate that a chosen size modifier or its contribution is inadequate; a pattern across maturation may indicate a corresponding problem with the maturation effect. The comparison should include the joint covariate range, since the models’ predictions can separate most where both modifiers depart from 1.

Check predicted clearance over the observed covariate combinations and over any intended prediction range. For an additive model, inspect the minimum predicted value as well as whether predictions remain positive. For either model, look for systematic discrepancies by size, maturation, or their combinations. A model that fits the central data range but yields implausible predictions at the edges may be unreliable for those covariates.

The main alternatives are not simply “correct” and “incorrect.” A multiplicative formulation is natural when effects are understood as relative scaling and positive predictions are important. An additive formulation is natural when effects are understood as absolute contributions and the additive predictor stays positive over the domain of interest. A more flexible formulation may be considered if neither captures the observed pattern, but added flexibility needs data to support it and should not be introduced solely to force predictions in a small sample.

The comparison also has limits. A numerical difference between predictions does not establish which biological mechanism is true. Data with limited coverage of size and maturation combinations may not distinguish the formulations well. Furthermore, a fitted coefficient is conditional on the chosen modifier definitions and reference. The model form should therefore be justified by biological reasoning, prediction checks, and the covariate range for which it will be used, rather than selected from a single summary of fit alone.

Common Mistake

Wrong reasoning: “The additive model has a separate size term and maturation term, so the size effect must be independent of maturation in every sense, and its predictions are automatically safe.” The first part confuses a constant absolute change with a constant relative change; the second ignores that the additive prediction can be zero or negative.

Correction: In the stated additive model, a fixed change in \(S\) produces the same absolute change in clearance at fixed coefficients, but its relative change depends on the starting clearance. Positivity must be checked across the intended covariate domain. The multiplicative model also has a specific interpretation: its absolute change with one modifier depends on the other modifier, while its relative scaling follows the product structure.