Shape (Saturation) Effects

Summary

The shape effect captures diminishing returns: at high spend, advertising saturates. Jin et al. model curvature with the Hill function from pharmacology, parameterized by a half-saturation point ${\mathcal{K}}_m$ and a slope/shape parameter ${\mathcal{S}}_m$, scaled by the coefficient ${\beta }_m$ (the “$\beta$Hill” form). Depending on ${\mathcal{S}}_m$ the curve is concave ($\mathcal{S}<1$) or S-shaped ($\mathcal{S}>1$). The flexible $\beta$Hill is poorly identifiable — very different parameter triples can produce near-identical curves over the observed spend range — so a parsimonious one-parameter reach transformation (fixing $\mathcal{S}=1$) is often preferred.

Overview

A linear response curve (Guadagni & Little 1983) cannot represent ad saturation. A curvature function maps transformed spend to a saturating response. The Hill function (Gesztelyi et al. 2012; Hill 1910), originally an empirical receptor-binding model, provides a flexible saturating form on $[0,\infty )$.

Main Content

Hill function (saturation)

$$\text{Hill}(x_{t,m};{\mathcal{K}}_m,{\mathcal{S}}_m)=\frac{1}{1+(x_{t,m}/{\mathcal{K}}_m{)}^{-{\mathcal{S}}_m}},x_{t,m}\ge 0,$$

where ${\mathcal{S}}_m>0$ is the shape parameter (a.k.a. slope) and ${\mathcal{K}}_m>0$ is the half-saturation point — so named because $\text{Hill}({\mathcal{K}}_m)=1/2$ for any ${\mathcal{S}}_m$. As $x\to \infty$, $\text{Hill}\to 1$. (Eq. 4)

Scaled shape transform ( $\beta$Hill)

To allow channel-specific maximum effects, multiply by the regression coefficient ${\beta }_m$:

$${\beta }_m{\text{Hill}}_m(x_{t,m})={\beta }_m-\frac{{\mathcal{K}}_m^{{\mathcal{S}}_m}{\beta }_m}{x_{t,m}^{{\mathcal{S}}_m}+{\mathcal{K}}_m^{{\mathcal{S}}_m}}.$$

${\beta }_m$ is the asymptotic (maximum) effect as spend grows large. Shape behavior in $\mathcal{S}$ (Figure 2, same $\mathcal{K},\beta$): $\mathcal{S}>1$ gives a convex-near-zero S-shape (red curve); $\mathcal{S}<1$ is more concave near zero and flattens faster (green); $\mathcal{S}=1$ is the intermediate concave case. (Eq. 5)

Reach transformation (one-parameter form, $\mathcal{S}=1$)

A parsimonious special case used by Jin, Shobowale, Koehler & Case (2012) relating reach $R$ to GRPs/impressions $G$:

$$R=a-\frac{b}{G+b/a}.$$

Setting $a=\beta$, $b=\mathcal{K}\beta$, $G=x$ makes this equal to $\beta \text{Hill}(x)$ with $\mathcal{S}=1$. Estimating the reach transformation is equivalent to fixing $\mathcal{S}=1$ in the Hill function — fewer parameters, better identifiability. (Eq. 6)

Identifiability problem

The three $\beta$Hill parameters $(\mathcal{K},\mathcal{S},\beta )$ are essentially unidentifiable in some regimes. (1) Within a finite observed range, a curve with $(\mathcal{K}=0.5,\mathcal{S}=1,\beta =0.3)$ is matched almost exactly by a very different triple $(\mathcal{K}=0.95,\mathcal{S}=0.748,\beta =0.393)$, diverging only for $x>1$. (2) When $\mathcal{K}$ lies outside the observed spend range, two curves can be nearly identical inside the range and diverge only outside it. Consequence: the model cannot extrapolate the response beyond observed spend, but the $\beta$Hill curve can still be estimated well within the observed range even when the individual parameters cannot. Other curvature forms — sigmoid/logistic, the integral of a normal, or monotone regression splines — are also possible but may share the identifiability issue.

Examples

Curve estimable, parameters not (Sec. 6)

In large-sample simulations the $\beta$Hill curves are recovered with very low bias and small variance, yet the posterior medians of $\mathcal{K}$, $\mathcal{S}$, $\beta$ vary widely — a direct manifestation of the near-unidentifiability above. In small samples the curves are systematically underestimated; Media 1 (true $\mathcal{S}=1$) has the largest bias because $\mathcal{S}=1$ makes the shape less identifiable. Relative bias of $\beta$Hill at $x=1$, two-year sample: Media 1 −32.5%, Media 2 −18.1%, Media 3 −25.3%; at sixty years all shrink to roughly −0.2% to +10%. See MMM Model Selection and Application.

Connections

• Formalizes the concave-vs-S-shaped response distinction in Shape of the Marketing Response Function.
• Applied after adstock in the combined model: Carryover (Adstock) Functional Forms, Bayesian Media Mix Modeling - Overview.
• Priors on $\mathcal{S}$ (gamma) and $\mathcal{K}$ (beta over observed range) and their sensitivity: Bayesian Estimation and Priors for MMM.
• The Hill/reach choice is one axis of the model-selection grid in MMM Model Selection and Application.

See Also

• Carryover (Adstock) Functional Forms
• Shape of the Marketing Response Function
• Bayesian Media Mix Modeling - Overview
• Index: Bayesian Media Mix Modeling