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.

Figure 2 parameter sets (p. 5; added in the 2026-10-13 audit)

Figure 2 plots $\beta \text{Hill}(x)$ for $x\in [0,1.5]$ with the following $(K,S,\beta )$:

Curve	Half-saturation $K$	Slope $S$	$\beta$	Role
black solid	0.5	1	0.3	reference ($S=1$)
black dashed	0.95	0.748	0.3925	near-identical to black solid for $x\le 1$, diverges for $x>1$ (identifiability case 1)
red	0.5	2	0.3	$S>1$: convex near 0, S-shape
green	0.5	0.5	0.3	$S<1$: more concave near 0, flattens faster
blue solid	1.5	2	0.8	$K$ outside observed range $(0,1)$
blue dashed	2	1.867	1.144	near-identical to blue solid within $(0,1)$ (identifiability case 2)

Footnote definitions used with the reach transform (Eq. 6): reach is the number of people in the target audience reached by a campaign divided by the audience size (footnote 1). GRP is impressions delivered to the target audience divided by the number of people in the target audience, times 100 (footnote 2).

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 (Table 4, p. 16):

	$x=0.5$, two years	$x=0.5$, sixty years	$x=1$, two years	$x=1$, sixty years
Media 1 ($S=1$)	-39.7%	-0.2%	-32.5%	-0.2%
Media 2	-25.6%	3.4%	-18.1%	3.3%
Media 3	-39.0%	10.4%	-25.3%	10.4%

The bias is worse at $x=0.5$ (mid-curve) than at full spend for every medium in the two-year arm. Full study design, per-parameter Fig. 7 readings and the mechanism: Jin 2017 MMM - Sample Size vs Bias Study (Two vs Sixty Years).

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
• Jin 2017 MMM - Sample Size vs Bias Study (Two vs Sixty Years) — curve vs parameter recovery
• Jin 2017 MMM - Prior Sensitivity Experiments on Beta and Half-Saturation — K outside the observed range
• Hooker Mentch Zhou 2021 - Partial Dependence Alternatives and Conclusions — partial dependence extrapolates at unsupported spend levels; additive models are safer for response curves
• Kumar et al - Additivity Axiom Limits for Non-Additive Models — saturating response makes Shapley attributions baseline-dependent
• Manheim Garrabrant - Extremal Goodhart, Model Insufficiency and Regime Change — extrapolating saturation curves beyond observed spend
• AMSS Traditional Media Module (Reach, Frequency, Hill Effect) — Hill function applied to per-person frequency
• Chan Perry 2017 - MMM Data Limitations (156 Rows, Correlated Channels, Narrow Spend Range) — narrow spend range leaves saturation unidentified
• Lewis Rao 2015 - The Advertiser’s Problem and Marginal ROI — concave response, optimal vs break-even spend