Variable Transformation in Marketing Mix Modeling (MMM)

Summary (Key Takeaways)

MMM transformations are designed to reflect real-world ad dynamics.
Adstock → captures lag/decay effects of advertising.
Saturation → models diminishing marginal returns.
Together, they allow MMM tools like Robyn to better align statistical models with actual marketing behaviour.

Marketing Mix Modeling (MMM) typically relies on two key hypotheses:

Lagged Effects (Adstock Hypothesis): Advertising effects carry over across time rather than being confined to a single period.
- Example: I see ads today, but make a purchase next week.
Diminishing Returns (Saturation Hypothesis): The marginal return from advertising decreases as spending increases.
- Example: Spending $1,000 on ads produces strong impact, but spending an additional $1,000 yields less incremental benefit.

To account for these, tools like Meta’s Robyn apply two transformations: Adstock and Saturation.

1. Adstock Transformation

Adstock models carryover effects and decay of advertising. There are multiple functional forms:

Geometric Adstock A simple recursive formulation:
$\text{media\_adstocked}_t = \text{media\_raw}_t + \text{decay\_rate} \cdot \text{media\_raw}_{t-1}$
- where t is a given time period.
- Decay rate ∈ [0,1], controlling how strongly past advertising persists.
Weibull PDF & CDF Adstock
- More flexible than geometric.
- Captures delayed peak effects and variable decay rates.
- Useful when the impact of advertising builds up over time before decaying.

Saturation reflects the theory of diminishing returns: each additional advertising unit produces smaller incremental response.

Functional form:
$\text{media\_saturated}_j = \frac{1}{1 + \Big(\frac{\gamma_j}{\text{media\_adstocked}_j}\Big)^{\alpha_j}}$
Parameters:
- $\gamma$ : half-saturation point (value at which response is 50%).
- $\alpha$ : controls the steepness of the curve.

This transformation ensures that extremely high media spends do not unrealistically scale ROI.