ROAS, mROAS, and Optimal Media Mix

Summary

From the fitted MMM, advertisers want attribution metrics: ROAS (return on ad spend — total incremental revenue per dollar of channel spend) and mROAS (marginal ROAS — extra revenue from one more unit of spend). Both are computed counterfactually by zeroing or perturbing a channel’s spend and comparing predicted sales, including the post-change period because carryover keeps affecting sales after the change. In the Bayesian framework, posterior samples of $\Phi$ are plugged in to get full posterior distributions of ROAS/mROAS. The same machinery yields an optimal media mix under a budget constraint — but its posterior has large variance (in one scenario, three modes), so the optimal allocation is often untrustworthy.

Overview

ROAS and mROAS can be computed overall or per channel; per-channel is what guides optimization. The paper assumes channels can be changed independently (additive media effects, no cross-channel demand spillover — otherwise metrics must be computed jointly). Metrics are evaluated over a selected change period $(t_0,t_1)$, framed by pre- and post-change periods each set to the carryover length $L$ so lagged effects are fully captured (Figure 3).

Main Content

ROAS (return on ad spend)

Let ${\hat{Y}}_t^m(\cdot ;\Phi )$ be predicted sales as a function of channel $m$‘s spend (sum of the non-noise terms in the model; apply the inverse transform if $y$ was log-sales). With historical spend $x_{t,m}$ and counterfactual spend ${\tilde{x}}_{t,m}$ (channel $m$ set to zero during the change period):

$${\text{ROAS}}_m=\frac{{\sum }_{t_0\le t\le t_1+L-1}[{\hat{Y}}_t^m(x_{t-L+1,m},\dots ,x_{t,m};\Phi )-{\hat{Y}}_t^m({\tilde{x}}_{t-L+1,m},\dots ,{\tilde{x}}_{t,m};\Phi )]}{{\sum }_{t_0\le t\le t_1}{x}_{t,m}}.$$

The numerator sums the sales difference over both the change period and the post-change period (up to $t_1+L-1$) to capture carryover. Because effects are additive, ${\text{ROAS}}_m$ depends only on channel $m$. (Eq. 10)

mROAS (marginal ROAS)

The additional revenue from a one-unit (here, +1%) increase in spend — the derivative of cumulative predicted sales w.r.t. cumulative spend. With ${\bar{x}}_{t,m}$ = historical spend increased by 1% during the change period:

$${\text{mROAS}}_m=\frac{{\sum }_{t_0\le t\le t_1+L-1}[{\hat{Y}}_t^m({\bar{x}}_{t-L+1,m},\dots ;\Phi )-{\hat{Y}}_t^m(x_{t-L+1,m},\dots ;\Phi )]}{0.01\times {\sum }_{t_0\le t\le t_1}{x}_{t,m}}.$$

Computed analytically for tractable models or numerically by a small perturbation. (Eq. 11)

Plug in posterior samples, not posterior means

In the Bayesian framework, each posterior sample ${\Phi }_j$ is plugged into Eqs. 10-11 to obtain posterior distributions of ROAS/mROAS (summarize by mean/median + credible interval). It is wrong to first average the parameters and then plug the means in — that ignores the correlation among parameters in $\Phi$ and can produce incorrect metrics.

Optimal media mix under a budget constraint

With total change-period budget $\mathcal{C}$, the optimal mix ${\mathbf{X}}^o=\{x_{t,m}^o\}$ solves

$$\max \sum _{t_0\le t\le t_1+L-1}{\hat{Y}}_t(\cdot ;\Phi )\text{s.t.}\sum _{t_0\le t\le t_1}\sum _m{x}_{t,m}=\mathcal{C}.\text{\hspace{1em}(12-13)}$$

The objective includes post-change sales (carryover). Solved by constrained optimization (e.g. Lagrange multipliers). To cut free parameters, fix the flight pattern (e.g. constant weekly spend $x_{t,m}=c_m/(t_1-t_0+1)$), reducing the constraint to ${\sum }_m{c}_m=\mathcal{C}$ (Eq. 14). Two Bayesian routes: (A) optimize the average predicted sales across all $J$ posterior samples (Eq. 15) — gives one stable optimal mix; (B) optimize each sample separately (Eq. 16) — gives a posterior distribution of the optimal mix that reveals its uncertainty.

Examples

Optimal-mix variance in simulation (Sec. 5.2)

Optimizing Media 1 vs Media 2 spend under a fixed weekly budget. Scenario I (budget = 1, near the historical average spend): the posterior of optimal Media-1 spend is unimodal and tight around the truth — both approaches land near the true optimum. Scenario II (budget = 0.5, predicting in the sparse part of observed data): the posterior of the optimal mix has three modes (some posterior samples peak at the low end, others at the high end), with much higher variation. Crucially, the variance in estimated sales across posterior samples is comparable to or larger than the variation in sales caused by changing the mix — so the optimal allocation is not trustworthy when extrapolating beyond well-observed spend. ROAS of Media 3 (weak signal) had very large extreme values and large mROAS bias; Media 2’s $\beta$Hill curve had the least bias, so its ROAS/mROAS were estimated best.

Connections

• Built on the posterior samples produced in Bayesian Estimation and Priors for MMM; the high optimal-mix variance is the downstream consequence of the prior-dominated, hard-to-identify parameters there and in Shape (Saturation) Effects.
• Carryover-period accounting depends on $L$ from Carryover (Adstock) Functional Forms.
• Applied to TV vs magazines in the shampoo case: MMM Model Selection and Application.
• Connects to optimal-budget/forecasting practice in Optimal Marketing Decisions and Forecasting and elasticity benchmarks in Advertising and Promotion Effects.

See Also

• Bayesian Estimation and Priors for MMM
• Bayesian Media Mix Modeling - Overview
• MMM Model Selection and Application
• Index: Bayesian Media Mix Modeling