How Marketing Mixed Modeling Can Improve Your ROI
How to plan your marketing spend while measuring the true effect of any marketing activity on your bottom line? The answer lies in Marketing Mix Modeling (MMM).
What are diminishing returns?
Diminishing returns describes the way each additional pound of marketing spend produces less incremental return than the last. The relationship between spend and response is a curve that flattens, not a line, which is why average return and marginal return differ.
In plain English
Spend on a channel and the first pounds reach the people most likely to respond. Keep spending and you reach people who are less interested, then people you have already reached several times, then people who were going to buy anyway. Response keeps rising and rises more slowly each time, so the relationship between spend and outcome is a curve that flattens rather than a straight line. That is diminishing returns, and it is the reason a channel can be worth funding and also be over-funded.
The practical consequence is the gap between average and marginal return. Average return divides total response by total spend and is what almost every dashboard reports. Marginal return is what the *next* pound would earn, and it is the only number relevant to a budget decision. On a saturating curve the marginal is always below the average, and far below it once you are deep into the flat region — so a channel showing a 4.0 average return can easily have a marginal return under 1.0, meaning the last tranche of spend is losing money inside a headline that looks excellent.
This is also what makes budget optimisation a real exercise rather than a ranking. If response were linear you would put everything into the highest-return channel, which is obviously wrong and is precisely what a linear model recommends. With curves, the optimum equalises marginal returns across channels: you keep moving money from the channel with the lower marginal return to the one with the higher until they meet. That allocation typically looks nothing like the ranking by average return.
Modelling the curve is a standard part of marketing mix modelling, and the two common forms are the Hill function and a negative-exponential. Both are S-shaped or concave, both are parameterised by a half-saturation point and a steepness, and both are applied after adstock — accumulate exposure over time first, then pass it through the curve. Some channels genuinely show an S-shape with a threshold below which almost nothing happens, which matters for television, where a very small budget can be worse than none at all.
The estimation problem is that a curve can only be identified where spend actually varied. If a channel has run between £90k and £110k a week for two years, the model has seen a narrow slice of the curve and its shape outside that band is an extrapolation dressed as a finding. This is the single biggest reason to deliberately vary spend — flighting, regional variation, or a geo experiment that turns spend up and down — since a plan built on the extrapolated part of a saturation curve is a plan built on the model's prior.
The formula
Two standard response curves and the two derived quantities that make them useful for allocation.
- Hill function
f(x) = x^α / ( x^α + κ^α )κ is the half-saturation point, α the steepness. α > 1 gives an S-shape with a threshold; α ≤ 1 gives pure concavity from the origin.
- Negative exponential
f(x) = 1 − e^{−x/κ}Concave everywhere, so no threshold. Simpler, one parameter, and usually adequate for channels with no evidence of an S-shape.
- Marginal versus average return
marginal = d(response)/d(spend); average = response / spendMarginal is always below average on a concave curve. Budget on the first and report the second only with that caveat.
- The optimal allocation
marginal_A = marginal_B = … = marginal_NEqualise marginal returns across channels, not average ones. Ranking by average return systematically over-funds whatever is already saturated — see the correlation calculator for why a linear fit misses the curve entirely.
Worked example
A fitted response curve for paid social has a half-saturation point of £180k a week. Current spend is £320k a week. The team is deciding whether to add £80k a week from another channel, and the dashboard reports a 3.4 average return on social.
The extra £80k on social would earn about £0.82 per pound. The same money on the alternative channel earns about £2.10.
The dashboard number and the decision number differ by a factor of three, and both are correct. A 3.4 average return is a true statement about the whole £320k, most of which sits on the steep part of the curve and earns very well; the marginal pound sits far out on the flat part and earns 1.15, falling to 0.82 by the time you have added the £80k. Ranking channels by the reported average would have moved money *towards* social, which is exactly backwards. The rule that falls out is to shift budget until marginal returns equalise — here that means moving money away from social until its marginal rises to meet the alternative's, which will lower social's average return and raise total response.
Common misconceptions
- “A channel with a 4.0 ROAS should get more budget than one with a 2.0.”
- Not without knowing where each sits on its curve. A saturated channel at 4.0 average can have a marginal return below 1.0, while an under-funded channel at 2.0 average may be earning 3.0 on the next pound. Average return describes money already spent; only marginal return speaks to money you are about to spend.
- “The model shows returns still positive at our spend level, so we should spend more.”
- Positive is not the same as better than the alternative. The question is whether the marginal return here exceeds the marginal return of every other use of the money, including channels not currently funded. A channel can be profitable at the margin and still be the wrong place for the next pound.
- “We can extrapolate the response curve to plan a much larger budget.”
- The curve is only identified over the range of spend the model has observed. Beyond that its shape is the prior speaking, not the data. If a plan requires doubling spend, vary spend deliberately first — flight it, or run a geo test at the higher level — so the curve is fitted where the decision lives.
Frequently asked questions
How do I find the saturation point of a channel?
Fit a response curve within a marketing mix model and read off the half-saturation parameter, but treat the answer as reliable only over the spend range the data actually covered. If spend has been near-constant, the curve is extrapolated rather than estimated. Deliberately varying spend — flighting, regional differences, or a geo test at deliberately higher and lower levels — is the only way to observe the curve where you need it.
Do all channels show an S-shaped response curve?
No, and assuming one can mislead. Channels driven by existing intent, such as paid search, are usually concave from the origin: the first pound already earns well. Broad-reach brand channels more often show a threshold below which spend achieves too little frequency to register, which is the S-shape. Which form fits should be a question the data answer, and it frequently cannot be settled without deliberate variation in spend.
How do response curves change budget allocation?
They turn allocation from a ranking exercise into an optimisation. With linear response the answer is always to put everything into the top channel; with curves the optimum equalises marginal returns across all of them, which almost always means funding more channels at lower levels than a ranking would suggest. It also produces a defensible answer to how large the total budget should be: expand while the worst marginal return still exceeds your cost of capital.
Calculate it
Knowing the term is the easy part
Applying it to a live measurement problem is the part that goes wrong. If you are designing an experiment, reading a result you do not trust, or trying to work out what your marketing actually caused, that is the work we do.