How do you model saturation and adstock?

A media response model — carryover then saturation — is fitted on 560 declared parameter combinations. The best explains 0.938098 of the variance. But eight combinations explain the data to within 1%, and among them the curvature runs from 0.600000 to 1.400000 and the half-saturation point from 25 to 65: the curve is not determined. What is determined is the carryover, recovered exactly at 0.600000, and the return on one extra unit of spend, between 1.478765 and 1.586695. Verdict: RESPONSE_CURVE_IDENTIFIED_FOR_REALLOCATION.

A media department must arbitrate a weekly budget. Its mix model explains the history very well and produces a smooth response curve, complete with a numbered saturation point. The question to ask before moving a single unit is this: is that curve the only one compatible with the data, or would other, very different curves explain them just as well? The answer completely changes what one is allowed to do with the model.

For these 156 weeks, this declared model and this declared grid of 560 combinations, which combinations of carryover, curvature and half-saturation explain the data to within 1% of the best? And do those combinations agree on the only quantity an arbitration really uses, the return on one extra unit of spend at the spend level currently observed? The question is not about goodness of fit, but about what the data actually determine.

Taking the best-fitting combination and reading its parameters as quantities is the central error of this family of models. Nothing in a good fit guarantees that the parameters are determined: carryover, curvature and half-saturation compensate for one another, so that very different curves almost coincide over the observed spend range. On this file, the best combination gives a curvature of 1.000000 and a half-saturation point of 45, while the values that actually generated the data are 1.80 and 55 — and their fit is only 2.556023% worse.

The method consists in not stopping at the best fit. One runs through a declared grid of combinations, fits each, then keeps every one that explains the data to within 1% of the best: that is the set the data cannot separate. Two things are then examined inside that set. Are the parameters tight? And above all, do they agree on the return of one extra unit of spend, which is the only quantity an arbitration consumes? A wide set of parameters can perfectly well agree on that return, and that is what decides.

Frozen before any reading: the gapless week numbering, the weekly media spend, the weekly revenue, the model form, the initialization of carryover at zero, the order of the transformations — carryover first, saturation second — the baseline made of a constant, a linear trend and two annual harmonics, the grid of 560 combinations, the 1% tolerance that defines the region, the spend level at which the marginal return is read, and the two thresholds. The sealed file holds 156 weeks. The case is synthetic and declared as such.

Carryover accumulates past spend: A(t) = x(t) + λ·A(t−1), with A(0) = 0 and λ the carry rate. Saturation turns that stock into a share between zero and one: S(t) = A(t)^β / (γ^β + A(t)^β), where β sets the curvature and γ is the level at which the share reaches exactly one half. That share enters linearly beside the baseline, and its coefficient gives the amplitude. The return of one extra unit is the derivative of the response with respect to the stock, read at the mean stock of the combination considered.

The declared computation runs seven steps: validate the schema and refuse any non-conforming or too short file; build the baseline columns; for each declared carry rate, build the carryover series once and record its mean level; for each of the 560 combinations, build the saturated regressor and solve the seven linear parameters in closed form by Gaussian elimination with partial pivoting, refusing a singular design; keep the combination with the smallest residual sum of squares and gather every one within 1% of it; inside that region, compute the spread of each parameter and the range of the marginal return, refusing a non-positive marginal return; apply the two declared thresholds, then the verdict rule.

Synthetic illustration — teaching values, not observed

On the sealed file: 156 weeks, 560 combinations. The best has a carryover of 0.600000, a curvature of 1.000000, a half-saturation point of 45 and an amplitude of 629.019162; it explains 0.938098 of the variance with a residual standard deviation of 25.033965. Eight combinations fall inside the 1% tolerance. In that region, the carryover runs from 0.600000 to 0.600000, a null range, check passed; the curvature runs from 0.600000 to 1.400000 and the half-saturation point from 25 to 65. The marginal return runs from 1.478765 to 1.586695, a ratio of 1.072987, check passed; the best combination gives 1.535957. The true parameters are not in the region, their fit being 2.556023% worse, and they imply a return of 1.539121. Verdict: RESPONSE_CURVE_IDENTIFIED_FOR_REALLOCATION.

The model assumes that carryover is geometric and without a delayed peak, that saturation follows this precise form, that the baseline captures trend and seasonality well, and that spend was not planned according to what one is trying to measure. That last assumption is the heaviest and the only one that really matters: in this synthetic case it is true by construction, in real life it is almost always false. Three things are not testable here: the functional form itself, what would happen at a spend level never observed, and the causal effect of moving budget.

The decisive diagnostic fits in one comparison. The goodness of fit is high, 0.938098, and says nothing: it would be almost identical for eight different combinations. The spread of the parameters inside the region says far more: the carryover is fixed, the curvature more than doubles, the half-saturation point runs from 25 to 65. And yet the range of the marginal return stays narrow, a ratio of 1.072987. As the case is synthetic, one can check what no real analysis allows: the true return, 1.539121, does fall inside that range, even though the true parameters are excluded from it.

Alternatives declared before results: impose sign constraints and estimate all parameters simultaneously rather than in two stages; tighten the grid around the best combination to see whether the region shrinks; replace geometric carryover by a form allowing a delayed peak; or replace the observational reading by a geographic experiment that varies spend in a controlled way. Each variant must be announced before reading and reported even if it widens the region.

The usable result is a local slope, not a curve. At the observed spend level, one extra unit returns about 1.54, and that value is robust to everything the data cannot settle. The curve is not determined: the best combination is wrong about the curvature and about the half-saturation point alike, and the fact that it fits better than the true values is a property of the noise, not a discovery. That is exactly why this dossier authorizes a marginal budget adjustment and nothing else.

Allowed: adjusting the budget at the margin around the observed level, relying on the marginal return and its range; publishing the size of the region and the spread of the parameters beside the goodness of fit, never one without the others; requiring any supplier to provide that region with every response curve delivered; and concluding that the carryover is determined here because spend arrives in sharp bursts. Also allowed: saying that this result holds only within the observed spend range.

Forbidden: quoting 1.000000 as the channel’s curvature or 45 as its saturation point, when eight combinations explain the data equally well and the true values lie elsewhere; extending the curve beyond the observed spend range, where the combinations in the region diverge; reading 0.938098 as proof that the model is correct; presenting the marginal return as the causal effect of a budget move; or choosing the tolerance after seeing the size of the region. Also forbidden: presenting this synthetic case as an observed measurement.

The reasonable decision is a modest budget move, calibrated on the marginal return and bounded by its range, inside the spend range already practised. Any wider ambition — finding the saturation point, defining an optimal budget, comparing two channels on their curves — requires varying spend on purpose, through a geographic experiment, rather than reading more finely a history that does not contain the information. The model indicates which direction to push; it does not say how far.

Use this protocol whenever a media response curve is delivered as the basis of an arbitration, and before any automated budget optimization. Avoid it when spend has varied little over the period, since the region will then be very wide and the protocol will merely record the fact. Avoid it too when spend was planned according to expected sales, which is the usual case on real data and introduces a bias this dossier does not address. And avoid it as a substitute for an experiment when the question concerns the effect of a budget change.

The CC0 CSV holds the 156 synthetic weeks. Python and R are the reference implementations; SPSS carries the same computation in a Python block; SAS uses the same grid but delegates the 560 fits to its own by-group regression procedure, and checks the two flags rather than every decimal. The final deliverable gathers the data, the variable dictionary, the declared model form, the order of the transformations, the grid, the region tolerance, the two thresholds, the seven numbered steps, the best combination, the size of the region and the spread of each parameter, the range of the marginal return, the three diagnostics specific to the synthetic case, both flags, the verdict and the software versions.

Jin, Wang, Sun, Chan and Koehler set out the carryover-then-saturation form and note that the parameters of the Hill function are essentially unidentifiable in some cases, that two very different curves nearly coincide over the observed range but diverge outside it, that no statistical method resolves this, and that the optimal mix drawn from the model has a large variance. Chan and Perry recall that the estimated relationship can change radically for small changes in the data, that advertisers often stay within a narrow spend range, and that observed budgets carry a selection bias. Chen, Zhang, Han and Lim describe the common practice of multi-stage estimation and propose estimating carryover, shape and scale jointly. Heusch proposes judging a model on its ability to recover a known truth rather than on its fit, and notes that synthetic generators produce exogenous spend. One caveat must be stated: of these four sources, only one is peer-reviewed. The two statements this dossier leans on most come from technical reports published openly by their authors, not from refereed articles; that is where they are stated publicly and precisely, and this dossier’s computation stands on its own. Founding references: Jin et al. (2017) and Chan & Perry (2017), Google technical reports, not peer-reviewed. Recent developments: Chen et al. (2021, arXiv preprint; published in the Journal of Applied Statistics, peer-reviewed) and Heusch (2026, arXiv preprint, not peer-reviewed).

1. Jin, Wang, Sun, Chan & Koehler (2017) ↗Full text verified, with a short excerpt located in the source.
2. Chan & Perry (2017) ↗Full text verified, with a short excerpt located in the source.
3. Chen, Zhang, Han & Lim (2021) ↗Full text verified, with a short excerpt located in the source.
4. Heusch (2026) ↗Full text verified, with a short excerpt located in the source.