Normalisation
Compare shapes on normalised spend and response before recalibrating parameters in the economic units used for the decision.
Response curves: a library of testable, comparable shapes.
Decision questionWhich curve shape should be kept, and over what range can it be used?
SUPPORTED DECISIONUse a curve only within its documented range, and keep the simplest shape whose marginal slope stays stable under validation and sensitivity.
Reproducible calculation
Specification and limits of the published calculation.
This file checks the local arithmetic of the teaching model. It does not turn the stated assumptions into evidence of market performance or causality.
- Technical specification of the response-curve library (opens in a new tab)
v1, 31 August 2026 · CC BY 4.0 · First-party specification of normalised response profiles, without market calibration or observed commercial return.
Key distinctions
Three conditions before calculating.
Observed range
Separate observed, tested and extrapolated ranges. A good fit inside the range does not validate a recommendation outside it.
Slope and uncertainty
The marginal slope at the contemplated budget and its uncertainty determine whether a ranking is stable enough to move spend.
Method
Read a curve as a decision hypothesis
- 01Choose shape constraints
Decide whether the mechanism requires monotonicity, saturation, a threshold or local changes.
- 02Fit coherent units
Express spend and response in units that are genuinely linked, and align the horizon with adstock.
- 03Validate the slope
Compare holdout error, parameter stability, residuals, derivative and half-saturation over several windows.
- 04Document the range
Publish observed minimum and maximum spend, the usable range and the update rule; anything beyond it is learning, not an established forecast.
ORIGINAL ASSET
Four normalised profiles on the same budget
Responses are indexed to 100 so that shape, half-saturation, marginal slope at the current budget and out-of-range behaviour can be compared.
050100150200€300k
1025456585100
83862788896
42658768692
63461778691
Observed support: 0–200 k€300 k€ = extrapolation
JSONDownload profiles and parameters↓ILLUSTRATIVE EXAMPLE · SIMULATED DATA
Same fit, €50.48k of divergence outside the support.
Five observations between €0k and €200k are fitted with exponential saturation and a Hill curve. Their validation error differs by only 1.6 points. The contemplated budget is €300k, while the maximum observed is €200k.
At €200k, the published responses are 108.08 and 103.22 units. At €300k, the exponential forecasts 118.78 units and Hill 112.47. With €8k contribution per unit, the extrapolation gap is (118.78 − 112.47) × 8 = €50.48k. The published slopes at €200k are 0.1692 and 0.1362 unit per €k; their uncertainty must be estimated before any ranking.
The library classifies €300k as outside the documented range. It recommends no spend above €200k without a test: a €20k increment can be used to learn the slope. The small difference in fit does not justify an untested extrapolation.
Acceptance conditions
What must be true to act.
- 01
The form has a justified mechanism and constraints.
- 02
Recommended spend stays inside the observed or tested range.
- 03
The marginal slope and its interval are published at the decision budget.
- 04
A material extrapolation is funded as sequential learning.
Limits
What this analysis does not prove.
- This teaching library does not replace market-specific calibration.
- Adstock, targeting and creative quality can shift a curve over time.
- Single-channel curves describe synergies and substitutions poorly.
References
Works cited.
- Little, J. D. C. (1979), “Aggregate Advertising Models: The State of the Art”, Operations Research, 27(4), 629–667.
- Jin, Y. et al. (2017), “Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects”, Google Research.
- Hill, A. V. (1910), “The Possible Effects of the Aggregation of the Molecules of Haemoglobin on Its Dissociation Curves”, Journal of Physiology, 40, iv–vii.