Response curves - help required plotting from 'first principles'

[gedkitchener-CT]

Approach A
From the pareto_media_transform_matrix.csv file, I am plotting decompMedia (y) vs. (x) adstockedMedia for a paid-media variable (for a single solID).

Approach B
Using the Hill Equation. With y = β (x^α / (x^α + γ^α)), I get the β, α, and γ values for the relevant paid-media variable and model, and plug them into the equation.

For example, with β = 50,000, α = 3.7, and γ = 0.6, I find that the fraction very quickly tends to 1.0 because x^α >> γ^α (even for relatively small values of x).

Issue
The two approaches (A and B) don't agree - have I misunderstood how the Hill equation is used perhaps? Any help would be great.

[gedkitchener-CT]

For example, is gamma scaled in some way? (e.g., by max of x)

[igorskokan]

Hey! γ is the inflection parameter in Hill function.
What units is x in? What are the typical values of x - is it very small, e.g. x ∈ <0,1> for example?

[gedkitchener-CT]

x is impressions or spend, so runs in the thousands, while γ is bounded by (0.3, 1.0), as per the suggestion in the docs.

This is why I'm confused - there's something I'm missing in how the Hill function is actually applied, i.e., γ values between (0.3, 1.0) cannot be appropriate for typical impression volumes or spend levels (what marketing team would aim for on the order of 1 impression per day?!?)

Hence my question about whether γ is perhaps scaled behind-the-scenes, to make it roughly the same magnitude as the metric of interest (e.g., spend level)?

[gedkitchener-CT]

As it turns out, my suspicions were confirmed.

But rather than γ being scaled, it is x that is scaled.

[igorskokan]

Thanks @gedkitchener-CT apologies if I misunderstood your original question. As per your own investigation the Hill saturation transformation normalises all independent vars to 0-1

[gedkitchener-CT]

Oh, I think I've found the answer in another discussion

all independent variables are transformed before the fitting ... the hill function we use for saturation transformation normalises all independent to 0-1

EDIT

Actually, it seems that it's indeed γ that is transformed (see transformation.R). That is to say, it's not x that is transformed.

gammaTrans <- round(quantile(seq(range(x)[1], range(x)[2], length.out = 100), gamma), 4)