Saturation curves

Diminishing returns to media. Adstock says when money works; saturation says how hard it works at the margin. Four curves are provided; they differ in whether the response is bounded, and in whether they allow an S-shape.

Usage

saturate_hill(x, half_max, shape = 1)

saturate_exponential(x, rate)

saturate_michaelis_menten(x, vmax = 1, km)

saturate_power(x, exponent)

saturate(x, type = c("hill", "exponential", "michaelis_menten", "power"), ...)

Arguments

x: Numeric vector of media, normally already adstocked. Values must be non-negative: all four curves are defined for spend, not for arbitrary reals.
half_max: The value of x at which the response reaches half its ceiling. Interpretable in the units of x, which is why it is used here in preference to the equivalent unnamed scale parameter.
shape: Hill exponent, positive. shape = 1 gives a concave curve with diminishing returns everywhere. shape > 1 gives an S-shape with an initial convex region, the usual representation of a threshold effect.
rate: Exponential rate parameter, positive. Larger values saturate sooner.
vmax: Michaelis–Menten asymptote: the response as x grows without bound.
km: Michaelis–Menten constant: the value of x at which the response reaches half of vmax.
exponent: Power exponent in (0, 1]. exponent = 1 is the identity (no saturation); smaller values bend the curve harder.
type: Which curve saturate() should dispatch to.
...: Passed to the individual curve function.

Value

A numeric vector the same length as x, in the same order.

Details

saturate_hill() and saturate_exponential() are bounded on [0, 1), so the fitted coefficient carries the channel's ceiling. saturate_michaelis_menten() is bounded by vmax. All three approach their ceiling smoothly at arbitrarily large x, including Inf, rather than overflowing. saturate_power() is unbounded but concave: response keeps growing, just ever more slowly. Unboundedness is not automatically wrong, but it does mean the model will happily extrapolate a return on a spend level never observed.

saturate_hill(x, half_max, shape = 1) and saturate_michaelis_menten(x, vmax = 1, km = half_max) are the same function. Both are provided because the two literatures name it differently and practitioners arrive expecting one or the other.

Transform order

Saturation is applied after adstock, not before. Saturating first would cap each period's spend in isolation and then let carryover accumulate the capped values past the cap, which defeats the point of having a ceiling. Use media_transform() to get the order right without having to remember it.

References

Hill, A. V. (1910). The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. The Journal of Physiology, 40(Suppl), iv–vii.

Examples

spend <- c(0, 25, 50, 100, 200, 400)

# Concave: diminishing returns from the first pound
round(saturate_hill(spend, half_max = 100), 3)
#> [1] 0.000 0.200 0.333 0.500 0.667 0.800

# S-shaped: a threshold below which media barely registers
round(saturate_hill(spend, half_max = 100, shape = 3), 3)
#> [1] 0.000 0.015 0.111 0.500 0.889 0.985

# The four curves side by side. Hill, exponential and Michaelis-Menten are
# bounded; power keeps growing, just ever more slowly.
round(rbind(
  hill        = saturate_hill(spend, half_max = 100),
  exponential = saturate_exponential(spend, rate = 0.007),
  michaelis   = saturate_michaelis_menten(spend, vmax = 1, km = 100),
  power       = saturate_power(spend, exponent = 0.5)
), 3)
#>             [,1]  [,2]  [,3]   [,4]   [,5]   [,6]
#> hill           0 0.200 0.333  0.500  0.667  0.800
#> exponential    0 0.161 0.295  0.503  0.753  0.939
#> michaelis      0 0.200 0.333  0.500  0.667  0.800
#> power          0 5.000 7.071 10.000 14.142 20.000

# Bounded means bounded: no overflow, even at the extremes
saturate_hill(c(1e300, Inf), half_max = 100, shape = 3)
#> [1] 1 1

# The dispatcher, for programmatic use
round(saturate(spend, type = "hill", half_max = 100), 3)
#> [1] 0.000 0.200 0.333 0.500 0.667 0.800