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Response curve for a channel

Source: R/contributions.R
response_curve.Rd

Predicted response across a range of spend, derived from the fitted saturation parameters. spend_for() is the inverse: the spend that achieves a target response.

Usage

response_curve(spend, coefficient, type = "hill", ...)

spend_for(target, coefficient, type = "hill", max_spend = NULL, ...)

Arguments

spend

Numeric vector of spend levels to evaluate. response_curve() only; spend_for() has no such argument.

coefficient

The channel's fitted coefficient.

type

Saturation curve type, as in saturate().

...

Passed to saturate(), for example half_max and shape.

target

Target response level. spend_for() only.

max_spend

Upper bound of the search interval for spend_for(). Defaults to 1000 times the curve's spend-scaled parameter (half_max, km, or log(2)/rate). saturate_power() has no such parameter, so max_spend is required there.

Value

response_curve() returns a data frame with columns spend, response and marginal, one row per element of spend and in the same order. spend_for() returns a single number, or NA_real_ with a warning when the target is not reachable within max_spend.

Details

These describe the response curve the model fitted, which is not the same thing as what would happen if you actually spent that much. The curve is identified only over the range of spend the data contains; asking a saturated Hill curve what happens at ten times the observed maximum returns a number, and that number is extrapolation. spend_for() returns NA rather than a fabricated answer when the target lies beyond max_spend, but it cannot tell you that a reachable target is outside the data's support. Check the observed spend range yourself.

See also

saturate(), mroi()

Examples

# A concave curve: marginal return falls from the first pound onward.
concave <- response_curve(
  spend = seq(0, 100000, by = 20000),
  coefficient = 5200, half_max = 45000, shape = 1
)
concave
#>   spend response   marginal
#> 1 0e+00    0.000 0.11555556
#> 2 2e+04 1600.000 0.05538475
#> 3 4e+04 2447.059 0.03238772
#> 4 6e+04 2971.429 0.02122466
#> 5 8e+04 3328.000 0.01497615
#> 6 1e+05 3586.207 0.01112974
all(diff(concave$marginal) < 0)
#> [1] TRUE

# An S-curve: marginal return RISES to the inflection point and only then
# falls. Below the peak the channel is under-funded, not saturated.
s_curve <- response_curve(
  spend = seq(0, 100000, by = 20000),
  coefficient = 5200, half_max = 45000, shape = 1.6
)
s_curve
#>   spend response     marginal
#> 1 0e+00    0.000 5.523462e-07
#> 2 2e+04 1115.858 7.011242e-02
#> 3 4e+04 2355.734 5.154110e-02
#> 4 6e+04 3188.033 3.289370e-02
#> 5 8e+04 3718.836 2.118571e-02
#> 6 1e+05 4066.624 1.418186e-02
s_curve$spend[which.max(s_curve$marginal)]
#> [1] 20000

# What spend achieves a response of 2000?
spend_for(target = 2000, coefficient = 5200, half_max = 45000, shape = 1.6)
#> [1] 33545.75

# An unbounded curve needs an explicit search range
spend_for(target = 200000, coefficient = 5200, type = "power",
          exponent = 0.5, max_spend = 1e6)
#> [1] 1479.29