Average and marginal return by counterfactual simulation

Computes a channel's average return and its marginal return by re-running the full media transform – carryover and saturation – on a counterfactually increased spend plan, rather than by differentiating the saturation curve at a single point. This is the definition of marginal ROI used in the Bayesian MMM literature (Jin et al., 2017) and in Google's Meridian: the incremental response from a small proportional increase in spend, divided by the incremental spend.

Usage

marginal_roi(
  spend,
  coefficient,
  adstock = list(kernel = "geometric", decay = 0.5),
  saturation = list(type = "none"),
  by = NULL,
  lift = 0.01,
  rows = NULL,
  extend = 0L
)

Arguments

spend: Numeric vector of raw spend for one channel, in time order.
coefficient: The channel's fitted coefficient on the transformed regressor.
adstock, saturation: Lists describing the transform, exactly as for media_transform(). They must match the transform the model was fitted on.
by: Optional grouping vector, as for media_transform().
lift: Proportional increase in spend used for the marginal return. The default 0.01 asks what a 1% larger budget would have returned.
rows: Optional logical or integer index of the periods whose spend is increased – the planning window. Defaults to every period. Response is always summed over the whole series, so carryover from the window into later periods is counted.
extend: Number of zero-spend periods appended to the end of the series (to each group, when by is supplied) before summing response, so that carryover from late spend is allowed to play out. The default 0 matches contributions() and roi(), which only count response inside the observed window; set it to effective_window() of the decay to measure long-run return.

Value

A one-row data frame with columns spend (total spend over the series), contribution (the channel's total response), roi (their ratio, the average return) and mroi (incremental response per unit of incremental spend, when the spend in rows is increased by lift).

Details

Why not just differentiate the saturation curve? mroi() does that, and it answers a narrower question: the slope of the curve at one level of transformed media. Turning it into a return on spend needs two further steps that are easy to get wrong.

Carryover. A unit spent this week enters this week's adstock with the kernel's first weight only – 1 - decay for a normalised geometric kernel – but it also enters every later week's adstock. Under a normalised kernel those weights sum to one, so the total marginal return is close to the curve's slope, not to the slope times 1 - decay. Multiplying by the first weight gives the return inside the week of spend and ignores the rest; for a slow channel that understates its marginal return several-fold, and a reallocation driven by it moves money away from exactly the channels whose effect arrives late.
Averaging over periods. The curve's slope at the mean adstocked level is not the mean of its slope across periods. Flighted media spends some weeks near zero and some near saturation, and for an S-shaped curve the two can differ by a large factor (Jensen's inequality).

Simulation handles both by construction, for any kernel and any curve media_transform() supports, and it is what makes average and marginal return directly comparable: both are totals over the same periods.

What this is not

The answer is only as good as the fitted curve, and a curve is identified only over the spend levels the data contains. A marginal return is a local quantity – a 1% change is the kind of question the data can speak to; a 50% change is extrapolation, which is why lift defaults to 0.01.

References

Jin, Y., Wang, Y., Sun, Y., Chan, D. and Koehler, J. (2017). Bayesian methods for media mix modeling with carryover and shape effects. Google Inc. https://research.google/pubs/pub46001/

Examples

data(mm_weekly)
north <- mm_weekly[mm_weekly$geo == "north", ]
truth <- attr(mm_weekly, "truth")

# Television: long carryover, S-shaped response
marginal_roi(
  north$tv, coefficient = truth$beta[["tv"]],
  adstock = list(decay = truth$decay[["tv"]]),
  saturation = list(half_max = truth$half_max[["tv"]],
                    shape = truth$shape[["tv"]])
)
#>    spend contribution      roi     mroi
#> 1 266978     302361.4 1.132533 1.083959

# Letting the carryover from the final weeks play out
marginal_roi(
  north$tv, coefficient = truth$beta[["tv"]],
  adstock = list(decay = truth$decay[["tv"]]),
  saturation = list(half_max = truth$half_max[["tv"]],
                    shape = truth$shape[["tv"]]),
  extend = effective_window(truth$decay[["tv"]], 0.99)
)
#>    spend contribution      roi     mroi
#> 1 266978     311695.6 1.167495 1.128674

# Only the last quarter's budget changes
last_q <- seq_len(nrow(north)) > nrow(north) - 13
marginal_roi(
  north$tv, coefficient = truth$beta[["tv"]],
  adstock = list(decay = truth$decay[["tv"]]),
  saturation = list(half_max = truth$half_max[["tv"]],
                    shape = truth$shape[["tv"]]),
  rows = last_q
)
#>    spend contribution      roi      mroi
#> 1 266978     302361.4 1.132533 0.7083583