Additive and Multiplicative Decomposition

By the end of this lesson, you should be able to: split a change into contributions that actually sum to the change, name the three standard conventions and say which one closes, explain where the interaction term comes from, and recognise the same Shapley value when it turns up again in attribution.

Two shapes of decomposition

Almost every "why did this move" question is one of two shapes.

Additive. The metric is a sum of parts. Active users are new plus retained plus resurrected. Total orders are the sum across countries. These are easy: the change in the total is the sum of the changes in the parts, exactly, always, with nothing to argue about. Lesson 3 did this.

Multiplicative. The metric is a product of factors. This one isn't easy, and it's where most contribution analysis quietly goes wrong.

$\text{GMV} = \text{Users} \times \text{Orders per user} \times \text{Average order value}$

Here's Basket, first four whole weeks against the last four:

GMV rose by $123,483 a week. How much of that was users, how much was frequency, and how much was basket size?

That sounds like a question with one answer. It has at least three.

Where the interaction term comes from

Take the simple two-factor case, revenue as users times conversion. Expand the change:

$\Delta R = \underbrace{\Delta U \cdot C_0}_{\text{users}} + \underbrace{U_0 \cdot \Delta C}_{\text{conversion}} + \underbrace{\Delta U \cdot \Delta C}_{\text{both at once}}$

The first two terms are the ones people expect. The third is the extra revenue that exists only because both factors moved together: the new users spending at the new conversion rate. It belongs to neither factor on its own, and it has to go somewhere.

The three standard conventions are three different answers to "where".

• Laspeyres. Hold everything else at its starting value. Each factor gets credit for what it would have done alone in the old world.
• Paasche. Hold everything else at its ending value. Each factor gets credit for what it does in the new world.
• Shapley. Average over every order in which the factors could have changed. Nobody goes first, so nobody gets the overlap.