Child pages
  • Contribution Analysis (5.3)

Contents

Context

Analysis | Report | Target Analysis | Contribution Analysis

Contribution Analysis was first designed to be used in the context of marketing mix analysis. It used counterfactuals for measuring the added value of each driver on the target variable (e.g. Sales) over the entire analyzed period (by using the model), as well as for each observation, i.e. each line in the dataset. 

In the initial version, the contributions of each driver were computed using the following two counterfactuals:

  • ModelWhat would have been the mean value of the target, had the driver been set to its min value, instead of its marginal probability distribution, all the other confounders being hold to their marginal probability distributions? 
  • Data (for each observation): What would have been the mean value of the target, had the driver been set to its min value, instead of the state in which it was observed, all the other confounders being set to their observed states? 

New Feature: Neutral State

For each driver, counterfactuals are now based on a specified Neutral State as opposed to the minimum value of the driver. By default, the minimum  Value of drivers is used as their Neutral State. However for some drivers, the minimum value may not reflect the neutral condition of a system. For instance, the lowest temperature ever recorded would not be an appropriate neutral condition of a domain. Rather, the average temperature might be a better Neutral State.

The  Reference State , which was first introduced in BayesiaLab 5.1, can now be used to specify the Neutral State of a driver.

New Feature: Type I and Type II Contributions

The measure of a driver's contribution is highly sensitive to the conditions used in the counterfactual:

  1. the state of the driver,
  2. the states of all the other confounders.

There are then multiple ways to compute contributions. 

 In the initial version of  Contribution Analysis , the estimations were based on  what we call now  Type I Contribution ,  i.e.:  

  1. Neutral State state for the driver
  2. observed values for the confounders.

We introduce in this new BayesiaLab release Type II Contribution, which is based on the following counterfactuals:

  • ModelWhat would have been the mean value of the target, had all the other confounders being set to their Neutral States, the driver being hold to its marginal probability distribution? 
  • Data (for each observation): What would have been the mean value of the target, had all the other confounders been set to their Neutral States, the driver being set to its observed state? 

i.e.:

  1. observed value for the driver
  2. Neutral States for the confounders

As illustrated in the examples below, the divergence between the results of these two types of contributions is directly dependent on the kind of relation that links the confounders and the target. 

Example

Let's use the following network made of 3 drivers and one target variable.

We will use 3 different deterministic functions to define the Sales value based on the driver values: Sum(), Max(), and Min(), and illustrate how to compute the Model Contributions for TV.

Note

The contributions are identical for the 3 drivers in these specific examples.

Sales = Sum(Radio, Online, TV)

Counterfactual: What would have been the value of Sales, had TV been set to 0, instead of its marginal probability distribution (all the other confounders being hold to their marginal probability distributions)?

Setting TV to 0 reduces Sales by 0.488.

The Model D ecomposition for TV is:

Initial Value of Sales - Final Value of Sales = 1.479 - 0.991 = 0.488

The Model Contribution is:

Model Decomposition / Initial Value of Sales = 0.488/1.479 = 33%

Counterfactual: What would have been the mean value of Sales, had all the other confounders being set to 0 (the driver being hold to its marginal probability distribution)? 

Setting TV to its original marginal probability distribution and setting all the other drivers to 0 increases the Sales by 0.512.

The Model D ecomposition for TV is:

Final Value of Sales - Value of Sales given all the drivers set to 0 = 0.512 - 0 = 0.512

The Model Contribution is:

Model Decomposition / Initial Value of Sales = 0.512/1.479 = 34%

Sales = Max(Radio, Online, TV)

Counterfactual: What would have been the value of Sales, had TV been set to 0, instead of its marginal probability distribution (all the other confounders being hold to their marginal probability distributions)?

Setting TV to 0 reduces Sales by 0.054.

The Model D ecomposition for TV is:

Initial Value of Sales - Final Value of Sales = 0.662 - 0.608 = 0.054

The Model Contribution is:

Model Decomposition / Initial Value of Sales = 0.054 /0.662 = 8%

Counterfactual: What would have been the mean value of Sales, had all the other confounders being set to 0 (the driver being hold to its marginal probability distribution)? 

Setting TV to its original marginal probability distribution and setting all the other drivers to 0 increases the Sales by 0.5.

The Model D ecomposition for TV is:

Final Value of Sales - Value of Sales given all the drivers set to 0 = 0.5 - 0 = 0.5

The Model Contribution is:

Model Decomposition / Initial Value of Sales = 0.5/0.662 = 75%

Sales = Min(Radio, Online, TV)

Counterfactual: What would have been the value of Sales, had TV been set to 0, instead of its marginal probability distribution (all the other confounders being hold to their marginal probability distributions)?

Setting TV to 0 reduces Sales by 0.337.

The Model D ecomposition for TV is:

Initial Value of Sales - Final Value of Sales = 0.337 - 0 = 0.337

The Model Contribution is:

Model Decomposition / Initial Value of Sales = 0.337/0.337 = 100%

Counterfactual: What would have been the mean value of Sales, had all the other confounders being set to 0 (the driver being hold to its marginal probability distribution)? 

Setting TV to its original marginal probability distribution and setting all the other drivers to 0 does not change Sales.

The Model D ecomposition for TV is:

Final Value of Sales - Value of Sales given all the drivers set to 0 = 0 - 0 = 0

The Model Contribution is:

Model Decomposition / Initial Value of Sales = 0%

New Feature: Base

In classical Linear Regression models, there are usually two additional terms in the function that links the Target and the Drivers:

  • the Intercept, the Target value when all the Drivers are set to 0
  • the Error, the difference between the observed value of the Target and its predicted value.

A Base contribution is now available in the Contribution Analysis tool. The objective is to combine the Intercept and the Error terms.

The Base mean is computed by setting all the Drivers (i.e. all the confounders that are observable) to their Neutral State, while

  • Model: holding fixed all the probability distributions of the confounder variables that are Not Observable .
  • Data  (for each observation): setting the confounder variables that are  Not Observable to their observed states.

The initial contribution of the Base is then defined by:

Base Contribution = Base Mean / Initial Target Value 

Note

The Base mean is used in Type II Contribution for computing the decompositions:

D = Final Target Value - Base mean

Note

The Base mean is also used in the computation of the Contributions (denominator)

d = Initial Target Value - Base mean

Example

Let's take back the Sum model we previously used for illustrating Type I and II Contributions. Let's suppose Online is beyond our control. We then set this confounder variable as Not Observable.

Setting the Drivers to 0 while holding fixed the probability distribution of Online returns a Sales value of 0.512.

The (Model) Base mean is then equal to 0.512, and its contribution is:

Base mean / Initial Value of Sales = 0.512/1.479 = 33%

Setting TV to its original marginal probability distribution and setting all the other drivers to 0 increases the Sales by 0.479.

The Model D ecomposition for TV is:

Final Value of Sales - Base Mean = 0.991 - 0.512 = 0.479

The Model Contribution is:

Model Decomposition / (Initial Value of Sales - Base Mean) = 0.479/0.967 = 50%

New Feature: Mean Contribution and Normalization

Four different contributions are now available for each driver: Type I and Type II, computed with the model and the data.

The three examples we used to illustrate Type I and Type II clearly show that Type I returns an underestimated contribution for Max type functions and an overestimated contribution for Min type functions. Conversely, Type II returns an underestimated contribution for Min type functions and an overestimated contribution for Max type functions. These two measures are only identical for the Sum type function.

The Mean Driver Contribution combines  Type I  and  Type II  by averaging the computed contributions.

Mean Contribution = (Type I + Type II) / 2

When the sum of all the drivers's contributions and the Base contribution is greater than 1, all the contributions are normalized to get a final sum equal to 1.

For each observation of the dataset, the mean contribution of each driver is equal to:

Mean Contribution = (Type I + Type II) / 2

When the sum of all the drivers's contributions and the Base contribution is greater than 1, all the contributions are normalized to get a final sum equal to 1.

When this sum is lower than 1, the complement to one is added to the Base contribution.

The Final Mean Contribution of a Driver is the average of its normalized contributions:

Final Mean Contribution = (Mean Contribution Model + Mean Contribution Data) / 2

After computing the sum of all the final driver mean contributions, the Final Base Contribution is defined as (1 - sum).

 

Example

Let's take again the network with the Sum function.

Update Feature: CSV File

Setting an output file allows saving the Decompositions and Contributions in a CSV file for every single observation, i.e. each line in the database.

This report comes now with:

  • the Normalized Base and Drivers' Mean Contributions
  • Type I Normalized Decompositions and Contributions 
  • Type II Normalized Decompositions and Contributions 
Example

New Feature: Curves

Curves allows generating a Stacked Curves Graph that illustrates, for each observation, the normalized Base and Drivers' Mean Contributions.

Example

  • The X-axis corresponds to the observations read in the original dataset
  • The Y-axis is the value of the Target
  • The first curve corresponds to the Base contribution for each observed Target value  :
  • The curves corresponding to the Drivers contributions are then stacked up as follows:
  • The final curves corresponds to the observed Target Values .

Note: Curve Selection

Clicking the boxes of the legend hides/shows the corresponding contributions


 

Note: Highlighting Contributions

 Hovering over the legend boxes increases the size of the associated points

Conversely, hovering over the data points increases the corresponding box legend.

Note: Zooming

 

Zooming in the graph can be done by left clicking on the graph to define zone to zoom in.

 

Double clicking on the graph allows to come back to the default zoom.

Note: Copy

Right clicking on the graph allows copying either the image (PDF) or the data points (Html, Plain Text).

New Feature: Synergies

The purpose of this new measure is to compare the contributions of jointly used Drivers versus the sum of their particular contributions.

Synergy computations are based on Type II contributions (model and data), and the search is restricted to pair of Drivers only.

The following counterfactuals are then used:

  • ModelWhat would have been the mean value of the target, had all the other confounders being set to their Neutral States, the pair of analyzed drivers being hold to their marginal probability distributions? 
  • Data (for each observation): What would have been the mean value of the target, had all the other confounders been set to their Neutral States,  the pair of drivers being set to their observed states? 

 

Example

Let's use the following network made of 3 drivers and one target variable.

We will use two different deterministic functions to define the Sales values based on the drivers: SumMin() and SumMax(), and illustrate how to compute the Synergy between Online and Radio .

Sales = Sum(Min(Online, Radio), TV)

Setting Online to its original marginal probability distribution does not change the Sales value. The Decomposition and Contribution are then equal to 0.

These values are identical for Radio.

Setting  Online  and Radio to their original marginal probability distributions increases the  Sales  value by 0.407.

The Model Synergy Decomposition for Online-Radio is:

Final Value of Sales - Base Mean - Decomposition_Online - Decomposition_Radio = 0.407 - 0 - 0 - 0 = 0.407

The Model Synergy for Online-Radio is:

Model Synergy Decomposition / (Initial Value of  Sales - Base Mean) =   0.407/0.885 = 46%

Sales = Sum(Max(Online, Radio), TV)

 

Setting Online to its original marginal probability distribution increases the Sales by 0.512.

The Model D ecomposition for Online is:

Final Value of Sales - Base Mean = 0.512 - 0 = 0.512

The Model Contribution is:

Model Decomposition / (Initial Value of  Sales - Base Mean) = 0.512/1.097 = 46%

Again, these values are identical for Radio.

Setting Online and Radio to their original marginal probability distributions increases the Sales value by 0.617.

The Model Synergy Decomposition for Online-Radio is:

Final Value of Sales - Base Mean - Decomposition_Online - Decomposition_Radio = 0.617 - 0 - 0.512 - 0.512 = -0.407

The Model Synergy for Online-Radio is:

Model Synergy Decomposition / (Initial Value of  Sales - Base Mean) = -0.407/1.097 = -37%