Types of Interaction Effects in Market Mix Modeling (MMM)

What is an Interaction Effect in Regression ?

 An interaction effect is the simultaneous effect of two or more independent variables on at least one dependent variable in which their joint effect is significantly greater (or significantly less) than the sum of the parts. It helps in understanding how two or more independent variables work in tandem to impact the dependent variable.

It is important to understand two components first– Main Effects and interaction effects.

Main Effects:

It is important to understand two components first– Main Effects and interaction effects.

Interaction Effect:

As mentioned above, the simultaneous effect of two or more independent variables on at least one dependent variable in which their joint effect is significantly greater (or significantly less) than the sum of the parts.

I will limit this article to discussing about interaction between two variables.

Interaction Effect can be between two: 

I. Categorical variables 

II. Continuous variables 

III. One categorical and one continuous variable

For each of these scenarios, the interpretation would vary slightly.

1. Between categorical variables:

Imagine someone is trying to lose weight. Weight Loss could be a result of exercising or following a diet plan or due to both working in tandem.

The above numbers indicate weight loss in kg.

What does the above result indicate?

i. It shows that exercising alone is more effective than diet plan and results in 5 kg weight loss

ii. Only exercising causes more weight loss as compared to a scenario when both exercising and diet plan are followed together(Your diet plan is not working :) )

What does the above result indicate?

It shows that the weight loss is higher when exercising and diet plan are implemented together. So, we can say that there is an interaction effect between exercising and diet plan.

2. Between continuous variables

Let us view a Regression equation showing both main effect and interaction effect components.

Y = β0 + β1* X1 + β2*X2 + β3* X1X2

The above equation is interpreted as follows:

i. β1 is the effect of X1 on Y when X2 equal to 0 i.e. one unit increase in X1 causes β1 unit increase in Y, when X2 equals 0.

ii. Similarly, β2 is the effect of X2 on Y when X1 equal to 0 i.e. one unit increase in X2 causes β2 unit increase in Y, when X1 equals 0.

iii. In case, neither X1 nor X2 is zero, the effect of X1 on Y depends on X2 and the effect of X2 on Y depends on X1.

To make it clearer, let us rewrite the above equation in another format.

Y = β0 + (β1 + β3* X2) X1 + β2*X2 

=> Y= β0 + β1* X1 + (β2 + β3* X1)X2

=> (β1 + β3* X2) is the effect of X1 on Y and it depends on the value of X2

=> (β2 + β3* X1)is the effect of X2 on Y and it depends on the value of X1

Please note that this article has been written w.r.t to inputs/variables used for Market Mix Modeling. The above concept is a likely scenario for MMM where the inputs could have a zero value.

For a scenario where input variables cannot be zero, some other measures are taken. An example could be a model where a person’s weight is considered as one of the regressors. A person’s weight cannot be zero :)

3. One continuous variable and one categorical variable

The interaction between one categorical variable and one continuous variable is similar to two continuous variables.

Let’s go back to our regression equation:

Y = β0 + β1* X1 + β2*X2 + β3* X1X2

Where X1 is categorical variable, say (Female = 1, Male = 0)

And X2 = Continuous variable

 When X1 = 0, Y = β0 + β2*X2 

=> One unit increase in X2 will cause β2 units increase in Y for males

When X1 = 1, Y = β0 + β1 + (β2 + β3)*X2 


=>One unit increase in X2 will cause β2 + β3 units increase in Y for females 


Effect of X2 on Y is higher for females than males (Please refer figure 1 below)

Figure 1


Interpretation of Interaction in MMM:


1. Both categorical variables:

 Let’s take two categorical variables — seasonality and some launch of product.

Assume that both seasonality and launch of product have a positive relationship with sales. Seasonality and product launch in their individual capacity will lead to sales. If there is an interaction effect between them, this might lead to incremental sales. 


Y = β0 + β1* Seasonality + β2*Product launch + β3* Seasonality * Product Launch


=> Y = β0 + β1 + β2 + β3 


where Seasonality and Product Launch = 1

In case there is no interaction, Y = β0 + β1 + β2

2. Both continuous variables:

Example of interaction between two continuous variable in a MMM could be — effect of TV advertisement and Digital ads together on sales.

So, when there is an interaction term, effect of TV ads on Sales depend on Digital ads and effect of Digital ads on Sales depends on TV ads. 

Y = β0 + β1* TV Ad + β2*Digital Ad+ β3* TV Ad * Digital Ad -> Positive interaction term

If the interaction term is positive, then the joint effect of these two variables is synergistic as it is leading to additional sales. It is suggested that both type of ads should be run simultaneously to get higher sales. 

Y = β0 + β1* TV Ad + β2*Digital Ad - β3* TV Ad * Digital Ad -> Negative interaction term

If the interaction term is negative, the interaction component takes away some part of Sales thus reducing the overall sales. In this scenario, it is suggested not to run both the campaigns simultaneously as it takes away the sales. (Your campaigns are creating confusion among customers: P)

Note that the main effects of these two inputs is positive but the combined effect has a negative Beta value resulting in reduction in total sales.

3. One continuous variable and one categorical variable

Where X1 is categorical variable, say Seasonality (1 if there is seasonality, 0 otherwise)

And X2 = Continuous variable: TV advertisement

Y = Sales

Sales are impacted by seasonality and TV advertisement individually and when they work together. 

Y = β0 + β1* Seasonality + β2* TV ad + β3* TV Ad * Seasonality

In this scenario, when seasonality component is there, then: 

Y = β0 + β1 + β2* TV ad + β3* TV Ad 

=> Y = β0 + β1 + (β2 + β3)* TV Ad

The interaction effect between TV and seasonality has led to additional sales.

This was a brief about Interaction effects between variables. Interaction effect is a vast topic in itself. There are more nuances to it.


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