Statistical test to establish that event B is caused by event A I want to have a statistical test to establish that event B is caused by event A
Here
event B is customers that are supposed to leaved leave for competitor stay in our company
event A is an attractive offer for some customers (some may leave, and some may stay)
If we know that some of our customers is going to go for the competitor, and we come up with an attractive offer for them to stay.
If in the end, we notice that there were fewer customers that are leaving for competitors (event B), is there a statistical test that helps to establish a relationship that it is the attractive offer (event A) that retains those customers that will otherwise leave (event B)?
 A: Proving Causation
One could argue that there is no statistical test that can establish causation. This can only be done by experimental design.
Causation by definition includes time ordering. Physics demands that the cause of an event must happen before the event. Thus you must create an experiment in which you control the cause and observe the effect.
You would then use statistical tests to support an argument that the effect is strongly associated with your cause.
Even that is not sufficient because you can't know, among all possibilities in the universe, that the effect you observed was really due to the cause. This is where the question becomes more philosophical.
If you have a very plausible mechanistic model that is repeatedly validated experimentally than you are on your way to establishing causation.
Your Question
If I understand correctly, you want to create a predictive model by which you can classify a customer preference as A or B based on some other variables, some of which you can control. There are a number of statistical models that allow you to test the association of these variables with the customer's choice. The most common is logistic regression. Others include descriminant analysis and cluster analysis.
In order to "prove" causation you would need to do the following:

*

*Use existing knowledge from marketing, psychology, etc. to establish a mechanism by which your incentives influence customer choice.

*Design well-controlled experiments in which you create various incentives (positive incentives, negative incentives, neutral incentives), and observe customer choice.

*Apply statistical methods to test the hypothesis that the incentives have no effect (null hypothesis) and see if you can reject that hypothesis.

I hope this helps!
