Effect of email offer - did buy /did not buy; and spending amount, Chi square, ANOVA, R, TSQL Customers who bought product A in the past are offered supplementary product B by email. They are split into 3 groups:


*

*Did not get email offer, (about 60.000 population) 

*Did get email offer and opened mail, (about 30.000 population)

*Did get email offer and did not open mail). (about 30.000 population)





*

*I know whether they bought something and for how much.

*I also know the size of these original groups.


What sort of test would I use to evaluate these hypotheses:


*

*The mailing had an effect - more customers came into shop (Bought
something T/F)

*The mailing had an effect - customers spend more. (Did spend x
money.)


It can be two separate analyses. My initial guesses are ANOVA or CHI square. I am using R / excel / tsql.
 A: Preliminary remarks
In general, I would be very careful with assuming your observations are independent and identically distributed. In particular, business count data ("how many products bought") usually does not follow normal distribution. 
Do you plan to take into consideration other factors that may affect your response variable ("they bought something and for how much")? I would suggest that in such business cases there might be multiple additional confounding variables: 


*

*how long the client is with you, 

*via what channel he reached the website, 

*what are his previous purchases,

*type of a product purchased,


etc. 
Inference idea
Having this stated, you can try to model the response variable with a model that incorporates (as explanatory variables) information about:


*

*the customer_group the customer was assigned to (levels: with_mail, without_mail etc.),

*other variables, for example such as outlined above,


and infer about statistical significance of being assigned to a particular level of a customer_group. 


*

*A technical remark: in particular, if there might be several different actions taken (long_email_sent, short_email_sent etc.), it is convenient to set the one which refers to "no action" (here: without_mail) as a reference level for such a factor variable. 


Modelling tools
The models you can think of are, starting from less and moving to more advanced: 


*

*Logistic Regression - model with a 0/1 response stating whether or not any product has been purchased by a customer,

*Poisson Regression - model with a count response stating how many products has been purchased (0 including),

*Negative Binomial regression / Zero-Inflated Poisson Regression - model with a count response stating how many products has been purchased (0 including), used to model count data that has an excess of zero counts (what is indeed typical in business cases),

*Generalized Linear Mixed Model (e.g. Poisson model with a random intercept) - model with a count response, which is adequate for clustered data (you can consider "clusters" as a different groups of products which differ from each other by general demand you observe).

