Self Selection bias for estimating a valiable correlated to the selector I am trying to find a way to see if a measured variable between two groups is significantly different. This would normally be done through a t-test if the two groups were randomly selected from the population. However, in my case the two group are not randomly selected. They are selected by a binary variable which is strongly correlated to the measured variable. 
As a toy example say we are trying to find if having a profile on an e-commerce site causes users to spend more. If we look at the two groups (with and without a profile) there is a statistically significant greater mean spend by the group who has a profile. However we know that users that expect to spend often are more likely to set up an account in the first place since they are more engaged. Is there a way to compensate for this so that one could confidently assert that encouraging users to create a profile is likely to increase revenue for the e-commerce site? 
 A: You're essentially asking about the problem of making a causal inference from observational data (since you mention in your comments that you don't want to perform experiments).
This problem is in general very tough, and the tools we have currently are not very solid or trustworthy. That being said, there are several different possible angles of attack, like  instrumental variables, regression discontinuity designs, difference-in-differences, etc.
For instance, in your ecommerce website example, if they introduced some feature at one point that made people more likely to fill out their profiles when they signed up, you could use this as an instrumental variable to see whether people who signed up right before the change vs. right after had different buying behavior.
These types of analysis are not nearly as trustworthy as a truly randomized experiment, since it's somewhat tricky to get them right and make sure the assumptions are satisfied. But they can provide better evidence for causation than a simple regression analysis.
