Since Prosper provides data on members and their friends who are also members, we can conduct a simple “social network” analysis. What is the value of a friend when getting approved for a loan through Prosper? I first determined how many borrowers were approved and how many borrowers were declined for a loan. Next, I determined how many approved friends each borrower had. From that data, we get the following contingency table of counts:

Now we can calculate the following probabilities: the probability that you are approved given that you have at least 1 approved friend, or P(A | F), where A = Approved and F = Has at least 1 approved friend. We can also calculate the probability that you are approved given that you have zero approved friends, or P(A | F’).
Following the rules of conditional probability we have P(A | F) = P(A ∩ F) / P(F).
Probability of being approved: P(A) = 37212 / 286791 = 0.129
Probability of having at least 1 approved friend: P(F) = 5692 / 286791 = 0.0198
Probability of being approved and having at least 1 approved friend: P(A ∩ F) = 2838 / 286791 = 0.0098
Probability of being approved given that you have at least 1 approved friend:
P(A ∩ F) / P(F) = 0.0098 / 0.0198 = 0.4949
Now we will also calculate the probability of being approved given that you do not have at least 1 friend:
Probability of being approved: P(A) = 0.129
Probability of having zero approved friends: (F’) = 281099/286791 = 0.980
Probability of being approved and having zero approved friends: P(A ∩ F’) = 34374 / 286791 = 0.119
Probability of being approved given that you have zero approved friends: P(A ∩ F’) / P(F’) = 0.119 / .980 = 0.12
Therefore:
P(A | F) = 0.49 (49% of applicants with at least one friend in the network were approved.)
P(A | F’) = 0.12 (12% of applicants with no friends in the network were approved.)
We can calculate a risk ratio from these two quantities:
Risk Ratio: P(A | F) / P(A | F’) = 4.08
Members with at least 1 approved friend are 4.08x more likely to be approved for a loan than members who have 0 approved friends
While this is an interesting statement, it does not mean that having an approved friend causes approval for a loan, nor does it mean that being approved for a loan causes one to have an approved friend. It is simply an observation of two correlated variables. In fact, I would be willing to bet that being approved for a loan actually causes one to have approved friends as a result of word of mouth referrals.
Dataspora leverages a proprietary platform that can distinguish correlation from causality between variables from massive data sets. This complex yet extremely important notion of causality vs. correlation applied to business intelligence will be discussed in further detail in a future post.



















