# Binary logistic regression with 3 similar outcomes

I was given three binary dependent variables, which are the following:

1) Paid in Full (Subject paid their full balance)
2) Settled in Full (Subject paid 80% or more of their balance in 1, 3, or 5 payments)
3) Rehab (Subject made 10 or more payments)


With these three binary dependent variables, the goal of this project is to build a predictive model to predict the odds of Paid in Full (PIF), Settled in Full (SIF), or Rehab. I have 250k records and ~500 potential predictor variables to work with.

My question is how to best capture these 3 types of payers. I was told to simply combine all three dependent variables into one dependent variable, where: Paid = 1 (PIF, SIF, or Rehab) and 0 = Not Paid (All other records), and use the pool of predictive variables to predict this single outcome using binary logistic regression. However, I do not believe this is the best approach as the factors that result in PIF likely differ from factors that result in SIF and/or Rehab.

Is there a better way to model this data than using only the single outcome variable/binary logistic regression?

Any references or explanations are greatly appreciated!

• Is there an ordering between 2 and 3? What about an ordered logistic regression? Feb 27, 2014 at 12:48
• Not especially, as Rehabs could theoretically pay ten dollars (\$1 at each of the 10 payments), whereas someone who Settled in Full has to have paid 80% or more of their balance in 1, 3, or 5 payments. Do you think multinomial logit would be a better approach in this case? Feb 27, 2014 at 13:16
• My worry is primarily with the interpretation of the multinomial logit model in this case. Feb 27, 2014 at 13:39
• I agree, but our model with simple interpretation (binary logit) is not predictive enough. We may need to sacrifice interpretation for prediction in this case, as I am trying to do whatever I can to improve model performance. I am going to give multinomial logit a try, but I am definitely interested in any other alternative approaches. Feb 27, 2014 at 13:42
• There are some other models for classification that you could look at - the Naive Bayes Classifier springs to mind. Of course, one must be wary of the independence assumptions, but you could give it a go. Feb 27, 2014 at 16:52

It sounds like multinomial logit should work. Your states are competing risks. Look at mlogit command in stata.

UPDATE: Here's the paper with an example of application of mlogit model Appendix A: Econometric Analysis of Mortgages

I think that Begg and Gray (1984) were the first to use in this setup, and the paper is referenced from the link above. As long as your states are mutually exclusive and exhaustive, this should work.

• Thank you for the lead. Do you have any experience performing this in SAS or R? Unfortunately, I do not have access to Stata. I will look into multinomial logit though. Thanks again! Feb 27, 2014 at 12:48
• I didn't use mlogit in sas or r, but I think the command name is the same. You can implement it yourself using 3 binomial logit models for each outcome then combining them with bayes theorem. It helps to understand how exactly mlogit works, a good exercise. Feb 27, 2014 at 13:48
• That does sound like a good exercise, can you provide any references to further explain how to combine the 3 binomial logit models using Baye's Theorem. This would be very helpful! Feb 27, 2014 at 13:51