Regression vs. Expected value for campaign results

Question

Let's say that I have 100k customers, and I can only send 10k letters for a loan, and I do have information from past campaigns and I know I can expect about 100 responders.

If I want to get in my next campaign the best possible outcome in terms of money (i.e. the responders who request the highest loans), what would be the difference between:

1) Creating a regression model using 0 for those who have not responded in past campaigins, and the loan amount for those who did, and select the 10k highest, or 2) Same as above, but also create a classification model and multiply the regression versus the probability of respond, and select the 10k highest

I'm confused as to what would be the proper procedure here Any help will be greatly appreciated!

Answer 1

There are special types of regression models for these kinds of problems.

The expected loan amount for a given individual is a function of both the probability that they respond and take out some type of loan and the loan amount they take out conditional on them responding. In mathematical terms, we can think of it as expected value of the loan amount and it can be expressed like this:

$E[loan$ $amount] = Prob(response) * E[amount | responded] + Prob(no$ $response) * E[amount | not$ $responded]$

(It seems like in your case you may be assuming that without receiving a letter an individual has a zero probability of taking out a loan, so you can drop everything after the $+$ operator, if that's the case.)

So you have to model both variables - probability and amount. The question is how. If you just go with the first approach you outlined, i.e., a basic regression for modeling counts like Poisson or Negative Binomial and include 0's in the model, then you will be assuming that the zeros and the nonzeros (positive values) come from the same data-generating process, governed by the same data distribution, and are influenced in the same manner by the same set of predictors. But models like hurdle models and zero-inflated models (e.g., zero-inflated Poisson) separately model the binary outcome of whether a count variable is zero-valued or positively-valued and the conditional distribution of positive values.

(If you want to know more hurdle and zero-inflated models, here is a couple of links to Q&A's on the Cross Validated site: https://stats.stackexchange.com/questions/81457/what-is-the-difference-between-zero-inflated-and-hurdle-models, https://stats.stackexchange.com/questions/279273/zero-inflated-distributions-what-are-they-really.)

You could also manually construct those separate binomial (logistic) and count models (using separate sets of predictors if you wanted) and try to combine the results. One issue though is that you would need to model amounts only for those individuals who have previously responded (as you are looking for expected amount conditional on responding) and it would be difficult for you to calculate/estimate expected amounts for individuals who have not previously responded and for any new customers.

Answer 2

You are trying to calculate the probability (0 to 1) of a person responding and taking out a loan. If you use regression you may get a negative probability. Using a classification model with loan amount (and any other covariates like age, income etc.) is the right method. This question and answer may help.