Selection Bias in Application Data
Selection bias is a known issue in data science, but its depth is not fully appreciated. The bias makes a data set far from an infallible neutral source from which models can learning. To blindly run analysis on biased data risks infecting the model with the same bias in the initial data and thereby perpetuating errors and discrimination. In this post, I hope to explain why selection bias is so challenging particularly in application data such as when applying for a financial loan.
Cicero wrote of the Greek poet and famed poet Diagoras of Melos. His friend, in an effort to convince him of the existence of gods points to painting of men saved from storms on the sea through prayer and said, “You think the gods have no care for man? Why, you can see from all these votive pictures here how many people have escaped the fury of storms at sea by praying to the gods who have brought them safe to harbor.” Diagoras countered “Yes, indeed, but where are the pictures of all those who suffered shipwreck and perished in the waves?”[^1]
A selection bias occurs whenever the data are not representative of the full underlying distribution. If you want to create a sample of all people who pray, but only include those who commission paintings of themselves surviving storm, then it is not covering the full distribution by missing the those who drowned. Just as Diagoras was not convinced by the paintings, so should we be skeptical by conclusions drawn from data with a selection bias.
Once you start looking for selection bias, it is easy to spot. Right now in the U.S. we are inundated with political polls in the run-up to the presidential election. These polls try to be representative of U.S. voters, but struggle with selection bias from whom is willing to answer the poll. Pollsters put in a lot of effort to minimize this through a range of statistical techniques. Typically, this involves segmenting the population into buckets, like white college educated males. Each responses is weighted depending on which buckets it belongs and how many likely voters are in that bucket. [^2] Occasionally, though, the effort to reduce bias can cause problems with small sample sizes. In one notable poll, the New York Times discovered a large weight was placed on a single voter that fell into small, hard to sample demographics. [^3]
Handling this bias is a huge part of the value added by fiverthirtyeight and the Upshot at the New York Times. Instead of working to reduce the polling bias, they model how the selection bias in different polls is likely correlated. For example, the selection bias in polls in Ohio are likely to be strongly correlated to the selection bias in Michigan, but less correlated in North Carolina because of the different demographics. So if Donald Trump outperforms the polls and wins in Ohio, he is more likely to also outperform in Michigan than in North Carolina.[^4]
Application data, whether for jobs, colleges, or loans, almost always have a huge selection bias. In all applications, it is much easier to obtain data from the people that were accepted. A university that wants to assess their undergraduate admissions could run a test to see how well their selection process correlates with graduation rate or GPA of the admitted students. But the same university could never run a test to see how well a student whom they rejected would have performed. The same core problem exists in financial data.
Selection Bias in Financial Data
I recently was asked to create a predictive model for financial loans. The data I was given for training consisted of about 100,000 U.K. loan applications. The data included whether each loan was either approved or denied and, if approved, whether or not it was repaid. The evaluation criteria for the predictive model I was tasked to build was simply +1 if the model gave a loan and it was repaid, -1 if the model gave a loan and it wasn’t repaid, and 0 if the model denied a loan.
Though this may seems like a standard machine learning based approach, there are fundamental problems. The data are subject to a massive selection bias in that there is only information on repayment for loans that were given. There are loan applicants for whom their ability to repay the loans is unknown because it is untested. This is apparent when evaluating the a model with the criteria above as there is no way to score what happens when the model gives a loan when the ability to pay is unknown. Handling this problem is an incredibly complex problem and key to building any prediction on loans.
The ultimate goal of loan modeling is to predict the probability that a loan would be repaid given input data \(x\): \(P(LoanRepaid | x)\). The input data includes information like current salary, savings account balance, loan purpose and other financial information. But it could also include demographic information like age, gender, and ethnicity as well geographical information like zip code.
The challenge is that this is not what we actually have in the data. Instead, we can only learn the probability that a loan was granted by the bank, \(P(LoanGranted | x)\), and the probability that it was repaid if a loan was granted \(P(LoanRepaid | LoanGranted=True, x)\). The additional conditional statement is a way of mathematically representing the selection bias.
A naive approach is to approximate \(P(LoanRepaid | x)\) as \(P(LoanRepaid | LoanGranted=True, x)\). This could done by throwing out training cases in which the loan was not given and learning to classify loans as either repaid or not repaid on the remaining data. This has the nice feature of avoiding the pesky uncertainty of whether or not loan would be repaid whenever the loan was denied. But has the bad feature of being wrong. A classifier trained only on data for which a loan was approved is worthless as that data that is not representative of the distribution of new loan applications.
In the data I analyzed, one binary feature whether or not the applicant
is_employed. Intuitively, this feature should be useful in determining if a loan would be repaid. However, when following the naive approach of only analyzing applications for which a loan was granted,
is_employed has almost no predictive power for the simple reason that damn near every loan was given to someone employed. In other words,
is_employed is very predictive of if a loan is given. But it is not useful for predicting if a loan is repaid because there are nearly no data on an unemployed applicant receiving a loan. The loan applicants who were denied a loan never had their ability to repay tested.
A different approach would be to model \( P(LoanRepaid | LoanGranted=True, x) \) the probability that a loan is repaid given that the bank would grant the loan and information \(x\) and \( P(LoanGranted | x) \), the probability that the bank would grant the loan given the information \( x \). This means building two separate models. The first model is exactly the same as the naive model above. The second model is trained to predict whether or not the bank from whom the training data came would issue a loan. Once The decision to grant a loan would occur only if there is a high probability that the loan was granted by the bank and a high probability that a loan would be repaid given it was granted. The exact decision process depends on the bank’s necessary repayment rate and on estimates of how likely people are to repay loans for whom the bank is likely to not grant a loan as well.
Mathematically, we can be more precise. The desired quantity is \(P(LoanRepaid | x)\). This can be factored in terms \(P(LoanRepaid | LoanGiven, x)\) and \(P(LoanGiven | x)\):
The first term is learnable with the data collected. The second term, however creates problems due to the selection bias in the data. Since there is no knowledge of the the probability that a loan is repaid if a loan was not given the second term is impossible to determine.
What this means in practice
The learning on the financial data is heavily dependent upon \P(LoanGranted | x)\, the probability that the bank the produced the initial data would grant a loan for an application with values \x\. This means that a diligent modeler, who wants to be as fair as possible for potential loan applicants, is forced to depend upon the loan history of the bank. Whatever method the bank was using to determine loans will necessarily infect any future model created from that biased data.
Consider a situation in which a bank was explicitly discriminatory in its loan practice and refused to give loans to immigrants. There are correlations between immigrants and other features. These features would then correlated to whether or not a loan is granted. For example, a postal code with a large immigrant population would then become strongly correlated with a loan being denied.
Since a new model needs to learn in part from how the bank gave loans, these correlations would also be learned. Without concerted work to correct the situation, any new model would ‘learn’ to discriminate against immigrants! Biases in data are infectious: erroneous assumptions or discriminatory behavior of loan decisions infect the data, which in turn infects the next model. Discriminatory practices from years ago can still leave fingerprints in the data and models today.
What can we do?
This is not an easy problem to solve. One possible solution is to intentionally include some randomization. If a small percentage of the time, a loan that would be denied is randomly approved. This would provided crucial data to explore the full distribution of loan repayments, but the cost of giving out loans with a potential higher rate of default.
Randomization of financial loans is not without precedence. In 2011, Nigeria launched the YouWin! competition for $50,000 cash grants to start a new business or expand an existing one. 729 of the grants were given out to a random selection of 1,841 semifinalist. The randomization of the award provided a powerful data for making future decisions about how effective the loans are and which potential recipients are most likely to succeed.[^5]
Randomization is not always a viable solution. In the case of university admissions, this would necessitate randomly admitting some students for whom the admission process says reject. This exploration of possible candidates that are being missed is expensive to the point of absurdity. In general, anytime there is a selection bias in which only the result of positive examples are observed and the cost of exploration is high, it will be impossible to get unbiased data.
I don’t have a good answer for how to solve these problems. But I do know that the first step is for data scientists to recognize biases in data and to consider that any data set was not created in a vacuum. Whenever there is a strong selection bias such as occurs job applications or loan applications, the data will always be biased. And these biases can have long lasting and severe consequences.
[^1] Hecht, Jennifer Michael (2003). “Whatever Happened to Zeus and Hera?, 600 BCE-1 CE”. Doubt: A History. Harper San Francisco. pp. 9–10