15, for any value of X, observations with positive errors will be included in the sample."/>
Figure 1.3 Sample selection bias. Dashed line for observations
If we have an umbrella problem and we only want to make a prediction of
Now let us return to the credit example, and suppose that we had lots of variables that included all possible lurking variables. Suppose we knew the model so that there was no garden of forking paths problem. Now, could we really get causal answers out of these data? Suppose we brought in a statistical expert on getting causal results from observational data. Could he do it? The answer is “no.”
Aside from the garden of forking paths, there is a more serious problem with these data, and it is rather subtle. Let us consider where our data came from. People apply for credit. Some are granted credit, while others are not. Of those who are granted credit, some default and some do not. It should be apparent that our data do not constitute a random sample from the population, but a selected sample from the population. In the present case, there is no data on those who were denied credit, some of whom would have defaulted and some of whom would not have defaulted. This is a general problem called sample selection bias, and it plagues observational data; in such a case, the sample is not representative of the population. Let us be more explicit. The population of credit applicants includes four types of persons:
1 Non‐defaulters who get credit.
2 Non‐defaulters who are denied credit.
3 Defaulters who get credit.
4 Defaulters who are denied credit.
Meanwhile, since the sample consists only of persons who have been granted credit, the sample consists only of categories (1) and (3) above and does not look like the population. Therefore, inferences from the sample do not extrapolate to the population, and using such a model would likely be a mistake.
This is a very important problem that bedevils the credit industry, and this problem even has a name: “reject inference,” which is how to conduct inference when there is no data on persons whose credit applications were rejected. Very sophisticated statistical machinery, far beyond the level of this book, has been unleashed on this problem, with only a modicum of success. Indeed, some credit card companies deliberately conduct designed experiments and issue credit to persons who otherwise would have been rejected in order to collect data from categories (2) and (4) so that they may extrapolate their results to the population. However, this is a very expensive solution to the problem, so these types of experiments are rarely performed. The credit card industry largely makes do with sophisticated statistical analyses to answer causal questions.
Finally, there is a modeling problem in the data to which we must draw attention. Suppose there was no reject inference problem, and we knew all the lurking variables. We might run a regression (perhaps a logistic regression, for those of you who know that method) with “default” on the loan as the dependent variable and “age” as one of the independent variables. Suppose further that the coefficient on age is positive and statistically significant. What does this mean?
Such a regression coefficient would not agree with what creditors generally know about the relationship between age and default probability. Based on years of empirical data, they know that young people tend to default more than older people. The nonlinearity in this relationship cannot be captured by a linear regression. What does this nonlinearity mean for our linear regression model? We now have a very substantial modeling problem, and we will get different answers depending on how we model the effect of age on defaults (Is it linear or nonlinear; if nonlinear, what type of nonlinearity?). Does a modeler really want results to be dependent on the choice of how the regression model is built? This is just part of what makes drawing causal inferences from observational data so fraught with danger. When we run experiments, we don't have to worry about any of these things.
Exercises
1 1.2.1 Consider the five examples of lurking variables. We already described an experiment and a manipulated variable for the fire damage example. Come up with experiments and manipulated variables to expose the falsity of the observational conclusion for the bombing and drownings examples (we don't know the lurking variable for the other two).
2 1.2.2 For the following cases of observational data, articulate the precise nature of the sample selection.We wish to determine the effect of education on income by running a regression. (Think of the minimum wage.)Mutual funds that have been in business longer tend to have higher returns than newer mutual funds. We “know” this because we collected observational data on mutual funds, regressed return on number of years in business, and found that funds with more years had higher returns. (What happened to mutual funds with low returns?)
3 1.2.3 In the text, we asked, “If it is really the case that persons with higher credit limits are less likely to default, can we decrease the default rate simply by giving everybody a higher credit limit?” Definitely not! Why not? What is the lurking variable?
4 Using the data set credit.csv , create the variable “percentage default by age” () and plot it against age (). The relationship between and is definitely nonlinear. Describe the nonlinearity and a reason for it.
5 We know that regression results are biased when data have been selected due to a condition on the dependent variable, e.g. . What happens when the data are selected due to a condition on ? Subset the data for those values and run the regression. Is the slope estimate biased? What about the intercept? What can you conclude?
1.3 Case: Salk Polio Vaccine Trials
By the 1950s, polio had killed hundreds of thousands worldwide and infected tens of thousands per year in the United States. Many victims who did not die were condemned to spend the rest of their lives in “iron lungs” since they were unable to breathe on their own. To say nothing of the misery wrought by the disease, the effect on the community was devastating: parents kept their children indoors all summer, playgrounds were vacant, and one sick student in a class was reason for many healthy students to stay home from school. Jonas Salk developed a vaccine in 1952, and in 1954 two separate field trials were conducted, involving nearly 2 million children and 300 000 volunteers in the United States, Canada, and Finland. These constituted the largest clinical trials in history.
The first trial monitored nearly a million first, second, and third graders. The
