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How to use conditional expectation language to detect BS comparisons

We talked in class that using language of conditional expectation and regression can help identify issues with causal statements or analysis. In this worksheet, we will go through that concept step by step in case you need more practice with it.

Example 1: How to help people with delinquent hospital bills

This example is drawn from a true story. At a university hospital in Virginia a hospital manager gather a team of workers (MDs, nurses, etc) to improve on the policy of the hospital. One main issue that they needed to tackle is the many patients that had delinquent hospital bills. Delinquent is a financial terms that means a bill is 90 days past due. This team was able to get lots of information about their patients and perform some analysis. Here is a summary of that exercise: “We started collecting information about people who have large unpaid medical bills, and we discover that among these individuals, the main common characteristics is that they are past due on their electrical bill. Hence we are creating a hospital policy to cover their electrical bill for the time they are in the hospital”

  1. What’s the comparison they are doing using conditional expectation?
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    Answer
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    E(Have Delinquent Electrical Bill?∣Delinquent hospital bill=1)E(Have\ Delinquent\ Electrical\ Bill? | Delinquent\ hospital\ bill=1)
  2. What is the causal effect they are trying to identify, and what would be the regression version of that causal statement?
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    Answer
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    With this exercise they are implying that having a delinquent bill has a causal effect on having a delinquent hospital bill. The regression version of that causal link would be Pr(Delinquent Hospital Bill)=α0+α1Pr(Delinquent Electrical Bill)Pr(Delinquent\ Hospital\ Bill)=\alpha_0+\alpha_1Pr(Delinquent\ Electrical\ Bill)
  3. Think about the differences between those two. What do you notice?
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    Think, then click here once you have thought about it!
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    This seems like a contradiction, these two things are not aligned with each other. Given their causal framework, they think that electrical bills affects the likelihood of having a delinquent hospital bill, and the exercise they came up with is exactly the opposite, they are conditioning on the dependent variable (on the variable of interest) and highlighted the main characteristics based on who had a delinquent hospital bill.
  4. What should they have done?
    5. ‣
    Think about it then click here
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    Well it depends on what they were trying to do. If they are trying to understand what are common characteristics of people who have delinquent bills, they can improve this exercise by also comparing it to characteristics of people who don’t have a delinquent bills. This would be at least more informative. E(X∣Delinquent hospital bill=1)−E(X∣Delinquent hospital bill=0)E(X | Delinquent\ hospital\ bill=1) - E(X | Delinquent\ hospital\ bill=0)

    However, if they are trying to understand if a delinquent bill causes a delinquent hospital bill, then following the regression, they should have done the following comparison

    E(Delinquent Hospital Bill∣Have Delinquent Electrical Bill?=1)−E(Delinquent Hospital Bill∣Have Delinquent Electrical Bill=0)E(Delinquent\ Hospital\ Bill | Have\ Delinquent\ Electrical\ Bill?=1) - E(Delinquent\ Hospital\ Bill|Have\ Delinquent\ Electrical\ Bill=0)

    But would this be informative?? Why would we think this is the “right” causal framework to analyze?

Example 2: Drinking and Birthweight

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A study compares pregnant women who drink about 1-3 drinks per drinking occasion vs. pregnant women who do not drink. They find that babies from the mothers who drink more were more likely to have lower birthweight.

What’s the exercise they are doing in conditional expectation?

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Answer
E(Birthweight ∣ Moderately Drinking=1)−E(Birthweight∣ Moderately Drinking=0)E(Birthweight\ |\ Moderately\ Drinking=1) - E(Birthweight|\ Moderately\ Drinking=0)

What’s the causal claim they are making expressed as a regression?

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Answer
Birthweight=α0+α1ModeratelyDrinkingα1<0Birthweight=\alpha_0+\alpha_1ModeratelyDrinking \\ \alpha_1<0

In this example this tracks well, the causal framework and the conditional expectation match, therefore the at least the comparison is sensible. The second step here would be to wonder, what are reasons that we may find an effect like that, while drinking having no effect on babies birthweight. This would give you a reason of why this comparison, although sound, it’s biased.

Example 3: Interviews to CEO

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A reporter is doing a piece on big tech CEOs (microsoft, facebook, etc), and notice a pattern among the people she interviewed. Most of them were college dropouts. Therefore someone could conclude that dropping out of college can increase the chance for someone to be a big tech CEO.

On your own or with your study partner, use conditional expectation language or regression to unveil if that makes sense or not.

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