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RMDA 1 Practice

The objective of this worksheet is to refresh concepts seen in RMDA 1 by practicing some questions.

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Health and Sanitation

Suppose you are interested in the effects of sanitation and hygiene recommendations on children's health. The main recommendation is to wash hands for about 20 seconds. You find that those who follow the recommendations have worse health outcomes than those who do. The following are potential hypothesis that could explain the fact above. How would you change the following statements to make sure they could be explaining the fact above?

  1. The water they are using is contaminated.
  2. There is no effect of handwashing on health outcomes.
  3. The kid who wash their hands tend to be the kids with worse health outcomes.
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Answer
  1. If kids who are following recommendations are using “more” water than those don’t because they are washing their hands, then the water being contaminated could affect more the kids who follow the recommendation vs. not.
  2. If there is no effect, then we shouldn’t expect a difference unless (3) is true.
  3. This could explain that pattern, even if there is a positive effect of handwashing.
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A Fish Oil a Day Keeps the Doctor Away

Many people around the world take fish oil supplements, which are rich in Omega-3 fatty acids, to improve their cardiac health. In the early 1970s, researchers found that Inuits living in Greenland had lower cholesterol levels than Inuits living in other countries. They posited that a fish-heavy diet in Greenland was a key reason for this difference. Today, the supplements are available over-the-counter and are relatively cheap, and this has contributed to their popularity. Yet, more recent evidence – including a large-scale randomized controlled trial described in a New York Times article – has cast some skepticism over the effectiveness of fish oil supplements.

  1. Why might those who take fish oil supplements live longer than those who do not take them even if fish oil has no causal effect on cardiac health?
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    Answer

    Those who take fish oil may be more health conscious so they are also taking other measures like frequent exercise and healthy eating which is leading them to live longer even though fish oil has no effect on cardiac health.

  2. Why might we expect selection bias to go the other way? Provide one specific reason why those who take fish oil supplements could die earlier than those who do not take them even if fish oil has no causal effect on cardiac health.
    3. ‣
    Answer

    Those who are taking fish oil might be doing this as a last ditch effort to improve their cardiac health because other means/treatments have no worked or been unsustainable. They will likely live a shorter life even though fish oil does not affect cardiac health.

  3. Now let’s define some variables to represent some hypothetical data:
      4. Di=1{individual i consumed fish oil supplment}Yi=years of life of individual iYi1=years of life of individual i if consumed fish oilYi0=years of life of individual i if not consumed fish oilD_i=\mathbb{1}\{individual\ i \ consumed\ fish\ oil\ supplment\} \\ Y_i=years\ of\ life\ of\ individual\ i \\ Y_i^1=years\ of\ life\ of\ individual\ i \ if\ consumed\ fish\ oil \\ Y_i^0=years\ of\ life\ of\ individual\ i \ if\ not\ consumed\ fish\ oil \\
    1. Express the following phrase in conditional expectation notation: “the average years of life of all individuals in the data”
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      Answer

      E[Yi]E[Y_i ]

    2. Express the following phrase in conditional expectation notation: “the average years of life of individuals consuming fish oil”
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      Answer

      E[Yi∣Di=1]E[Y_i| D_i=1]

    3. What does the following notation mean in words? E[Yi1−Yi0]E[Y_i^1-Y_i^0]
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      Answer

      The causal effect of consuming fish oil on years of life. Notice that saying “Average difference in lifespan for an individual who consumed fish oil versus if that same individual did not.” may be correct but is not sufficient, because given the index that we have, this is not only a difference, but a causal effect.

    4. What does the following notation mean in words? E[Yi1−Yi0∣Di=1]E[Y_i^1-Y_i^0|D_i=1]
      5. ‣
      Answer

      The average causal effect of consuming fish oil on years of life (only for those who end up consuming fish oil)

    5. Why might the following statement be true? Give one specific reason. E[Yi1−Yi0∣Di=1]>E[Yi1−Yi0]E[Y_i^1-Y_i^0|D_i=1]>E[Y_i^1-Y_i^0]
      6. ‣
      Answer

      This statement posits that the causal effect of fish oils on years of life is larger for those who end up consuming fish oil than for all individuals. This implies that the causal effect is larger for those who end up consuming fish oil than for those who do not consume fish oil.

      This could be true if the individuals who consume fish oil derive larger benefits from doing so because of innate characteristics that allow them to maximize the benefits of fish oils – e.g., they are more motivated to take them every day, they take other supplements that interact positively with fish oils, etc.

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Interpreting Coefficients 1

  1. For the following questions, refer to the following equation and its respective graph
Y=α0+α1Treatment+ϵY=\alpha_0+\alpha_1Treatment+\epsilon
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  1. What’s the value of α^0 and α^1\hat\alpha_0 \ and\ \hat\alpha_1?
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    Answer
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    α^0=0  and  α^1=2\hat\alpha_0=0\ \ and\ \ \hat\alpha_1=2
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  2. What’s the value of α^0 and α^1\hat\alpha_0 \ and\ \hat\alpha_1?
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    Answer
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    α^0=4 and α^1=−10\hat\alpha_0=4 \ and\ \hat\alpha_1=-10
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Interpreting Coefficients 2

In an effort to better understand the effects of “Get-out-the-vote” messages on voter turnout, Gerber and Green (2005) conducted an RCT involving approximately 30,000 individuals in New Haven, CT, in 1998. One of the treatments was randomly assigned in person visits in which a volunteer visited the person's home and encouraged him or her to vote. Table 3 reflects the findings from the RCT.

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  1. What’s the marginal effect of being assign to in person contact on voting?
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    Answer
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    0.47-0.45=0.02, it’s 2 percentage points. (Y is a binary variable in 0/1)
  2. Fill in the values of the following tables using the values from the table above
    3. RegressionsActually ContactedVoted in 1998Assigned to in-person contactβ1α1Constantβ0α0\begin{array}{c:c:c} \colorbox{yellow}{Regressions} & \text{Actually Contacted} & \text{Voted in 1998} \\ \hline \\ \text{Assigned to in-person contact} & \beta_1 & \alpha_1 \\ & \\ Constant & \beta_0 & \alpha_0 \end {array}
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    Answer
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    β1=0.25, β0=0.03, α1=0.02, α0=0.45\beta_1=0.25,\ \beta_0=0.03,\ \alpha_1=0.02,\ \alpha_0=0.45
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Interpreting Coefficients 3

Use the figure below to answer: What’s the sign of β2\beta_2 in the following equation? Y=β0+β1X+β2X2Y=\beta_0+\beta_1X+\beta_2X^2

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Answer
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It’s negative. This is because since there is a local max, this means that the second derivative must be negative, because the rate of change of the first derivative is decreasing. The first derivative represents the slope at any point, so in order to see that the slope is decreasing, pick a point of x (say X=25) and see what the slope is, in this case it will be a positive number. Now pick a point that’s higher than your original point (say x=75), and now see the slope of this point, notice that now the slope is negative. Therefore the slope went from a positive number to a negative number, so the slope is decreasing, which means the first-derivative is decreasing, which means the second derivative is negative, and since the second derivative is the sign of β2\beta_2 we then know that β2\beta_2 is negative.

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