Ethics and causation

Class

Author

Joshua Loftus

Note

This is the student version of the class sheet. The exercises are for your own time; the class works through everything under “In class”. Answers are revealed in class.

How this sheet works

Exercises rederive or apply the lecture’s results. Do them on your own time; they are not worked in class. The class works the discussions and the problems under “In class,” in that order. Discussions are in groups of three or four, then with the room. Each problem is gone through as soon as the room has tried it.

Exercises

The exercises can be done by hand, except the last three, which use R. Attempt each part before you open a hint.

WarningHints

Do not open a hint unless you have been stuck for at least a few minutes, and open them in order.

Exercise 1. Omitted-variable bias

In the linear model of the notes, \(Z := U_Z\), \(X := \alpha Z + U_X\), \(Y := \beta X + \gamma Z + U_Y\), with independent mean-zero noise, take \(\alpha = 2\), \(\gamma = 1\), \(\beta = 0.5\) and \(\operatorname{Var}(Z) = \operatorname{Var}(U_X) = 1\).

  1. Compute \(\operatorname{Var}(X)\), \(\operatorname{Cov}(X, Z)\) and the slope of \(Y\) on \(X\) alone.

  2. Repeat with \(\gamma = -1\).

  3. Repeat with \(\gamma = 1\) and \(\beta = 0\). What would someone with data on \(X\) and \(Y\) alone conclude?

  4. Is the error in (a) a bias or a variance? What happens to it as the sample grows?

\(Z\) and \(U_X\) are independent, so \(\operatorname{Var}(X) = \alpha^2 \operatorname{Var}(Z) + \operatorname{Var}(U_X)\).

Exercise 2. Choosing an adjustment set

  1. List the backdoor paths from \(X\) to \(Y\).

  2. Which of these sets satisfy the backdoor criterion: the empty set, \(\{Z\}\), \(\{M\}\), \(\{Z, M\}\), \(\{W\}\), \(\{Z, W\}\)?

  3. In a linear model, what does the coefficient on \(X\) estimate if you adjust for \(\{Z, M\}\)?

  4. If more than one of the sets in (b) satisfies the criterion, which would you use, and why?

A backdoor path starts with an arrow into \(X\). How many arrows point into \(X\)?

Exercise 3. The graph behind an argument

A university says: “Students who use the library more get better grades. We will extend the library’s opening hours, because using the library improves grades.”

  1. Draw the causal graph the argument presupposes, and mark the action.

  2. Add one variable that would make the observed difference larger than the effect, with its arrows.

  3. What comparison would support the university’s claim?

The university’s evidence compares students. Its action changes opening hours. Are those the same thing?

Exercise 4. Hormone therapy

The notes’ section on hormone therapy and heart disease gives three studies of whether to prescribe hormone therapy to prevent heart disease. Two of them are Grodstein and colleagues (1996) and Hernán and colleagues (2008).

  1. Write the decision with \(X\) and \(Y\). Say what the treatment and the outcome are, and write the quantity a doctor needs with \(do(\cdot)\).

  2. The 1996 estimate (0.39) and the 2008 intention-to-treat estimates (1.42 over the first two years, 0.96 over the whole follow-up) come from the same cohort, the Nurses’ Health Study. Which comparison did each make?

  3. Draw a graph under which the 1996 estimate is confounded.

  4. Which comparison matches the decision a doctor makes, and why?

For (a), what does the doctor choose, and which outcome is the choice about? For (b), ask which two groups each analysis compares.

Exercise 5. Prediction and intervention, in R

Open a fresh R script. You will need the tidyverse.

n <- 10000
# program 1: X causes Y
X1 <- rnorm(n); Y1 <- 2 * X1 + rnorm(n)
# program 2: Z causes both
Z2 <- rnorm(n); X2 <- Z2 + rnorm(n); Y2 <- Z2 + rnorm(n)
# program 3: X causes Y through M
X3 <- sqrt(2) * rnorm(n); M3 <- X3 + sqrt(2) * rnorm(n); Y3 <- 0.5 * M3 + rnorm(n)
  1. Write down the slope of \(Y\) on \(X\) you expect in each program. Then fit lm and compare.
  2. For each program, intervene: set \(X\) to 1 for everyone and run the statements after the one for \(X\) again, then the same with \(X\) set to 0. Compute the difference in the means of \(Y\).
  3. In program 2, fit lm(Y2 ~ X2 + Z2). In program 3, fit lm(Y3 ~ X3 + M3). Compare the coefficients on \(X\) with your answers to 2.

In each program, which statements after the one for \(X\) use \(X\), directly or through a variable computed from it? Those are the ones whose values change when you set \(X\).

Exercise 6. Adjustment on data, in R

The table of problem 1, one row for each rider.

cells <- data.frame(Z = c(1, 1, 1, 1, 0, 0, 0, 0),
                    W = c(1, 1, 0, 0, 1, 1, 0, 0),
                    T = c(1, 0, 1, 0, 1, 0, 1, 0),
                    n = c(70, 30, 100, 100, 80, 120, 50, 450),
                    accidents = c(21, 12, 10, 15, 4, 18, 1, 18))
# In base R, T means TRUE; inside dplyr verbs, T is the column
riders <- bind_rows(
  cells |> mutate(Y = 1, count = accidents),
  cells |> mutate(Y = 0, count = n - accidents)
) |>
  select(Z, W, T, Y, count) |>
  tidyr::uncount(count)
nrow(riders)
[1] 1000
  1. Compute the share with an accident among riders who took the course and among those who did not, from riders, and the difference.
  2. Compute the adjusted difference: the difference within each combination of \(Z\) and \(W\), averaged with weights \(P(Z = z, W = w)\). Then adjust for \(Z\) alone, and for \(W\) alone.
  3. A randomized version: give each of the 1,000 riders the course with probability 0.3, whatever their group, and draw whether they have an accident from the table’s rates. Compute the difference. Repeat 200 times. What is the average difference, and how much does it vary?

For 2: riders |> group_by(Z, W) |> summarize(...) gives one row for each group of riders; compute the difference and the group’s share of all riders there.

Exercise 7. Intervening in a program, in R

The job center from the lecture, as a program. As in the lecture, everyone referred attends and nobody else does. Each line generates one variable from the ones above it.

n <- 10000
Z <- rbinom(n, 1, 0.5)                        # worked last year
X <- rbinom(n, 1, ifelse(Z == 1, 0.8, 0.2))   # attended the course
Y <- rbinom(n, 1, 0.3 + 0.1 * X + 0.3 * Z)    # in work six months later
mean(Y[X == 1]) - mean(Y[X == 0])             # attenders against the others
  1. Run it. Compare the difference with the lecture’s first job-center table (64% against 36%).
  2. Compute the adjusted difference: the difference between attenders and non-attenders within \(Z = 1\) and within \(Z = 0\), averaged with weights mean(Z == 1) and mean(Z == 0).
  3. Intervene. Replace the line for X by X <- rep(1, n), run the line for Y again, and record mean(Y). Do the same with X <- rep(0, n). Compare the difference with your answer to 2.
  4. The center changes its referrals to 0.95 for \(Z = 1\) and 0.05 for \(Z = 0\). Before you run anything, write down what you expect for the differences in 1 and in 3. Then run it.

For step 3, replace the line for X and leave the line for Z alone. Then run the line for Y again, so that it uses the new X.

In class

Discussion 1: absurd trolley problems

In your group, open Absurd Trolley Problems on one device and play for about six minutes, choosing together. Note the scenarios where the group disagreed, and the ones you personally found hard to choose.

  1. Take one scenario where the group disagreed. Which framework was each side using?
  2. Did any of your choices depend on a prediction of what would happen? How sure of it were you?

Discussion 2: a model that decides about people

Think of a scenario in which a predictive algorithm is deployed to make, or help to make, decisions with large effects on people: which applicants get a loan, which patients a GP sees first, which benefit claims are checked for fraud. Choose one in your group.

  1. Describe the scenario: who uses the model, what it predicts, from what data, and what is decided with it.
  2. Decide what features or mechanisms would have to be in place to satisfy the four Belmont–Menlo principles (National Commission 1979; Dittrich and Kenneally 2012) and the five Cs of Patil, Mason and Loukides (2018): consent; clarity; consistency and trust; control and transparency; consequences. Say which principle or C each one serves.
  3. Which of your requirements is about an effect, so that checking it needs a causal claim?

At the end, each group describes its scenario and its requirements to the other groups.

The adjustment formula, from memory

Before the problems, from memory: what does the adjustment formula average, and with what weights?

Problem 1. A safety course for riders

A food-delivery platform offers its riders a road-safety course. It pushes the course hardest to new riders and to riders who work mostly at night. New riders have more accidents than experienced riders, by day and by night, and night work raises accidents for both. New riders, who book their shifts last, more often work at night. Assume that nothing else affects both taking the course and having an accident. The platform’s records for 1,000 riders, with \(Z = 1\) for a rider who had joined in the three months before the course was offered, \(W = 1\) for one who then worked mostly at night, \(T = 1\) for one who took the course and \(Y = 1\) for one who had an accident in the six months after:

new (\(Z\)) at night (\(W\)) course (\(T\)) riders accidents
yes yes yes 70 21
yes yes no 30 12
yes no yes 100 10
yes no no 100 15
no yes yes 80 4
no yes no 120 18
no no yes 50 1
no no no 450 18

  1. Draw the graph with \(T\), \(Y\), \(Z\) and \(W\). Which set of variables satisfies the backdoor criterion for the effect of \(T\) on \(Y\)?

  2. Compute the share who had an accident among riders who took the course and among those who did not.

  3. Compute \(P(Y = 1 \mid do(T := 1))\) and \(P(Y = 1 \mid do(T := 0))\) with the adjustment formula, and the average treatment effect.

  4. A manager proposes to drop the course, citing (b). An analyst who adjusts only for whether riders are new finds that the course lowers the chance of an accident by about 2.5 percentage points. What would you tell each of them?

Problem 2. Backdoor paths

In each graph, \(T\) is the treatment and \(Y\) the outcome.

A path is blocked by a set of variables if somewhere along it

  • a fork, \(\leftarrow V \to\), or a chain, \(\to V \to\) or \(\leftarrow V \leftarrow\), has its middle variable \(V\) in the set, or
  • a collider, \(\to V \leftarrow\), has neither \(V\) nor any descendant of \(V\) in the set.

The second rule is the reading’s (Barocas, Hardt and Narayanan 2023, pp. 115–116): a path through a collider is blocked when you adjust for nothing, and adjusting for the collider opens it.

  1. For each graph, list the backdoor paths from \(T\) to \(Y\). Which of them are open when you adjust for nothing?

  2. For each graph, find a smallest set of variables that satisfies the backdoor criterion for the effect of \(T\) on \(Y\). Is it the only one of that size?

  3. In graph 2, an analyst adjusts for \(C\), because \(C\) was measured before \(T\) and “adjusting for more can’t hurt.” Which path does that open? Suppose \(A\), \(B\) and \(C\) are yes-or-no variables, \(A\) and \(B\) are independent, and \(A = 1\) or \(B = 1\) is enough to make \(C = 1\), as either disease was enough for admission to hospital in the reading’s example (p. 113). Among people with \(C = 1\), are \(A\) and \(B\) associated, and in which direction?

  4. In graph 3, only \(C\) was measured. Can the effect of \(T\) on \(Y\) be estimated by adjusting for measured variables?

Discussion 3: the case for one framework

The room splits into three groups. Each group develops an argument for one statement and presents it to the other groups.

  1. Ethical data science and AI development requires us to lean mostly on consequentialist ethics.
  2. Ethical data science and AI development requires us to lean mostly on deontological ethics.
  3. Ethical data science and AI development requires us to lean mostly on virtue ethics.

You have 15 minutes to develop your argument and 2 minutes to present it. Think about how to structure it to convince the listener, and answer the strongest objection to it. The group arguing for consequentialism should also answer this question: does it become more or less defensible once you have to write the causal model down?

Then everyone votes for the account they find most convincing, which need not be the one their group argued for: consequentialist, deontological or virtue ethics for data science and AI, a combination, or none of them?

If there is time

  • One sentence, individually: what is the difference between \(P(Y = 1 \mid T = 1)\) and \(P(Y = 1 \mid do(T := 1))\)?
  • The reading (p. 113) warns against judging an admissions test by how well it predicts success among the students who were admitted. Draw the graph. Where is the collider?
  • “Correlation (more generally, association) is not causation.” This is true and important, but can it be misused?
  • Can causality be useful in the context of reproducibility? In designing and communicating studies?
  • A company tests two versions of its app on its users, without telling them, to see which version keeps them using the app longer. Which of the Belmont–Menlo principles does this touch? How would each framework judge it?
  • Find a sentence in a news report of an observational study that avoids saying “causes” but implies a cause. What graph does it assume?

For next time

Bring one score or algorithm used to make decisions about people. Write down what it measures and what it stands in for. Every group’s examples will go on the screen.

References

  • American Statistical Association, Committee on Professional Ethics, “Ethical Guidelines for Statistical Practice” (2018), Principle A, as quoted in Loftus, section 3.3.3. The current version (2022), https://www.amstat.org/your-career/ethical-guidelines-for-statistical-practice, asks instead that a statistician represent “their capabilities and activities honestly” (Principle A).
  • Barocas, Solon, Moritz Hardt and Arvind Narayanan, Fairness and Machine Learning: Limitations and Opportunities (MIT Press, 2023), ch. 5, “Causality,” pp. 105–116, https://fairmlbook.org/pdf/fairmlbook.pdf.
  • Dittrich, David and Erin Kenneally, “The Menlo Report: Ethical Principles Guiding Information and Communication Technology Research” (US Department of Homeland Security, Science and Technology Directorate, August 2012), https://www.caida.org/catalog/papers/2012_menlo_report_actual_formatted/menlo_report_actual_formatted.pdf.
  • Grodstein, Francine, Meir J. Stampfer, JoAnn E. Manson, Graham A. Colditz, Walter C. Willett, Bernard Rosner, Frank E. Speizer and Charles H. Hennekens, “Postmenopausal estrogen and progestin use and the risk of cardiovascular disease” (New England Journal of Medicine 335(7), 1996, 453–461), https://doi.org/10.1056/NEJM199608153350701.
  • Hernán, Miguel A., Alvaro Alonso, Roger Logan, Francine Grodstein, Karin B. Michels, Walter C. Willett, JoAnn E. Manson and James M. Robins, “Observational studies analyzed like randomized experiments: an application to postmenopausal hormone therapy and coronary heart disease” (Epidemiology 19(6), 2008, 766–779), https://doi.org/10.1097/EDE.0b013e3181875e61.
  • Larrazabal, Agostina J., Nicolás Nieto, Victoria Peterson, Diego H. Milone and Enzo Ferrante, “Gender imbalance in medical imaging datasets produces biased classifiers for computer-aided diagnosis” (Proceedings of the National Academy of Sciences 117(23), 2020, 12592–12594), https://doi.org/10.1073/pnas.1919012117.
  • Loftus, Joshua, Modern Statistics 4 Data Science, ch. 3, “Ethical data science,” sections 3.2–3.3, https://joshualoftus.com/ms4ds/ethical-data-science.html.
  • National Commission for the Protection of Human Subjects of Biomedical and Behavioral Research, “The Belmont Report: Ethical Principles and Guidelines for the Protection of Human Subjects of Research” (Federal Register 44(76), 18 April 1979, 23192–23197), https://www.govinfo.gov/content/pkg/FR-1979-04-18/pdf/FR-1979-04-18.pdf.
  • Obermeyer, Ziad, Brian Powers, Christine Vogeli and Sendhil Mullainathan, “Dissecting racial bias in an algorithm used to manage the health of populations” (Science 366(6464), 2019, 447–453), https://doi.org/10.1126/science.aax2342.
  • Patil, DJ, Hilary Mason and Mike Loukides, “The five Cs” (O’Reilly Radar, 24 July 2018), https://www.oreilly.com/radar/the-five-cs/.