Fairness I: observational

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 problem under “In class,” in that order, then what it can of “If there is time.” Discussions are in groups of three or four, then with the room. The problem, and the coding, are gone through as soon as the room has tried them.

Every item is about a rule that selects some people, or places: for detention, for a fraud check, for a treatment, for patrols. The same table of who was selected and what happened supports different numbers, and a disagreement about fairness is often a disagreement about which number a fair rule has to equalize. So each item ends by asking who the error falls on, who gains from the number that was reported, or who the choice of criterion could be justified to. The arithmetic narrows the answer; it does not give it.

Notation as in the lecture. \(Y = 1\) is the bad outcome; \(\hat Y = 1\) means flagged, or selected: picked out by the rule. Where a rule selects people for something good, an offer or a loan, the coding runs the other way, as in the book’s examples: \(Y = 1\) is success and \(\hat Y = 1\) is the offer. \(A\) is the group and \(p = P(Y = 1)\) the prevalence within a group. Within a group, \(\text{FPR} = P(\hat Y = 1 \mid Y = 0)\), \(\text{FNR} = P(\hat Y = 0 \mid Y = 1)\), \(\text{TPR} = 1 - \text{FNR}\), \(\text{PPV} = P(Y = 1 \mid \hat Y = 1)\) and \(\text{NPV} = P(Y = 0 \mid \hat Y = 0)\).

Exercises

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. Two groups, twenty people

Group a has ten people, and five of them reoffend. Group b has ten people, and two of them reoffend. A rule flags six people in group a (four who reoffend and two who do not) and three in group b (two who reoffend and one who does not).

  1. For each group, write the two-by-two table of flag against outcome, and compute FPR, FNR, PPV, NPV and the share flagged.

  2. Which of independence, separation and sufficiency does the rule satisfy, and which does it fail?

  3. Check the identity \(\text{FPR} = \frac{p}{1-p} \cdot \frac{1-\text{PPV}}{\text{PPV}} \cdot (1-\text{FNR})\) in each group with your numbers.

  4. In group a, stop flagging one of the two people who do not reoffend, and leave the rest as they are. Compute group a’s FPR and PPV again. Could any choice of who to flag in group a give it the same FPR as group b? Then flag only three people in group a, as many of them reoffenders as possible, so that both groups have the same share flagged. What happens to the two error rates?

  5. Suppose instead the rule is perfect in both groups. Which criterion does it fail, and why does that depend on the base rates?

  6. For each change in (d), say who gains, who loses, and who you would have to explain the change to.

Group b has eight people who do not reoffend. An FPR is a count out of the people who do not reoffend in that group.

For (b), sufficiency for a flag needs the same PPV and the same NPV.

Exercise 2. Chouldechova’s identity

  1. Starting from \(\text{PPV} = \frac{p\,(1-\text{FNR})}{p\,(1-\text{FNR}) + (1-p)\,\text{FPR}}\), derive \(\text{FPR} = \frac{p}{1-p} \cdot \frac{1-\text{PPV}}{\text{PPV}} \cdot (1-\text{FNR})\), then solve it for FNR.

  2. Two groups, \(a\) and \(b\), have prevalences \(p_a \neq p_b\). A rule has the same PPV in both groups, strictly between 0 and 1, and the same FNR in both. Find \(\text{FPR}_a / \text{FPR}_b\) in terms of \(p_a\) and \(p_b\), and evaluate it with COMPAS’s prevalences, \(0.514\) and \(0.394\).

  3. At the county’s cutoff COMPAS had PPV \(0.630\) and \(0.591\) and FNR \(0.280\) and \(0.477\), and the observed ratio of FPRs was \(1.91\). Write that ratio as a product of three factors, one for each term of the identity, and compute each. Which factor takes the ratio above your answer to (b)?

  4. A vendor reports equal predictive values and a journalist reports unequal false positive rates for the same rule. Which of them has made an arithmetic mistake? Who is each number useful to?

For (a), take the odds against a flag being right: \((1 - \text{PPV}) / \text{PPV}\).

For (b), write the identity for each group and divide.

Exercise 3. Separation and sufficiency together

  1. A rule has the same FPR, FNR, PPV and NPV in two groups, with \(0 < \text{FPR} < 1\). Show that the two groups have the same prevalence.

  2. The reversed rule \(\hat Y = 1 - Y\) flags the people who do not reoffend and no one else. Compute its four rates in any group. Which criteria does it satisfy, whatever the base rates? Which condition in (a) rules it out?

  3. An auditor finds a rule that satisfies separation and sufficiency in two groups. What should they check before calling it fair, and who would be misled if they did not?

For (a), if \(\text{TPR} > 0\), solve the identity for \(p/(1-p)\). If \(\text{TPR} = 0\), look at NPV instead.

With \(\text{TPR} = 0\), the people not flagged are all of the reoffenders and a share \(1 - \text{FPR}\) of the others, so \(\text{NPV} = \frac{(1-p)(1-\text{FPR})}{(1-p)(1-\text{FPR}) + p}\).

Exercise 4. Two calibrated scores

Take the two groups of exercise 1, with prevalences \(0.5\) and \(0.2\).

  1. Score 1 gives everyone the overall base rate, \(0.35\). Show that it is calibrated in the whole population. Is it calibrated by group?

  2. Score 2 gives each person in group a their true outcome, 0 or 1, and each person in group b the value \(0.2\). Show that it is calibrated within each group.

  3. Select everyone whose score 2 is at least \(0.5\). Compute the share selected, FPR, FNR and NPV by group. Which of the three criteria, if any, does the selection satisfy?

  4. Score 2 is calibrated by group, which the replies to ProPublica treated as the test of fairness. Who does score 2 serve, and who would you have to explain it to?

Calibrated means: among the people given score \(r\), a share \(r\) have \(Y = 1\). Check it for each value the score takes.

In class

Discussion 1: the rule you brought

Use the rule or score you brought, which selects some people and has two ways of being wrong. If you did not bring one, choose: a spam filter; a university’s offer rule; a bank’s credit decision; a hospital’s triage score; an airport security check.

  1. Name the two errors and who pays for each. Of FPR, FNR, PPV and NPV, which would each of those people want equal across groups, and which would the decision maker want equal?
  2. ProPublica wrote that Black defendants who did not reoffend were nearly twice as likely as white defendants to be labeled higher risk. Which criterion is that sentence about? The replies said that the score was equally accurate for both groups. Which criterion is that about?
  3. For your rule, suppose the base rates differ between two groups. Which of the two criteria in question 2 would you give up, and who would you explain it to?

Problem 1. The fraud checks

The Department for Work and Pensions scores requests for a universal credit advance, and refers those it scores high risk to a caseworker, who approves or declines them. For non-UK against UK nationals, its fairness assessment reports two ratios: requests referred \(2.27\) times as often, and referrals declined, which it counts as correct, \(0.97\) times as often, not significantly different from 1. It reports no error rate. Take a decline as \(Y = 1\) and a referral as \(\hat Y = 1\), and write \(f\) for the share of a group’s requests that are referred.

  1. Show that correct referrals are a share \(f \cdot \text{PPV} = p \cdot \text{TPR}\) of all of a group’s requests, and incorrect referrals a share \(f\,(1 - \text{PPV}) = (1 - p)\,\text{FPR}\).

  2. Non-UK against UK, what is the ratio of correct referrals per request? Show that the ratio of incorrect referrals per request is above \(2.27\).

  3. Suppose both groups had the same TPR and the same FPR. Write the two ratios in (b) in terms of the prevalences \(p_{\text{non-UK}}\) and \(p_{\text{UK}}\). Can both be above 1? What do you conclude?

  4. Show that \(\text{FPR} = f\,(1 - \text{PPV}) / (1 - p)\). Taking the two PPVs as equal, show that the ratio of non-UK to UK FPRs is at least \(2.27\,(1 - p_{\text{UK}})\). If fewer than one UK request in ten would be declined on review, how much more often are non-UK requests that a caseworker would approve referred?

  5. The \(0.97\) is not significantly different from 1. Does your conclusion in (c) depend on it?

  6. The department judged the model “reasonable and proportionate” to keep using. Which number would a claimant who did nothing wrong want to see, and who could have published it?

Discussion 2: three views

  1. The middle view says to treat seemingly dissimilar people similarly when the dissimilarity is the result of something that should be discounted. Give one setting where that seems reasonable and one where it seems less defensible, and say what differs between them. Who decides what should be discounted?
  2. For two of the three criteria, say which view of equality of opportunity it fits, and where the fit fails.
  3. In one sentence: what do you take away from the impossibility results?

Discussion 3: predictive policing

Lum and Isaac, “To predict and serve?” (Significance, 2016), ran PredPol’s published predictive policing algorithm on Oakland police records of drug crimes. Drug arrests in 2010 were concentrated in two areas of the city. Estimates of drug use, from a national survey applied to a synthetic population of the city, were spread across it, and roughly equal by race. Run on the police records, the algorithm would have targeted Black residents at roughly twice the rate of white residents, although it uses no demographic data. In a simulation in which 20 percent more crimes are recorded wherever police are sent, the model grew more confident that crime was where it had already sent them.

  1. Suppose the model is checked against records of the same kind as those it was trained on. Could it look accurate, even calibrated? What would a difference in base rates between neighborhoods, or between racial groups, be measuring?
  2. List as many reasons as you can why two groups’ recorded rates of an outcome could differ when their actual rates do not. Which of them could apply to re-arrest for COMPAS, and to declined requests in the fraud checks?
  3. Have you met a loop like this before in the course? What is the same here, and what is new?
  4. A defense of risk scores goes: the score only reflects real differences in base rates, and ignoring them would be unfair. What would you say back? If base rates are partly the product of earlier decisions, who should bear the cost of correcting them?

Discussion 4: two open questions

Take one of the two.

  1. A county that uses a risk score like COMPAS before trial is deciding whether to keep it. Make the strongest case you can for keeping it and the strongest case for dropping it. Which is stronger, and what would change your mind? Who would the county have to justify its decision to?
  2. In each of today’s examples, a decision about one person rested on what is true of people like them: which job ads they see, whether their request for an advance is referred for checks, whether they are held before trial, how well their kidneys are judged to work. Is that ever unfair to the person decided about, and does your answer differ among the four?

If there is time

Coding: COMPAS from ProPublica’s file

You need dplyr and ggplot2. ProPublica published the data behind its analysis: one row per defendant, with names and dates of birth as well as scores and outcomes. Keep only the three columns the analysis needs. decile_score is the COMPAS decile, 1 to 10, and two_year_recid is 1 if the person was arrested again within two years.

library(dplyr)
library(ggplot2)
compas <- read.csv("https://raw.githubusercontent.com/propublica/compas-analysis/master/compas-scores-two-years.csv") |>
  filter(race %in% c("African-American", "Caucasian")) |>
  select(race, decile_score, two_year_recid)

rates_at <- function(cutoff) {
  compas |>
    mutate(flag = decile_score >= cutoff) |>
    group_by(race) |>
    summarize(FPR = mean(flag[two_year_recid == 0]),
              FNR = # ...
              PPV = # ...
              selected = mean(flag),
              .groups = "drop") |>
    mutate(cutoff = cutoff)
}
rates_at(5)
  1. Complete FNR and PPV. Each is a mean over the right people, like the FPR line. Check rates_at(5) against the lecture: FPR 45 and 23 percent, PPV 63 and 59 percent.
  2. Compute the rates at every cutoff from 2 to 10, with bind_rows(lapply(2:10, rates_at)), and plot FPR, then FNR, against the cutoff with one line per group. Is there a cutoff at which the two groups have the same FPR? The same FNR?
  3. A defendant whose decile is exactly 5 has been flagged and asks what the flag means for someone like them. Compute the share re-arrested among defendants at decile 5, by group, and compare it with the PPV. What does the flag tell them, and what does it leave out?
cutoff <- c("African-American" = # ...
            "Caucasian" = # ...
compas |>
  mutate(flag = decile_score >= cutoff[race]) |>
  group_by(race) |>
  summarize(FPR = mean(flag[two_year_recid == 0]),
            FNR = # ...
            PPV = # ...
            selected = mean(flag))
  1. The code above gives each group its own cutoff. Find a pair of cutoffs at which the two groups’ FPRs and FNRs are both within two percentage points. What has happened to the PPVs? To the shares selected?
  2. At your pair of cutoffs, describe a Black defendant and a white defendant with the same decile who are treated differently. Which criterion did you satisfy, which did you give up, and who do you owe the explanation to: the defendant who is now flagged, the one who is not, or the judge who reads the same score two ways? Is the prediction now fairer?

Which criterion

  1. Of independence, separation and sufficiency, which is the better fairness measure for COMPAS, and why? On that measure, is COMPAS fair? Is your answer the same for the fraud checks?
  2. Equal predictive values and equal error rates cannot both hold when base rates differ, except for a perfect rule or one that is wrong about everyone. How should a county, or a department, decide which to require and which to give up? Possible grounds: who bears the cost; which error is worse here; the history that produced the base rates; the purpose the rule serves. Which would you put first, and which would you refuse to use?
  3. Whichever you chose, name the people who lose by it and write the sentence you would say to them.

The kidney equation

Kidney function is estimated from blood creatinine, age and sex. In 2021 the US equation also multiplied the estimate by 1.159 for Black patients, who in the data it was fitted to had higher creatinine at the same measured function. In a validation set it overestimated Black participants’ measured function by a median of 3.7 (ml/min per 1.73 m²). Refitted without race, it underestimated Black participants’ by 3.6 and overestimated others’ by 3.9.

  1. Neither equation is right on average for Black patients. Which error is worse here, an estimate too high or one too low, and who is it worse for?
  2. US transplant programs restored waiting time to Black candidates whose disease the old equation made look milder. Who pays for that correction, and is it owed?
  3. A second blood marker, cystatin C, narrows the gap without race. Suppose some hospitals can measure it and others cannot. Should the equation without the second marker use race in the hospitals that cannot? Who decides, and who is it explained to?

For next time

Bring one decision in which an attribute is used, or deliberately not used, with one route by which it could affect the outcome, and whether that route is fair. One sentence each.

References