The story everyone tells runs one sentence long: the bombers came back with holes in the wings, and Wald said to armour the engines, where the holes were not. The memoranda Wald actually wrote are a hundred pages of equations, the worked examples are marked hypothetical, and armour is mentioned once. The idea survives the retelling. The method does not, and the method is the part you can use.
In 1943 Abraham Wald, working in the Statistical Research Group at Columbia, wrote a series of memoranda for the people deciding how to protect aircraft. The first, issued as SRG memo 85, became Part I of the collected text CITED. It opens not with a picture of a damaged wing but with definitions: let Pi be the probability that a plane is downed by i hits; let Ai be the number of planes that came back with exactly i hits; let N be the number that went out.
That third quantity is the whole idea. The data in the hangar — hits counted on returning aircraft — are a sample from the planes that survived. Wald never treats them as a sample from the planes that flew. Everything that follows is an argument about the gap between the two: the planes that did not come back left no holes to count, and the missing holes are information.
Suppose, as Wald does first, that every hit independently fails to down the plane with the same probability q. A plane that received i hits came back with probability qi. So the observed share of returners with i hits, ai = Ai/N, is the share that received i hits, shrunk by qi. Undo the shrinking and the shares must add up to the planes that were hit at all:
One equation, one unknown, and Wald proves it has exactly one root between zero and one CITED. His confidence-limit example (p. 71) gives N = 500 and returners with 0 to 5 hits of 400, 40, 25, 5, 3 and 2. The script solves the equation by bisection:
| quantity | value | source |
|---|---|---|
| root q | 0.850 | ch01-verify.py [2]; Wald prints .850 |
| chance one hit downs the plane, 1 − q | 0.150 | [2] |
| planes that did not return | 25 of 500 | [2] |
Twenty-five planes are gone and have nothing to say for themselves; the equation recovers what they would have said from the 475 that came back. MODEL — the answer is only as good as the assumption that hits are independent and alike, and Wald spends much of the text relaxing exactly that assumption.
Part V asks the question the story is about: which part of the plane is most vulnerable? Wald's worked example (pp. 63–65) splits a plane of 130 sq ft into engines, fuselage, fuel system and the rest, and records 202 hits on the returners. If a hit lands on a part in proportion to its area γ(i), and the returners show a share of hits δ(i) on that part, then a part whose hits are under-represented among survivors must be one where hits bring planes down. The rule that reproduces every entry of his table is
| part | area share γ | share of holes δ | holes on returners | downed by one hit | Wald's table |
|---|---|---|---|---|---|
| engines | 0.269 | 0.193 | 39 | 0.390 | .39 |
| fuselage | 0.346 | 0.386 | 78 | 0.052 | .05 |
| fuel system | 0.154 | 0.153 | 31 | 0.152 | .15 |
| other parts | 0.231 | 0.267 | 54 | 0.015 | .02 |
Read the table the naive way — protect where the holes are — and you armour the fuselage: 78 of the 202 holes, the most of any part. Read it Wald's way and the fuselage is nearly the safest place to be hit (0.05). The engines are 26.9% of the area but only 19.3% of the holes; that shortfall is the missing planes, and it makes a hit on an engine almost eight times as likely to down the aircraft as a hit on the fuselage COMPUTED. The fuel system is the control: its share of holes matches its share of area, and its vulnerability comes out equal to the plane's average.
Evans and Rosenthal open their chapter on inference (§5.1, p. 254) with the Stanford heart-transplant study: did patients who received a new heart live longer than those who did not? Their tables record, for each treated patient, the days waited for a heart and the days lived after it, so the treated group's survival is the sum of the two CITED.
That sum carries Wald's gap inside it. To be counted among the treated, a patient had to survive the wait. A candidate who died on day ten never received a heart and so is counted as a control. The treated group is, by construction, the patients who came back from the first part of the mission. Comparing its lifelengths with the controls' without accounting for this credits the transplant with the waiting-room survival that let patients in. This paragraph draws the parallel; the textbook's own treatment of the covariates is in its Chapter 10 and is not reproduced here. MODEL
This volume promised to use its instruments on the series before using them on anyone else. The series has a hangar too: the site. What a reader sees is the pages that are still there.
| count, from git | value |
|---|---|
| HTML paths tracked today | 855 |
| HTML paths ever added | 877 |
| added, and no longer tracked | 66 |
Sixty-six pages were published at some path and are gone from it COMPUTED. Some were renames, some duplicates, and some were retractions — the withdrawn g6_equals_schumann is the case Chapter 2 audits. The script counts paths, not reasons, and does not sort them. What it establishes is the Wald point for a body of work: a reader who judges this series by its surviving pages is looking at the returners. The claims that failed badly enough to be removed have left no holes to count, unless something outside the site — docs/audit-log.md, the git history, docs/ml-evidence/ — keeps them.
That is why those records exist, and it is the practical conclusion of the chapter. A body of work that wants to be read correctly keeps its downed planes on file.