FAILURE MAP
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FA-74526 / Experiment statistics / Open access

Per-user ratio metric variance: The covariance term is added · case 01

Standard errors are too large because positively correlated clicks and sessions no longer cancel.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The delta-method numerator uses + 2 R cov instead of - 2 R cov.

VERIFIED REPAIR

Subtract 2 R cov_xy.

Unsuccessful approach: Subtracting R cov (without the factor two) only half corrects the term.

Case contract

Each user contributes [clicks, sessions]; the unit of randomisation is the user. The arm ratio is sum(clicks) / sum(sessions). Its delta-method variance is (var_x - 2 R cov_xy + R^2 var_y) / (n mean_y^2) with sample (n - 1) moments over users. Arms with fewer than two users or no sessions -> None. Return [R_control, R_treatment, standard error of the difference] rounded to 6.

Why this case matters

Click-through per session is analysed per user; treating sessions as independent understates variance.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(users_c, users_t):
    def stats(users):
        n = len(users)
        if n < 2:
            return None
        xs = [u[0] for u in users]
        ys = [u[1] for u in users]
        mx, my = sum(xs) / n, sum(ys) / n
        if my == 0:
            return None
        r = mx / my
        vx = sum((x - mx) ** 2 for x in xs) / (n - 1)
        vy = sum((y - my) ** 2 for y in ys) / (n - 1)
        cxy = sum((x - mx) * (y - my) for x, y in zip(xs, ys)) / (n - 1)
        var = (vx + 2 * r * cxy + r * r * vy) / (n * my * my)
        return r, var
    a, b = stats(users_c), stats(users_t)
    if a is None or b is None:
        return None
    return [round(a[0], 6), round(b[0], 6), round(math.sqrt(max(a[1] + b[1], 0.0)), 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('user aggregate sample 1', [[[1, 2], [9, 10]], [[0, 0], [0, 2], [7, 10]]], [0.833333, 0.583333, 0.201748]),
  ('user aggregate sample 2',
   [[[0, 0], [1, 5], [1, 2], [0, 0]], [[0, 0], [1, 1], [7, 10], [10, 10]]],
   [0.285714, 0.857143, 0.153928]),
  ('user aggregate sample 3', [[[0, 10], [1, 1], [0, 2], [0, 0], [4, 5], [0, 0]], [[0, 0], [0, 0]]], None)],
 [('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 6',
   [[[1, 1], [5, 5], [1, 1], [3, 5], [10, 10]], [[0, 1], [1, 5], [0, 0], [0, 10], [1, 5], [1, 1]]],
   [0.909091, 0.136364, 0.126043]),
  ('user aggregate sample 7',
   [[[0, 1], [5, 10], [1, 5]], [[1, 2], [1, 2], [0, 1], [3, 5], [5, 5]]],
   [0.375, 0.666667, 0.184982]),
  ('user aggregate sample 8', [[[3, 5], [0, 1]], [[6, 10], [1, 1], [0, 5]]], [0.5, 0.4375, 0.270448])],
 [('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 11',
   [[[7, 10], [0, 0]], [[1, 5], [0, 1], [0, 5], [1, 1], [1, 1], [2, 10]]],
   [0.7, 0.217391, 0.075178]),
  ('user aggregate sample 12',
   [[[0, 5], [5, 10], [0, 2], [0, 0]], [[5, 5], [0, 1], [10, 10], [7, 10], [1, 1], [0, 1]]],
   [0.294118, 0.821429, 0.203966]),
  ('user aggregate sample 13',
   [[[0, 0], [1, 5], [0, 2], [0, 2], [1, 2], [7, 10]], [[2, 10], [2, 10], [1, 1], [0, 0], [1, 1]]],
   [0.428571, 0.272727, 0.181987])],
 [('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 16',
   [[[0, 0], [1, 1], [5, 5], [1, 1], [0, 1], [0, 10]], [[0, 0], [0, 2], [0, 0], [0, 10]]],
   [0.388889, 0.0, 0.306461]),
  ('user aggregate sample 17',
   [[[0, 10], [0, 2]], [[3, 10], [6, 10], [0, 0], [5, 10]]],
   [0.0, 0.466667, 0.083148]),
  ('user aggregate sample 20',
   [[[0, 1], [2, 2], [0, 2], [2, 5], [0, 1]], [[2, 2], [0, 0], [6, 10], [8, 10], [0, 0]]],
   [0.363636, 0.727273, 0.177734])],
 [('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 21',
   [[[4, 5], [2, 2], [2, 2]], [[2, 5], [2, 2], [5, 5], [7, 10]]],
   [0.888889, 0.727273, 0.137879]),
  ('user aggregate sample 22',
   [[[0, 0], [2, 2], [8, 10], [0, 1], [2, 10]], [[0, 0], [1, 1], [0, 2], [0, 0]]],
   [0.521739, 0.333333, 0.421009]),
  ('user aggregate sample 27',
   [[[0, 0], [1, 2], [3, 10]], [[2, 10], [2, 5], [1, 2], [0, 2]]],
   [0.333333, 0.263158, 0.085843])]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
heavy user dominates the ratio of sums[0.52381, 0.318182, 0.985831][0.52381, 0.318182, 0.132002]Failed
correlated clicks and sessions[0.538462, 0.7, 0.538368][0.538462, 0.7, 0.172146]Failed
users with zero sessions count[0.625, 0.4, 0.693357][0.625, 0.4, 0.161613]Failed
single user armNoneNonePassed
no sessions in controlNoneNonePassed
user aggregate sample 1[0.833333, 0.583333, 1.594684][0.833333, 0.583333, 0.201748]Failed
user aggregate sample 2[0.285714, 0.857143, 0.96205][0.285714, 0.857143, 0.153928]Failed
user aggregate sample 3NoneNonePassed

SHA-256 / a8ada9db3ea1407ad9bb62e6d1a0ca23e6be433302b32b0a0bc17f268fac790e

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(users_c, users_t):
    def stats(users):
        n = len(users)
        if n < 2:
            return None
        xs = [u[0] for u in users]
        ys = [u[1] for u in users]
        mx, my = sum(xs) / n, sum(ys) / n
        if my == 0:
            return None
        r = mx / my
        vx = sum((x - mx) ** 2 for x in xs) / (n - 1)
        vy = sum((y - my) ** 2 for y in ys) / (n - 1)
        cxy = sum((x - mx) * (y - my) for x, y in zip(xs, ys)) / (n - 1)
        var = (vx - r * cxy + r * r * vy) / (n * my * my)
        return r, var
    a, b = stats(users_c), stats(users_t)
    if a is None or b is None:
        return None
    return [round(a[0], 6), round(b[0], 6), round(math.sqrt(max(a[1] + b[1], 0.0)), 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('user aggregate sample 1', [[[1, 2], [9, 10]], [[0, 0], [0, 2], [7, 10]]], [0.833333, 0.583333, 0.201748]),
  ('user aggregate sample 2',
   [[[0, 0], [1, 5], [1, 2], [0, 0]], [[0, 0], [1, 1], [7, 10], [10, 10]]],
   [0.285714, 0.857143, 0.153928]),
  ('user aggregate sample 3', [[[0, 10], [1, 1], [0, 2], [0, 0], [4, 5], [0, 0]], [[0, 0], [0, 0]]], None)],
 [('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 6',
   [[[1, 1], [5, 5], [1, 1], [3, 5], [10, 10]], [[0, 1], [1, 5], [0, 0], [0, 10], [1, 5], [1, 1]]],
   [0.909091, 0.136364, 0.126043]),
  ('user aggregate sample 7',
   [[[0, 1], [5, 10], [1, 5]], [[1, 2], [1, 2], [0, 1], [3, 5], [5, 5]]],
   [0.375, 0.666667, 0.184982]),
  ('user aggregate sample 8', [[[3, 5], [0, 1]], [[6, 10], [1, 1], [0, 5]]], [0.5, 0.4375, 0.270448])],
 [('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 11',
   [[[7, 10], [0, 0]], [[1, 5], [0, 1], [0, 5], [1, 1], [1, 1], [2, 10]]],
   [0.7, 0.217391, 0.075178]),
  ('user aggregate sample 12',
   [[[0, 5], [5, 10], [0, 2], [0, 0]], [[5, 5], [0, 1], [10, 10], [7, 10], [1, 1], [0, 1]]],
   [0.294118, 0.821429, 0.203966]),
  ('user aggregate sample 13',
   [[[0, 0], [1, 5], [0, 2], [0, 2], [1, 2], [7, 10]], [[2, 10], [2, 10], [1, 1], [0, 0], [1, 1]]],
   [0.428571, 0.272727, 0.181987])],
 [('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 16',
   [[[0, 0], [1, 1], [5, 5], [1, 1], [0, 1], [0, 10]], [[0, 0], [0, 2], [0, 0], [0, 10]]],
   [0.388889, 0.0, 0.306461]),
  ('user aggregate sample 17',
   [[[0, 10], [0, 2]], [[3, 10], [6, 10], [0, 0], [5, 10]]],
   [0.0, 0.466667, 0.083148]),
  ('user aggregate sample 20',
   [[[0, 1], [2, 2], [0, 2], [2, 5], [0, 1]], [[2, 2], [0, 0], [6, 10], [8, 10], [0, 0]]],
   [0.363636, 0.727273, 0.177734])],
 [('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 21',
   [[[4, 5], [2, 2], [2, 2]], [[2, 5], [2, 2], [5, 5], [7, 10]]],
   [0.888889, 0.727273, 0.137879]),
  ('user aggregate sample 22',
   [[[0, 0], [2, 2], [8, 10], [0, 1], [2, 10]], [[0, 0], [1, 1], [0, 2], [0, 0]]],
   [0.521739, 0.333333, 0.421009]),
  ('user aggregate sample 27',
   [[[0, 0], [1, 2], [3, 10]], [[2, 10], [2, 5], [1, 2], [0, 2]]],
   [0.333333, 0.263158, 0.085843])]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
heavy user dominates the ratio of sums[0.52381, 0.318182, 0.505998][0.52381, 0.318182, 0.132002]Failed
correlated clicks and sessions[0.538462, 0.7, 0.30771][0.538462, 0.7, 0.172146]Failed
users with zero sessions count[0.625, 0.4, 0.373865][0.625, 0.4, 0.161613]Failed
single user armNoneNonePassed
no sessions in controlNoneNonePassed
user aggregate sample 1[0.833333, 0.583333, 0.81626][0.833333, 0.583333, 0.201748]Failed
user aggregate sample 2[0.285714, 0.857143, 0.499155][0.285714, 0.857143, 0.153928]Failed
user aggregate sample 3NoneNonePassed

SHA-256 / 34ef6297c136812e7ba153307261dd67bf13f20dcb78f1f2e8f5c2a730d62b3b

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(users_c, users_t):
    def stats(users):
        n = len(users)
        if n < 2:
            return None
        xs = [u[0] for u in users]
        ys = [u[1] for u in users]
        mx, my = sum(xs) / n, sum(ys) / n
        if my == 0:
            return None
        r = mx / my
        vx = sum((x - mx) ** 2 for x in xs) / (n - 1)
        vy = sum((y - my) ** 2 for y in ys) / (n - 1)
        cxy = sum((x - mx) * (y - my) for x, y in zip(xs, ys)) / (n - 1)
        var = (vx - 2 * r * cxy + r * r * vy) / (n * my * my)
        return r, var
    a, b = stats(users_c), stats(users_t)
    if a is None or b is None:
        return None
    return [round(a[0], 6), round(b[0], 6), round(math.sqrt(max(a[1] + b[1], 0.0)), 6)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('user aggregate sample 1', [[[1, 2], [9, 10]], [[0, 0], [0, 2], [7, 10]]], [0.833333, 0.583333, 0.201748]),
  ('user aggregate sample 2',
   [[[0, 0], [1, 5], [1, 2], [0, 0]], [[0, 0], [1, 1], [7, 10], [10, 10]]],
   [0.285714, 0.857143, 0.153928]),
  ('user aggregate sample 3', [[[0, 10], [1, 1], [0, 2], [0, 0], [4, 5], [0, 0]], [[0, 0], [0, 0]]], None)],
 [('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 6',
   [[[1, 1], [5, 5], [1, 1], [3, 5], [10, 10]], [[0, 1], [1, 5], [0, 0], [0, 10], [1, 5], [1, 1]]],
   [0.909091, 0.136364, 0.126043]),
  ('user aggregate sample 7',
   [[[0, 1], [5, 10], [1, 5]], [[1, 2], [1, 2], [0, 1], [3, 5], [5, 5]]],
   [0.375, 0.666667, 0.184982]),
  ('user aggregate sample 8', [[[3, 5], [0, 1]], [[6, 10], [1, 1], [0, 5]]], [0.5, 0.4375, 0.270448])],
 [('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 11',
   [[[7, 10], [0, 0]], [[1, 5], [0, 1], [0, 5], [1, 1], [1, 1], [2, 10]]],
   [0.7, 0.217391, 0.075178]),
  ('user aggregate sample 12',
   [[[0, 5], [5, 10], [0, 2], [0, 0]], [[5, 5], [0, 1], [10, 10], [7, 10], [1, 1], [0, 1]]],
   [0.294118, 0.821429, 0.203966]),
  ('user aggregate sample 13',
   [[[0, 0], [1, 5], [0, 2], [0, 2], [1, 2], [7, 10]], [[2, 10], [2, 10], [1, 1], [0, 0], [1, 1]]],
   [0.428571, 0.272727, 0.181987])],
 [('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('single user arm', [[[1, 2]], [[1, 2], [2, 3]]], None),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 16',
   [[[0, 0], [1, 1], [5, 5], [1, 1], [0, 1], [0, 10]], [[0, 0], [0, 2], [0, 0], [0, 10]]],
   [0.388889, 0.0, 0.306461]),
  ('user aggregate sample 17',
   [[[0, 10], [0, 2]], [[3, 10], [6, 10], [0, 0], [5, 10]]],
   [0.0, 0.466667, 0.083148]),
  ('user aggregate sample 20',
   [[[0, 1], [2, 2], [0, 2], [2, 5], [0, 1]], [[2, 2], [0, 0], [6, 10], [8, 10], [0, 0]]],
   [0.363636, 0.727273, 0.177734])],
 [('heavy user dominates the ratio of sums',
   [[[1, 1], [10, 20]], [[2, 2], [5, 20]]],
   [0.52381, 0.318182, 0.132002]),
  ('correlated clicks and sessions',
   [[[1, 2], [2, 4], [3, 6], [1, 1]], [[2, 2], [4, 5], [1, 3]]],
   [0.538462, 0.7, 0.172146]),
  ('users with zero sessions count',
   [[[0, 0], [2, 4], [3, 4]], [[1, 2], [1, 2], [0, 1]]],
   [0.625, 0.4, 0.161613]),
  ('no sessions in control', [[[0, 0], [0, 0]], [[1, 2], [2, 3]]], None),
  ('identical arms', [[[1, 4], [3, 4]], [[1, 4], [3, 4]]], [0.5, 0.5, 0.353553]),
  ('user aggregate sample 21',
   [[[4, 5], [2, 2], [2, 2]], [[2, 5], [2, 2], [5, 5], [7, 10]]],
   [0.888889, 0.727273, 0.137879]),
  ('user aggregate sample 22',
   [[[0, 0], [2, 2], [8, 10], [0, 1], [2, 10]], [[0, 0], [1, 1], [0, 2], [0, 0]]],
   [0.521739, 0.333333, 0.421009]),
  ('user aggregate sample 27',
   [[[0, 0], [1, 2], [3, 10]], [[2, 10], [2, 5], [1, 2], [0, 2]]],
   [0.333333, 0.263158, 0.085843])]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
heavy user dominates the ratio of sums[0.52381, 0.318182, 0.132002][0.52381, 0.318182, 0.132002]Passed
correlated clicks and sessions[0.538462, 0.7, 0.172146][0.538462, 0.7, 0.172146]Passed
users with zero sessions count[0.625, 0.4, 0.161613][0.625, 0.4, 0.161613]Passed
single user armNoneNonePassed
no sessions in controlNoneNonePassed
user aggregate sample 1[0.833333, 0.583333, 0.201748][0.833333, 0.583333, 0.201748]Passed
user aggregate sample 2[0.285714, 0.857143, 0.153928][0.285714, 0.857143, 0.153928]Passed
user aggregate sample 3NoneNonePassed

SHA-256 / b53ae4ad19d50f0f4f37a183b32272172d7fba190c52c4022dd96bdc246de3a7

Verification & scope

A deterministic toy experiment-analysis model with a stipulated contract; results are rounded and are not a substitute for a validated statistics package. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:48:57.649282+00:00.

Case digest / 50fffa8202fb3bc1c7018521c504a741c0c963be08cd4def4918da3c887c643a