FA-74711 / Experiment statistics / Open access

Cluster randomisation design effect: Effective sample is multiplied by the design effect · case 01

Clustering appears to increase information instead of reducing it.

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ROOT CAUSE

Effective n is total * deff.

VERIFIED REPAIR

Divide the total by the design effect.

Unsuccessful approach: Dividing the number of clusters confuses clusters with units.

Case contract

For clusters (e.g. companies) randomised as units: mean size m, population variance of sizes / m^2 = cv^2, design effect = 1 + ((cv^2 + 1) m - 1) icc. Effective sample = total units / design effect; clusters needed = ceil(n_required * design effect / m). No clusters or zero mean -> None. Return [round(deff, 6), round(effective n, 6), clusters needed].

Why this case matters

B2B experiments randomise by account; ignoring clustering makes results look far more certain.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(cluster_sizes, icc, n_required):
    k = len(cluster_sizes)
    if k == 0:
        return None
    total = sum(cluster_sizes)
    mbar = total / k
    if mbar == 0:
        return None
    var = sum((s - mbar) ** 2 for s in cluster_sizes) / k
    cv2 = var / mbar ** 2
    deff = 1 + ((cv2 + 1) * mbar - 1) * icc
    return [round(deff, 6), round(total * deff, 6), math.ceil(n_required * deff / mbar)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 1', [[40, 20, 40, 20, 20, 40], 0.01, 500], [1.323333, 136.020151, 23]),
  ('cluster sample 2', [[3, 10, 10, 20], 0.05, 1000], [1.65814, 25.932679, 155]),
  ('cluster sample 3', [[40, 10, 5], 0.01, 500], [1.303636, 42.189679, 36])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 3', [[40, 10, 5], 0.01, 500], [1.303636, 42.189679, 36]),
  ('cluster sample 4', [[20, 10, 20, 5, 5, 3], 0.2, 100], [3.844444, 16.387283, 37]),
  ('cluster sample 6', [[40], 0.2, 1000], [8.8, 4.545455, 220])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 10', [[40], 0.0, 500], [1.0, 40.0, 13]),
  ('cluster sample 11', [[40], 0.01, 500], [1.39, 28.776978, 18]),
  ('cluster sample 17', [[20, 40, 3, 20, 5], 0.05, 100], [2.332955, 37.720409, 14])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 16', [[3, 20, 10, 5, 10, 40], 0.0, 500], [1.0, 88.0, 35]),
  ('cluster sample 17', [[20, 40, 3, 20, 5], 0.05, 100], [2.332955, 37.720409, 14]),
  ('cluster sample 25', [[10, 5, 3, 40], 0.05, 100], [2.444828, 23.723554, 17])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 21', [[40, 10, 5], 0.05, 100], [2.518182, 21.841155, 14]),
  ('cluster sample 24', [[5, 10], 0.2, 100], [2.466667, 6.081081, 33]),
  ('cluster sample 36', [[5, 10, 20, 3], 0.01, 1000], [1.130526, 33.612663, 120])]]
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 fixture	Actual	Expected	Outcome
unequal cluster sizes inflate the design effect	`[2.54375, 203.5, 64]`	`[2.54375, 31.449631, 64]`	Failed
equal clusters	`[1.9, 57.0, 57]`	`[1.9, 15.789474, 57]`	Failed
zero icc means no inflation	`[1.0, 43.0, 5]`	`[1.0, 43.0, 5]`	Passed
single cluster	`[5.8, 145.0, 24]`	`[5.8, 4.310345, 24]`	Failed
small clusters	`[1.7, 10.2, 29]`	`[1.7, 3.529412, 29]`	Failed
cluster sample 1	`[1.323333, 238.2, 23]`	`[1.323333, 136.020151, 23]`	Failed
cluster sample 2	`[1.65814, 71.3, 155]`	`[1.65814, 25.932679, 155]`	Failed
cluster sample 3	`[1.303636, 71.7, 36]`	`[1.303636, 42.189679, 36]`	Failed

SHA-256 / 61c1e73d0d9605f1f8f200d4a81f5f30da9e40d58b32b5d2c5fb71fedbb08182

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(cluster_sizes, icc, n_required):
    k = len(cluster_sizes)
    if k == 0:
        return None
    total = sum(cluster_sizes)
    mbar = total / k
    if mbar == 0:
        return None
    var = sum((s - mbar) ** 2 for s in cluster_sizes) / k
    cv2 = var / mbar ** 2
    deff = 1 + ((cv2 + 1) * mbar - 1) * icc
    return [round(deff, 6), round(k / deff, 6), math.ceil(n_required * deff / mbar)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 1', [[40, 20, 40, 20, 20, 40], 0.01, 500], [1.323333, 136.020151, 23]),
  ('cluster sample 2', [[3, 10, 10, 20], 0.05, 1000], [1.65814, 25.932679, 155]),
  ('cluster sample 3', [[40, 10, 5], 0.01, 500], [1.303636, 42.189679, 36])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 3', [[40, 10, 5], 0.01, 500], [1.303636, 42.189679, 36]),
  ('cluster sample 4', [[20, 10, 20, 5, 5, 3], 0.2, 100], [3.844444, 16.387283, 37]),
  ('cluster sample 6', [[40], 0.2, 1000], [8.8, 4.545455, 220])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 10', [[40], 0.0, 500], [1.0, 40.0, 13]),
  ('cluster sample 11', [[40], 0.01, 500], [1.39, 28.776978, 18]),
  ('cluster sample 17', [[20, 40, 3, 20, 5], 0.05, 100], [2.332955, 37.720409, 14])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 16', [[3, 20, 10, 5, 10, 40], 0.0, 500], [1.0, 88.0, 35]),
  ('cluster sample 17', [[20, 40, 3, 20, 5], 0.05, 100], [2.332955, 37.720409, 14]),
  ('cluster sample 25', [[10, 5, 3, 40], 0.05, 100], [2.444828, 23.723554, 17])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 21', [[40, 10, 5], 0.05, 100], [2.518182, 21.841155, 14]),
  ('cluster sample 24', [[5, 10], 0.2, 100], [2.466667, 6.081081, 33]),
  ('cluster sample 36', [[5, 10, 20, 3], 0.01, 1000], [1.130526, 33.612663, 120])]]
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 fixture	Actual	Expected	Outcome
unequal cluster sizes inflate the design effect	`[2.54375, 1.572482, 64]`	`[2.54375, 31.449631, 64]`	Failed
equal clusters	`[1.9, 1.578947, 57]`	`[1.9, 15.789474, 57]`	Failed
zero icc means no inflation	`[1.0, 2.0, 5]`	`[1.0, 43.0, 5]`	Failed
single cluster	`[5.8, 0.172414, 24]`	`[5.8, 4.310345, 24]`	Failed
small clusters	`[1.7, 1.176471, 29]`	`[1.7, 3.529412, 29]`	Failed
cluster sample 1	`[1.323333, 4.534005, 23]`	`[1.323333, 136.020151, 23]`	Failed
cluster sample 2	`[1.65814, 2.412342, 155]`	`[1.65814, 25.932679, 155]`	Failed
cluster sample 3	`[1.303636, 2.301255, 36]`	`[1.303636, 42.189679, 36]`	Failed

SHA-256 / 6d1806dd682ab5ea7a8f96b47381c36ddb3dea5686c98deee080b840119c2a6d

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(cluster_sizes, icc, n_required):
    k = len(cluster_sizes)
    if k == 0:
        return None
    total = sum(cluster_sizes)
    mbar = total / k
    if mbar == 0:
        return None
    var = sum((s - mbar) ** 2 for s in cluster_sizes) / k
    cv2 = var / mbar ** 2
    deff = 1 + ((cv2 + 1) * mbar - 1) * icc
    return [round(deff, 6), round(total / deff, 6), math.ceil(n_required * deff / mbar)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 1', [[40, 20, 40, 20, 20, 40], 0.01, 500], [1.323333, 136.020151, 23]),
  ('cluster sample 2', [[3, 10, 10, 20], 0.05, 1000], [1.65814, 25.932679, 155]),
  ('cluster sample 3', [[40, 10, 5], 0.01, 500], [1.303636, 42.189679, 36])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 3', [[40, 10, 5], 0.01, 500], [1.303636, 42.189679, 36]),
  ('cluster sample 4', [[20, 10, 20, 5, 5, 3], 0.2, 100], [3.844444, 16.387283, 37]),
  ('cluster sample 6', [[40], 0.2, 1000], [8.8, 4.545455, 220])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 10', [[40], 0.0, 500], [1.0, 40.0, 13]),
  ('cluster sample 11', [[40], 0.01, 500], [1.39, 28.776978, 18]),
  ('cluster sample 17', [[20, 40, 3, 20, 5], 0.05, 100], [2.332955, 37.720409, 14])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 16', [[3, 20, 10, 5, 10, 40], 0.0, 500], [1.0, 88.0, 35]),
  ('cluster sample 17', [[20, 40, 3, 20, 5], 0.05, 100], [2.332955, 37.720409, 14]),
  ('cluster sample 25', [[10, 5, 3, 40], 0.05, 100], [2.444828, 23.723554, 17])],
 [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),
  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),
  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),
  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),
  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),
  ('cluster sample 21', [[40, 10, 5], 0.05, 100], [2.518182, 21.841155, 14]),
  ('cluster sample 24', [[5, 10], 0.2, 100], [2.466667, 6.081081, 33]),
  ('cluster sample 36', [[5, 10, 20, 3], 0.01, 1000], [1.130526, 33.612663, 120])]]
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 fixture	Actual	Expected	Outcome
unequal cluster sizes inflate the design effect	`[2.54375, 31.449631, 64]`	`[2.54375, 31.449631, 64]`	Passed
equal clusters	`[1.9, 15.789474, 57]`	`[1.9, 15.789474, 57]`	Passed
zero icc means no inflation	`[1.0, 43.0, 5]`	`[1.0, 43.0, 5]`	Passed
single cluster	`[5.8, 4.310345, 24]`	`[5.8, 4.310345, 24]`	Passed
small clusters	`[1.7, 3.529412, 29]`	`[1.7, 3.529412, 29]`	Passed
cluster sample 1	`[1.323333, 136.020151, 23]`	`[1.323333, 136.020151, 23]`	Passed
cluster sample 2	`[1.65814, 25.932679, 155]`	`[1.65814, 25.932679, 155]`	Passed
cluster sample 3	`[1.303636, 42.189679, 36]`	`[1.303636, 42.189679, 36]`	Passed

SHA-256 / f13c78976cea82124e8e4d57259293e4d4c5753aa5dbcbf72a2d518feaf61da5

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 2026-09-29T14:48:59.242104+00:00.

Case digest / 0fd516c31838006d56a07071f2e49f535fe053567570ee712c2ef220da2480ce