{"abstract":"Design effects are understated when a few accounts are much larger than the rest.","category":"Experiment statistics","checks":8,"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].","contract_signature":"cluster_sizes, icc, n_required","evaluation_group":"w2-experiment-statistics-cluster-design-effect","failed_approach":"Dropping the +1 inside the factor removes the base cluster-size term.","family":"w2-experiment-statistics-cluster-design-effect-cluster-size-variation","id":"FA-74706","implementations":{"attempt":{"sha256":"d96058dbb1f654632797ec6a9c9d89e76bcbc07e0a863a6b63e6ce63eb1f30b3","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(cluster_sizes, icc, n_required):\n    k = len(cluster_sizes)\n    if k == 0:\n        return None\n    total = sum(cluster_sizes)\n    mbar = total / k\n    if mbar == 0:\n        return None\n    var = sum((s - mbar) ** 2 for s in cluster_sizes) / k\n    cv2 = var / mbar ** 2\n    deff = 1 + (cv2 * mbar - 1) * icc\n    return [round(deff, 6), round(total / deff, 6), math.ceil(n_required * deff / mbar)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),\n  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),\n  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),\n  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),\n  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),\n  ('cluster sample 1', [[40, 20, 40, 20, 20, 40], 0.01, 500], [1.323333, 136.020151, 23]),\n  ('cluster sample 2', [[3, 10, 10, 20], 0.05, 1000], [1.65814, 25.932679, 155]),\n  ('cluster sample 3', [[40, 10, 5], 0.01, 500], [1.303636, 42.189679, 36])],\n [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),\n  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),\n  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),\n  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),\n  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),\n  ('cluster sample 4', [[20, 10, 20, 5, 5, 3], 0.2, 100], [3.844444, 16.387283, 37]),\n  ('cluster sample 6', [[40], 0.2, 1000], [8.8, 4.545455, 220]),\n  ('cluster sample 8', [[5, 5, 3, 5], 0.2, 1000], [1.733333, 10.384615, 386])],\n [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),\n  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),\n  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),\n  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),\n  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),\n  ('cluster sample 11', [[40], 0.01, 500], [1.39, 28.776978, 18]),\n  ('cluster sample 17', [[20, 40, 3, 20, 5], 0.05, 100], [2.332955, 37.720409, 14]),\n  ('cluster sample 24', [[5, 10], 0.2, 100], [2.466667, 6.081081, 33])],\n [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),\n  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),\n  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),\n  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),\n  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),\n  ('cluster sample 16', [[3, 20, 10, 5, 10, 40], 0.0, 500], [1.0, 88.0, 35]),\n  ('cluster sample 25', [[10, 5, 3, 40], 0.05, 100], [2.444828, 23.723554, 17]),\n  ('cluster sample 39', [[5, 10], 0.01, 500], [1.073333, 13.975155, 72])],\n [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),\n  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),\n  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),\n  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),\n  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),\n  ('cluster sample 21', [[40, 10, 5], 0.05, 100], [2.518182, 21.841155, 14]),\n  ('cluster sample 36', [[5, 10, 20, 3], 0.01, 1000], [1.130526, 33.612663, 120]),\n  ('cluster sample 49', [[20, 10, 20, 20, 10, 3], 0.01, 1000], [1.159759, 71.56659, 84])]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"f47f45bff178e4d0c0c7de87eca77a1fa753e6b6484d01698e2f348b9771cefa","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(cluster_sizes, icc, n_required):\n    k = len(cluster_sizes)\n    if k == 0:\n        return None\n    total = sum(cluster_sizes)\n    mbar = total / k\n    if mbar == 0:\n        return None\n    var = sum((s - mbar) ** 2 for s in cluster_sizes) / k\n    cv2 = var / mbar ** 2\n    deff = 1 + (mbar - 1) * icc\n    return [round(deff, 6), round(total / deff, 6), math.ceil(n_required * deff / mbar)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),\n  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),\n  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),\n  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),\n  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),\n  ('cluster sample 1', [[40, 20, 40, 20, 20, 40], 0.01, 500], [1.323333, 136.020151, 23]),\n  ('cluster sample 2', [[3, 10, 10, 20], 0.05, 1000], [1.65814, 25.932679, 155]),\n  ('cluster sample 3', [[40, 10, 5], 0.01, 500], [1.303636, 42.189679, 36])],\n [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),\n  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),\n  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),\n  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),\n  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),\n  ('cluster sample 4', [[20, 10, 20, 5, 5, 3], 0.2, 100], [3.844444, 16.387283, 37]),\n  ('cluster sample 6', [[40], 0.2, 1000], [8.8, 4.545455, 220]),\n  ('cluster sample 8', [[5, 5, 3, 5], 0.2, 1000], [1.733333, 10.384615, 386])],\n [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),\n  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),\n  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),\n  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),\n  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),\n  ('cluster sample 11', [[40], 0.01, 500], [1.39, 28.776978, 18]),\n  ('cluster sample 17', [[20, 40, 3, 20, 5], 0.05, 100], [2.332955, 37.720409, 14]),\n  ('cluster sample 24', [[5, 10], 0.2, 100], [2.466667, 6.081081, 33])],\n [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),\n  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),\n  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),\n  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),\n  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),\n  ('cluster sample 16', [[3, 20, 10, 5, 10, 40], 0.0, 500], [1.0, 88.0, 35]),\n  ('cluster sample 25', [[10, 5, 3, 40], 0.05, 100], [2.444828, 23.723554, 17]),\n  ('cluster sample 39', [[5, 10], 0.01, 500], [1.073333, 13.975155, 72])],\n [('unequal cluster sizes inflate the design effect', [[5, 10, 20, 45], 0.05, 500], [2.54375, 31.449631, 64]),\n  ('equal clusters', [[10, 10, 10], 0.1, 300], [1.9, 15.789474, 57]),\n  ('zero icc means no inflation', [[3, 40], 0.0, 100], [1.0, 43.0, 5]),\n  ('single cluster', [[25], 0.2, 100], [5.8, 4.310345, 24]),\n  ('small clusters', [[2, 4], 0.3, 50], [1.7, 3.529412, 29]),\n  ('cluster sample 21', [[40, 10, 5], 0.05, 100], [2.518182, 21.841155, 14]),\n  ('cluster sample 36', [[5, 10, 20, 3], 0.01, 1000], [1.130526, 33.612663, 120]),\n  ('cluster sample 49', [[20, 10, 20, 20, 10, 3], 0.01, 1000], [1.159759, 71.56659, 84])]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"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.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"w2-experiment-statistics-cluster-design-effect-cluster-size-variation","generated_at":"2026-09-29T14:48:59.200167+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"B2B experiments randomise by account; ignoring clustering makes results look far more certain.","root_cause":"The inflation term uses (m - 1) without the cv^2 adjustment.","sha256":"59018392ee744c8475a91200398ba3ef6ed7f049b9c5a5ef52b24f616d4c8736","title":"Cluster randomisation design effect: Unequal cluster sizes are ignored · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":39.564,"exit_code":1,"observations":[{"actual":[1.54375,51.821862,39],"check":"unequal cluster sizes inflate the design effect","expected":[2.54375,31.449631,64],"passed":false},{"actual":[0.9,33.333333,27],"check":"equal clusters","expected":[1.9,15.789474,57],"passed":false},{"actual":[1.0,43.0,5],"check":"zero icc means no inflation","expected":[1.0,43.0,5],"passed":true},{"actual":[0.8,31.25,4],"check":"single cluster","expected":[5.8,4.310345,24],"passed":false},{"actual":[0.8,7.5,14],"check":"small clusters","expected":[1.7,3.529412,29],"passed":false},{"actual":[1.023333,175.895765,18],"check":"cluster sample 1","expected":[1.323333,136.020151,23],"passed":false},{"actual":[1.12064,38.370947,105],"check":"cluster sample 2","expected":[1.65814,25.932679,155],"passed":false},{"actual":[1.120303,49.09386,31],"check":"cluster sample 3","expected":[1.303636,42.189679,36],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"unequal cluster sizes inflate the design effect\", \"actual\": [1.54375, 51.821862, 39], \"expected\": [2.54375, 31.449631, 64], \"passed\": false}, {\"check\": \"equal clusters\", \"actual\": [0.9, 33.333333, 27], \"expected\": [1.9, 15.789474, 57], \"passed\": false}, {\"check\": \"zero icc means no inflation\", \"actual\": [1.0, 43.0, 5], \"expected\": [1.0, 43.0, 5], \"passed\": true}, {\"check\": \"single cluster\", \"actual\": [0.8, 31.25, 4], \"expected\": [5.8, 4.310345, 24], \"passed\": false}, {\"check\": \"small clusters\", \"actual\": [0.8, 7.5, 14], \"expected\": [1.7, 3.529412, 29], \"passed\": false}, {\"check\": \"cluster sample 1\", \"actual\": [1.023333, 175.895765, 18], \"expected\": [1.323333, 136.020151, 23], \"passed\": false}, {\"check\": \"cluster sample 2\", \"actual\": [1.12064, 38.370947, 105], \"expected\": [1.65814, 25.932679, 155], \"passed\": false}, {\"check\": \"cluster sample 3\", \"actual\": [1.120303, 49.09386, 31], \"expected\": [1.303636, 42.189679, 36], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":36.519,"exit_code":1,"observations":[{"actual":[1.95,41.025641,49],"check":"unequal cluster sizes inflate the design effect","expected":[2.54375,31.449631,64],"passed":false},{"actual":[1.9,15.789474,57],"check":"equal clusters","expected":[1.9,15.789474,57],"passed":true},{"actual":[1.0,43.0,5],"check":"zero icc means no inflation","expected":[1.0,43.0,5],"passed":true},{"actual":[5.8,4.310345,24],"check":"single cluster","expected":[5.8,4.310345,24],"passed":true},{"actual":[1.6,3.75,27],"check":"small clusters","expected":[1.7,3.529412,29],"passed":false},{"actual":[1.29,139.534884,22],"check":"cluster sample 1","expected":[1.323333,136.020151,23],"passed":false},{"actual":[1.4875,28.907563,139],"check":"cluster sample 2","expected":[1.65814,25.932679,155],"passed":false},{"actual":[1.173333,46.875,32],"check":"cluster sample 3","expected":[1.303636,42.189679,36],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"unequal cluster sizes inflate the design effect\", \"actual\": [1.95, 41.025641, 49], \"expected\": [2.54375, 31.449631, 64], \"passed\": false}, {\"check\": \"equal clusters\", \"actual\": [1.9, 15.789474, 57], \"expected\": [1.9, 15.789474, 57], \"passed\": true}, {\"check\": \"zero icc means no inflation\", \"actual\": [1.0, 43.0, 5], \"expected\": [1.0, 43.0, 5], \"passed\": true}, {\"check\": \"single cluster\", \"actual\": [5.8, 4.310345, 24], \"expected\": [5.8, 4.310345, 24], \"passed\": true}, {\"check\": \"small clusters\", \"actual\": [1.6, 3.75, 27], \"expected\": [1.7, 3.529412, 29], \"passed\": false}, {\"check\": \"cluster sample 1\", \"actual\": [1.29, 139.534884, 22], \"expected\": [1.323333, 136.020151, 23], \"passed\": false}, {\"check\": \"cluster sample 2\", \"actual\": [1.4875, 28.907563, 139], \"expected\": [1.65814, 25.932679, 155], \"passed\": false}, {\"check\": \"cluster sample 3\", \"actual\": [1.173333, 46.875, 32], \"expected\": [1.303636, 42.189679, 36], \"passed\": false}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}