{"abstract":"Clustering appears to increase information instead of reducing it.","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":"Dividing the number of clusters confuses clusters with units.","family":"w2-experiment-statistics-cluster-design-effect-effective-sample","id":"FA-74711","implementations":{"attempt":{"sha256":"6d1806dd682ab5ea7a8f96b47381c36ddb3dea5686c98deee080b840119c2a6d","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 + 1) * mbar - 1) * icc\n    return [round(deff, 6), round(k / 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 3', [[40, 10, 5], 0.01, 500], [1.303636, 42.189679, 36]),\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 [('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 10', [[40], 0.0, 500], [1.0, 40.0, 13]),\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 [('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 17', [[20, 40, 3, 20, 5], 0.05, 100], [2.332955, 37.720409, 14]),\n  ('cluster sample 25', [[10, 5, 3, 40], 0.05, 100], [2.444828, 23.723554, 17])],\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 24', [[5, 10], 0.2, 100], [2.466667, 6.081081, 33]),\n  ('cluster sample 36', [[5, 10, 20, 3], 0.01, 1000], [1.130526, 33.612663, 120])]]\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":"61c1e73d0d9605f1f8f200d4a81f5f30da9e40d58b32b5d2c5fb71fedbb08182","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 + 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 3', [[40, 10, 5], 0.01, 500], [1.303636, 42.189679, 36]),\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 [('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 10', [[40], 0.0, 500], [1.0, 40.0, 13]),\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 [('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 17', [[20, 40, 3, 20, 5], 0.05, 100], [2.332955, 37.720409, 14]),\n  ('cluster sample 25', [[10, 5, 3, 40], 0.05, 100], [2.444828, 23.723554, 17])],\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 24', [[5, 10], 0.2, 100], [2.466667, 6.081081, 33]),\n  ('cluster sample 36', [[5, 10, 20, 3], 0.01, 1000], [1.130526, 33.612663, 120])]]\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-effective-sample","generated_at":"2026-09-29T14:48:59.242104+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":"Effective n is total * deff.","sha256":"5f64c0029c27d9e91b15a9fd327528d4e69ba4b70cb349be0a6ef5aa21ec3430","title":"Cluster randomisation design effect: Effective sample is multiplied by the design effect · 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":40.613,"exit_code":1,"observations":[{"actual":[2.54375,1.572482,64],"check":"unequal cluster sizes inflate the design effect","expected":[2.54375,31.449631,64],"passed":false},{"actual":[1.9,1.578947,57],"check":"equal clusters","expected":[1.9,15.789474,57],"passed":false},{"actual":[1.0,2.0,5],"check":"zero icc means no inflation","expected":[1.0,43.0,5],"passed":false},{"actual":[5.8,0.172414,24],"check":"single cluster","expected":[5.8,4.310345,24],"passed":false},{"actual":[1.7,1.176471,29],"check":"small clusters","expected":[1.7,3.529412,29],"passed":false},{"actual":[1.323333,4.534005,23],"check":"cluster sample 1","expected":[1.323333,136.020151,23],"passed":false},{"actual":[1.65814,2.412342,155],"check":"cluster sample 2","expected":[1.65814,25.932679,155],"passed":false},{"actual":[1.303636,2.301255,36],"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\": [2.54375, 1.572482, 64], \"expected\": [2.54375, 31.449631, 64], \"passed\": false}, {\"check\": \"equal clusters\", \"actual\": [1.9, 1.578947, 57], \"expected\": [1.9, 15.789474, 57], \"passed\": false}, {\"check\": \"zero icc means no inflation\", \"actual\": [1.0, 2.0, 5], \"expected\": [1.0, 43.0, 5], \"passed\": false}, {\"check\": \"single cluster\", \"actual\": [5.8, 0.172414, 24], \"expected\": [5.8, 4.310345, 24], \"passed\": false}, {\"check\": \"small clusters\", \"actual\": [1.7, 1.176471, 29], \"expected\": [1.7, 3.529412, 29], \"passed\": false}, {\"check\": \"cluster sample 1\", \"actual\": [1.323333, 4.534005, 23], \"expected\": [1.323333, 136.020151, 23], \"passed\": false}, {\"check\": \"cluster sample 2\", \"actual\": [1.65814, 2.412342, 155], \"expected\": [1.65814, 25.932679, 155], \"passed\": false}, {\"check\": \"cluster sample 3\", \"actual\": [1.303636, 2.301255, 36], \"expected\": [1.303636, 42.189679, 36], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":38.378,"exit_code":1,"observations":[{"actual":[2.54375,203.5,64],"check":"unequal cluster sizes inflate the design effect","expected":[2.54375,31.449631,64],"passed":false},{"actual":[1.9,57.0,57],"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":[5.8,145.0,24],"check":"single cluster","expected":[5.8,4.310345,24],"passed":false},{"actual":[1.7,10.2,29],"check":"small clusters","expected":[1.7,3.529412,29],"passed":false},{"actual":[1.323333,238.2,23],"check":"cluster sample 1","expected":[1.323333,136.020151,23],"passed":false},{"actual":[1.65814,71.3,155],"check":"cluster sample 2","expected":[1.65814,25.932679,155],"passed":false},{"actual":[1.303636,71.7,36],"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\": [2.54375, 203.5, 64], \"expected\": [2.54375, 31.449631, 64], \"passed\": false}, {\"check\": \"equal clusters\", \"actual\": [1.9, 57.0, 57], \"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\": [5.8, 145.0, 24], \"expected\": [5.8, 4.310345, 24], \"passed\": false}, {\"check\": \"small clusters\", \"actual\": [1.7, 10.2, 29], \"expected\": [1.7, 3.529412, 29], \"passed\": false}, {\"check\": \"cluster sample 1\", \"actual\": [1.323333, 238.2, 23], \"expected\": [1.323333, 136.020151, 23], \"passed\": false}, {\"check\": \"cluster sample 2\", \"actual\": [1.65814, 71.3, 155], \"expected\": [1.65814, 25.932679, 155], \"passed\": false}, {\"check\": \"cluster sample 3\", \"actual\": [1.303636, 71.7, 36], \"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."}}