{"abstract":"Intervals collapse for small variances and explode for large ones.","category":"Experiment statistics","checks":8,"contract":"lift = mean_t / mean_c - 1. With variances of the means v_t = var_t/n_t and v_c = var_c/n_c, the delta-method variance is v_t/mean_c^2 + mean_t^2 v_c / mean_c^4 and the interval is lift +/- z * sqrt(variance). mean_c = 0 or nonpositive n -> None. Return [lift, low, high] rounded to 6 places.","contract_signature":"mean_c, var_c, n_c, mean_t, var_t, n_t, z","evaluation_group":"w2-experiment-statistics-relative-lift-ci","failed_approach":"Taking sqrt(z * variance) puts z under the square root.","family":"w2-experiment-statistics-relative-lift-ci-half-width","id":"FA-74426","implementations":{"attempt":{"sha256":"2d94e5605db4da49fd5876485ad53acbd9091c30dc2c4197119286eb129cda06","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(mean_c, var_c, n_c, mean_t, var_t, n_t, z):\n    if mean_c == 0 or n_c <= 0 or n_t <= 0:\n        return None\n    lift = mean_t / mean_c - 1\n    se_t = var_t / n_t\n    se_c = var_c / n_c\n    v = se_t / mean_c ** 2 + mean_t ** 2 * se_c / mean_c ** 4\n    half = math.sqrt(z * v)\n    return [round(lift, 6), round(lift - half, 6), round(lift + half, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('ten percent lift', [10.0, 4.0, 400, 11.0, 4.0, 400, 1.96], [0.1, 0.070863, 0.129137]),\n  ('treatment below control', [2.0, 1.0, 100, 1.5, 1.0, 100, 1.96], [-0.25, -0.3725, -0.1275]),\n  ('equal means still carry control noise',\n   [4.0, 9.0, 1000, 4.0, 1.0, 1000, 1.645],\n   [0.0, -0.041125, 0.041125]),\n  ('small means', [0.5, 0.25, 250, 0.6, 0.3, 250, 1.96], [0.2, -0.001413, 0.401413]),\n  ('zero control mean', [0.0, 1.0, 10, 1.0, 1.0, 10, 1.96], None),\n  ('summary statistic sample 1', [4.0, 4.0, 1000, 0.45, 16.0, 250, 1.96], [-0.8875, -1.01151, -0.76349]),\n  ('summary statistic sample 2', [10.0, 9.0, 400, 3.0, 1.0, 250, 1.645], [-0.7, -0.712769, -0.687231]),\n  ('summary statistic sample 3', [2.0, 1.0, 400, 4.0, 1.0, 250, 1.645], [1.0, 0.90268, 1.09732])],\n [('ten percent lift', [10.0, 4.0, 400, 11.0, 4.0, 400, 1.96], [0.1, 0.070863, 0.129137]),\n  ('treatment below control', [2.0, 1.0, 100, 1.5, 1.0, 100, 1.96], [-0.25, -0.3725, -0.1275]),\n  ('equal means still carry control noise',\n   [4.0, 9.0, 1000, 4.0, 1.0, 1000, 1.645],\n   [0.0, -0.041125, 0.041125]),\n  ('small means', [0.5, 0.25, 250, 0.6, 0.3, 250, 1.96], [0.2, -0.001413, 0.401413]),\n  ('zero control mean', [0.0, 1.0, 10, 1.0, 1.0, 10, 1.96], None),\n  ('summary statistic sample 4', [4.0, 1.0, 400, 2.2, 4.0, 100, 1.645], [-0.45, -0.533024, -0.366976]),\n  ('summary statistic sample 6', [10.0, 9.0, 1000, 2.2, 4.0, 1000, 1.645], [-0.78, -0.790956, -0.769044]),\n  ('summary statistic sample 7', [2.0, 9.0, 400, 10.5, 1.0, 100, 1.645], [4.25, 3.59708, 4.90292])],\n [('ten percent lift', [10.0, 4.0, 400, 11.0, 4.0, 400, 1.96], [0.1, 0.070863, 0.129137]),\n  ('treatment below control', [2.0, 1.0, 100, 1.5, 1.0, 100, 1.96], [-0.25, -0.3725, -0.1275]),\n  ('equal means still carry control noise',\n   [4.0, 9.0, 1000, 4.0, 1.0, 1000, 1.645],\n   [0.0, -0.041125, 0.041125]),\n  ('small means', [0.5, 0.25, 250, 0.6, 0.3, 250, 1.96], [0.2, -0.001413, 0.401413]),\n  ('zero control mean', [0.0, 1.0, 10, 1.0, 1.0, 10, 1.96], None),\n  ('summary statistic sample 11', [4.0, 4.0, 400, 10.5, 16.0, 1000, 1.96], [1.625, 1.48222, 1.76778]),\n  ('summary statistic sample 12', [10.0, 9.0, 400, 0.45, 1.0, 250, 1.96], [-0.955, -0.967467, -0.942533]),\n  ('summary statistic sample 13', [0.5, 1.0, 1000, 10.5, 4.0, 100, 1.645], [20.0, 17.718248, 22.281752])],\n [('ten percent lift', [10.0, 4.0, 400, 11.0, 4.0, 400, 1.96], [0.1, 0.070863, 0.129137]),\n  ('treatment below control', [2.0, 1.0, 100, 1.5, 1.0, 100, 1.96], [-0.25, -0.3725, -0.1275]),\n  ('equal means still carry control noise',\n   [4.0, 9.0, 1000, 4.0, 1.0, 1000, 1.645],\n   [0.0, -0.041125, 0.041125]),\n  ('small means', [0.5, 0.25, 250, 0.6, 0.3, 250, 1.96], [0.2, -0.001413, 0.401413]),\n  ('zero control mean', [0.0, 1.0, 10, 1.0, 1.0, 10, 1.96], None),\n  ('summary statistic sample 16', [4.0, 9.0, 100, 4.0, 4.0, 250, 1.645], [0.0, -0.133893, 0.133893]),\n  ('summary statistic sample 17', [4.0, 1.0, 100, 4.0, 16.0, 1000, 1.645], [0.0, -0.066312, 0.066312]),\n  ('summary statistic sample 18', [2.0, 1.0, 400, 10.5, 4.0, 1000, 1.645], [4.25, 4.027915, 4.472085])],\n [('ten percent lift', [10.0, 4.0, 400, 11.0, 4.0, 400, 1.96], [0.1, 0.070863, 0.129137]),\n  ('treatment below control', [2.0, 1.0, 100, 1.5, 1.0, 100, 1.96], [-0.25, -0.3725, -0.1275]),\n  ('equal means still carry control noise',\n   [4.0, 9.0, 1000, 4.0, 1.0, 1000, 1.645],\n   [0.0, -0.041125, 0.041125]),\n  ('small means', [0.5, 0.25, 250, 0.6, 0.3, 250, 1.96], [0.2, -0.001413, 0.401413]),\n  ('zero control mean', [0.0, 1.0, 10, 1.0, 1.0, 10, 1.96], None),\n  ('summary statistic sample 21', [0.5, 1.0, 400, 0.45, 1.0, 100, 1.96], [-0.1, -0.529862, 0.329862]),\n  ('summary statistic sample 22', [0.5, 4.0, 1000, 10.5, 4.0, 1000, 1.96], [20.0, 14.787726, 25.212274]),\n  ('summary statistic sample 25', [4.0, 9.0, 400, 10.5, 4.0, 250, 1.96], [1.625, 1.422351, 1.827649])]]\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":"19ceee1f1621eaa685e8842d2dc271d25c1a74edfec30a2466bc7b55bb9cee48","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(mean_c, var_c, n_c, mean_t, var_t, n_t, z):\n    if mean_c == 0 or n_c <= 0 or n_t <= 0:\n        return None\n    lift = mean_t / mean_c - 1\n    se_t = var_t / n_t\n    se_c = var_c / n_c\n    v = se_t / mean_c ** 2 + mean_t ** 2 * se_c / mean_c ** 4\n    half = z * v\n    return [round(lift, 6), round(lift - half, 6), round(lift + half, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[('ten percent lift', [10.0, 4.0, 400, 11.0, 4.0, 400, 1.96], [0.1, 0.070863, 0.129137]),\n  ('treatment below control', [2.0, 1.0, 100, 1.5, 1.0, 100, 1.96], [-0.25, -0.3725, -0.1275]),\n  ('equal means still carry control noise',\n   [4.0, 9.0, 1000, 4.0, 1.0, 1000, 1.645],\n   [0.0, -0.041125, 0.041125]),\n  ('small means', [0.5, 0.25, 250, 0.6, 0.3, 250, 1.96], [0.2, -0.001413, 0.401413]),\n  ('zero control mean', [0.0, 1.0, 10, 1.0, 1.0, 10, 1.96], None),\n  ('summary statistic sample 1', [4.0, 4.0, 1000, 0.45, 16.0, 250, 1.96], [-0.8875, -1.01151, -0.76349]),\n  ('summary statistic sample 2', [10.0, 9.0, 400, 3.0, 1.0, 250, 1.645], [-0.7, -0.712769, -0.687231]),\n  ('summary statistic sample 3', [2.0, 1.0, 400, 4.0, 1.0, 250, 1.645], [1.0, 0.90268, 1.09732])],\n [('ten percent lift', [10.0, 4.0, 400, 11.0, 4.0, 400, 1.96], [0.1, 0.070863, 0.129137]),\n  ('treatment below control', [2.0, 1.0, 100, 1.5, 1.0, 100, 1.96], [-0.25, -0.3725, -0.1275]),\n  ('equal means still carry control noise',\n   [4.0, 9.0, 1000, 4.0, 1.0, 1000, 1.645],\n   [0.0, -0.041125, 0.041125]),\n  ('small means', [0.5, 0.25, 250, 0.6, 0.3, 250, 1.96], [0.2, -0.001413, 0.401413]),\n  ('zero control mean', [0.0, 1.0, 10, 1.0, 1.0, 10, 1.96], None),\n  ('summary statistic sample 4', [4.0, 1.0, 400, 2.2, 4.0, 100, 1.645], [-0.45, -0.533024, -0.366976]),\n  ('summary statistic sample 6', [10.0, 9.0, 1000, 2.2, 4.0, 1000, 1.645], [-0.78, -0.790956, -0.769044]),\n  ('summary statistic sample 7', [2.0, 9.0, 400, 10.5, 1.0, 100, 1.645], [4.25, 3.59708, 4.90292])],\n [('ten percent lift', [10.0, 4.0, 400, 11.0, 4.0, 400, 1.96], [0.1, 0.070863, 0.129137]),\n  ('treatment below control', [2.0, 1.0, 100, 1.5, 1.0, 100, 1.96], [-0.25, -0.3725, -0.1275]),\n  ('equal means still carry control noise',\n   [4.0, 9.0, 1000, 4.0, 1.0, 1000, 1.645],\n   [0.0, -0.041125, 0.041125]),\n  ('small means', [0.5, 0.25, 250, 0.6, 0.3, 250, 1.96], [0.2, -0.001413, 0.401413]),\n  ('zero control mean', [0.0, 1.0, 10, 1.0, 1.0, 10, 1.96], None),\n  ('summary statistic sample 11', [4.0, 4.0, 400, 10.5, 16.0, 1000, 1.96], [1.625, 1.48222, 1.76778]),\n  ('summary statistic sample 12', [10.0, 9.0, 400, 0.45, 1.0, 250, 1.96], [-0.955, -0.967467, -0.942533]),\n  ('summary statistic sample 13', [0.5, 1.0, 1000, 10.5, 4.0, 100, 1.645], [20.0, 17.718248, 22.281752])],\n [('ten percent lift', [10.0, 4.0, 400, 11.0, 4.0, 400, 1.96], [0.1, 0.070863, 0.129137]),\n  ('treatment below control', [2.0, 1.0, 100, 1.5, 1.0, 100, 1.96], [-0.25, -0.3725, -0.1275]),\n  ('equal means still carry control noise',\n   [4.0, 9.0, 1000, 4.0, 1.0, 1000, 1.645],\n   [0.0, -0.041125, 0.041125]),\n  ('small means', [0.5, 0.25, 250, 0.6, 0.3, 250, 1.96], [0.2, -0.001413, 0.401413]),\n  ('zero control mean', [0.0, 1.0, 10, 1.0, 1.0, 10, 1.96], None),\n  ('summary statistic sample 16', [4.0, 9.0, 100, 4.0, 4.0, 250, 1.645], [0.0, -0.133893, 0.133893]),\n  ('summary statistic sample 17', [4.0, 1.0, 100, 4.0, 16.0, 1000, 1.645], [0.0, -0.066312, 0.066312]),\n  ('summary statistic sample 18', [2.0, 1.0, 400, 10.5, 4.0, 1000, 1.645], [4.25, 4.027915, 4.472085])],\n [('ten percent lift', [10.0, 4.0, 400, 11.0, 4.0, 400, 1.96], [0.1, 0.070863, 0.129137]),\n  ('treatment below control', [2.0, 1.0, 100, 1.5, 1.0, 100, 1.96], [-0.25, -0.3725, -0.1275]),\n  ('equal means still carry control noise',\n   [4.0, 9.0, 1000, 4.0, 1.0, 1000, 1.645],\n   [0.0, -0.041125, 0.041125]),\n  ('small means', [0.5, 0.25, 250, 0.6, 0.3, 250, 1.96], [0.2, -0.001413, 0.401413]),\n  ('zero control mean', [0.0, 1.0, 10, 1.0, 1.0, 10, 1.96], None),\n  ('summary statistic sample 21', [0.5, 1.0, 400, 0.45, 1.0, 100, 1.96], [-0.1, -0.529862, 0.329862]),\n  ('summary statistic sample 22', [0.5, 4.0, 1000, 10.5, 4.0, 1000, 1.96], [20.0, 14.787726, 25.212274]),\n  ('summary statistic sample 25', [4.0, 9.0, 400, 10.5, 4.0, 250, 1.96], [1.625, 1.422351, 1.827649])]]\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-relative-lift-ci-half-width","generated_at":"2026-09-29T14:48:56.641931+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Online experiment readouts drive launch decisions; a silent formula slip flips conclusions.","root_cause":"half = z * variance instead of z * sqrt(variance).","sha256":"6bfc06fbd4dff32ee343e7d2b70c45a51425c78b0c0df017a9f26d9fcbe6f5ca","title":"Relative lift interval: The half-width multiplies z by the variance · 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":40.081,"exit_code":1,"observations":[{"actual":[0.1,0.079188,0.120812],"check":"ten percent lift","expected":[0.1,0.070863,0.129137],"passed":false},{"actual":[-0.25,-0.3375,-0.1625],"check":"treatment below control","expected":[-0.25,-0.3725,-0.1275],"passed":false},{"actual":[0.0,-0.032064,0.032064],"check":"equal means still carry control noise","expected":[0.0,-0.041125,0.041125],"passed":false},{"actual":[0.2,0.056133,0.343867],"check":"small means","expected":[0.2,-0.001413,0.401413],"passed":false},{"actual":null,"check":"zero control mean","expected":null,"passed":true},{"actual":[-0.8875,-0.976079,-0.798921],"check":"summary statistic sample 1","expected":[-0.8875,-1.01151,-0.76349],"passed":false},{"actual":[-0.7,-0.709955,-0.690045],"check":"summary statistic sample 2","expected":[-0.7,-0.712769,-0.687231],"passed":false},{"actual":[1.0,0.924122,1.075878],"check":"summary statistic sample 3","expected":[1.0,0.90268,1.09732],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"ten percent lift\", \"actual\": [0.1, 0.079188, 0.120812], \"expected\": [0.1, 0.070863, 0.129137], \"passed\": false}, {\"check\": \"treatment below control\", \"actual\": [-0.25, -0.3375, -0.1625], \"expected\": [-0.25, -0.3725, -0.1275], \"passed\": false}, {\"check\": \"equal means still carry control noise\", \"actual\": [0.0, -0.032064, 0.032064], \"expected\": [0.0, -0.041125, 0.041125], \"passed\": false}, {\"check\": \"small means\", \"actual\": [0.2, 0.056133, 0.343867], \"expected\": [0.2, -0.001413, 0.401413], \"passed\": false}, {\"check\": \"zero control mean\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"summary statistic sample 1\", \"actual\": [-0.8875, -0.976079, -0.798921], \"expected\": [-0.8875, -1.01151, -0.76349], \"passed\": false}, {\"check\": \"summary statistic sample 2\", \"actual\": [-0.7, -0.709955, -0.690045], \"expected\": [-0.7, -0.712769, -0.687231], \"passed\": false}, {\"check\": \"summary statistic sample 3\", \"actual\": [1.0, 0.924122, 1.075878], \"expected\": [1.0, 0.90268, 1.09732], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":38.46,"exit_code":1,"observations":[{"actual":[0.1,0.099567,0.100433],"check":"ten percent lift","expected":[0.1,0.070863,0.129137],"passed":false},{"actual":[-0.25,-0.257656,-0.242344],"check":"treatment below control","expected":[-0.25,-0.3725,-0.1275],"passed":false},{"actual":[0.0,-0.001028,0.001028],"check":"equal means still carry control noise","expected":[0.0,-0.041125,0.041125],"passed":false},{"actual":[0.2,0.179302,0.220698],"check":"small means","expected":[0.2,-0.001413,0.401413],"passed":false},{"actual":null,"check":"zero control mean","expected":null,"passed":true},{"actual":[-0.8875,-0.895346,-0.879654],"check":"summary statistic sample 1","expected":[-0.8875,-1.01151,-0.76349],"passed":false},{"actual":[-0.7,-0.700099,-0.699901],"check":"summary statistic sample 2","expected":[-0.7,-0.712769,-0.687231],"passed":false},{"actual":[1.0,0.994243,1.005758],"check":"summary statistic sample 3","expected":[1.0,0.90268,1.09732],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"ten percent lift\", \"actual\": [0.1, 0.099567, 0.100433], \"expected\": [0.1, 0.070863, 0.129137], \"passed\": false}, {\"check\": \"treatment below control\", \"actual\": [-0.25, -0.257656, -0.242344], \"expected\": [-0.25, -0.3725, -0.1275], \"passed\": false}, {\"check\": \"equal means still carry control noise\", \"actual\": [0.0, -0.001028, 0.001028], \"expected\": [0.0, -0.041125, 0.041125], \"passed\": false}, {\"check\": \"small means\", \"actual\": [0.2, 0.179302, 0.220698], \"expected\": [0.2, -0.001413, 0.401413], \"passed\": false}, {\"check\": \"zero control mean\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"summary statistic sample 1\", \"actual\": [-0.8875, -0.895346, -0.879654], \"expected\": [-0.8875, -1.01151, -0.76349], \"passed\": false}, {\"check\": \"summary statistic sample 2\", \"actual\": [-0.7, -0.700099, -0.699901], \"expected\": [-0.7, -0.712769, -0.687231], \"passed\": false}, {\"check\": \"summary statistic sample 3\", \"actual\": [1.0, 0.994243, 1.005758], \"expected\": [1.0, 0.90268, 1.09732], \"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."}}