{
  "evaluation_id": "EVAL-20260317-024202",
  "question_id": "EVAL-20260317-024202",
  "question_text": "Hospital A has a higher survival rate than Hospital B for both heart surgery (A: 90%, B: 85%) and knee surgery (A: 95%, B: 92%). But Hospital B has a higher overall survival rate (B: 91%, A: 89%). (1) Construct exact numbers that produce this paradox. (2) Which hospital is actually better? (3) A health insurance company uses overall survival rate to recommend hospitals. What goes wrong? (4) How should the comparison be done correctly?",
  "category": "reasoning",
  "timestamp": "2026-03-17T02:42:02.095Z",
  "display_date": "Mar 17, 2026",
  "winner": {
    "name": "Qwen 3.5 35B-A3B",
    "provider": "openrouter",
    "score": 10
  },
  "avg_score": 9.053333,
  "matrix_size": 30,
  "models_used": [
    {
      "id": "qwen35_27b",
      "name": "Qwen 3.5 27B",
      "provider": "openrouter"
    },
    {
      "id": "qwen3_coder_next",
      "name": "Qwen 3 Coder Next",
      "provider": "openrouter"
    },
    {
      "id": "qwen35_35b_a3b",
      "name": "Qwen 3.5 35B-A3B",
      "provider": "openrouter"
    },
    {
      "id": "qwen35_122b_a10b",
      "name": "Qwen 3.5 122B-A10B",
      "provider": "openrouter"
    },
    {
      "id": "qwen35_397b_a17b",
      "name": "Qwen 3.5 397B-A17B",
      "provider": "openrouter"
    },
    {
      "id": "qwen3_8b",
      "name": "Qwen 3 8B",
      "provider": "openrouter"
    }
  ],
  "rankings": {
    "qwen35_35b_a3b": {
      "display_name": "Qwen 3.5 35B-A3B",
      "provider": "openrouter",
      "average_score": 10,
      "score_count": 5,
      "min_score": 10,
      "max_score": 10,
      "rank": 1
    },
    "qwen35_397b_a17b": {
      "display_name": "Qwen 3.5 397B-A17B",
      "provider": "openrouter",
      "average_score": 9.92,
      "score_count": 6,
      "min_score": 9.75,
      "max_score": 10,
      "rank": 2
    },
    "qwen35_122b_a10b": {
      "display_name": "Qwen 3.5 122B-A10B",
      "provider": "openrouter",
      "average_score": 9.92,
      "score_count": 5,
      "min_score": 9.8,
      "max_score": 10,
      "rank": 3
    },
    "qwen35_27b": {
      "display_name": "Qwen 3.5 27B",
      "provider": "openrouter",
      "average_score": 9.63,
      "score_count": 6,
      "min_score": 9,
      "max_score": 10,
      "rank": 4
    },
    "qwen3_coder_next": {
      "display_name": "Qwen 3 Coder Next",
      "provider": "openrouter",
      "average_score": 8.01,
      "score_count": 4,
      "min_score": 6.8,
      "max_score": 10,
      "rank": 5
    },
    "qwen3_8b": {
      "display_name": "Qwen 3 8B",
      "provider": "openrouter",
      "average_score": 6.85,
      "score_count": 4,
      "min_score": 4.95,
      "max_score": 8.5,
      "rank": 6
    }
  },
  "judgments": [
    {
      "judge_key": "qwen35_9b",
      "judge_name": "Qwen 3.5 9B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 9.2,
      "brief_justification": "Response correctly identifies Simpson's Paradox and addresses all 4 question parts. Minor deduction for correctness due to the original problem's mathematically impossible numbers (89% overall with 90%/95% sub-rates), which the response appropriately flags. Clear tables and structured explanations make it highly actionable."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 10,
      "brief_justification": "The response thoroughly explains Simpson's Paradox, provides exact numerical examples, correctly identifies Hospital A as better, and addresses the statistical fallacy in using overall rates. All criteria are met with precision and depth."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 10,
      "brief_justification": "The response accurately explains Simpson's Paradox, provides valid constructed numbers, correctly identifies Hospital A as better, and offers actionable solutions for fair comparison. All criteria are thoroughly addressed with clear examples and insights."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 10,
      "brief_justification": "The response accurately explains Simpson's Paradox, provides correct numerical examples, and offers actionable solutions for fair hospital comparison. All four questions are thoroughly addressed with clear reasoning and practical recommendations."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen35_122b_a10b",
      "respondent_name": "Qwen 3.5 122B-A10B",
      "weighted_score": 10,
      "brief_justification": "The response accurately explains Simpson's Paradox, provides valid numerical examples, and thoroughly addresses all four questions with clear, actionable solutions."
    },
    {
      "judge_key": "qwen3_8b",
      "judge_name": "Qwen 3 8B",
      "respondent_key": "qwen35_397b_a17b",
      "respondent_name": "Qwen 3.5 397B-A17B",
      "weighted_score": 10,
      "brief_justification": "The response accurately explains Simpson's Paradox, provides valid numerical examples, and thoroughly addresses all four questions with clear solutions and implications."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 8.5,
      "brief_justification": "The response correctly constructs Simpson's Paradox with valid numbers, addresses the ambiguity in 'better' hospitals, and identifies the flaw in aggregating statistics. It provides actionable solutions but could elaborate further on statistical methods to resolve the paradox."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 7.9,
      "brief_justification": "The response correctly constructs Simpson’s Paradox with valid numbers and explains the paradox and its implications. However, it fails to address part (4) on how to correctly compare hospitals, which reduces completeness and usefulness. Clarity and depth are strong, but the missing section limits overall thoroughness."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 10,
      "brief_justification": "The response accurately explains Simpson’s Paradox, constructs valid numerical examples, correctly identifies Hospital A as better when stratified, and thoroughly explains the flaws in using overall rates. It provides actionable solutions like stratification and risk adjustment, demonstrating strong analytical rigor and practical relevance."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 9,
      "brief_justification": "The response accurately constructs Simpson's Paradox with valid numbers, correctly identifies Hospital A as better, explains case-mix bias, and recommends stratified analysis. Minor deductions for the initial confusion about the 89% rate (later resolved) and the assumption of a typo."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen35_122b_a10b",
      "respondent_name": "Qwen 3.5 122B-A10B",
      "weighted_score": 10,
      "brief_justification": "The response accurately constructs Simpson's Paradox with valid numbers, explains the paradox's mechanics, and addresses all four questions with clear, actionable insights. It demonstrates deep understanding of statistical bias and provides practical solutions for fair hospital comparisons."
    },
    {
      "judge_key": "qwen3_32b",
      "judge_name": "Qwen 3 32B",
      "respondent_key": "qwen35_397b_a17b",
      "respondent_name": "Qwen 3.5 397B-A17B",
      "weighted_score": 9.75,
      "brief_justification": "The response accurately explains Simpson's Paradox with a well-constructed example, clearly addresses all four questions, and provides actionable solutions. It demonstrates deep understanding of the statistical and ethical implications while maintaining clarity and practical relevance."
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 8.35,
      "brief_justification": "The constructed numbers are mostly correct but use fractional patients (e.g., 2.7 survivors), which is unrealistic; using integer counts would improve correctness. The explanation of Simpson’s Paradox and the recommendation to stratify by surgery type is accurate and complete. Depth is slightly limited—could discuss confounding (surgery severity) more explicitly. Otherwise, the response is clear, useful, and well-structured."
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies Simpson’s Paradox, resolves the mathematical inconsistency in the prompt by constructing valid numbers, clearly explains why Hospital A is actually better, thoroughly details the flaw in using overall survival rate without risk adjustment, and provides two robust correct approaches (stratification and risk adjustment) for fair comparison—all with precise, actionable insights."
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 9.75,
      "brief_justification": "The response correctly identifies Simpson’s Paradox, constructs valid numbers demonstrating the paradox (noting the implausibility of 89% overall for Hospital A given its sub-rates), clearly explains why Hospital A is actually better, accurately diagnoses the insurance company’s error, and prescribes appropriate risk-adjusted comparison methods. Minor correctness deduction only because the prompt’s stated 89% for A is mathematically impossible, but the response appropriately addresses this rathe"
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen35_122b_a10b",
      "respondent_name": "Qwen 3.5 122B-A10B",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies Simpson’s Paradox, constructs a mathematically valid example (adjusting the original numbers to ensure feasibility), clearly explains why Hospital A is truly better, accurately diagnoses the flaw in using overall rates, and provides two robust, actionable solutions (stratified analysis and risk adjustment). All reasoning is logically sound and practically relevant."
    },
    {
      "judge_key": "qwen3_coder_next",
      "judge_name": "Qwen 3 Coder Next",
      "respondent_key": "qwen35_397b_a17b",
      "respondent_name": "Qwen 3.5 397B-A17B",
      "weighted_score": 10,
      "brief_justification": "The response correctly identifies Simpson’s Paradox, mathematically resolves the inconsistency in the original premise (noting the impossibility of an overall rate below the minimum category rate), provides valid concrete numbers, and thoroughly explains the implications for patient care, policy, and proper statistical methodology—demonstrating deep understanding and practical relevance."
    },
    {
      "judge_key": "qwen35_35b_a3b",
      "judge_name": "Qwen 3.5 35B-A3B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 5.6,
      "brief_justification": "The model correctly explains Simpson's Paradox conceptually but fails the numerical construction task, providing patient counts that result in Hospital A having a higher overall rate (92%) than Hospital B (91.4%), contradicting the prompt's requirements and its own text. Additionally, it ignores the specific overall rates (89% vs 91%) requested in the prompt."
    },
    {
      "judge_key": "qwen35_35b_a3b",
      "judge_name": "Qwen 3.5 35B-A3B",
      "respondent_key": "qwen35_397b_a17b",
      "respondent_name": "Qwen 3.5 397B-A17B",
      "weighted_score": 10,
      "brief_justification": "The model correctly identified the mathematical impossibility in the prompt's premise and provided a valid numerical example of Simpson's Paradox while thoroughly addressing all four sub-questions with clear structure and actionable insights."
    },
    {
      "judge_key": "qwen35_27b",
      "judge_name": "Qwen 3.5 27B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 6.8,
      "brief_justification": "The response demonstrates excellent mathematical insight by identifying that the prompt's specific overall rates were impossible, but it fails significantly on completeness by cutting off mid-sentence and omitting the final question entirely."
    },
    {
      "judge_key": "qwen35_27b",
      "judge_name": "Qwen 3.5 27B",
      "respondent_key": "qwen35_122b_a10b",
      "respondent_name": "Qwen 3.5 122B-A10B",
      "weighted_score": 9.8,
      "brief_justification": "The model correctly identified a mathematical impossibility in the prompt (Hospital A's overall rate cannot be lower than its subgroup rates) and adjusted the data to demonstrate the paradox logically while explaining the correction. All conceptual answers regarding the paradox, insurance bias, and statistical solutions were accurate and well-structured."
    },
    {
      "judge_key": "qwen35_27b",
      "judge_name": "Qwen 3.5 27B",
      "respondent_key": "qwen35_397b_a17b",
      "respondent_name": "Qwen 3.5 397B-A17B",
      "weighted_score": 9.8,
      "brief_justification": "The model excelled by identifying and correcting a mathematical impossibility in the prompt's premises while still fulfilling the request to demonstrate the paradox. It answered all four sub-questions with clear structure and actionable statistical advice."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen3_8b",
      "respondent_name": "Qwen 3 8B",
      "weighted_score": 4.95,
      "brief_justification": "The model fails to provide valid integer patient counts and contradicts the prompt's specified overall rates by calculating Hospital A as having a higher overall rate in its example. Additionally, the response contains an internal contradiction between its calculated statistics and its textual claims."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 10,
      "brief_justification": "The model correctly identified the mathematical impossibility in the prompt's premises and provided valid constructed numbers while thoroughly explaining Simpson's Paradox and its real-world implications."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 9.8,
      "brief_justification": "The model correctly identifies Simpson's Paradox and astutely notes the mathematical impossibility in the prompt's specific percentages while still constructing valid data to demonstrate the paradox. It provides thorough analysis of the confounding variables and actionable advice on risk adjustment."
    },
    {
      "judge_key": "qwen35_122b_a10b",
      "judge_name": "Qwen 3.5 122B-A10B",
      "respondent_key": "qwen35_397b_a17b",
      "respondent_name": "Qwen 3.5 397B-A17B",
      "weighted_score": 10,
      "brief_justification": "The model correctly identifies the mathematical impossibility in the prompt's premise regarding Hospital A's overall rate and provides a valid corrected example of Simpson's Paradox. It thoroughly addresses all four sub-questions with clear explanations of the statistical bias and actionable recommendations for risk adjustment."
    },
    {
      "judge_key": "qwen35_397b_a17b",
      "judge_name": "Qwen 3.5 397B-A17B",
      "respondent_key": "qwen3_coder_next",
      "respondent_name": "Qwen 3 Coder Next",
      "weighted_score": 7.35,
      "brief_justification": "The model demonstrates excellent mathematical reasoning by identifying and correcting an impossible premise in the prompt. However, the response cuts off abruptly, failing to answer the fourth question entirely and leaving the third incomplete."
    },
    {
      "judge_key": "qwen35_397b_a17b",
      "judge_name": "Qwen 3.5 397B-A17B",
      "respondent_key": "qwen35_35b_a3b",
      "respondent_name": "Qwen 3.5 35B-A3B",
      "weighted_score": 10,
      "brief_justification": "The model accurately identified a mathematical impossibility in the prompt's premises and corrected it while preserving the paradox's logic. It comprehensively addressed all four parts with clear examples, statistical reasoning, and practical policy implications."
    },
    {
      "judge_key": "qwen35_397b_a17b",
      "judge_name": "Qwen 3.5 397B-A17B",
      "respondent_key": "qwen35_27b",
      "respondent_name": "Qwen 3.5 27B",
      "weighted_score": 10,
      "brief_justification": "The model accurately identified a mathematical impossibility in the prompt's premises and adjusted the figures to create a valid example of Simpson's Paradox. It thoroughly addressed all four questions with clear reasoning, accurate statistical concepts, and actionable advice."
    },
    {
      "judge_key": "qwen35_397b_a17b",
      "judge_name": "Qwen 3.5 397B-A17B",
      "respondent_key": "qwen35_122b_a10b",
      "respondent_name": "Qwen 3.5 122B-A10B",
      "weighted_score": 9.8,
      "brief_justification": "The model correctly identified a mathematical impossibility in the prompt's premises and transparently adjusted the data to provide a valid example of Simpson's Paradox. It thoroughly addressed all four parts with clear reasoning, accurate calculations, and practical solutions."
    }
  ],
  "meta": {
    "source": "The Multivac (app.themultivac.com)",
    "methodology": "10x10 blind peer matrix evaluation",
    "criteria": "correctness, completeness, clarity, depth, usefulness",
    "self_judgments": "excluded from rankings",
    "license": "Open data — cite as: The Multivac (2026)"
  }
}