{ "evaluation_id": "EVAL-20260318-162808", "question_id": "EVAL-20260318-162808", "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": "2026-03-18T16:28:08.039Z", "display_date": "Mar 18, 2026", "winner": { "name": "Claude Sonnet 4.6", "provider": "openrouter", "score": 9.71 }, "avg_score": 8.373333, "matrix_size": 20, "models_used": [ { "id": "minimax_01", "name": "MiniMax-01", "provider": "openrouter" }, { "id": "judge_claude_sonnet", "name": "Claude Sonnet 4.6", "provider": "openrouter" }, { "id": "judge_gpt54", "name": "GPT-5.4", "provider": "openrouter" } ], "rankings": { "judge_claude_sonnet": { "display_name": "Claude Sonnet 4.6", "provider": "openrouter", "average_score": 9.71, "score_count": 7, "min_score": 8.2, "max_score": 10, "rank": 1 }, "judge_gpt54": { "display_name": "GPT-5.4", "provider": "openrouter", "average_score": 9.33, "score_count": 6, "min_score": 7.95, "max_score": 9.8, "rank": 2 }, "minimax_01": { "display_name": "MiniMax-01", "provider": "openrouter", "average_score": 6.08, "score_count": 7, "min_score": 3.95, "max_score": 7.7, "rank": 3 } }, "judgments": [ { "judge_key": "minimax_m27", "judge_name": "MiniMax M2.7", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 6.2, "brief_justification": "The response correctly identifies Simpson's paradox conceptually but contains a critical error in part (1) - the constructed numbers show Hospital A with HIGHER overall survival (94.5% vs 85.7%), opposite to what the paradox requires (B having 91% vs A's 89%). Parts (2)-(4) provide conceptually correct but unverified analysis based on incorrect numbers." }, { "judge_key": "minimax_m27", "judge_name": "MiniMax M2.7", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 10, "brief_justification": "This response provides a comprehensive, mathematically rigorous explanation of Simpson's Paradox with perfectly constructed example numbers, clear causal reasoning, real-world healthcare implications, and multiple correct methodological approaches including stratified comparison, standardization, and risk-adjusted outcomes." }, { "judge_key": "minimax_m27", "judge_name": "MiniMax M2.7", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 9.75, "brief_justification": "The response accurately identifies an internal inconsistency in the original problem and provides a mathematically corrected Simpson's paradox example with precise numbers. It thoroughly explains all four parts of the question, uses clear structure, demonstrates deep understanding of confounding and case-mix issues, and offers practical solutions for proper hospital comparison." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 7, "brief_justification": "The response addresses all parts of the question and provides a reasonable explanation of Simpson's paradox. However, there is a critical error: the constructed numbers do not actually produce the paradox stated in the question (where B has higher overall survival). In the constructed example, Hospital A ends up with higher overall survival (94.5% vs 85.7%), which contradicts the premise. Additionally, the response states 'Hospital B has a higher overall survival rate (85.7% vs. 94.5%)' which is" }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 10, "brief_justification": "Excellent response that correctly identifies Simpson's Paradox, provides valid numerical examples with working math, correctly identifies Hospital A as superior, explains the insurance company's error using a causal diagram, and offers three proper methodological approaches (stratified, standardized, risk-adjusted) with real-world healthcare applications." }, { "judge_key": "minimax_m25", "judge_name": "MiniMax M2.5", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 9.8, "brief_justification": "The response correctly identifies the mathematical impossibility in the original problem (89% cannot be below both 90% and 95%), provides corrected exact numbers demonstrating Simpson's paradox, explains that Hospital A is better for both procedures, identifies the confounding issue with overall rates, and recommends stratified/risk-adjusted comparison. Slight deduction for not exactly matching the requested 89%/91% numbers, though the correction is mathematically necessary." }, { "judge_key": "minimax_m21", "judge_name": "MiniMax M2.1", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 4.95, "brief_justification": "Critical error: the constructed numbers show Hospital A with higher overall survival (94.5% vs 85.7%), which contradicts the question's premise that Hospital B has higher overall survival (91% vs 89%). The paradox is not demonstrated correctly." }, { "judge_key": "minimax_m21", "judge_name": "MiniMax M2.1", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 10, "brief_justification": "Comprehensive and accurate demonstration of Simpson's Paradox with exact numbers, algebraic proof, causal analysis, and multiple correct comparison methods including real-world healthcare applications." }, { "judge_key": "minimax_m2", "judge_name": "MiniMax M2", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 6.25, "brief_justification": "The response correctly explains Simpson's paradox and gives sound recommendations, but contains a critical error: the constructed numbers produce the OPPOSITE of what the question asked. The question specifies Hospital A has lower overall (89% vs 91%), but the response shows Hospital A with higher overall (94.5% vs 85.7%). The logic and recommendations are correct, but the example doesn't match the stated paradox." }, { "judge_key": "minimax_m2", "judge_name": "MiniMax M2", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 10, "brief_justification": "Excellent response with mathematically precise numerical examples, clear algebraic derivation of why the paradox occurs, and comprehensive coverage of all four questions. Provides multiple correct comparison methods (stratified, standardized, risk-adjusted) with real-world healthcare applications." }, { "judge_key": "minimax_m2", "judge_name": "MiniMax M2", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 9.55, "brief_justification": "The response correctly identifies that the original percentages are mathematically inconsistent (89% overall cannot be below both 90% and 95%), provides corrected exact numbers demonstrating Simpson's paradox, and thoroughly addresses all four parts of the question with clear explanations of confounding, case mix, and risk-adjusted comparisons." }, { "judge_key": "minimax_m1", "judge_name": "MiniMax M1", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 7.7, "brief_justification": "The response correctly explains Simpson's paradox and provides valid recommendations, but contains a critical error: the constructed numbers produce the opposite result from what was stated (A ends up with higher overall, not B). The explanation of the phenomenon and correct comparison methods are accurate and well-presented." }, { "judge_key": "minimax_m1", "judge_name": "MiniMax M1", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 10, "brief_justification": "Excellent response with correct mathematical construction of Simpson's Paradox, clear tables and algebraic explanation, insightful causal analysis, and comprehensive coverage of multiple correct comparison methods (stratified, standardized, risk-adjusted) used in real healthcare settings." }, { "judge_key": "minimax_m1", "judge_name": "MiniMax M1", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 9.35, "brief_justification": "Excellent response that correctly identifies Simpson's paradox, provides exact workable numbers (after noting the original 89% was mathematically impossible), explains which hospital is better for each procedure, details the insurance company's error, and recommends stratified/risk-adjusted comparison methods." }, { "judge_key": "minimax_01", "judge_name": "MiniMax-01", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 9.8, "brief_justification": "The response provides a thorough, accurate, and insightful explanation of Simpson's Paradox, including mathematical reasoning, practical implications, and actionable solutions, with only minor improvements needed in structure for enhanced clarity." }, { "judge_key": "minimax_01", "judge_name": "MiniMax-01", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 9.55, "brief_justification": "The response correctly explains Simpson's paradox, provides a detailed and accurate mathematical construction, and offers a nuanced analysis of the problem with actionable recommendations for improvement." }, { "judge_key": "judge_claude_sonnet", "judge_name": "Claude Sonnet 4.6", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 6.5, "brief_justification": "The response correctly identifies Simpson's Paradox and provides good qualitative analysis, but the constructed numbers in part (1) fail to match the required conditions: Hospital A ends up with 94.5% overall vs Hospital B's 85.7%, which is the opposite of what the problem requires (B should have higher overall rate of 91% vs A's 89%). The paradox is not actually demonstrated numerically, which is the core task." }, { "judge_key": "judge_claude_sonnet", "judge_name": "Claude Sonnet 4.6", "respondent_key": "judge_gpt54", "respondent_name": "GPT-5.4", "weighted_score": 7.95, "brief_justification": "The response correctly identifies the mathematical inconsistency in the original problem (89% overall cannot be below both 90% and 95% subgroup rates), which is a genuinely important insight. However, it misses that the problem could be valid if there are additional surgery types or patient categories beyond just heart and knee surgery — a common way Simpson's paradox is constructed. The corrected example with exact numbers is well-constructed and clearly demonstrates the paradox. The analysis o" }, { "judge_key": "judge_gpt54", "judge_name": "GPT-5.4", "respondent_key": "minimax_01", "respondent_name": "MiniMax-01", "weighted_score": 3.95, "brief_justification": "The response clearly explains the general idea of Simpson’s paradox, but its constructed numbers do not satisfy the stated overall rates and in fact show Hospital A with the higher overall survival, contradicting the question. It also fails to state the key conclusion that A is better after proper stratification/case-mix adjustment and that using unadjusted overall rates confounds quality with procedure mix." }, { "judge_key": "judge_gpt54", "judge_name": "GPT-5.4", "respondent_key": "judge_claude_sonnet", "respondent_name": "Claude Sonnet 4.6", "weighted_score": 8.2, "brief_justification": "The response correctly identifies Simpson's paradox, explains the case-mix issue well, and gives a valid numeric construction showing the reversal, though it does not match the exact requested overall rates of 89% for A and 91% for B. It is clear, insightful, and practically useful in recommending stratified or risk-adjusted comparisons." } ], "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)" } }