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  "Description": "Comprehensive functions to calculate sample size and power\nfor clinical trials with two co-primary endpoints. The package\nsupports five endpoint combinations: two continuous endpoints\n(Sozu et al. 2011 <doi:10.1080/10543406.2011.551329>), two\nbinary endpoints using asymptotic methods (Sozu et al. 2010\n<doi:10.1002/sim.3972>) and exact methods (Homma and Yoshida\n2025 <doi:10.1177/09622802251368697>), mixed continuous and\nbinary endpoints (Sozu et al. 2012\n<doi:10.1002/bimj.201100221>), and mixed count and continuous\nendpoints (Homma and Yoshida 2024 <doi:10.1002/pst.2337>). All\nmethods appropriately account for correlation between endpoints\nand provide both sample size and power calculation\ncapabilities.",
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        "Background",
        "Clinical Context",
        "Why Mixed Endpoints?",
        "Statistical Framework",
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        "Correlation Structure: Biserial Correlation",
        "Hypothesis Testing",
        "Test Statistics",
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        "Sample Size Determination",
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        "Replicating Sozu et al. (2012) Supporting Information Table 5",
        "Basic Usage Examples",
        "Example 1: Balanced Design",
        "Example 2: Effect of Correlation",
        "Example 3: Comparison of Test Methods",
        "Example 4: Unbalanced Allocation",
        "Power Verification",
        "Practical Recommendations",
        "Design Considerations",
        "When to Use This Method",
        "Challenges and Considerations",
        "References"
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        "Sample Size Determination",
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        "Example: Calculate Correlation Bounds",
        "Replicating Homma and Yoshida (2024) Table 1 (Case B)",
        "Basic Usage Examples",
        "Example 1: Balanced Design",
        "Example 2: Effect of Correlation",
        "Example 3: Effect of Dispersion Parameter",
        "Example 4: Unbalanced Allocation",
        "Power Verification",
        "Practical Recommendations",
        "Design Considerations",
        "When to Use This Method",
        "Challenges and Considerations",
        "References"
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        "Introduction",
        "What are Co-Primary Endpoints?",
        "Statistical Properties",
        "Hypotheses Structure",
        "Statistical Framework",
        "Intersection-Union Test (IUT)",
        "Type I Error Control",
        "Overall Power",
        "Impact of Correlation",
        "Supported Endpoint Types",
        "1. Two Continuous Endpoints",
        "2. Two Binary Endpoints (Asymptotic Approximation)",
        "3. Two Binary Endpoints (Exact Methods)",
        "4. Mixed Continuous and Binary Endpoints",
        "5. Mixed Count and Continuous Endpoints",
        "Why Does Correlation Matter?",
        "Choosing the Right Method",
        "Decision Guidelines",
        "Sample Size Calculation Approach",
        "Detailed Vignettes",
        "References"
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        "Clinical Context",
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        "Statistical Framework",
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        "Correlation Structure",
        "Hypothesis Testing",
        "Test Statistics",
        "Method 1: Standard Normal Approximation (AN)",
        "Method 2: Normal Approximation with Continuity Correction (ANc)",
        "Method 3: Arcsine Transformation (AS)",
        "Method 4: Arcsine Transformation with Continuity Correction (ASc)",
        "Joint Distribution and Correlation",
        "Power Calculation",
        "Sample Size Calculation",
        "Replicating Sozu et al. (2010) Table III",
        "Key Findings",
        "Basic Usage Examples",
        "Example 1: Equal Effect Sizes",
        "Example 2: Unequal Effect Sizes",
        "Example 3: Effect of Correlation",
        "Example 4: Unbalanced Allocation",
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        "Practical Recommendations",
        "Design Considerations",
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        "Method 2: Fisher's Exact Test (Fisher)",
        "Method 3: Fisher's Mid-P Test (Fisher-midP)",
        "Method 4: Z-pooled Exact Unconditional Test (Z-pool)",
        "Method 5: Boschloo's Exact Unconditional Test (Boschloo)",
        "Exact Power Calculation",
        "Power Formula",
        "Sample Size Calculation",
        "Replicating Homma and Yoshida (2025) Table 4",
        "Practical Examples",
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        "Example 2: Sample Size Calculation",
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        "Impact of Correlation",
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        "Comparison: Exact vs Asymptotic",
        "Example 5: Exact vs AN Method",
        "Practical Recommendations",
        "Test Method Selection",
        "When to Use Each Method",
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        "Allocation Ratio",
        "Computational Considerations",
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        "References"
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        "Overview",
        "Background and Motivation",
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        "Clinical Examples",
        "Statistical Framework",
        "Model and Assumptions",
        "Effect Size Parameterization",
        "Hypothesis Testing",
        "Test Statistics",
        "Joint Distribution",
        "Power Formula",
        "Sample Size Calculation",
        "Basic Example",
        "Impact of Correlation",
        "Visualization with plot()",
        "Replicating Sozu et al. (2011) Table 1",
        "Power Calculation",
        "Power for a Given Sample Size",
        "Power Verification",
        "Unified Interface",
        "Unknown Variance Case",
        "Practical Considerations",
        "Correlation Estimation",
        "Sensitivity Analysis",
        "References"
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