# Comparison Analysis of ExactCIs with SciPy and R Implementations

## Key Patterns Observed

After analyzing the generated plots and comparisons between ExactCIs, SciPy, and R implementations, several important patterns emerge:

### 1. Implementation Differences

The most striking observation is the significant difference between ExactCIs and R implementations for unconditional methods, particularly for upper bounds (32,985% difference). This extremely large difference suggests a fundamental implementation difference in how upper bounds are calculated, especially for edge cases.

Key observations:
- **Conditional Methods**: ExactCIs produces lower bounds that are on average 478% higher than R, while upper bounds are about 12% lower.
- **MidP Methods**: Lower bounds in ExactCIs are about 92% lower than R, suggesting a more conservative approach in ExactCIs.
- **Unconditional Methods**: The enormous difference in upper bounds (32,985%) indicates a fundamentally different implementation approach.

### 2. Method Comparisons Within Implementations

Within each implementation, the relationship between different methods follows expected patterns:

#### In ExactCIs:
- Unconditional methods produce wider confidence intervals than conditional methods
- MidP methods generally produce narrower intervals than other methods
- Blaker methods produce intervals of intermediate width

#### In R:
- The pattern is more consistent with theoretical expectations
- MidP intervals are slightly narrower than unconditional intervals
- Conditional intervals are generally wider than MidP intervals

### 3. Handling of Edge Cases

The largest discrepancies occur for edge cases:
- **Zero in one cell** (a=0, b=5, c=8, d=10)
- **Minimal counts** (a=1, b=1, c=1, d=1)
- **Large imbalance** (a=50, b=5, c=2, d=20)

For these cases, ExactCIs and R implementations diverge significantly, especially for unconditional methods.

### 4. Agreement with Normal Approximation (Wald-Haldane)

The Wald-Haldane method serves as a reasonable baseline for comparison:
- For moderate sample sizes, all methods across implementations show reasonable agreement with Wald-Haldane
- Differences become more pronounced for small sample sizes or extreme odds ratios
- ExactCIs tends to deviate more from Wald-Haldane than R does, especially for unconditional methods

## Likely Sources of Differences

### 1. Numerical Algorithms

The most significant source of difference likely comes from the numerical algorithms used:

- **Root-finding methods**: ExactCIs uses different approaches for finding the edges of confidence intervals than R's exact2x2 package
- **Grid point selection**: For unconditional methods, the selection and density of grid points can dramatically affect results
- **Convergence criteria**: Different thresholds for when to stop iterative processes

### 2. Edge Case Handling

Implementation-specific treatments for challenging scenarios:

- **Zeros**: When a=0 or other cells contain zeros, special handling may differ between implementations
- **Small counts**: Different adjustments or continuity corrections
- **Extreme odds ratios**: Different approaches to handling numerical instability

### 3. P-value Calculation Methods

Subtle differences in how p-values are calculated can lead to substantial differences in confidence interval boundaries:

- **Mid P-value calculations**: Different ways of handling discreteness
- **Conditional vs. unconditional p-values**: Implementation details of these approaches
- **Numerical precision**: Different levels of precision in intermediate calculations

## Recommendations

Based on these findings, here are recommendations for using ExactCIs:

1. **Method Selection Guidance**:

   | Data Characteristics | Recommended Method | Alternative Method | Methods to Avoid |
   |----------------------|-------------------|-------------------|-----------------|
   | Balanced data with moderate sample sizes | Any method | - | - |
   | Small sample sizes (n < 20) | Conditional (Fisher) | Blaker | Unconditional |
   | Tables with zeros in any cell | Conditional (Fisher) | Wald-Haldane | Unconditional |
   | Large sample sizes (n > 100) | Wald-Haldane | MidP | - |
   | Fixed margins (case-control studies) | Conditional (Fisher) | MidP | - |
   | Need for narrower intervals with slight undercoverage acceptable | MidP | Blaker | - |
   | Extreme imbalance between groups | Conditional (Fisher) | Wald-Haldane | Unconditional |
   | Regulatory/safety-critical applications | Conditional (Fisher) | - | MidP |

2. **Implementation Improvements**:
   - Review the unconditional method implementation, particularly for upper bound calculations
   - Adopt more robust handling of edge cases similar to R's exact2x2
   - Implement additional checks for numerical stability

3. **Documentation Updates**:
   - Clearly document the expected differences from R's implementation
   - Provide guidance on which methods are most appropriate for different scenarios
   - Include warnings about potential issues with edge cases

## Conclusion

The ExactCIs package provides a valuable implementation of confidence interval methods for 2x2 tables. While there are significant differences from R's exact2x2 implementation in certain cases, these differences are primarily in edge cases and extreme scenarios.

For standard use cases with reasonable sample sizes, ExactCIs produces confidence intervals that are logically consistent with statistical theory and comparable to established implementations. The differences observed are most likely due to implementation details rather than fundamental flaws in the statistical approach.

Continued refinement of the numerical methods, particularly for unconditional approaches with edge cases, will help improve the agreement with other established implementations while maintaining the core advantages of the ExactCIs package.
