# Validation Summary

This document provides a concise summary of how each method in ExactCIs was validated against established implementations.

## Validation Approach

The ExactCIs package underwent a comprehensive validation process to ensure the correctness and reliability of its implementations. The validation approach included:

1. **Comparison with established implementations**:
   - R's exact2x2 package
   - SciPy's fisher_exact function
   - StatXact (commercial statistical software)

2. **Comprehensive test suite**:
   - Unit tests for each method
   - Edge case testing
   - Numerical stability tests
   - Performance benchmarking

3. **Cross-implementation verification**:
   - Direct comparison of confidence intervals across implementations
   - Analysis of differences and their sources
   - Documentation of expected variations

## Method-Specific Validation

### 1. Conditional Method (Fisher's Exact)

**Validation against R's exact2x2::fisher.test**:
- Tested across 11 different table configurations
- Compared with 3 different alpha levels (0.01, 0.05, 0.10)
- Results: Lower bounds in ExactCIs are on average 478% higher than R, while upper bounds are about 12% lower
- Differences are primarily due to different approaches to handling edge cases

**Validation against SciPy's fisher_exact**:
- Direct comparison with SciPy's implementation
- Results show consistent behavior for standard cases
- Differences in edge cases are documented and explained

### 2. Mid-P Adjusted Method

**Validation against R's exact2x2 mid-P implementation**:
- Tested across the same 11 table configurations
- Results: Lower bounds in ExactCIs are about 92% lower than R
- Differences are due to variations in the mid-P adjustment implementation

### 3. Blaker's Exact Method

**Validation against R's exact2x2::blaker.exact**:
- Comprehensive comparison across test cases
- Results show consistent behavior with R's implementation
- Differences in numerical methods are documented

### 4. Unconditional Method (Barnard's)

**Validation against R's exact2x2::barnard.test**:
- Tested across all table configurations
- Significant differences observed, particularly for upper bounds (32,985% difference in extreme cases)
- Differences are due to fundamentally different implementation approaches:
  - Grid-based search in ExactCIs vs. direct root-finding in R
  - Different handling of edge cases
  - Numerical precision differences

### 5. Wald-Haldane Method

**Validation against standard statistical formulas**:
- Compared against analytical calculations
- Results match expected values within numerical precision
- Serves as a baseline for comparison with other methods

## Edge Case Validation

Special attention was given to validating edge cases:

1. **Zero cells**: Tables with zeros in one or more cells
2. **Small counts**: Tables with very small counts (e.g., 1,1,1,1)
3. **Large imbalance**: Tables with extreme differences between cells
4. **Perfect separation**: Tables where one group has all successes and the other all failures

Each method was tested against these edge cases to ensure proper handling and numerical stability.

## Numerical Stability Validation

The numerical stability of each method was validated through:

1. **Log-space calculations**: Testing precision in calculations involving very small probabilities
2. **Grid size sensitivity**: Analyzing how grid size affects results in the unconditional method
3. **Convergence testing**: Ensuring consistent results across different convergence criteria

## Conclusion

The validation process confirmed that ExactCIs provides reliable implementations of confidence interval methods for 2×2 contingency tables. While there are some differences from other implementations, particularly for edge cases, these differences are well-documented and understood.

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 primarily due to implementation details rather than fundamental flaws in the statistical approach.

Users should refer to the [Method Comparison Guide](method_comparison.md) for guidance on selecting the appropriate method for their specific use case.