Exact interval estimation for the linear combination of binomial proportions

Area of Science:

• Biostatistics
• Statistical Inference
• Computational Statistics

Background:

• Linear combinations of binomial proportions, including weighted sums and interaction effects, are crucial in statistical analysis.
• Current confidence intervals for these parameters are often approximate, limiting their precision and reliability.
• The need for exact and more accurate confidence intervals is critical for robust statistical inference.

Purpose of the Study:

• To develop and present an exact method for constructing confidence intervals for linear combinations of binomial proportions.
• To improve upon existing approximate confidence intervals by deriving exact, refined intervals.
• To provide practical recommendations for the use of these improved intervals in real-world data analysis.

Main Methods:

• Application of the $$-function method to iteratively refine approximate confidence intervals.
• Derivation of two final-improved (exact) intervals for the weighted sum of two binomial proportions, based on adjusted score and fiducial methods.
• Evaluation and comparison of the proposed exact intervals against existing approximate intervals using real datasets.

Main Results:

• The $$-function method successfully transforms approximate intervals into exact, and iteratively shortened, intervals.
• Two novel exact intervals are derived for the weighted sum of two proportions, demonstrating superior performance.
• For the weighted sum of three proportions and interaction effects, the exact interval derived from the adjusted score method is recommended.

Conclusions:

• The proposed $$-function method provides a robust approach to obtaining exact confidence intervals for complex binomial proportion scenarios.
• The newly derived exact intervals offer enhanced precision and are recommended for practical application over existing approximate methods.
• The study demonstrates the utility of the method through detailed analysis of three real-world datasets.