Exact Confidence Intervals for Linear Combinations of Multinomial Probabilities

Abstract

Linear combinations of multinomial probabilities, such as those resulting from contingency tables, are of use when evaluating classification system performance. While large sample inference methods for these combinations exist, small sample methods exist only for regions on the multinomial parameter space instead of the linear combinations. However, in medical classification problems it is common to have small samples necessitating a small sample confidence interval on linear combinations of multinomial probabilities. Therefore, in this paper we derive an exact confidence interval, through the use of fiducial inference, for linear combinations of multinomial probabilities. Simulation demonstrates the presented interval's adherence to exact coverage. Additionally, an adjustment to the exact interval is provided, giving shorter lengths while still achieving better coverage than large sample methods. Computational efficiencies in estimation of the exact interval are achieved through the application of a fast Fourier transform and combining a numerical solver and stochastic optimizer to find solutions. The exact confidence interval presented in this paper allows for comparisons between diagnostic methods previously unavailable, demonstrated through an example of diagnosing chronic allograph nephropathy in post kidney transplant patients.