An Algebra of Intervals

This chapter, new to the Second Edition, develops a systematic algebraic framework for constructing confidence intervals for complex functions of Binomial and Multinomial proportions, extending Newcombe’s difference method into a general-purpose method known as ‘MOVER’. Going beyond simple proportions, the chapter demonstrates how to create reliable confidence intervals for statistical measures commonly used in corpus linguistics research. The chapter establishes the monotonic transformation theorem, showing how confidence intervals for functions of single proportions like the odds or logit of a proportion, can be obtained by applying the same function to interval bounds. For non-monotonic functions like entropy, the chapter provides methods for handling turning points. The core innovation generalises Newcombe’s difference interval to create a universal framework for confidence intervals for combinations of independent properties. This enables closed-form intervals for ratios (including risk ratios and odds ratios), sums, products, logarithms and power functions of independent proportions. A crucial distinction is made between independent and dependent properties. This requires us to perform the analytical reduction of formulae into a canonical form where each sampled property is cited only once. For Multinomial variables, a specialised summation method accounts for constraints by scaling interval widths appropriately. Performance evaluation reveals that MOVER consistently produces well-behaved intervals, representing significant improvement over traditional methods. This systematic approach transforms confidence interval calculation from ad hoc processes into principled algebraic exercises.