Competition between Choices Over Time

This chapter examines analysis of linguistic variation expressed as time series data, focusing on competitive systems where alternate forms undergo change over time. It introduces the logistic curve (S-curve) as a mathematical model for bounded binary competition, demonstrating how it emerges naturally from probabilistic constraints governing linguistic alternation. The chapter explores relationships between the logistic function and confidence interval estimation, showing how Wilson score intervals become symmetric when transformed to logit space, providing theoretical foundation for logistic regression techniques. The chapter presents a novel logit-Wilson regression method that accounts for variable uncertainty of observations by weighting data points according to their Wilson interval widths in logit space. This approach is demonstrated through two corpus-based case studies: the decline of to -infinitive perfect constructions in American English, and competition among catenative verbs over two centuries using Corpus of Historical American English (COHA) data. The chapter addresses critical limitations of logistic modelling, particularly the mathematical impossibility of fitting multiple competing forms simultaneously to logistic curves against the same baseline. Hierarchical alternation is one potential solution, but the author points out parallels with the three-body problem in physics, where multiple interacting elements may produce non-linear, chaotic behaviour. This leads us to conclude that logistic regression should be treated as an exploratory tool for identifying patterns and estimating rates of change, rather than as definitive model of linguistic variation.