  Binary regression models are ubiquitous in virtually every scientific field.
Frequently, traditional generalized linear models fail to capture the
variability in the probability surface that gives rise to the binary
observations and novel methodology is required. This has generated a
substantial literature comprised of binary regression models motivated by
various applications. We describe a novel organization of generalizations to
traditional binary regression methods based on the familiar three-part
structure of generalized linear models (random component, systematic component,
link function). This new perspective facilitates both the comparison of
existing approaches, and the development of novel, flexible models with
interpretable parameters that capture application-specific data generating
mechanisms. We use our proposed organizational structure to discuss some
concerns with certain existing models for binary data based on quantile
regression. We then use the framework to develop several new binary regression
models tailored to occupancy data for European red squirrels (Sciurus
vulgaris).
