Tutorial on removing the shackles of regression analysis: How to stay true to your theory of binary response probabilities

Statistical analyses of data often add some additional constraints to a theory and leave out others, so as to convert the theory into a testable hypothesis. In the case of binary data, such as yes/no responses, or such as the presence/absence of a symptom or a behavior, theories often actually predict that certain response probabilities change monotonically in a specific direction and/or that certain response probabilities are bounded from above or below in specific ways. A regression analysis is not really true to such a theory in that it may leave out parsimonious constraints and in that extraneous assumptions like linearity or log-linearity, or even the assumption of a functional relationship, are dictated by the method rather than the theory. That mismatch may well bias the results of empirical analysis and jeopardize attempts at meaningful replication of psychological research. This tutorial shows how contemporary order-constrained methods can shed more light on such questions, using far weaker auxiliary assumptions, while also formulating more detailed, nuanced, and concise hypotheses, and allowing for quantitative model selection.