Nonlinearity

Nonlinearity means that a relationship fails the relevant linearity conditions. It can involve changing marginal effects, interactions, or other departures from an additive proportional map, while applied modeling uses several distinct linearity conventions.

A linear map preserves addition and scalar multiplication. In applied regression, a model can be linear in its parameters while nonlinear in an input, and an affine straight-line model includes an intercept. These distinctions matter when deciding which superposition or scaling rule is valid.

Nonlinear responses include saturation, changing marginal effects, interactions, and thresholds, but none is present in every nonlinear model. Local linear approximations can still be useful over a bounded range. Estimate the relevant response and its uncertainty rather than assume that a small intervention must have a large effect or that nonlinearity implies chaos.

When to use it

When analyzing data; when building predictive models; when evaluating statistical claims; when quantifying uncertainty.

How it can help

Define the relationship and range, inspect response and interactions, and test local approximations before extrapolating. Let evidence determine the shape.