From Linear to Nonlinear

Moving from linear to nonlinear modeling means checking whether a constant-slope or additive approximation misses consequential curvature, thresholds, or interactions in the relevant range. The appropriate form depends on the relationship and the decision.

A response is nonlinear in an input when a constant slope cannot describe it across the relevant range. Diminishing gains, thresholds, and interactions can all matter, but they imply different mechanisms. Before searching for a dramatic tipping point, ask which relationship bends, where it bends, and whether the decision requires more than a local straight-line approximation.

An intercept creates a common trap: a cost of ten plus two per item is an affine straight line, yet doubling item count does not double total cost. Conversely, checking just one doubled input cannot establish linearity everywhere. Compare changes around several relevant operating points and distinguish a observed curve from a tested explanation of its cause.

When to use it

When measured responses depart from a plausible straight-line or additive model enough to affect a decision, or when mechanism knowledge predicts consequential thresholds, saturation, or interactions.

How it can help

Inspect incremental effects at several operating points and account for fixed baselines. Compare candidate shapes and mechanisms against observations. Use a nonlinear model when its added detail improves the decision, while retaining useful local linear approximations.