Isotonic Regression

Isotonic regression finds the monotone (non-decreasing or non-increasing) step function of a predictor that minimizes the squared error, computed with the pool adjacent violators algorithm.

단조(isotonic) 회귀는 설명변수에 대해 증가(또는 감소)만 하는 계단함수 가운데 오차제곱합이 가장 작은 것을 인접 위반 합치기 알고리즘으로 구하는 방법입니다.

I only know that the outcome should never decrease as the dose increases; how can I fit that without choosing a curve shape?

용량이 늘수록 결과가 줄지 않는다는 것만 알 때, 곡선 모양을 정하지 않고 어떻게 적합하나요?

Isotonic regression fits the best-fitting step function that only goes up (or only goes down) with the predictor, pooling neighbouring values whenever they violate the order. It imposes monotonicity without any other assumption about shape, so it is useful for dose-response curves and calibration.

When to use Isotonic Regression

• Theory says the relationship is monotone but its shape is unknown
• You need to calibrate predicted probabilities or scores
• Parametric curves would impose an unjustified shape

When to use something else

• The relationship may rise and then fall; use LOESS smoothing regression instead
• You need coefficients and hypothesis tests for the effect; use polynomial or spline regression instead

Assumptions and requirements

• The mean of the outcome is monotone in the predictor
• Observations are independent
• Enough data at each level to estimate the steps
• The direction (increasing or decreasing) follows the sign of Spearman's rho unless specified

What the analysis reports

• Fitted step function with the range of each step
• Number of steps
• Variance explained
• Direction of monotonicity and Spearman's rho

How to report it

An isotonic (monotone increasing) regression of [outcome] on [predictor] produced [value] steps and explained [value]% of the variance (Spearman's ρ = [value], p = [value]).

Related methods

• LOESS Smoothing Regression
• Spearman Rank Correlation

Run Isotonic Regression on Quorant