High-Breakdown Robust Regression (MM and LTS)

High-breakdown robust regression (MM and LTS) estimates linear coefficients that stay reliable even when up to half of the observations are outliers or bad leverage points.

고붕괴점 강건회귀(MM·LTS)는 관측의 절반 가까이가 이상값이나 나쁜 지렛점이어도 흔들리지 않는 선형회귀 계수를 추정합니다.

My regression has several extreme cases; how can I estimate coefficients that are not pulled by them?

회귀분석에 극단적인 사례가 몇 개 있는데, 그 사례에 끌려가지 않는 계수를 어떻게 구하나요?

MM-estimation starts from a 50% breakdown S-estimate of the residual scale and then refines the coefficients with a Tukey bisquare loss tuned to 95% efficiency under normal errors. Least trimmed squares instead minimizes the sum of the smallest half of squared residuals and is then reweighted by ordinary least squares on the retained cases.

When to use High-Breakdown Robust Regression (MM and LTS)

• Residual plots or leverage diagnostics show a cluster of outlying cases.
• You want coefficients that ordinary least squares would give without the contaminating cases, without deleting them by hand.
• You need to flag vertical outliers and bad leverage points objectively.

When to use something else

• Only the standard errors are a concern under heteroskedasticity; use regression with robust standard errors.
• Outliers are only mild and in the outcome alone; Huber M-regression is simpler.

Assumptions and requirements

• A linear model holds for the majority of the data.
• Fewer than half of the observations are contaminated.
• Errors of the clean majority are roughly symmetric.

What the analysis reports

• Robust coefficients with asymptotic standard errors and OLS coefficients for comparison
• Robust residual scale and tuning constants
• Counts of vertical outliers, good and bad leverage points (MCD distances)
• Final robustness weights and listed outlying rows

How to report it

An MM-estimator regression (bisquare, 95% efficiency) of [outcome] (N = [value]) gave a robust scale of [value]; for [predictor], b = [value], SE = [value], p = [value], compared with b = [value] by OLS.

Related methods

• Robust Regression (Huber)
• Robust Multivariate Outlier Detection (MCD)
• Regression with Robust Standard Errors

Run High-Breakdown Robust Regression (MM and LTS) on Quorant