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Diagnose outliers and influence in linear regression

Last updated: May 21, 2026

Quick Overview

This question evaluates a data scientist's competency in diagnosing outliers, high-leverage points, influential observations, and Cook's distance in ordinary least squares regression, including recognition of robust alternatives and rationale for reporting model-review decisions.

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Diagnose outliers and influence in linear regression

Citadel

Oct 13, 2025, 9:49 PM

hardData ScientistTechnical ScreenMachine Learning

OLS Diagnostics: Outliers, Leverage, Influence, and Cook's Distance

Context

You are fitting an ordinary least squares (OLS) linear regression with an intercept. Let X be the n×p design matrix (p includes the intercept), y the response, and the OLS fit is ŷ = Xβ̂ with residuals e = y − ŷ.

Tasks

Define outliers, high-leverage points, and influential observations in OLS.
Derive Cook's distance and explain how it relates to leverage and (studentized) residuals.
Provide a step-by-step diagnostic workflow (plots and statistics) to detect each of the above.
Demonstrate how conclusions can change if you remove the top-1 influential point, and propose robust alternatives (e.g., Huber, Tukey biweight, RANSAC).
Explain how to report and justify decisions about such points in a model review.

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