Diagnose outliers and influence in linear regression

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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.

Diagnose outliers and influence in linear regression

Company: Citadel

Role: Data Scientist

Category: Machine Learning

Difficulty: hard

Interview Round: Technical Screen

In ordinary least squares, define outliers, high-leverage points, and influential observations. Derive Cook’s distance and explain its relationship to leverage and studentized residuals. Then: (1) Describe a step-by-step diagnostic workflow (plots and statistics) to detect each. (2) Show how conclusions can change if you remove the top-1 influential point, and propose robust alternatives (e.g., Huber, Tukey biweight, RANSAC). (3) Explain how to report and justify decisions about such points in a model review.

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.

Read the full Citadel Data Scientist interview experience this question came from

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Citadel
Oct 13, 2025
hardData ScientistTechnical ScreenMachine Learning
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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

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