Regression leverage
Regression leverage hᵢᵢ measures how far an observation's predictor values lie from the centre of the regressor space.
FrameworkInfluential observations
See it move
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The hat matrix H maps outcomes onto fitted values, ŷ = Hy, and its diagonal element hᵢᵢ is the leverage of observation i. Leverage satisfies 0 ≤ hᵢᵢ ≤ 1 and sums to k + 1, giving an average of (k+1)/n. A point with hᵢᵢ above twice that average sits far from the rest of the predictors and is flagged, though it only distorts the fit alongside a large residual.
Where it fits
SubjectData Analysis & StatisticsAdvancedTopicRegression Diagnostics & ProblemsAdvanced
The formula
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Variables
- hat (projection) matrix (dimensionless)
- n × (k+1) design matrix of predictors including intercept column (dimensionless)
- leverage of observation i; i-th diagonal element of H (dimensionless)
ŷ = Hy. Leverage satisfies 0 ≤ hᵢᵢ ≤ 1 and Σhᵢᵢ = k + 1. Average leverage is (k+1)/n; observations with hᵢᵢ > 2(k+1)/n are commonly flagged.