Influential observation

In statistics, an influential observation is an observation for a statistical calculation whose deletion from the dataset would noticeably change the result of the calculation. In particular, in regression analysis an influential point is one whose deletion has a large effect on the parameter estimates.

Various methods have been proposed for measuring influence. Assume an estimated regression ${\displaystyle \mathbf {y} =\mathbf {X} \mathbf {b} +\mathbf {e} }$, where ${\displaystyle \mathbf {y} }$ is an n×1 column vector for the response variable, ${\displaystyle \mathbf {X} }$ is the n×k design matrix of explanatory variables (including a constant), ${\displaystyle \mathbf {e} }$ is the n×1 residual vector, and ${\displaystyle \mathbf {b} }$ is a k×1 vector of estimates of some population parameter ${\displaystyle \mathbf {\beta } \in \mathbb {R} ^{k}}$. Also define ${\displaystyle \mathbf {H} \equiv \mathbf {X} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}}$, the projection matrix of ${\displaystyle \mathbf {X} }$. Then we have the following measures of influence:

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