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. 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 y = X b + e {displaystyle mathbf {y} =mathbf {X} mathbf {b} +mathbf {e} } , where y {displaystyle mathbf {y} } is an n×1 column vector for the response variable, X {displaystyle mathbf {X} } is the n×k design matrix of explanatory variables (including a constant), e {displaystyle mathbf {e} } is the n×1 residual vector, and b {displaystyle mathbf {b} } is a k×1 vector of estimates of some population parameter β ∈ R k {displaystyle mathbf {eta } in mathbb {R} ^{k}} . Also define H ≡ X ( X T X ) − 1 X T {displaystyle mathbf {H} equiv mathbf {X} left(mathbf {X} ^{mathsf {T}}mathbf {X} ight)^{-1}mathbf {X} ^{mathsf {T}}} , the projection matrix of X {displaystyle mathbf {X} } . Then we have the following measures of influence: An outlier may be defined as a data point that differs significantly from other observations.A high-leverage point are observations made at extreme values of the independent variables.Both types of atypical observations will force the regression line to be close to the point. In Anscombe's quartet, the bottom right image has a point with high leverage and the bottom left image has an outlying point.