Английская Википедия: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.[1] In particular, in regression analysis an influential observation is one whose deletion has a large effect on the parameter estimates.[2]
Assessment
Various methods have been proposed for measuring influence.[3][4] Assume an estimated regression <math>\mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{e}</math>, where <math>\mathbf{y}</math> is an n×1 column vector for the response variable, <math>\mathbf{X}</math> is the n×k design matrix of explanatory variables (including a constant), <math>\mathbf{e}</math> is the n×1 residual vector, and <math>\mathbf{b}</math> is a k×1 vector of estimates of some population parameter <math>\mathbf{\beta} \in \mathbb{R}^{k}</math>. Also define <math>\mathbf{H} \equiv \mathbf{X} \left(\mathbf{X}^{\mathsf{T}} \mathbf{X} \right)^{-1} \mathbf{X}^{\mathsf{T}}</math>, the projection matrix of <math>\mathbf{X}</math>. Then we have the following measures of influence:
- <math>\text{DFBETA}_{i} \equiv \mathbf{b} - \mathbf{b}_{(-i)} = \frac{\left( \mathbf{X}^{\mathsf{T}} \mathbf{X} \right)^{-1} \mathbf{x}_{i}^{\mathsf{T}} e_{i}}{1 - h_{ii}}</math>, where <math>\mathbf{b}_{(-i)}</math> denotes the coefficients estimated with the i-th row <math>\mathbf{x}_{i}</math> of <math>\mathbf{X}</math> deleted, <math>h_{ii} = \mathbf{x}_{i} \left( \mathbf{X}^{\mathsf{T}} \mathbf{X} \right)^{-1} \mathbf{x}_{i}^{\mathsf{T}}</math> denotes the i-th value of matrix's <math>\mathbf{H}</math> main diagonal. Thus DFBETA measures the difference in each parameter estimate with and without the influential point. There is a DFBETA for each variable and each observation (if there are N observations and k variables there are N·k DFBETAs).[5] Table shows DFBETAs for the third dataset from Anscombe's quartet (bottom left chart in the figure):
| x | y | intercept | slope |
| 10.0 | 7.46 | -0.005 | -0.044 |
| 8.0 | 6.77 | -0.037 | 0.019 |
| 13.0 | 12.74 | -357.910 | 525.268 |
| 9.0 | 7.11 | -0.033 | 0 |
| 11.0 | 7.81 | 0.049 | -0.117 |
| 14.0 | 8.84 | 0.490 | -0.667 |
| 6.0 | 6.08 | 0.027 | -0.021 |
| 4.0 | 5.39 | 0.241 | -0.209 |
| 12.0 | 8.15 | 0.137 | -0.231 |
| 7.0 | 6.42 | -0.020 | 0.013 |
| 5.0 | 5.73 | 0.105 | -0.087 |
Outliers, leverage and influence
An outlier may be defined as a data point that differs significantly from other observations.[6][7] A high-leverage point are observations made at extreme values of independent variables.[8] Both types of atypical observations will force the regression line to be close to the point.[2] In Anscombe's quartet, the bottom right image has a point with high leverage and the bottom left image has an outlying point.
See also
References
Further reading
- ↑ Шаблон:Citation.
- ↑ 2,0 2,1 Ошибка цитирования Неверный тег
<ref>; для сносокEverittне указан текст - ↑ Шаблон:Cite web
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
