Influential observation

{{Short description|Observation that would cause a large change if deleted}}

File:Anscombe's quartet 3.svg the two datasets on the bottom both contain influential points. All four sets are identical when examined using simple summary statistics, but vary considerably when graphed. If one point is removed, the line would look very different.]]

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.{{citation|title=Elementary Statistics for Geographers|first1=James E.|last1=Burt|first2=Gerald M.|last2=Barber|first3=David L.|last3=Rigby|publisher=Guilford Press|year=2009|isbn=9781572304840|page=513|url=https://books.google.com/books?id=p7YMOPuu8ugC&pg=PA513}}. In particular, in regression analysis an influential observation is one whose deletion has a large effect on the parameter estimates.

Assessment

Various methods have been proposed for measuring influence.{{cite web |first=Larry |last=Winner |title=Influence Statistics, Outliers, and Collinearity Diagnostics |date=March 25, 2002 |url=http://stat.ufl.edu/~winner/sta6127/influence.doc }}{{cite book |last1=Belsley |first1=David A. |last2=Kuh |first2=Edwin |last3=Welsh |first3=Roy E. | year=1980 |title=Regression Diagnostics: Identifying Influential Data and Sources of Collinearity |publisher=John Wiley & Sons |location=New York |series=Wiley Series in Probability and Mathematical Statistics |isbn=0-471-05856-4 |pages=11–16 |url=https://books.google.com/books?id=GECBEUJVNe0C&pg=PA11 }} Assume an estimated regression $\mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{e}$ , where $\mathbf{y}$ is an n×1 column vector for the response variable, $\mathbf{X}$ is the n×k design matrix of explanatory variables (including a constant), $\mathbf{e}$ is the n×1 residual vector, and $\mathbf{b}$ is a k×1 vector of estimates of some population parameter $\mathbf{\beta} \in \mathbb{R}^{k}$ . Also define $\mathbf{H} \equiv \mathbf{X} \left(\mathbf{X}^{\mathsf{T}} \mathbf{X} \right)^{-1} \mathbf{X}^{\mathsf{T}}$ , the projection matrix of $\mathbf{X}$ . Then we have the following measures of influence:

$\text{DFBETA}_{i} \equiv \mathbf{b} - \mathbf{b}_{(-i)} = \frac{\left( \mathbf{X}^{\mathsf{T}} \mathbf{X} \right)^{-1} \mathbf{x}_{i}^{\mathsf{T}} e_{i}}{1 - h_{ii}}$ , where $\mathbf{b}_{(-i)}$ denotes the coefficients estimated with the i-th row $\mathbf{x}_{i}$ of $\mathbf{X}$ deleted, $h_{ii} = \mathbf{x}_{i} \left( \mathbf{X}^{\mathsf{T}} \mathbf{X} \right)^{-1} \mathbf{x}_{i}^{\mathsf{T}}$ denotes the i-th value of matrix's $\mathbf{H}$ 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).{{cite web |title=Outliers and DFBETA |url=http://www.albany.edu/faculty/kretheme/PAD705/SupportMat/DFBETA.pdf |url-status=live |archive-date=May 11, 2013 |archive-url=https://web.archive.org/web/20130511013229/http://www.albany.edu/faculty/kretheme/PAD705/SupportMat/DFBETA.pdf }} Table shows DFBETAs for the third dataset from Anscombe's quartet (bottom left chart in the figure):

class="wikitable" style="text-align: center; margin-left:auto; margin-right:auto;" border="1" \| 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

{{ordered list

| item1_value=2 | 1 = DFFITS - difference in fits

| item2_value=3 | 2 = Cook's D measures the effect of removing a data point on all the parameters combined.{{cite book | last = Everitt | first = Brian | title = The Cambridge Dictionary of Statistics | publisher = Cambridge University Press | location = Cambridge, UK New York | year = 1998 | isbn = 0-521-59346-8 | url = https://archive.org/details/cambridgediction00ever_0 }}

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Outliers, leverage and influence

An outlier may be defined as a data point that differs markedly from other observations.{{Cite journal |last=Grubbs |first=F. E. |date=February 1969 |title=Procedures for detecting outlying observations in samples |journal=Technometrics |volume=11 |issue=1 |pages=1–21 |doi= 10.1080/00401706.1969.10490657|quote=An outlying observation, or "outlier," is one that appears to deviate markedly from other members of the sample in which it occurs.}}{{cite book |last=Maddala |first=G. S. |author-link=G. S. Maddala |chapter=Outliers |title=Introduction to Econometrics |location=New York |publisher=MacMillan |edition=2nd |year=1992 |isbn=978-0-02-374545-4 |pages=[https://archive.org/details/introductiontoec00madd/page/89 89] |quote=An outlier is an observation that is far removed from the rest of the observations. |chapter-url=https://books.google.com/books?id=nBS3AAAAIAAJ&pg=PA89 |url=https://archive.org/details/introductiontoec00madd/page/89 }}

A high-leverage point are observations made at extreme values of independent variables.{{cite book |last=Everitt |first=B. S. |year=2002 |title=Cambridge Dictionary of Statistics |publisher=Cambridge University Press |isbn=0-521-81099-X }}

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.

Influential observation

Assessment

Outliers, leverage and influence

See also

References

Further reading