Editing Outlier (section)

== Working with outliers ==
The choice of how to deal with an outlier should depend on the cause. Some estimators are highly sensitive to outliers, notably [[estimation of covariance matrices]].

=== Retention ===
Even when a normal distribution model is appropriate to the data being analyzed, outliers are expected for large sample sizes and should not automatically be discarded if that is the case.<ref name="karch2023">{{cite journal |last1=Karch |first1=Julian D. |title=Outliers may not be automatically removed. |journal=Journal of Experimental Psychology: General |date=2023 |volume=152 |issue=6 |pages=1735–1753 |doi=10.1037/xge0001357|pmid=37104797 |s2cid=258376426 |url=https://psyarxiv.com/47ezg/ |hdl=1887/4103722 |hdl-access=free }}</ref>  Instead, one should use a method that is robust to outliers to model or analyze data with naturally occurring outliers.<ref name="karch2023"/>

=== Exclusion ===
When deciding whether to remove an outlier, the cause has to be considered.  As mentioned earlier, if the outlier's origin can be attributed to an experimental error, or if it can be otherwise determined that the outlying data point is erroneous, it is generally recommended to remove it.<ref name="karch2023"/><ref name="bakker2014">{{cite journal |last1=Bakker |first1=Marjan |last2=Wicherts |first2=Jelte M. |title=Outlier removal, sum scores, and the inflation of the type I error rate in independent samples t tests: The power of alternatives and recommendations. |journal=Psychological Methods |date=2014 |volume=19 |issue=3 |pages=409–427 |doi=10.1037/met0000014|pmid=24773354 }}</ref> However, it is more desirable to correct the erroneous value, if possible. 

Removing a data point solely because it is an outlier, on the other hand,  is a controversial practice, often frowned upon by many scientists and science instructors, as it typically invalidates statistical results.<ref name="karch2023"/><ref name="bakker2014"/> While mathematical criteria provide an objective and quantitative method for data rejection, they do not make the practice more scientifically or methodologically sound, especially in small sets or where a normal distribution cannot be assumed. Rejection of outliers is more acceptable in areas of practice where the underlying model of the process being measured and the usual distribution of measurement error are confidently known. 

The two common approaches to exclude outliers are [[truncation (statistics)|truncation]] (or trimming) and [[Winsorising]]. Trimming discards the outliers whereas Winsorising replaces the outliers with the nearest "nonsuspect" data.<ref>{{cite book |title=Data Analysis: A Statistical Primer for Psychology Students |pages=24–25 |first=Edward L. |last=Wike |date=2006 |publisher=Transaction Publishers |isbn=9780202365350}}</ref> Exclusion can also be a consequence of the measurement process, such as when an experiment is not entirely capable of measuring such extreme values, resulting in [[censoring (statistics)|censored]] data.<ref>{{cite journal |title=Simplified estimation from censored normal samples |first=W. J. |last=Dixon |journal=The Annals of Mathematical Statistics |volume=31 |number=2 |date=June 1960 |pages=385–391 |url=http://projecteuclid.org/download/pdf_1/euclid.aoms/1177705900 |doi=10.1214/aoms/1177705900|doi-access=free }}</ref>

In [[Regression analysis|regression]] problems, an alternative approach may be to only exclude points which exhibit a large degree of influence on the estimated coefficients, using a measure such as [[Cook's distance]].<ref>Cook, R. Dennis (Feb 1977). "Detection of Influential Observations in Linear Regression". Technometrics (American Statistical Association) 19 (1): 15–18.</ref>

If a data point (or points) is excluded from the [[data analysis]], this should be clearly stated on any subsequent report.

=== Non-normal distributions ===
The possibility should be considered that the underlying distribution of the data is not approximately normal, having "[[fat tails]]". For instance, when sampling from a [[Cauchy distribution]],<ref>Weisstein, Eric W. [http://mathworld.wolfram.com/CauchyDistribution.html Cauchy Distribution. From MathWorld--A Wolfram Web Resource]</ref> the sample variance increases with the sample size, the sample mean fails to converge as the sample size increases, and outliers are expected at far larger rates than for a normal distribution. Even a slight difference in the fatness of the tails can make a large difference in the expected number of extreme values.

=== Set-membership uncertainties ===
A [[set estimation|set membership approach]] considers that the uncertainty corresponding to the ''i''th measurement of an unknown random vector ''x'' is represented by a set ''X''<sub>i</sub> (instead of a probability density function). If no outliers occur, ''x'' should belong to the intersection of all ''X''<sub>i</sub>'s. When outliers occur, this intersection could be empty, and we should relax a small number of the sets ''X''<sub>i</sub> (as small as possible) in order to avoid any inconsistency.<ref>{{cite journal|last1=Jaulin|first1=L.|
title=Probabilistic set-membership approach for robust regression|
journal=Journal of Statistical Theory and Practice|volume=4|pages=155–167|
year=2010|
url=http://www.ensta-bretagne.fr/jaulin/paper_probint_0.pdf|doi=10.1080/15598608.2010.10411978|s2cid=16500768}}</ref> This can be done using the notion of ''q''-[[relaxed intersection]]. As illustrated by the figure, the ''q''-relaxed intersection corresponds to the set of all ''x'' which belong to all sets except ''q'' of them. Sets ''X''<sub>i</sub> that do not intersect the  ''q''-relaxed intersection could be suspected to be outliers.

[[File:Wiki q inter def.jpg|thumb|Figure 5. ''q''-relaxed intersection of 6 sets for ''q''=2 (red), ''q''=3 (green), ''q''= 4 (blue), ''q''= 5 (yellow).]]

=== Alternative models ===
In cases where the cause of the outliers is known, it may be possible to incorporate this effect into the model structure, for example by using a [[hierarchical Bayes model]], or a [[mixture model]].<ref>Roberts, S. and Tarassenko, L.: 1995, A probabilistic resource allocating network for novelty detection. Neural Computation 6, 270–284.</ref><ref>{{Cite journal |last= Bishop |first=C. M. |date= August 1994 |title= Novelty detection and Neural Network validation |journal= IEE Proceedings - Vision, Image, and Signal Processing|volume=141 |issue=4 |pages= 217–222 |doi=10.1049/ip-vis:19941330 |doi-broken-date=7 December 2024 }}</ref>