World Library  
Flag as Inappropriate
Email this Article

Homoscedasticity

Article Id: WHEBN0000294428
Reproduction Date:

Title: Homoscedasticity  
Author: World Heritage Encyclopedia
Language: English
Subject: F-test, Analysis of variance, Homogeneity (statistics), Heteroscedasticity, Cochran's C test
Collection: Statistical Deviation and Dispersion
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Homoscedasticity

Plot with random data showing homoscedasticity.

In statistics, a sequence or a vector of random variables is homoscedastic if all random variables in the sequence or vector have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The spellings homoskedasticity and heteroskedasticity are also frequently used.[1]

The assumption of homoscedasticity simplifies mathematical and computational treatment. Serious violations in homoscedasticity (assuming a distribution of data is homoscedastic when in reality it is heteroscedastic ) may result in overestimating the goodness of fit as measured by the Pearson coefficient.

Contents

  • Assumptions of a regression model 1
  • Testing 2
  • Homoscedastic distributions 3
  • See also 4
  • References 5

Assumptions of a regression model

As used in describing simple linear regression analysis, one assumption of the fitted model (to ensure that the least-squares estimators are each a best linear unbiased estimator of the respective population parameters, by the Gauss–Markov theorem) is that the standard deviations of the error terms are constant and do not depend on the x-value. Consequently, each probability distribution for y (response variable) has the same standard deviation regardless of the x-value (predictor). In short, this assumption is homoscedasticity. Homoscedasticity is not required for the estimates to be unbiased, consistent, and asymptotically normal.[2]

Testing

Residuals can be tested for homoscedasticity using the Breusch–Pagan test, which regresses squared residuals on the independent variables. Since the Breusch–Pagan test is sensitive to departures from normality, the Koenker–Basset or 'generalized Breusch–Pagan' test is used for general purposes. Testing for groupwise heteroscedasticity requires the Goldfeld–Quandt test.

Homoscedastic distributions

Two or more normal distributions, N(\mu_i,\Sigma_i), are homoscedastic if they share a common covariance (or correlation) matrix, \Sigma_i = \Sigma_j,\ \forall i,j. Homoscedastic distributions are especially useful to derive statistical pattern recognition and machine learning algorithms. One popular example is Fisher's linear discriminant analysis.

The concept of homoscedasticity can be applied to distributions on spheres.[3]

See also

References

  1. ^ For the Greek etymology of the term, see McCulloch, J. Huston (1985). "On Heteros*edasticity".  
  2. ^ Achen, Christopher H.; Shively, W. Phillips (1995), Cross-Level Inference, University of Chicago Press, pp. 47–48,  .
  3. ^ Hamsici, Onur C.; Martinez, Aleix M. (2007) "Spherical-Homoscedastic Distributions: The Equivalency of Spherical and Normal Distributions in Classification", Journal of Machine Learning Research, 8, 1583-1623
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from World Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.