“Comparison of general and generalized linear models”的意思、由来-开放百科全书

The general linear model (GLM)^[1]^[2] and the generalized linear model (GLiM)^[3]^[4] are two commonly used families of statistical methods to relate some number of continuous and/or categorical predictors to a single outcome variable.

The main difference between the two approaches is that the GLM strictly assumes that the residuals will follow a conditionally normal distribution^[2], while the GLiM loosens this assumption and allows for a variety of other distributions from the exponential family for the residuals^[3]. Of note, the GLM is a special case of the GLiM in which the distribution of the residuals follow a conditionally normal distribution.

The distribution of the residuals largely depends on the type and distribution of the outcome variable; different types of outcome variables lead to the variety of models within the GLiM family. Commonly used models in the GLiM family include binary logistic regression^[5] for binary or dichotomous outcomes, Poisson regression^[6] for count outcomes, and linear regression for continuous, normally distributed outcomes. This means that GLiM may be spoken of as a general family of statistical models or as specific models for specific outcome types.

General linear model

Generalized linear model

Typical estimation method

Least squares, best linear unbiased prediction

Maximum likelihood or Bayesian

Examples

ANOVA, ANCOVA, linear regression

linear regression, logistic regression, Poisson regression, gamma regression,^[7] general linear model

Extensions and related methods

MANOVA, MANCOVA, linear mixed model

generalized linear mixed model (GLMM), generalized estimating equations (GEE)

R package and function

[https://stat.ethz.ch/R-manual/R-devel/library/stats/html/lm.html lm()] in stats package (base R)

[https://stat.ethz.ch/R-manual/R-devel/library/stats/html/glm.html glm()] in stats package (base R)

Matlab function

mvregress()

glmfit()

SAS procedures

[https://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#glm_toc.htm PROC GLM], [https://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#reg_toc.htm PROC REG]

[https://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#genmod_toc.htm PROC GENMOD], [https://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#logistic_toc.htm PROC LOGISTIC] (for binary & ordered or unordered categorical outcomes)

Stata command

regress

glm

SPSS command

[https://stats.idre.ucla.edu/spss/output/regression-analysis/ regression], [https://stats.idre.ucla.edu/spss/library/spss-librarymanova-and-glm-2/ glm]

genlin, logistic

Wolfram Language & Mathematica function

LinearModelFit[]^[8]

GeneralizedLinearModelFit[]^[9]

EViews command

ls^[10]

glm^[11]

1. ^Neter, J., Kutner, M. H., Nachtsheim, C. J., & Wasserman, W. (1996). Applied linear statistical models (Vol. 4, p. 318). Chicago: Irwin.
2. ^¹Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences.
3. ^¹{{Citation|last=McCullagh|first=P.|title=An outline of generalized linear models|date=1989|work=Generalized Linear Models|pages=21–47|publisher=Springer US|isbn=9780412317606|last2=Nelder|first2=J. A.|doi=10.1007/978-1-4899-3242-6_2}}
4. ^Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications.
5. ^Hosmer Jr, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied logistic regression (Vol. 398). John Wiley & Sons.
6. ^Gardner, W., Mulvey, E. P., & Shaw, E. C. (1995). Regression analyses of counts and rates: Poisson, overdispersed Poisson, and negative binomial models. Psychological bulletin, 118(3), 392.
7. ^{{cite book|title=Generalized Linear Models, Second Edition|last=McCullagh|first=Peter|author2=Nelder, John|publisher=Boca Raton: Chapman and Hall/CRC|year=1989|isbn=978-0-412-31760-6|ref=McCullagh1989|authorlink=Peter McCullagh|authorlink2=John Nelder}}
8. ^LinearModelFit, Wolfram Language Documentation Center.
9. ^GeneralizedLinearModelFit, Wolfram Language Documentation Center.
10. ^ls, EViews Help.
11. ^glm, EViews Help.

References

{{cite book | last = McCullagh | first = Peter | authorlink= Peter McCullagh |author2=Nelder, John |authorlink2=John Nelder | title = Generalized Linear Models, Second Edition | publisher = Boca Raton: Chapman and Hall/CRC | year = 1989 | isbn = 978-0-412-31760-6 |ref=McCullagh1989}}

1 : Generalized linear models