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User:Sjcphd/sandbox Comparison of general and generalized linear models

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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
Exemplars ANOVA, ANCOVA, linear regression linear regression, logistic regression, Poisson regression, gamma regression,[3] 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 lm() in stats package (base R) glm() in stats package (base R)
Matlab function mvregress() glmfit()
SAS procedures PROC GLM, PROC REG PROC GENMOD, PROC LOGISTIC (for binary & ordered or unordered categorical outcomes)
Stata command regress glm
SPSS command regression, glm genlin, logistic
Wolfram Language & Mathematica function LinearModelFit[][7] GeneralizedLinearModelFit[][8]
EViews command ls[9] glm[10]

References[edit]

  1. ^ Neter, J., Kutner, M. H., Nachtsheim, C. J., & Wasserman, W. (1996). Applied linear statistical models (Vol. 4, p. 318). Chicago: Irwin.
  2. ^ a b Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences.
  3. ^ a b c McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. ISBN 0-412-31760-5.
  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. ^ LinearModelFit, Wolfram Language Documentation Center.
  8. ^ GeneralizedLinearModelFit, Wolfram Language Documentation Center.
  9. ^ ls, EViews Help.
  10. ^ glm, EViews Help.

Additional resources[edit]

Category:Generalized linear models


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Category:Statistical methods Category:General linear model

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