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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[5]. 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[6] for binary or dichotomous outcomes, Poisson regression[7] 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
Special cases ANOVA, ANCOVA, MANOVA, MANCOVA, linear regression, mixed model linear regression, logistic regression, Poisson regression, gamma regression,[8] general linear model
Function in R lm() glm()
Function in Matlab mvregress() glmfit()
Procedure in SAS PROC GLM, PROC MIXED PROC GENMOD, PROC GLIMMIX, PROC LOGISTIC (for regression with categorical variables)
Command in Stata regress glm
Command in SPSS regression, glm genlin, logistic regression
Function in Wolfram Language & Mathematica LinearModelFit[][9] GeneralizedLinearModelFit[][10]
Command in EViews ls[11] glm[12]
  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. ^ McCullagh, P.; Nelder, J. A. (1989), "An outline of generalized linear models", Generalized Linear Models, Springer US, pp. 21–47, ISBN 9780412317606, retrieved 2018-12-19
  4. ^ Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications.
  5. ^ Cite error: The named reference :0 was invoked but never defined (see the help page).
  6. ^ Hosmer Jr, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied logistic regression (Vol. 398). John Wiley & Sons.
  7. ^ 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.
  8. ^ McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. ISBN 0-412-31760-5.
  9. ^ LinearModelFit, Wolfram Language Documentation Center.
  10. ^ GeneralizedLinearModelFit, Wolfram Language Documentation Center.
  11. ^ ls, EViews Help.
  12. ^ glm, EViews Help.

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