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]- ^ Neter, J., Kutner, M. H., Nachtsheim, C. J., & Wasserman, W. (1996). Applied linear statistical models (Vol. 4, p. 318). Chicago: Irwin.
- ^ a b Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences.
- ^ a b c McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. ISBN 0-412-31760-5.
- ^ Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications.
- ^ Hosmer Jr, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied logistic regression (Vol. 398). John Wiley & Sons.
- ^ 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.
- ^ LinearModelFit, Wolfram Language Documentation Center.
- ^ GeneralizedLinearModelFit, Wolfram Language Documentation Center.
- ^ ls, EViews Help.
- ^ glm, EViews Help.
Additional resources
[edit]- McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. ISBN 0-412-31760-5.
Category:Generalized linear models
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