The general linear model is called ‘linear' in terms of what parameters?
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Before digital computers, statistics textbooks spoke of three procedures—regression, the
analysis of variance (ANOVA), and the analysis of covariance (ANCOVA)—as if they were
different entities designed for different types of problems. These distinctions were useful at the
time because different time saving computational methods could be developed within each
technique. Early statistical software emulated these texts and developed separate routines to deal
with this classically defined triumvirate.
In the world of mathematics, however, there is no difference between traditional
regression, ANOVA, and ANCOVA. All three are subsumed under what is called the general
linear model or GLM. Indeed, some statistical software contain a single procedure that can
perform regression, ANOVA, and ANCOVA (e.g., PROC GLM in SAS). Failure to recognize
the universality of the GLM often impedes quantitative analysis, and in some cases, results in a
misunderstanding of statistics. One major shortcoming in contemporary statistical analysis in
neuroscience—that if you have groups, then ANOVA is the appropriate procedure—can be
traced directly to this misunderstanding.
That said, modern statistical software still contain separate procedures for regression and
ANOVA. The difference in these procedures should not be seen in terms of “this procedure is
right for this type of data set,” but rather in terms of convenience of use. That is, for a certain
type of data, it is more convenient to use an ANOVA procedure to fit a GLM than a regression
procedure.
The organization of the next three chapters follows these principles. In the current
chapter, we outline the GLM, provide the criteria for fitting a GLM to data, and the major
statistics used to assess the fit of a model. We end the chapter by outlining the assumptions of
the GLM. This chapter is expressly theoretical and can be skipped by those with a more
pragmatic interested in regression and ANOVA. The next two chapters treat, respectively,
regression and ANOVA/ANCOVA.
analysis of variance (ANOVA), and the analysis of covariance (ANCOVA)—as if they were
different entities designed for different types of problems. These distinctions were useful at the
time because different time saving computational methods could be developed within each
technique. Early statistical software emulated these texts and developed separate routines to deal
with this classically defined triumvirate.
In the world of mathematics, however, there is no difference between traditional
regression, ANOVA, and ANCOVA. All three are subsumed under what is called the general
linear model or GLM. Indeed, some statistical software contain a single procedure that can
perform regression, ANOVA, and ANCOVA (e.g., PROC GLM in SAS). Failure to recognize
the universality of the GLM often impedes quantitative analysis, and in some cases, results in a
misunderstanding of statistics. One major shortcoming in contemporary statistical analysis in
neuroscience—that if you have groups, then ANOVA is the appropriate procedure—can be
traced directly to this misunderstanding.
That said, modern statistical software still contain separate procedures for regression and
ANOVA. The difference in these procedures should not be seen in terms of “this procedure is
right for this type of data set,” but rather in terms of convenience of use. That is, for a certain
type of data, it is more convenient to use an ANOVA procedure to fit a GLM than a regression
procedure.
The organization of the next three chapters follows these principles. In the current
chapter, we outline the GLM, provide the criteria for fitting a GLM to data, and the major
statistics used to assess the fit of a model. We end the chapter by outlining the assumptions of
the GLM. This chapter is expressly theoretical and can be skipped by those with a more
pragmatic interested in regression and ANOVA. The next two chapters treat, respectively,
regression and ANOVA/ANCOVA.
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