14: Generalized Linear Models
- Page ID
- 57767
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Until this point, we have been applying the classical linear model (CLM) to our problems of modeling a dependent variable. It is the model \(\mathbf{Y} = \mathbf{XB} + \mathbf{E}\), with Normal errors. While this model is quite prevalent in the literature, it does not always do a good job of approximating reality.
In this chapter, we introduce the generalized linear model (GLM) paradigm and start to show its versatility. We also repeat much of the previous chapter, but from a different perspective, one of paying attention to the data-generating process.
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In our regression examples thus far, we have been dealing with continuous dependent variables. The classical linear model (CLM) requires this because the dependent variable needs to be conditionally distributed according to the Normal (a.k.a. Gaussian) distribution. Chapters 3 through 5 discussed this in detail.
Chapter 8: Fixing the Violations examined how we can handle a couple of types of violations of these assumptions, focusing on the case where the dependent variable is bounded. When the dependent variable is bounded, it cannot be distributed Normal. (Why? What is the support of the Normal distribution?) As such, if your dependent variable is bounded, you will have to transform that variable into an unbounded analogue. Once this is done, one might be able to use the methods of the usual CLM paradigm.
We have, however, encountered some difficulties with this transformation method. In each of our examples from Chapter 8, the dependent variable was bounded — but was never equal to its bound. This was necessary. If the dependent variable ever is equal to its bound, then the transformation function you use will return an infinite value (either \(-\infty\) or \(+\infty\)).
In this part of the book, we will extend the classical linear model (CLM) to be more general, and we will introduce a unifying framework allowing us to fit many different types of dependent variables — both continuous and discrete.
Note that before the publication of the seminal
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John A. Nelder and Robert W. M. Wedderburn (1972). "Generalized Linear Models." Journal of the Royal Statistical Society: Series A (General). 135(3), 370–384.
doi: 10.2307/2344614
each different type of problem tended to have a different type of solution. Logistic regression was a nice procedure, but so was probit regression, and solving one used a totally different technique than the other. This is what John Nelder and Robert William Maclagan Wedderburn added to statistics with their GLM paradigm: not that these problems can be solved, but that these problems can be solved using the same process.
