15: Binary Dependent Variables
- Page ID
- 57775
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Thus far, we have examined linear regression where the dependent variable is continuous — either unbounded or bounded. These cases cover a wide variety of instances — but not all. Examples of dependent variables we can now use include heights, incomes, vote proportions, distances, and so forth.
However, we do not yet have the ability to handle dependent variables which are discrete. Such variables include dichotomous variables (presence of a characteristic), count variables (ages, deaths, numbers of fires), ordinal variables (importance level), and nominal variables (different outcomes). These types of variables are all limited in that there are adjacent outcomes. This chapter deals with modeling dichotomous (binary) random variables.}
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The previous chapter introduced the generalized linear model paradigm (GLM). Modeling with GLMs requires that we specify three things:
- the conditional distribution of the dependent variable (thus a formula for the expected value of the dependent variable), \(\mu\);
- the linear predictor, \(\eta = \mathbf{X}\mathbf{B}\); and
- a (bijective) function linking the two, \(g(\mu) = \eta\).
In that chapter, we showed that the Classical Linear Model is just a special case of the Generalized Linear Model. Specifically, the CLM is just a GLM using the Gaussian (Normal) distribution and the identity link. In this chapter, we cover the case of dichotomous (binary) dependent variables. In the following pages, we determine the appropriate distribution and the canonical link function.