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the total variation in the dependent variable that is "explained" (accounted for) by
variation in the independent variables.
Discrete Choice Models
Multiple regression (above) is generally used when the response variable is continuous
and has an unbounded range. Often the response variable may not be continuous but
rather discrete. While mathematically it is feasible to apply multiple regression to
discrete ordered dependent variables, some of the assumptions behind the theory of
multiple linear regression no longer hold, and there are other techniques such as
discrete choice models which are better suited for this type of analysis. If the dependent
variable is discrete, some of those superior methods are logistic regression, multinomial
logit and probit models. Logistic regression and probit models are used when the
dependent variable is binary.
Logistic regression
In a classification setting, assigning outcome probabilities to observations can be
achieved through the use of a logistic model, which is basically a method which
transforms information about the binary dependent variable into an unbounded
continuous variable and estimates a regular multivariate model (See Allison's Logistic
Regression for more information on the theory of logistic regression).
The Wald and likelihood-ratio test are used to test the statistical significance of each
coefficient b in the model (analogous to the t tests used in OLS regression; see above).
A test assessing the goodness-of-fit of a classification model is the "percentage
correctly predicted".
Multinomial Logistic Regression
An extension of the binary logit model to cases where the dependent variable has more
than 2 categories is the multinomial logit model. In such cases collapsing the data into
two categories might not make good sense or may lead to loss in the richness of the
data. The multinomial logit model is the appropriate technique in these cases, especially
when the dependent variable categories are not ordered (for examples colors like red,
blue, green). Some authors have extended multinomial regression to include feature
selection/importance methods such as random multinomial logit.
Probit Regression
Probit models offer an alternative to logistic regression for modeling categorical
dependent variables. Even though the outcomes tend to be similar, the underlying
distributions are different. Probit models are popular in social sciences like economics.
A good way to understand the key difference between probit and logit models is to
assume that the dependent variable is driven by a latent variable z, which is a sum of a
linear combination of explanatory variables and a random noise term.
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