Data Mining for Business Analytics by Galit Shmueli & Peter C. Bruce & Peter Gedeck & Nitin R. Patel

Author:Galit Shmueli & Peter C. Bruce & Peter Gedeck & Nitin R. Patel
Language: eng
Format: epub
ISBN: 9781119549864
Publisher: Wiley
Published: 2019-10-17T00:00:00+00:00

10.5 Logistic Regression for Multi-class Classification

The logistic model for a binary outcome can be extended for more than two classes. Suppose that there are m classes. Using a logistic regression model, for each record we would have m probabilities of belonging to each of the m classes. Since the m probabilities must add up to 1, we need estimate only m − 1 probabilities.

Ordinal Classes

Ordinal classes are classes that have a meaningful order. For example, in stock recommendations, the three classes buy, hold, and sell can be treated as ordered. As a simple rule, if classes can be numbered in a meaningful way, we consider them ordinal. When the number of classes is large (typically, more than 5), we can treat the outcome variable as continuous and perform multiple linear regression. When m = 2, the logistic model described above is used. We therefore need an extension of the logistic regression for a small number of ordinal classes (3 ⩽ m ⩽ 5). There are several ways to extend the binary-class case. Here, we describe the proportional odds or cumulative logit method. For other methods, see Hosmer and Lemeshow (2000).

For simplicity of interpretation and computation, we look at cumulative probabilities of class membership. For example, in the stock recommendations, we have m = 3 classes. Let us denote them by 1 = buy, 2 = hold, and 3 = sell. The probabilities estimated by the model are P(Y ⩽ 1), (the probability of a buy recommendation) and P(Y ⩽ 2) (the probability of a buy or hold recommendation). The three noncumulative probabilities of class membership can easily be recovered from the two cumulative probabilities:

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