1.6.6: Illustration- Insurance in Ruritania

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The decision to buy life insurance is related to several variables, including age and income. We would like to explore this relationship in Ruritania.

Since the dependent variable is dichotomous (life insurance purchased or life insurance not purchased), we need a new type of regression to ensure that our predictions make sense. One option is called logistic regression. Using this regression, we find significant positive relationships between the person's age and the likelihood to buy life insurance, as well as between the person's income and the likelihood to buy life insurance.

Additionally, we predict that the Knox graduate on the Council of Ministers, a 65-year-old making $125,000 annually, has a 74% chance of having life insurance.

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In this example, we had to use a different type of regression because the dependent variable was dichotomous, could only take on two values. This type of regression is known collectively as logistic regression, even though the link function can be almost any that map the real line to the interval $(0,1)$. Such functions include the venerable logit function (inverse of the logistic function). It also includes the probit (used in a lot of medical studies) and the cauchit (used in some financial studies to allow for highly variable events).

It was this class of problems that forced Nelder and associates to formulate an over-arching framework for regression. He called it the "generalized linear model" (GLM). While he rues the name to this day, he created it to signify that this class of regression problems is actually just a generalization of the class that can be solved using ordinary least squares regression. In other words, OLS regression is a special case of GLM regression.

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