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Types of Regression Models

For continuous variables, linear regression models can be used. Sometimes the outcome scale is transformed, by taking the logarithm, for example, to achieve a better compliance with model assumptions.

For dichotomous variables, logistic regression models are popular (Hosmer and Lemeshow, 1989). They have largely replaced discriminant models in medical applications. A logistic link function is used in the regression formula, such that the regression formula can be written as:

Logit[Prob(outcome)] = a + b₁x₁ + b₂x₂ + … + b_ix_i.

Here, a is the intercept, b1 to bi are regression coefficients for i covariables x₁ to x_i, similar to other regression models. The logit indicates the natural logarithm of the odds of the probability p that the outcome occurs: log(p/(1-p)). Odds ratios can be calculated by exponentiating the coefficients: OR=exp(b_i). The relationship between the probability of the outcome and the logit is a characteristic curve.

Figure 4.1: Logistic Link Function Illustration of the logistic link function. The relationship between the probability of an outcome and the logit of the probability is a characteristic curve. The logit is calculated as: ln(probability/(1-probability)). When the logit is 0, the probability is 50%.