5 Binary Logistic Regression

Binary logistic regression models the probability that a characteristic is present (i.e., “success”), given the values of explanatory variables \(x_{1}, \ldots, x_{k}\). We denote this by \(\pi\left(x_{1}, \ldots, x_{k}\right)=P\left(\right.\) success \(\left.\mid x_{1}, \ldots, x_{k}\right)\) or simply by \(\pi\) for convenience–but it should be understood that \(\pi\) will in general depend on one or more explanatory variables. For example, a physician may be interested in estimating the proportion of diabetic persons in a population. Naturally, she knows that all sections of the population do not have an equal probability of “success”, i.e., being diabetic. Certain conditions, such as advanced age and hypertension, would likely contribute to a higher proportion.