Logistic Regression Model

The logit model, also known as logistic regression, is used to model the probability of an event occurring (for example, \( Y=1 \)) as a function of explanatory variables \( X \). Its main equation is defined as:

$$ P(Y=1 \mid X) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 X_1 + \cdots + \beta_k X_k)}} $$

To better understand the model, we introduce the concept of odds, defined as the ratio between the probability that the event occurs and the probability that it does not occur:

$$ \text{odds} = \frac{P(Y=1 \mid X)}{1 - P(Y=1 \mid X)} $$

By using the logit function (or log-odds), we can linearize the problem as follows:

$$ \ln \left(\frac{P(Y=1 \mid X)}{1 - P(Y=1 \mid X)}\right) = \beta_0 + \beta_1 X_1 + \cdots + \beta_k X_k $$

In this way, logistic regression allows us to estimate the parameters \( \beta_0, \beta_1, \dots, \beta_k \) that best describe the relationship between the explanatory variables and the probability of the event occurring.