What type of regression is used when the response variable is categorical?

Logistic regression is specifically designed for situations where the response variable is categorical, which means it can take on a limited, fixed number of possible outcomes. This type of regression is particularly useful in binary classifications, where the outcome can be one of two categories, like 'yes' or 'no', 'success' or 'failure', etc. Logistic regression uses a logistic function to model the probability of the default class, providing a direct way to estimate the likelihood of the categorical response occurring based on predictor variables.

In contrast, linear regression is suited for continuous response variables and assumes a linear relationship between the input variables and continuous outcomes. Multi-way regression is not a standard term but may refer to multiple regression techniques that could involve multiple predictor variables; however, it still typically pertains to continuous outcomes. Variance does not relate to regression types; rather, it is a statistical measure of the dispersion among data points. Therefore, logistic regression is the appropriate choice when dealing with categorical responses.