Logistic regression (logit model) is a generalized linear model used for binomial regression. Logistic regression applies maximum likelihood estimation after transforming the dependent into a logit variable. A logit is the natural log of the odds of the dependent equaling a certain value or not (usually 1 in binary logistic models, the highest value in multinomial models). In this way, logistic regression estimates the odds of a certain event (value) occurring.
Logistic regression has many analogies to ordinary least squares (OLS) regression. Unlike OLS regression, however, logistic regression does not assume linearity of relationship between the raw values of the independent variables and the dependent, does not require normally distributed variables, does not assume homoscedasticity, and in general has less stringent requirements.
Compared with linear discriminant analysis, logistic regression has several advantages:
- It is more robust: the independent variables don't have to be normally distributed, or have equal variance in each group
- It does not assume a linear relationship between the independent variables and dependent variable.
- It may handle nonlinear effects since one can add explicit interaction and power terms.
However, it requires much more data to achieve stable, meaningful results.
Logistic regression also has strong connections with neural network and maximum entropy modeling. For example, binary logistic regression is equivalent to a one-layer, single-output neural network with a logistic activation function trained under log loss. Similarly, multinomial logistic regression is equivalent to a one-layer, softmax-output neural network.
Logistic regression estimation also obeys the maximum entropy principle, and thus logistic regression is sometimes called "maximum entropy modeling", and the resulting classifier the "maximum entropy classifier".