Logit

Logit

Logit-probit

Logit refers to a function used in statistics and mathematics to model odds and probabilities in logistic regression and other forms of mathematical modeling. The logit function is the logarithm of the odds ratio, or in other words, the log-odds. It plays a crucial role in various statistical methods, especially in the field of machine learning and data analysis.

Definition

The logit function, denoted as logit(p), is defined for a probability p (where 0 < p < 1) as:

logit(p) = log(\frac{p}{1-p})

where log denotes the natural logarithm. The function maps probabilities from (0, 1) to (-∞, +∞), making it particularly useful for transforming the bounded probability scale into an unbounded scale that can be modeled more easily with linear techniques.

Applications

The logit function is primarily used in logistic regression, a statistical method for analyzing datasets in which there are one or more independent variables that determine an outcome. The outcome is measured with a dichotomous variable (where there are only two possible outcomes). In logistic regression, the logit of the probability of the outcome is modeled as a linear combination of the independent variables.

Logistic regression is widely used in various fields such as medicine, social sciences, engineering, and marketing to predict the likelihood of events, such as disease occurrence, consumer purchase behavior, or machine failure, based on relevant predictors.

Inverse Logit

The inverse of the logit function is known as the logistic function or the expit function. It is defined as:

expit(x) = \frac{1}{1 + e^{-x}}

where e is the base of the natural logarithm. The logistic function maps values from (-∞, +∞) back to (0, 1), making it useful for converting the log-odds output by logistic regression models back into probabilities.

Relation to Other Functions

The logit function is closely related to other functions used in statistics, such as the probit function, which is used in probit regression. While the logit function uses the logistic distribution to model probabilities, the probit function uses the normal distribution. Both functions serve similar purposes but make different assumptions about the distribution of the error terms in the model.

See Also

• Logistic regression
• Probit model
• Odds ratio
• Linear regression
• Statistical model