LogOdds

Logodds, often called the logit, is the natural logarithm of the odds of an event. If p is the probability of the event, the odds are p/(1−p). The log-odds is log(p/(1−p)), where log denotes the natural logarithm. This transform maps probabilities from the interval (0,1) to the real numbers (−∞, ∞) and is monotonic, turning multiplicative changes in odds into additive changes in log-odds.

The inverse transformation is the logistic function: p = 1 / (1 + e^{−logodds}). This relationship underpins the logistic model, a common tool for binary outcomes.

In logistic regression, the log-odds of the probability of the positive outcome is modeled as a linear combination of predictor variables: log(p/(1−p)) = β0 + β1 x1 + ... + βk xk. The regression coefficients represent the change in log-odds per unit change in the corresponding predictor. Exponentiating a coefficient yields the odds ratio: exp(βj) is the multiplicative change in odds for a one-unit increase in xj.

Because p cannot be exactly 0 or 1, log-odds are undefined at those extremes. In practice, probabilities may be bounded away from 0 and 1 or a small adjustment is used. Log-odds are widely used in statistics, epidemiology, and machine learning as a convenient link function and a way to interpret binary outcomes.