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Refers generally to statistical procedures that utilize the logistic function, most commonly various forms of logistic regression

The logistic function is $$ f(x) = \frac{1}{1+e^{-x}}, $$ which maps real numbers to $(0,1)$. One common use of the logistic function is logistic regression, which is a standard method of quantifying the effect of a set of predictors $\{X_1, ..., X_p\}$ on a binary outcome, $Y$. The model can be written as

$$ P(Y=1|X) = f(\beta_0 + \beta_1X_1 + ... + \beta_p X_p) = \frac{1}{1 + e^{-(\beta_0 + \beta_1X_1 + ... + \beta_p X_p)}}$$

The logistic regression model has the nice property that the exponentiated regression coefficients can be interpreted as odds ratios associated with a one unit increase in the predictor.

Often we consider the odds in favor of $Y=1$ given $X$:

$$\text{odds} = \frac{P(Y=1|X)}{P(Y=0|X)} = \frac{P(Y=1|X)}{1 - P(Y=1|X)} = e^{\beta_0 + \beta_1X_1 + ... + \beta_p X_p}$$

The odds ratio associated with a one unit increase in some predictor, $X_i$, is therefore written as:

$$\frac{\text{odds}(x_i+1)}{\text{odds}(x_i)} = \frac{e^{\beta_0 + \beta_1X_1 + ...+ \beta_i(X_i+1) + ... + \beta_p X_p}}{e^{\beta_0 + \beta_1X_1 + ...+ \beta_iX_i + ... + \beta_p X_p}} = e^{\beta_i}$$

A second use of the logistic function (but unrelated to logistic regression) is the logistic distribution, which has $f(x)$ as its quantile function.