Definition: Logistic regression
Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Categorical data ▷ Logistic regression ▷ Definition
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Metadata: ID: D76  shortcut: logreg  author: JoramSoch  date: 20200628, 20:51.
Definition: A logistic regression model is given by a set of binary observations $y_i \in \left\lbrace 0, 1 \right\rbrace, i = 1,\ldots,n$, a set of predictors $x_j \in \mathbb{R}^n, j = 1,\ldots,p$, a base $b$ and the assumption that the logodds are a linear combination of the predictors:
\[\label{eq:logreg} l_i = x_i \beta + \varepsilon_i, \; i = 1,\ldots,n\]where $l_i$ are the logodds that $y_i = 1$
\[\label{eq:logodds} l_i = \log_b \frac{\mathrm{Pr}(y_i = 1)}{\mathrm{Pr}(y_i = 0)}\]and $x_i$ is the $i$th row of the $n \times p$ matrix
\[\label{eq:X} X = \left[ x_1, \ldots, x_p \right] \; .\]Within this model,

$y$ are called “categorical observations” or “dependent variable”;

$X$ is called “design matrix” or “set of independent variables”;

$\beta$ are called “regression coefficients” or “weights”;

$\varepsilon_i$ is called “noise” or “error term”;

$n$ is the number of observations;

$p$ is the number of predictors.
 Wikipedia (2020): "Logistic regression"; in: Wikipedia, the free encyclopedia, retrieved on 20200628; URL: https://en.wikipedia.org/wiki/Logistic_regression#Logistic_model.
Metadata: ID: D76  shortcut: logreg  author: JoramSoch  date: 20200628, 20:51.