Logistic Regression

A particular combination of values in $x_{1{j}},\ldots,x_{k{j}}$ produces binary responses of either "Yes" or "No" ("1" or "0") for

different combinations, where the variables for $x_{ij}$ 's are called explanatory variables or "predictors." For each combination, the responses are counted as $N_{1j}$ and $N_{0j}$ , and summarized in the following table.

Predictor	Yes count	No count
$x_{11},\ldots,x_{k1}$	$N_{11}$	$N_{01}$
$\vdots$	$\vdots$
$x_{1n},\ldots,x_{kn}$	$N_{1n}$	$N_{0n}$

The regression model

$\displaystyle \mathrm{logit}(p_{j}) = \beta_0 + \beta_1 x_{1j} + \cdots + \beta_k x_{kj}$

is called a logistic regression, where

$\displaystyle \mathrm{logit}(s) = \log\left(\frac{s}{1-s}\right)$

called the logit function. Here $p_j$

represents the probability (or the proportion) of "Yes" in the following table.

Predictor	Probability of Yes
$x_{11},\ldots,x_{k1}$	$p_{1}$
$\vdots$	$\vdots$
$x_{1n},\ldots,x_{kn}$	$p_{n}$