e-Statistics

Logistic Regression

A particular combination of values in $ x_{1i},\ldots,x_{ki}$ produces binary responses of either "yes" or "no" ("1" or "0") for $ n$ different combinations, where the variables for $ x_{ij}$'s are called explanatory variables or "predictor." For each combination, the responses are counted as $ N_{1i}$ and $ N_{0i}$, and summarized in the following table.

Predictor Response $ N_{1}$ Response $ N_{0}$
$ 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_i$ represents the probability (or the proportion) of "yes" response in the following table.

Predictor Proportion of "yes"
$ x_{11},\ldots,x_{k1}$ $ p_{1}$
$ \vdots$ $ \vdots$
$ x_{1n},\ldots,x_{kn}$ $ p_{n}$


© TTU Mathematics