Editing Logistic regression (section)

===Definition of the logistic function===
An explanation of logistic regression can begin with an explanation of the standard [[logistic function]]. The logistic function is a [[sigmoid function]], which takes any [[Real number|real]] input <math>t</math>, and outputs a value between zero and one.<ref name=Hosmer/> For the logit, this is interpreted as taking input [[log-odds]] and having output [[probability]]. The ''standard'' logistic function <math>\sigma:\mathbb R\rightarrow (0,1)</math> is defined as follows:

:<math>\sigma (t) = \frac{e^t}{e^t+1} = \frac{1}{1+e^{-t}}</math>

A graph of the logistic function on the ''t''-interval (−6,6) is shown in Figure 1.

Let us assume that <math>t</math> is a linear function of a single [[dependent and independent variables|explanatory variable]] <math>x</math> (the case where <math>t</math> is a ''linear combination'' of multiple explanatory variables is treated similarly). We can then express <math>t</math> as follows:

:<math>t = \beta_0 + \beta_1 x</math>

And the general logistic function <math>p:\mathbb R \rightarrow (0,1)</math> can now be written as:

:<math>p(x) = \sigma(t)= \frac {1}{1+e^{-(\beta_0 + \beta_1 x)}}</math>

In the logistic model, <math>p(x)</math> is interpreted as the probability of the dependent variable <math>Y</math> equaling a success/case rather than a failure/non-case. It is clear that the [[Dependent and independent variables|response variables]] <math>Y_i</math> are not identically distributed: <math>P(Y_i = 1\mid X)</math> differs from one data point <math>X_i</math> to another, though they are independent given [[design matrix]] <math>X</math> and shared parameters <math>\beta</math>.<ref name = "Freedman09" />