linguistic:voice_change, structural:summarization, information:extraction, stylistic:tone
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 input t {\displaystyle t}, and outputs a value between zero and one. For the logit, this is interpreted as taking input log-odds and having output probability. The standard logistic function σ: R → (0, 1) {\displaystyle \sigma :\mathbb {R} \rightarrow (0,1)} is defined as follows:

A graph of the logistic function on the t -interval (−6,6) is shown in Figure 1.

Let us assume that t {\displaystyle t} is a linear function of a single explanatory variable x {\displaystyle x} (the case where t {\displaystyle t} is a linear combination of multiple explanatory variables is treated similarly). We can then express t {\displaystyle t} as follows:

And the general logistic function p: R → (0, 1) {\displaystyle p:\mathbb {R} \rightarrow (0,1)} can now be written as:

In the logistic model, p (x) {\displaystyle p(x)} is interpreted as the probability of the dependent variable Y {\displaystyle Y} equaling a success/case rather than a failure/non-case. It is clear that the response variables Y i {\displaystyle Y_{i}} are not identically distributed: P (Y i = 1 ∣ X) {\displaystyle P(Y_{i}=1\mid X)} differs from one data point X i {\displaystyle X_{i}} to another, though they are independent given design matrix X {\displaystyle X} and shared parameters β {\displaystyle \beta }.