Form of Logit for Logistic Curve

Theorem

Let $p$ denote the probability of the occurrence of an event.

Let $p$ satisfy a logistic relationship with an explanatory variable $x$ of the form:

$p = \dfrac 1 {1 + \map \exp {-\paren {\alpha + \beta x} } }$

Let $Y$ be the logit of $p$.

Then:

$Y = \alpha + \beta x$

By definition, the logit of $p$ is given by:

$Y = \map \ln {\dfrac p {1 - p} }$

In order to simplify the algebra, let $c = -\paren {\alpha + \beta x}$.

Then we have:

$\ds Y$

$=$

$\ds \map \ln {\dfrac {\frac 1 {1 + \exp c} } {1 - \frac 1 {1 + \exp c} } }$

$\ds $

$=$

$\ds \map \ln {\dfrac {\frac 1 {1 + \exp c} } {\frac {1 + \exp c - 1} {1 + \exp c} } }$

putting everything in the lower fraction over a common denominator

$\ds $

$=$

$\ds \map \ln {\dfrac 1 {\exp c} }$

simplification, and multiplying top and bottom by $1 + \exp c$

$\ds $

$=$

$\ds -\map \ln {\exp c}$

$\ds $

$=$

$\ds -c$

Definition of Logarithm

$\ds $

$=$

$\ds \alpha + \beta x$

substituting for $c$ and simplifying

$\blacksquare$