Editing Logit (section)

== Definition ==
If {{mvar|p}} is a [[probability]], then {{math| {{nowrap|''p''/(1 &minus; ''p'')}} }} is the corresponding [[odds]]; the {{math|logit}} of the probability is the logarithm of the odds, i.e.:
: <math>\operatorname{logit}(p)=\ln\left( \frac{p}{1-p} \right) =\ln(p)-\ln(1-p)=-\ln\left( \frac{1}{p}-1\right)=2\operatorname{atanh}(2p-1). </math>

The base of the [[logarithm]] function used is of little importance in the present article, as long as it is greater than 1, but the [[natural logarithm]] with base {{mvar|[[e (mathematical constant)|e]]}} is the one most often used. The choice of base corresponds to the choice of [[logarithmic unit]] for the value: base&nbsp;2 corresponds to a [[shannon (unit)|shannon]], base&nbsp;{{mvar|e}} to a [[nat (unit)|nat]], and base&nbsp;10 to a [[hartley (unit)|hartley]]; these units are particularly used in information-theoretic interpretations. For each choice of base, the logit function takes values between negative and positive infinity.

The [[logistic function|“logistic” function]] of any number <math>\alpha</math> is given by the inverse-{{math|logit}}:
: <math>\operatorname{logit}^{-1}(\alpha) = \operatorname{logistic}(\alpha) = \frac{1}{1 + \exp(-\alpha)} = \frac{\exp(\alpha)}{ \exp(\alpha) + 1} = \frac{\tanh(\frac{\alpha}{2})+1}{2}</math>

The difference between the {{math|logit}}s of two probabilities is the logarithm of the [[odds ratio]] ({{mvar|R}}), thus providing a shorthand for writing the correct combination of odds ratios [[additive function|only by adding and subtracting]]:
: <math>\ln(R)=\ln\left( \frac{p_1/(1-p_1)}{p_2/(1-p_2)} \right) =\ln\left( \frac{p_1}{1-p_1} \right) - \ln\left(\frac{p_2}{1-p_2}\right) = \operatorname{logit}(p_1)-\operatorname{logit}(p_2)\,.</math>

The [[Taylor series]] for the logit function is given by:
:<math>\operatorname{logit}(x)=2\sum_{n=0}^\infty \frac{(2x-1)^{2n+1}}{2n+1}.</math>