Editing Logit (section)

== Comparison with probit ==
[[File:Logit-probit.svg|right|300px|thumb|Comparison of the logit function with a scaled [[probit]] (i.e. the inverse [[cumulative distribution function|CDF]] of the [[normal distribution]]), comparing <math>\operatorname{logit}(x)</math> vs. <math>\tfrac{\Phi^{-1}(x)}{\,\sqrt{\pi/8\,}\,}</math>, which makes the slopes the same at the {{mvar|y}}-origin.]]

Closely related to the {{math|logit}} function (and [[logit model]]) are the [[probit function]] and [[probit model]]. The {{math|logit}} and {{math|probit}} are both [[sigmoid function]]s with a domain between 0 and 1, which makes them both [[quantile function]]s – i.e., inverses of the [[cumulative distribution function]] (CDF) of a [[probability distribution]]. In fact, the {{math|logit}} is the [[quantile function]] of the [[logistic distribution]], while the {{math|probit}} is the quantile function of the [[normal distribution]]. The {{math|probit}} function is denoted <math>\Phi^{-1}(x)</math>, where <math>\Phi(x)</math> is the [[cumulative distribution function|CDF]] of the standard normal distribution, as just mentioned:
: <math>\Phi(x) = \frac 1 {\sqrt{2\pi}}\int_{-\infty}^x  e^{-y^2/2} dy.</math>

As shown in the graph on the right, the {{math|logit}} and {{math|probit}} functions are extremely similar when the {{math|probit}} function is scaled, so that its slope at {{math|''y'' {{=}} 0}} matches the slope of the {{math|logit}}. As a result, [[probit model]]s are sometimes used in place of [[logit model]]s because for certain applications (e.g., in [[item response theory]]) the implementation is easier.<ref>{{cite book |first=James H. |last=Albert |chapter=Logit, Probit, and other Response Functions |title=Handbook of Item Response Theory |volume=Two |location= |publisher=Chapman and Hall |year=2016 |isbn= 978-1-315-37364-5|pages=3–22 |doi=10.1201/b19166-1 |url=https://books.google.de/books?id=NWymCwAAQBAJ&pg=PA3 }}</ref>