Editing Logistic regression (section)

==History==
A detailed history of the logistic regression is given in {{harvtxt|Cramer|2002}}. The logistic function was developed as a model of [[population growth]] and named "logistic" by [[Pierre François Verhulst]] in the 1830s and 1840s, under the guidance of [[Adolphe Quetelet]]; see {{slink|Logistic function|History}} for details.{{sfn|Cramer|2002|pp=3–5}} In his earliest paper (1838), Verhulst did not specify how he fit the curves to the data.<ref>{{cite journal|first= Pierre-François |last=Verhulst |year= 1838| title = Notice sur la loi que la population poursuit dans son accroissement | journal = Correspondance Mathématique et Physique |volume = 10| pages = 113–121
|url = https://books.google.com/books?id=8GsEAAAAYAAJ | format = PDF| access-date = 3 December 2014}}</ref><ref>{{harvnb|Cramer|2002|p=4|ps=, "He did not say how he fitted the curves."}}</ref> In his more detailed paper (1845), Verhulst determined the three parameters of the model by making the curve pass through three observed points, which yielded poor predictions.<ref>{{cite journal|first= Pierre-François |last=Verhulst |year= 1845| title = Recherches mathématiques sur la loi d'accroissement de la population | journal = Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles |volume = 18 | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN129323640_0018&DMDID=dmdlog7| access-date = 2013-02-18|trans-title= Mathematical Researches into the Law of Population Growth Increase}}</ref>{{sfn|Cramer|2002|p=4}}

The logistic function was independently developed in chemistry as a model of [[autocatalysis]] ([[Wilhelm Ostwald]], 1883).{{sfn|Cramer|2002|p=7}} An autocatalytic reaction is one in which one of the products is itself a [[catalyst]] for the same reaction, while the supply of one of the reactants is fixed. This naturally gives rise to the logistic equation for the same reason as population growth: the reaction is self-reinforcing but constrained.

The logistic function was independently rediscovered as a model of population growth in 1920 by [[Raymond Pearl]] and [[Lowell Reed]], published as {{harvtxt|Pearl|Reed|1920}}, which led to its use in modern statistics. They were initially unaware of Verhulst's work and presumably learned about it from [[L. Gustave du Pasquier]], but they gave him little credit and did not adopt his terminology.{{sfn|Cramer|2002|p=6}} Verhulst's priority was acknowledged and the term "logistic" revived by [[Udny Yule]] in 1925 and has been followed since.{{sfn|Cramer|2002|p=6–7}} Pearl and Reed first applied the model to the population of the United States, and also initially fitted the curve by making it pass through three points; as with Verhulst, this again yielded poor results.{{sfn|Cramer|2002|p=5}}

In the 1930s, the [[probit model]] was developed and systematized by [[Chester Ittner Bliss]], who coined the term "probit" in {{harvtxt|Bliss|1934}}, and by [[John Gaddum]] in {{harvtxt|Gaddum|1933}}, and the model fit by [[maximum likelihood estimation]] by [[Ronald A. Fisher]] in {{harvtxt|Fisher|1935}}, as an addendum to Bliss's work. The probit model was principally used in [[bioassay]], and had been preceded by earlier work dating to 1860; see {{slink|Probit model|History}}. The probit model influenced the subsequent development of the logit model and these models competed with each other.{{sfn|Cramer|2002|p=7–9}}

The logistic model was likely first used as an alternative to the probit model in bioassay by [[Edwin Bidwell Wilson]] and his student [[Jane Worcester]] in {{harvtxt|Wilson|Worcester|1943}}.{{sfn|Cramer|2002|p=9}} However, the development of the logistic model as a general alternative to the probit model was principally due to the work of [[Joseph Berkson]] over many decades, beginning in {{harvtxt|Berkson|1944}}, where he coined "logit", by analogy with "probit", and continuing through {{harvtxt|Berkson|1951}} and following years.<ref>{{harvnb|Cramer|2002|p=8|ps=, "As far as I can see the introduction of the logistics as an alternative to the normal probability function is the work of a single person, Joseph Berkson (1899–1982), ..."}}</ref> The logit model was initially dismissed as inferior to the probit model, but "gradually achieved an equal footing with the probit",{{sfn|Cramer|2002|p=11}} particularly between 1960 and 1970. By 1970, the logit model achieved parity with the probit model in use in statistics journals and thereafter surpassed it. This relative popularity was due to the adoption of the logit outside of bioassay, rather than displacing the probit within bioassay, and its informal use in practice; the logit's popularity is credited to the logit model's computational simplicity, mathematical properties, and generality, allowing its use in varied fields.{{sfn|Cramer|2002|p=10–11}}

Various refinements occurred during that time, notably by [[David Cox (statistician)|David Cox]], as in {{harvtxt|Cox|1958}}.<ref name=wal67est>{{cite journal|last1=Walker|first1=SH|last2=Duncan|first2=DB|title=Estimation of the probability of an event as a function of several independent variables|journal=Biometrika|date=1967|volume=54|issue=1/2|pages=167–178|doi=10.2307/2333860|jstor=2333860}}</ref>

The multinomial logit model was introduced independently in {{harvtxt|Cox|1966}} and {{harvtxt|Theil|1969}}, which greatly increased the scope of application and the popularity of the logit model.{{sfn|Cramer|2002|p=13}} In 1973 [[Daniel McFadden]] linked the multinomial logit to the theory of [[discrete choice]], specifically [[Luce's choice axiom]], showing that the multinomial logit followed from the assumption of [[independence of irrelevant alternatives]] and interpreting odds of alternatives as relative preferences;<ref>{{cite book
|chapter=Conditional Logit Analysis of Qualitative Choice Behavior
|chapter-url=https://eml.berkeley.edu/reprints/mcfadden/zarembka.pdf
|archive-url=https://web.archive.org/web/20181127110612/https://eml.berkeley.edu/reprints/mcfadden/zarembka.pdf
|archive-date=2018-11-27
|access-date=2019-04-20
|first=Daniel |last=McFadden
|author-link=Daniel McFadden
|editor=P. Zarembka
|title=Frontiers in Econometrics
|pages=105–142
|publisher=Academic Press
|location=New York
|year=1973
}}</ref> this gave a theoretical foundation for the logistic regression.{{sfn|Cramer|2002|p=13}}