We call our S-shaped response curve a logistic ogive. How did this term originate?
S-shaped curves are also called sigmoid curves or sigmoidal functions, from the Greek "s", sigma. In his "Mathematical Researches into the Law of Population Growth Increase" (Nouveaux Memoires de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles, 18, Art. 1, 1-45, 1845), Pierre Francois Verhulst (1804-1849), Professor of Analysis at the Belgian Military College, examines population growth in Belgium. He discovers that sigmoid curves are useful for describing population growth. Following Malthus, Verhulst hypothesizes that small populations increase geometrically, because the supply of resources exceeds demand. Then, as supply and demand balance, population growth is constant. Finally, as demand exceeds supply, population growth decreases at the same rate that it had increased. Verhulst describes this process with an equation that enables him to predict when a population will reach any given size (see Verhulst's Figure):
t = log10( p/ ( m/n - p ) ) / m
where, for Belgium, with 1830 population of 4,247,113,
t is the time in years, since the mid-point of the curve (computed to
be in 1807)
p is the population count
1/m is a population growth coefficient of 87.885
m/n is the projected maximum population of 6,583,700.
Verhulst writes "We will give the name logistic [logistique] to the curve" (1845 p.8). Though he does not explain this choice, there is a connection with the logarithmic basis of the function. Logarithm was coined by John Napier (1550-1617) from Greek logos (ratio, proportion, reckoning) and arithmos (number). Logistic comes from the Greek logistikos (computational). In the 1700's, logarithmic and logistic were synonymous. Since computation is needed to predict the supplies an army requires, logistics has come to be also used for the movement and supply of troops.
Chemists, in their own work, discovered that the sigmoid curve describes an autocatalytic reaction. In autocatalysis, a product of the chemical reaction increases the speed of the reaction. But the reaction ceases when there is no chemical left to react. Consequently chemists called their sigmoid curve, the autocatalytic curve.
Raymond Pearl (1879-1940) renewed Verhulst's terminology. His "Studies in Human Biology" (Baltimore: Williams & Wilkins, 1924) review Verhulst's population growth work. In later work, Pearl exhorts researchers to use logistic curve or function in preference to autocatalytic curve because the latter is tied to a physical process. Pearl's student, Joseph Berkson, popularized the term logistic in his application of the sigmoid curve to bio-assay quantal response.
Berkson coined logit (pronouced "low-jit") in 1944 as a contraction of logistic unit to indicate the unit of measurement (J. Amer. Stat. Soc. 39:357-365). Georg Rasch derives logit as a contraction of logistic transform (1980, p.80). Ben Wright derives logit as a contraction of log-odds unit. Logit is also used to characterize the logistic function in the way that probit (probability unit, coined by Chester Bliss about 1934), characterizes the cumulative normal function.
Ogive (pronounced oh-jive) indicates the shape of the logistic curve. An ogive is an architectural shape - a pointed arch. Modern mathematical use of ogive began in an 1875 paper by Francis Galton (Phil. Mag. 49:33-46). Galton's curve is of finite length and comes to a point (see Galton's Figure). Galton's curve has been idealized and its axes rotated to produce smooth shapes like Verhulst's Figure. The name ogive has been retained, even though the curve now resembles the architectural ogee (pronounced oh-jee).
The combination, logistic ogive, to identify the logistic curve is recent. George Udny Yule discusses the logistic curve at length in his 1924 Presidential Address to the Royal Statistical Society (J. Roy. Stat. Soc. 88:1-62, 1925). This address includes tables of the numerical values of the logistic function, and suggests three methods for estimating the parameters of a logistic curve. But Yule does not use the term ogive. W.D. Ashton also fails to mention ogive in his 1972 monograph on the logistic curve, entitled "The Logit Transformation" (London: Griffin). Perhaps it is Rasch's student, Erling Andersen, who introduces the use, logistic ogive (Psychometrika 42:69-81, 1977).
Why logistic ogive and not autocatalytic curve?, J Linacre … Rasch Measurement Transactions, 1993, 6:4 p. 260-1
Rasch Publications | ||||
---|---|---|---|---|
Rasch Measurement Transactions (free, online) | Rasch Measurement research papers (free, online) | Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch | Applying the Rasch Model 3rd. Ed., Bond & Fox | Best Test Design, Wright & Stone |
Rating Scale Analysis, Wright & Masters | Introduction to Rasch Measurement, E. Smith & R. Smith | Introduction to Many-Facet Rasch Measurement, Thomas Eckes | Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. | Statistical Analyses for Language Testers, Rita Green |
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar | Journal of Applied Measurement | Rasch models for measurement, David Andrich | Constructing Measures, Mark Wilson | Rasch Analysis in the Human Sciences, Boone, Stave, Yale |
in Spanish: | Análisis de Rasch para todos, Agustín Tristán | Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez |
Forum | Rasch Measurement Forum to discuss any Rasch-related topic |
Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement
Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.
Coming Rasch-related Events | |
---|---|
Dec. 6-8, 2017, Wed.-Fri. | In-person workshop: Introductory Rasch Analysis using RUMM2030, Leeds, UK (M. Horton), Announcement |
Jan. 5 - Feb. 2, 2018, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
Jan. 10-16, 2018, Wed.-Tues. | In-person workshop: Advanced Course in Rasch Measurement Theory and the application of RUMM2030, Perth, Australia (D. Andrich), Announcement |
Jan. 17-19, 2018, Wed.-Fri. | Rasch Conference: Seventh International Conference on Probabilistic Models for Measurement, Matilda Bay Club, Perth, Australia, Website |
Jan. 22-24, 2018, Mon-Wed. | In-person workshop: Rasch Measurement for Everybody en español (A. Tristan, Winsteps), San Luis Potosi, Mexico. www.ieia.com.mx |
April 10-12, 2018, Tues.-Thurs. | Rasch Conference: IOMW, New York, NY, www.iomw.org |
April 13-17, 2018, Fri.-Tues. | AERA, New York, NY, www.aera.net |
May 25 - June 22, 2018, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
June 29 - July 27, 2018, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com |
July 25 - July 27, 2018, Wed.-Fri. | Pacific-Rim Objective Measurement Symposium (PROMS), (Preconference workshops July 23-24, 2018) Fudan University, Shanghai, China "Applying Rasch Measurement in Language Assessment and across the Human Sciences" www.promsociety.org |
Aug. 10 - Sept. 7, 2018, Fri.-Fri. | On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com |
Oct. 12 - Nov. 9, 2018, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
The URL of this page is www.rasch.org/rmt/rmt64k.htm
Website: www.rasch.org/rmt/contents.htm