"Over the last 150 years or so, considerable research has been carried out regarding the theory, methodology, and applications of the logistic distribution. We have made a sincere effort in this volume to consolidate most of these contributions and simultaneously present new developments in this area" announces editor N. Balakrishnan in the Preface to the 601 page "Handbook of the Logistic Distribution" (New York: Marcel Dekker, 1992).
The handbook skims over the early history of the logistic distribution, which can be found in more detail in RMT 6:4 p. 260-1. The text is then chiefly concerned with the mathematical and statistical properties of the function as applied to linear observations.
Some interesting reformulations of the logistic cumulative distribution function, cdf, are presented. (The cdf is never referred to as the logistic ogive.)
logit(x) = exp(x)/(1+exp(x)) = 1/(1+exp(-x)) = 0.5(1+tanh(0.5x))
This function also approximates the cdf for Student's t distribution with 9 degrees of freedom.
The "standardized" logistic cdf is
stlogit(y) = 1/(1+exp(-pi.y/sqrt(3))
A population distributed according to this function would have mean 0, and standard deviation 1. This curve closely approximates the unit normal ogive. The biggest discrepancy is .023 at y=0.7. An even closer approximation, with no discrepancy greater than .01, is given by
stlogit(y) = 1/(1+exp(-1.70y))
This equivalence expedites Cohen's (1979) PROX estimation algorithm, which contains the coefficient 1.70**2 = 2.89.
Chapter 13, by D'Agostino and Massaro, is devoted to "Goodness-of-fit Tests". They recommend graphical techniques, particularly plotting the empirical distribution function, known to us as the item characteristic curve, and the logistic ogive on the same axes. For an example see RMT 6:2 p. 209-210.
Just one section, 18.1, p. 495-512, discusses "Applications to Ordered Categorical Data", but even this is limited to the analysis of two-way contingency tables by partitioning the logistic distribution (Symanowksi & Koehler, 1989) or by using logistic regression (Agresti, 1984, Cox and Snell, 1989).
Conspicuous by their absence from this compendium are Rasch and IRT methodologies. The only indication of these is in the sentence: "The logistic function has also been used in studies of physiochemical phenomenon ..., geological studies ..., and psychological studies by Birnbaum and Dudman (1963), Lord (1965), Sanathanan (1974), and Formann (1982)" (p. 475). This brings to our attention, yet again, the great divide between descriptive statistics and measurement.
Agresti A. 1984. Analysis of Ordinal Categorical Data. New York: Wiley
Birnbaum A & Dudman J. 1963. Logistic order statistics. Ann. Math. Statist. 34, 659-663.
Cohen, L. 1979. Approximate expressions for parameter estimates in the Rasch model. British Journal of Mathematical and Statistical Psychology 32: 113-120.
Cox DR & Snell EJ. 1989. Analysis of Binary Data. 2nd Ed. London: Chapman and Hall
Formann AK. 1982. Linear logistic latent class analysis. Biomet. J., 24, 171-190.
Lord FM. 1965. A note on the normal ogive or logistic curve in item analysis. Psychometrika 30, 371-372.
Sanathanan L. 1974. Some properties of the logistic model for dichotomous response. J. Amer. Statist. Assoc. 69, 744-749.
Symanowksi JT & Koehler KJ. 1989. A bivariate logistic distribution with applications to categorical responses. Tech. Report No. 89-29. Department of Statistics, Iowa State University, Ames, Iowa.
Handbook of the logistic distribution. Balakrishnan N. Rasch Measurement Transactions, 1994, 8:2 p.367
|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|
|Aug. 14 - 16, 2019. Wed.-Fri.||An Introduction to Rasch Measurement: Theory and Applications (workshop led by Richard M. Smith) https://www.hkr.se/pmhealth2019rs|
|August 25-30, 2019, Sun.-Fri.||Pacific Rim Objective Measurement Society (PROMS) 2019, Surabaya, Indonesia https://proms.promsociety.org/2019/|
|Oct. 11 - Nov. 8, 2019, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|Nov. 3 - Nov. 4, 2019, Sun.-Mon.||International Outcome Measurement Conference, Chicago, IL,http://jampress.org/iomc2019.htm|
|Jan. 24 - Feb. 21, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|May 22 - June 19, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 26 - July 24, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
|Aug. 7 - Sept. 4, 2020, Fri.-Fri.||On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com|
|Oct. 9 - Nov. 6, 2020, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 25 - July 23, 2021, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
The URL of this page is www.rasch.org/rmt/rmt82n.htm