"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.
References:
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
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