Abstract
Brannigan andWanner argue that the empirical distribution of multiple grades can be more adequately explained in terms of a negative contagious poisson model. This alternative is based on a Zeitgeist theory which places emphasis on the role of communication in scientific discovery. Nonetheless, a detailed analysis indicates the following: (a) mathematically, the simple Poisson is the limiting case of the contagious Poisson when the contagion parameter approaches zero; (b) empirically, the mean and variance are so nearly equal (i. e., the contagion effect is very small) that predictions from the contagious Poisson are virtually equivalent to those of the simple Poisson; (c) in particular, both distributions predict that multiples are less common than singletons and even nulltons, the latter occurring with a probability of over one third (thereby implying that chance plays a much bigger part than Zeitgeist or maturational theories would suggest); (d) estimates from theSimonton, Merton, andOgburn-Thomas data sets all concur that the contagion effect is not only small, but positive besides, yielding a modest positive contagious Poisson that contradicts the principal tenet of the communication interpretation.
This is a preview of subscription content, access via your institution.
Notes and references
A. BRANNIGAN, R. A. WANNER, Historical distributions of multiple discoveries and theories of scientific change,Social Studies of Science, 13 (1983) 417.
D. DE SOLLA PRICE,Little Science, Big Science, Columbia University Press, New York, 1963
D. K. SIMONTON, Independent discovery in science and technology: A closer look at the Possion distribution,Social Studies of Science, 8 (1978) 521; D. K. SIMONTON, Multiple discovery and invention: Zeitgeist, genius, or chance?,Journal of Personality and Social Psychology, 37 (1979) 1603.
A. L. KROEBER, The superorganic,American Anthropologist, 19 (1917) 163.; L. T. WHITE,The Science of Culture, Farrar, Straus and Giroux, New York, 1969.
W. F. OGBURN, D. S. THOMAS, Are inventions inevitable? A note on social evolution,Political Science Quarterly, 37 (1922) 83.
D. K. SIMONTON (1978), op. cit. p. 523.
D. K. SIMONTON (1979), op. cit., fn. 1 p. 1606.
W. FELLER,An Introduction to Probability Theory and its Applications, Wiley, New York, 1968, 3 rd ed., Vol. 1, p. 281.
D. K. SIMONTON (1979), op. cit. Independent discovery in science and technology: A closer look at the Possion distribution,Social Studies of Science, 8
I have placed the two formulae in the same mathematical notation in order to facilitate direct comparison. Eq. (2) still differs from that shown inBrannigan andWanner, op. cit., 425, which they took from J. S. COLEMAN,Introduction to Mathematical Sociology, Free Press, New York, 1964, p. 300. First, I corrected an error in their equation by makingx=i. Second, I madet=l since that is the usual form taken by the formula when employed empirically.
D. K. SIMONTON (1979), op. cit., p. 1610.
D. K. SIMONTON (1979), op. cit., p. 1609, made an arithmetic error in calculatingx 2 and the corresponding degrees of freedom. BRANNIGAN and WANNER, op. cit. note 1, Historical distributions of multiple discoveries and theories of scientific change,Social Studies of Science, 426, incorrectly gave the probability level for the Poissonx 2 asp<0.1 rather than asp>0.1.
BRANNIGAN and WANNER, op. cit. (1983) p. 434, fn. 33, chose ‘not to report frequencies for these cells since it is difficult to know exactly what they refer to empirically, particularly in the case of the null set’. This choice is unfortunate since it gives the reader less basis for concluding that the simple and contagious Poisson distributions yield extremely similar distributions, with comparable theoretical conclusions. Even if the null set is difficult to comprehend-especially by those who subscribe to the doctrine of the inevitability of scientific history-the set of singletons is certainly grasped with ease. To leave out the predicted frequencies for singletons is to omit recognition of the fact that even the contagious Poisson yields a higher proportion of singletons than multiples. It is peculiar that Brannigan and Wanner exhibit so little faith in the Poisson expected frequencies for Grades 0 and 1 and yet they employ these very same frequencies to estimate the parameters of their contagious Poisson just as if the frequencies were reasonable and valid estimates. They cannot have it both ways. If they chose to exploit the expected frequencies this way they must accept the theoretical consequences. On the other hand, if they decide to dismiss the singletons and nulltons, to be consistent they must give up trying to estimate a model that has no estimators for truncated distributions.
A. BRANNIGAN, R. A. WANNER, op. cit., pp. 426–27.
D. K. SIMONTON,Genius, Creativity, and Leadership: Historiometric Inquiries, Harvard University Press, Cambridge, 1984, pp. 204–5.
A. BRANNIGAN, R. A. WANNER, Multiple discoveries in science: A test of the communication theory,Canadian Journal of Sociology, 8 (1983) 135.
Ibid., 135.
W. F. OGBURN, D. S. THOMAS, op. cit..
R. K. MERTON, Singletons and Multiples in Scientific Discovery: A Chapter in the Sociology of Science,Proceedings of the American Philosophical Society, 105 (1961), 470–486.
D. K. SIMONTON (1979), op. cit., p. 1606, fn. 1.
D. K. SIMONTON (1979), op. cit., pp. 1613–15.
A. BRANNIGAN, R. A. WANNER, op. cit. note 17. The proportions of variance explained for multiple grade is usually less than 10 percent, and the proportions for time interval is normally less than one third. More critically, even if the percentage of variance explanied approached 100% that would only show that the parameters of the probability model are determined by sociocultural factors. The Poisson distribution was once shown to predict the number of Prussian officers killed by the kick of a horse. Now sociocultural changes have altered the parameters defining the exposure of officers to horses, especially Prussian officers, who have ceased to exist. Yet that change does not imply that the incidence of equine-caused deaths in the Prussian military was totally determined by the Zeitgeist. The fact remains that some Prussian officers were less lucky than others.
The urn model presented in D. K. SIMONTON, op. cit. note 16, deserves further mathematical development, though at this time it appears mathematically intractable. This model permits the random draw of balls (representing discoveries) in a manner consistent with a chance theory, allows certain scientists to draw more balls than other scientists in a manner consistent with a genius theory, assumes that once a ball of a particular color is taken out all remaining balls of the same color begin to vanish in a negative contagious process that is consistent with the Brannigan-Wanner communication theory, and, finally, assigns the balls a certain temporal ordering or vertical stratification in the urn such that the Zeitgeist can participate in the form of necessary though not sufficient causes.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Simonton, D.K. Multiples, poisson distributions, and chance: An analysis of the Brannigan-Wanner model. Scientometrics 9, 127–137 (1986). https://doi.org/10.1007/BF02017236
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02017236
Keywords
- Poisson Distribution
- Empirical Distribution
- Scientific Discovery
- Poisson Model
- Multiple Grade