Lotkas Inverse Square Law of Scientific Productivity: Its Methods and Statistics

📝 Abstract

This brief communication analyzes the statistics and methods Lotka used to derive his inverse square law of scientific productivity from the standpoint of modern theory. It finds that he violated the norms of this theory by extremely truncating his data on the right. It also proves that Lotka himself played an important role in establishing the commonly used method of identifying power-law behavior by the R^2 fit to a regression line on a log-log plot that modern theory considers unreliable by basing the derivation of his law on this very method.

💡 Analysis

This brief communication analyzes the statistics and methods Lotka used to derive his inverse square law of scientific productivity from the standpoint of modern theory. It finds that he violated the norms of this theory by extremely truncating his data on the right. It also proves that Lotka himself played an important role in establishing the commonly used method of identifying power-law behavior by the R^2 fit to a regression line on a log-log plot that modern theory considers unreliable by basing the derivation of his law on this very method.

📄 Content

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LOTKA’S INVERSE SQUARE LAW OF SCIENTIFIC PRODUCTIVITY: ITS METHODS AND STATISTICS

Stephen J. Bensman LSU Libraries (Retired) Louisiana State University Baton Rouge, LA 70803 USA E-mail: notsjb@lsu.edu

Lawrence J. Smolinsky Department of Mathematics Louisiana State University Baton Rouge, LA 70803 USA E-mail: smolinsk@math.lsu.edu

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Abstract This brief communication analyzes the statistics and methods Lotka used to derive his inverse square law of scientific productivity from the standpoint of modern theory. It finds that he violated the norms of this theory by extremely truncating his data on the right. It also proves that Lotka himself played an important role in establishing the commonly used method of identifying power-law behavior by the R^2 fit to a regression line on a log-log plot that modern theory considers unreliable by basing the derivation of his law on this very method.

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Introduction

In recent years power-law distributions as scientific models have come under intensive scrutiny.
Of primary importance in this have been the papers by Newman (2005) and Clauset, Shalizi, and Newman (2009). In his paper Newman (2005, p. 323) states that when the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law. Lotka’s Inverse Square Law of Scientific Productivity—historically the first law of scientometrics—is precisely such a law. Power-law distributions have the general shape of negative exponential J-curves with a long tail to the right, and a key characteristic of them is a surfeit of observations at the right tip of what is termed this “heavy tail.” Clauset, Shalizi, and Newman (2009) focus on the problems of identifying power-law distributions, and they aver that “the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds” (p. 661). They also assert that the commonly used method of identifying power-law distributions by logging the variables on both axes of the graph and then using regression analysis to measure the R^2 linear fit to the resulting regression line or trendline is unreliable. It will be seen in the subsequent analysis of Lotka’s methods and data that his inverse square law of scientific productivity suffers from all these problems and that Lotka himself used the method of the linear fit on the log-log plot to derive his law of scientific productivity. Lotka’s Law

Lotka’s Inverse Square Law of Scientific Productivity is eponymously named after Alfred J. Lotka, and it is the first scientometric or informetric law. To obtain the data for deriving his law, Lotka (1926) made a count of the number of personal names in the 1907-1916 decennial index of Chemical Abstracts against which there appeared 1, 2, 3, etc. entries, covering only the letters A and B of the alphabet. He also applied a similar process to the name index in Felix Auerbach’s Geschichtstafeln der Physik (Leipzig: J. A. Barth, 1910), which dealt with the entire range of the history of physics through

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1900. By using the latter source, Lotka hoped to take into account not only the volume of production but also quality, since it listed only the outstanding contributions in physics. In making these counts Lotka credited only the senior author in joint publications. On the basis of this data, Lotka derived what he termed an “inverse square law“, according to which of any set of authors, ca. 60% produce one paper, whereas the percent producing 2 is 1 /2^2 or ca. 25%, the percent producing 3 equals 1/ 3^2 or ca. 11.1%, the percent producing 4 is 1/ 4^2 or ca. 6.3%, etc. Thus, of 1000 authors, 600 produce 1 paper, 250 produce 2 papers, 111 produce 3 papers, and 63 produce 4 papers. Lotka (1926, p. 320) defined the general formula for the relation he found between the frequency of y of persons making x contributions as: xny = const. Most interestingly, Lotka (1926, p. 320) referred to the exponent as the “slope,” and he stated that, as determined by “least squares,” the “slope of the curve” to his Chemical Abstracts data was found to be 1.888 and to his Auerbach data 2.021. The source of these statements can be located in Figure 1 below, which replicates the two histograms found in his article. Part A presents the percent chemists and physicists on the y-axis and the number of mentions up to 10 on the x-axis. However, on the histogram in Part B both the axes are logged, and the histogram is designed to show closeness of the linear fit to resulting regression lines determined by least-squares analysis. It will be seen below that the absolute values of the s

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