Empirically assessing noisy necessary conditions with activation functions
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Empirically assessing noisy necessary conditions with activation functions Wolfgang Messner1 Received: 30 December 2019 / Accepted: 12 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Ceiling lines were recently proposed to identify necessary conditions as constraints on the outcome in a scatterplot. However, these lines do not work very well on large data sets with random observation error. This paper suggests an alternate way of empirically assessing probabilistic necessary conditions in large and noisy data sets using sigmoidal activation functions, which describe the propensity of outcome at different levels of the independent variable. Keywords Activation function · Large survey data · Measurement error · Necessary conditions · Regression analysis
1 Introduction In 1823, the German mathematician Johann Carl Friedrich Gauss developed the normal surface of n correlated variates. He interpreted it as one of the several parameters in his distributional equations, but did not have any particular interest in correlation as a distinct idea. Similarly, when the French physicist Auguste Bravais developed the bivariate normal distribution in 1846, he did not recognize the importance of correlation as a measure of association between variables. It was the English statistician Sir Francis Galton, who defined regression and correlation as statistical concepts in 1885. He plotted the frequencies of combinations of childrens’ and parents’ heights, smoothed the results, and drew lines through points with equal frequency, the isodensity contour lines from the bivariate normal distribution. Together with the English mathematician J. D. Hamilton Dickson, Galton derived the theoretical formula for
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10287-02 0-00377-2) contains supplementary material, which is available to authorized users.
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Wolfgang Messner [email protected] Darla Moore School of Business, University of South Carolina, 1014 Greene Street, Columbia, SC 29208, USA
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the bivariate normal distribution. A decade later, in 1895, Karl Pearson developed the index that is still used today to measure correlation, Pearson’s r (Rodgers and Nicewander 1998, 59–60). Today’s majority of business research across disciplines focusses on factors that drive certain outcomes. Researchers use correlation, regression, structural equation modeling, and other related methodologies to probe their research models. These research methodologies aim to establish additive sufficiency. They identify variables contributing on average to an outcome, and then use these variables to predict the outcome. While researchers today take the existence of the correlation coefficient as a given, they rarely realize that, before Galton and Pearson, the only way to establish a relationship between variables was to discuss cause and effect. Notwithstanding the correlation coefficient’s popularity, Carroll (1961, 347) called it one of the most f
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