Maximum value of the standardized log of odds ratio and celestial mechanics
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The odds ratio (OR) is a widely used measure of the effect size in observational research. ORs reflect statistical association between a binary outcome, such as the presence of a health condition, and a binary predictor, such as an exposure to a pollutant. Statistical significance and interval estimates are often computed for the logarithm of OR, ln(OR), and depend on the asymptotic standard error of ln(OR). For a sample of size N, the standard error can be written as a ratio of sigma over square root of N, where sigma is the population standard deviation of ln(OR). The ratio of ln(OR) over sigma is a standardized effect size. Unlike correlation, that is another familiar standardized statistic, the standardized ln(OR) cannot reach values of minus one or one. We find that its maximum possible value is given by the Laplace Limit Constant, (LLC=0.6627...), that appears as a condition in solutions to Kepler equation -- one of the central equations in celestial mechanics. The range of the standardized ln(OR) is bounded by minus LLC to LLC, reaching its maximum for ln(OR) 4.7987. This range has implications for analysis of epidemiological associations, affecting the behavior of the reasonable prior distribution for the standardized ln(OR).
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