Abstract
Understanding the spatial and temporal distributions and fluctuations of living populations is a central goal in ecology and demography. A scaling pattern called Taylor's law has been used to quantify the distributions of populations. Taylor's law asserts a linear relationship between the logarithm of the mean and the logarithm of the variance of population size. Here, extending previous work, we use generalized least-squares models to describe three types of Taylor's law. These models incorporate the temporal and spatial autocorrelations in the mean-variance data. Moreover, we analyze three purely statistical models to predict the form and slope of Taylor's law. We apply these descriptive and predictive models of Taylor's law to the county population counts of the United States decennial censuses (1790–2010). We find that the temporal and spatial autocorrelations strongly affect estimates of the slope of Taylor's law, and generalized least-squares models that take account of these autocorrelations are often superior to ordinary least-squares models. Temporal and spatial autocorrelations combine with demographic factors (e.g., population growth and historical events) to influence Taylor's law for human population data. Our results show that the assumptions of a descriptive model must be carefully evaluated when it is used to estimate and interpret the slope of Taylor's law.
Highlights
According to Akaike Information Criterion corrected for sample size (AICc), spatially correlated generalized least-squares (GLS) linear regression is superior to the corresponding ordinary least-squares (OLS) linear regression in describing the mean-variance relationship in most recent censuses (Tables 1 and S1)
Our descriptive models of Taylor’s law (TL) show that, when temporal and spatial autocorrelations in the data violate the assumption of the OLS models, GLS provides a better description of the meanvariance relationship and yields different TL slope estimates
For all three types of TL, we find that the best linear regression model yields TL’s slope that is significantly higher than two in some censuses or states
Summary
US census data and Taylor’s law County population count (number of individuals living in each county, historical or existing) in the United States was obtained from the decennial census from 1790 to 2010 [25]. In each census, we calculate a spatial mean and a spatial variance of county population counts across all counties within each state. If the logarithm of the spatial variance is well approximated by a linear function of the logarithm of the spatial mean across all states within a census, the spatial hierarchical TL holds (approximately). For each state, we calculate a temporal mean and a temporal variance of county population counts across all censuses within each county. If the logarithm of the temporal variance is (approximately) a linear function of the logarithm of the temporal mean across all counties within a state, the temporal TL holds (approximately) for county populations within that state. If each set of logarithmic mean and logarithmic variance pairs follows a quadratic relationship, the corresponding QTL holds. Xu and Cohen [22] tested these three types of TL using the ordinary least-squares (OLS) regression model with uncorrelated error ε (normally distributed, with zero mean and constant variance): log ðvarianceÞ 1⁄4 a þ b log ðmeanÞ þ : ðEq 1Þ
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