Analyzing and interpreting spatial and temporal variability of the United States county population distributions using Taylor's law

Abstract

We study the spatial and temporal variation of the human population in the United States (US) counties from 1790 to 2010, using an ecological scaling pattern called Taylor's law (TL). TL states that the variance of population abundance is a power function of the mean population abundance. Despite extensive studies of TL for non-human populations, testing and interpreting TL using data on human populations are rare. Here we examine three types of TL that quantify the spatial and temporal variation of US county population abundance. Our results show that TL and its quadratic extension describe the mean-variance relationship of county population distribution well. The slope and statistics of TL reveal economic and demographic trends of the county populations. We propose TL as a useful statistical tool for analyzing human population variability. We suggest new ways of using TL to select and make population projections.

Spatial and temporal autocorrelations affect Taylor's law for US county populations: Descriptive and predictive models

“…First, for the spatial hierarchical TL, starting from the 1900 census (except the 1910 census), spatially correlated GLS models describe the mean-variance relationship of county population count better than the OLS regression ( S1 Table ). The assumption of uncorrelated errors in OLS is violated due to large spatial autocorrelation between states since 1900 (see S9 Table in [ 22 ]), which may be explained by the high population mobility contributed by several factors (e.g., employment, family, education attainment) [ 33 , 34 ]. For the censuses when GLS is the better linear regression model, the slope of TL estimated from GLS is higher (and significantly higher in four censuses) than that from OLS ( S1 and S3 Tables).…”

“…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 ]. Detailed descriptive statistics of the census data are given in Xu and Cohen [ 22 ]. As in [ 22 ], here "states" refers to states, territories, or equivalent primary administrative subdivisions of the United States, and "counties" refers to counties, parishes, or equivalent primary administrative subdivisions of any "state".…”

“…We reexamine the three types of TL by fitting generalized least-squares (GLS) regressions to the log(mean)-log(variance) data. This practice is warranted because the residuals of the OLS regressions had significant spatial and temporal autocorrelations (see S9-S14 Tables in [ 22 ]), which violated the assumption of uncorrelated errors [ 26 ]. These GLS regressions incorporate various autocorrelation error structures and may better describe the mean-variance relationships (Eqs 1 and 2 ).…”

“…Recent works have also tested and usually confirmed TL for age-specific annual human death rates in 12 countries [ 23 , 24 ]. Using the decennial census data, Xu and Cohen [ 22 ] found that US county population size obeys TL or its quadratic generalization, and compared the estimated TL slopes from different countries. They showed that significant temporal and spatial autocorrelation is present in the ordinary regression model residuals, which violated the assumptions of the ordinary regression models used for testing TL (i.e., a linear relationship between log(mean) and log(variance)).…”

“…Motivated by these gaps, here we extend the work by Xu and Cohen [ 22 ] to analyze the descriptive and predictive models of TL using US county population data. We have two goals.…”