Calculating the Williamson index (population-weighted coefficient of variation)
williamson (x, weighting, coefnorm = FALSE, wmean = FALSE, na.rm = TRUE)a numeric vector
mandatory: a numeric vector containing weighting data (usually regional population)
logical argument that indicates if the function output is the standardized cv (\(0 < v* < 1\)) or not (\(0 < v < \infty\)) (default: coefnorm = FALSE)
logical argument that indicates if the weighted mean is used when calculating the weighted coefficient of variation
logical argument that whether NA values should be extracted or not
Single numeric value. If coefnorm = FALSE the function returns the non-standardized cv (\(0 < v < \infty\)). If coefnorm = TRUE the standardized cv (\(0 < v* < 1\)) is returned.
The Williamson index (Williamson 1965) is a population-weighted coefficient of variation.
The coefficient of variation, \(v\), is a dimensionless measure of statistical dispersion (\(0 < v < \infty\)), based on variance and standard deviation, respectively. The cv (variance, standard deviation) can be weighted by using a second weighting vector. As there is more than one way to weight measures of statistical dispersion, this function uses the formula for the weighted cv (\(v_w\)) from Sheret (1984). The cv can be standardized, while this function uses the formula for the standardized cv (\(v*\), with \(0 < v* < 1\)) from Kohn/Oeztuerk (2013). The vector x is automatically treated as a sample (such as in the base sd function), so the denominator of variance is \(n-1\), if it is not, set is.sample = FALSE.
Gluschenko, K. (2018): “Measuring regional inequality: to weight or not to weight?” In: Spatial Economic Analysis, 13, 1, p. 36-59.
Lessmann, C. (2005): “Regionale Disparitaeten in Deutschland und ausgesuchten OECD-Staaten im Vergleich”. ifo Dresden berichtet, 3/2005. https://www.cesifo-group.de/DocDL/ifodb_2005_3_25-33.pdf.
Huang, Y./Leung, Y. (2009): “Measuring Regional Inequality: A Comparison of Coefficient of Variation and Hoover Concentration Index”. In: The Open Geography Journal, 2, p. 25-34.
Kohn, W./Oeztuerk, R. (2013): “Statistik fuer Oekonomen. Datenanalyse mit R und SPSS”. Berlin: Springer.
Portnov, B.A./Felsenstein, D. (2010): “On the suitability of income inequality measures for regional analysis: Some evidence from simulation analysis and bootstrapping tests”. In: Socio-Economic Planning Sciences, 44, 4, p. 212-219.
Sheret, M. (1984): “The Coefficient of Variation: Weighting Considerations”. In: Social Indicators Research, 15, 3, p. 289-295.
Williamson, J. G. (1965): “Regional Inequality and the Process of National Development: A Description of the Patterns”. In: Economic Development and Cultural Change, 13, 4/2, p. 1-84.
# NOT RUN {
data(GoettingenHealth2)
# districts with healthcare providers and population size
williamson((GoettingenHealth2$phys_gen/GoettingenHealth2$pop),
GoettingenHealth2$pop)
# }
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