williamson: Williamson index
In REAT: Regional Economic Analysis Toolbox

Description Usage Arguments Details Value Author(s) References See Also Examples

Calculating the Williamson index (population-weighted coefficient of variation)

1	williamson (x, weighting, coefnorm = FALSE, wmean = FALSE, na.rm = TRUE)

`x`	a `numeric` vector
`weighting`	mandatory: a `numeric` vector containing weighting data (usually regional population)
`coefnorm`	logical argument that indicates if the function output is the standardized cv (0 < v < 1) or not (0 < v < ∞*) (default: `coefnorm = FALSE`)
`wmean`	logical argument that indicates if the weighted mean is used when calculating the weighted coefficient of variation
`na.rm`	logical argument that whether NA values should be extracted or not

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 < ∞), 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.

Single numeric value. If coefnorm = FALSE the function returns the non-standardized cv (0 < v < ∞). If coefnorm = TRUE the standardized cv (0 < v* < 1) is returned.

Thomas Wieland

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.ifo.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.

gini, herf, hoover, cv, disp

data(GoettingenHealth2)
# districts with healthcare providers and population size

williamson((GoettingenHealth2$phys_gen/GoettingenHealth2$pop), 
GoettingenHealth2$pop)