Comparing the Aitchison distance and the angular distance for use as inequality or disproportionality measures for votes and seats

Colignatus, Thomas (2018): Comparing the Aitchison distance and the angular distance for use as inequality or disproportionality measures for votes and seats.

Abstract

Votes and seats satisfy only two of seven criteria for application of the Aitchison distance. Vectors of votes and seats, say for elections for political parties the House of Representatives, can be normalised to 1 or 100%, and then have the outward appearance of compositional data. The Aitchison geometry and distance for compositional data then might be considered for votes and seats too. However, there is an essential zero when a party gets votes but doesn't gain a seat, and a zero gives an undefined logratio. In geology, changing from weights to volumes affects the percentages but not the Aitchison distance. For votes and seats there are no different scales or densities per party component however, and thus reportioning (perturbation) would be improper. Another key issue is subcompositional dominance. For votes {10, 20, 70} and seats {20, 10, 70} it is essential that we consider three parties. For a disproportionality measure we would value it positively that there is a match on 70. The Aitchison distance looks at the ratio {10, 20, 70} / {20, 10, 70} = {1/2, 2, 1} and then neglects a ratio equal to 1. In this case it essentially compares the subcompositions, i.e. votes {10, 20} and seats {20, 10}, rescales to {1/3, 2/3} and {2/3, 1/3}, and finds high disproportionality. This means that it essentially looks at a two party outcome instead of a three party outcome. It follows that votes and seats are better served by another distance measure. Suggested is the angular distance and the Sine-Diagonal Inequality / Disproportionality (SDID) measure based upon this. Users may of course apply both the angular and the Aitchison measures while being aware of the crucial differences in properties.

• Item Type: MPRA Paper
• Institution: Thomas Cool Consultancy & Econometrics
• Language: English
• Keywords: Votes, Seats, Electoral System, Distance, Disproportionality, Aitchison Geometry, Angular Distance, Sine-Diagonal Inequality / Disproportionality, Loosemore-Hanby, Gallagher, Descriptive Statistics, Education, Reportion
• Subjects: A - General Economics and Teaching > A1 - General Economics > A10 - General; D - Microeconomics > D6 - Welfare Economics > D63 - Equity, Justice, Inequality, and Other Normative Criteria and Measurement; D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations; D - Microeconomics > D7 - Analysis of Collective Decision-Making > D72 - Political Processes: Rent-Seeking, Lobbying, Elections, Legislatures, and Voting Behavior

References

Colignatus is the name in science of Thomas Cool, econometrician and teacher of mathematics, Scheveningen, Holland, http://econpapers.repec.org/RAS/pco170.htm

This paper is also available at: https://www.wolframcloud.com/objects/thomas-cool/Voting/2018-01-18-Aitchison.nb (updated but still the same name).

Aitchison, J., and J.A.C. Brown (1957), "The lognormal distribution", CUP

Aitchison, J. (undated, perhaps latest version 2002), "A concise guide to compositional data analysis", http://ima.udg.edu/activitats/codawork05/A_concise_guide_to_compositional_data_analysis.pdf

Barcelo-Vidal, C., and J.A. Martin-Fernandez (2016), "The mathematics of compositional analysis", Austrian J. of Statistics, Vol 45, No 4, http://dx.doi.org/10.17713/ajs.v45i4.142

Blackwell, M., J. Honaker and G. King (2017), "A Unified Approach to Measurement Error and Missing Data: Overview and Applications", Sociological Methods and Research, 46, 3, Pp. 303-341, https://gking.harvard.edu/publications/multiple-overimputation-unified-approach-measurement-error-and-missing-data

Colignatus, Th. (2001, 2014), "Voting theory for democracy", https://zenodo.org/record/291985

Colignatus, Th. (2017a), "Comparing votes and seats with cosine, sine and sign, with attention for the slope and enhanced sensitivity to disproportionality", https://mpra.ub.uni-muenchen.de/81389/

Colignatus, Th. (2017b), "One woman, one vote. Though not in the USA, UK and France", https://mpra.ub.uni-muenchen.de/82513/

Colignatus, Th. (2017c), "Statistics, slope, cosine, sine, sign, significance and R-squared", https://boycottholland.wordpress.com/2017/10/21/statistics-slope-cosine-sine-sign-significance-and-r-squared/

Colignatus, Th. (2017d), "Karl Pearson's curious construction of spurious correlation", https://boycottholland.wordpress.com/2017/11/19/karl-pearsons-curious-construction-of-spurious-correlation/

Colignatus, Th. (2018), "Measures of policy distance and inequality / disproportionality of votes and seats", MPRA_paper_84324.pdf, https://www.wolframcloud.com/objects/thomas-cool/Voting/2018-02-02-PolicyDistance.nb

Cramer, J.S. (1975), "Empirical econometrics", North-Holland

Dongen, S. van & A.J. Enright (2012), "Metric distances derived from cosine similarity and Pearson and Spearman correlations", https://arxiv.org/abs/1208.3145

Gelman, A. and G. King (1994), "A Unified Method of Evaluating Electoral Systems and Redistricting Plans", American Journal of Political Science, 38, Pp. 514\[Dash]554, https://gking.harvard.edu/files/abs/writeit-abs.shtml

Golder, M. & Stramski, J. (2010), "Ideological congruence and electoral institutions", American Journal of Political Science, 54(1), 90-106

Honaker, J., J. Katz and G. King (2002), "A Fast, Easy, and Efficient Estimator for Multiparty Electoral Data", Political Analysis 10(1): 84-100, http://gking.harvard.edu/files/gking/files/trip.pdf?m=1360038989

Honaker, J. and D. Linzer (2006), "Analysis of Voteshares Across Party Systems", draft for APSA, https://www.semanticscholar.org/paper/Analysis-of-Voteshares-Across-Party-Systems-Honaker-Linzer/4f901713cfb120a0bc4a21150c3b5c3a9c85c2fe

Hron, K. (2008), "Remark on properties of bases for additive logratio transformations of compositional data", Acta Univ. Palacki. Olomuc. Fac. rer. nat., Mathematica 47, 77-82, http://kma.upol.cz/data/xinha/ULOZISTE/ActaMath/0708.pdf

Katz, J.N, and G. King (1999), "A statistical model for multiparty electoral data", American political science review, Vol 93, No 1, p15-32,https://gking.harvard.edu/files/gking/files/multiparty.pdf

Kropko, J. (2008), "Choosing Between Multinomial Logit and Multinomial Probit Models for Analysis of Unordered Choice Data", master's thesis at Chapel Hill, https://cdr.lib.unc.edu/indexablecontent/uuid:008129bb-c121-47ca-9671-3396eb655b2c

Maligranda, L. (2006), "Simple norm inequalities", Amer. Math. Monthly 113 (3), 256 \[Dash] 260

Martin-Fernandez, J.A., and C. Barcelo-Vidal, V. Pawlowsky-Glahn (undated), "Measures of difference for compositional data and hierarchical clusering methods", http://ima.udg.edu/~barcelo/index_archivos/Measures_of_difference__Clustering.pdf

Pawlowsky-Glahn, V., J.J. Egozcue, and R. Meziat (2007), "The statistical analysis of compositional data: The Aitchison geometry", (sheets), https://laboratoriomatematicas.uniandes.edu.co/cursocoda/04-Vera-geometry.pdf

Pawlowsky-Glahn, V., and J.J. Egozcue (2011), "The closure problem: one hundred years of debate", http://slon.diamo.cz/hpvt/2011/_Mat/M%2002.pdf

Theil, H. (1971), "Principles of econometrics", Wiley

Tolosana, R. (2008), "Compositional Data Analysis in a Nutshell", http://www.sediment.uni-goettingen.de/staff/tolosana/extra/CoDaNutshell.pdf

Vallis, R.A., H. Nunez, and J. C. Martin (2008), "Why, and How, Should Geologists Use Compositional Data Analysis", https://en.wikibooks.org/wiki/Why,_and_How,_Should_Geologists_Use_Compositional_Data_Analysis

Wang, H., Q. Liu, H.M.K. Mok, L. Fu, W. Man Tse (2007), "A hyperspherical transformation forcasting model for compositional data", Eur. J. of Oper. Res. 179, 459-468, https://pdfs.semanticscholar.org/67e5/fa4a7cc9c833109722eed64fc3611c4e15f4.pdf