It is all About Dissimilarity: Party System Characteristics and Their Proper Measurement

In this article we propose a new conceptualization of the crucial party system characteristics: disproportionality, electoral volatility, territorial heterogeneity and inter-election incongruence. We argue that these characteristics can be studied as dissimilarities between vectors of votes or seats. We present different specifications of vectors in order to address various research questions important for students of parties and party systems. Subsequently, developing the analyses of Monroe (1994) and Taagepera and Grofman (2003), we present nine measures of vectors’ dissimilarity: index of dissimilarity, Gallagher’s least squares measure and its transformations, cosine measure, Gini coefficient, Kullback-Leibler divergence (relative entropy), weighted variance and weighted standard deviation of ratios. We discuss their utility in empirical studies of main party system characteristics, using several dimensions of comparison, based on the formal postulates. We also add two new postulates concerning measure’s decomposability: horizontal (sumtype) and vertical (variance-type)

Two conditions for the application of Lorenz curve and Gini coefficient to voting and allocated seats

The Lorenz curve and Gini coefficient are applied here to measure and graph disproportionality in outcomes for multiseat elections held in 2017. The discussion compares Proportional Representation (PR) in Holland (PR Gini 3.6%) with District Representation (DR) in France (41.6%), UK (15.6%) and Northern Ireland (NI) (36.7%). In France the first preferences of voters for political parties show from the first round in the two rounds run-off election. In UK and NI the first preferences of voters are masked because of strategic voting in the single round First Past the Post system. Thus the PR Gini values for UK and NI must be treated with caution. Some statements in the voting literature hold that the Lorenz and Gini statistics are complex to construct and calculate for voting. Instead, it appears that the application is actually straightforward. These statistics appear to enlighten the difference between PR and DR, and they highlight the disproportionality in the latter. Two conditions are advised to enhance the usefulness of the statistics and the comparability of results: (1) Order the political parties on the ratio (rather than the difference) of the share of seats to the share of votes, (2) Use turnout as the denominator for the shares, and thus include the invalid and wasted vote (no seats received) as a party of their own. The discussion also touches upon the consequences of disproportionality by DR. Quite likely Brexit derives from the UK system of DR and the discontent about (mis-) representation. Likely voting theorists from countries with DR have a bias towards DR and they are less familiar with the better democratic qualities of PR

One Man, One Vote Part 2: Measurement of Malapportionment and Disproportionality and the Lorenz Curve

The main objective of this paper is to explore and estimate the departure from the “One Man, One Vote” principle in the context of political representation and its consequences for distributive politics. To proceed to the measurement of the inequalities in the representation of territories (geographical under/over representation) or opinions/parties (ideological under/over representation), we import (with some important qualifications and adjustments) the Lorenz curve which is an important tool in the economics of income distribution. We consider subsequently some malapportionment and disproportionality indices. It is applied to the 2010 Electoral College and the French parliamentary and local elections with a special attention to the electoral reform of 2015. In these applications, the Lorenz curve ordering is almost conclusive and consequently the Gini and DK indices are aligned and complement the almost complete ranking derived from Lorenz

Comparing votes and seats with a diagonal (dis-) proportionality measure, using the slope-diagonal deviation (SDD) with cosine, sine and sign

When v is a vector of votes for parties and s is a vector of their seats gained in the House of Commons or the House of Representatives - with a single zero for the lumped category of "Other", of the wasted vote for parties that got votes but no seats - and when V = 1'v is total turnout and S = 1's the total number of seats, and w = v / V and z = s / S, then k = Cos[w, z] is a symmetric measure of similarity of the two vectors, θ = ArcCos[k] is the angle between the two vectors, and Sin[θ] is a measure of disproportionality along the diagonal. The geometry that uses Sin appears to be less sensitive than voters, representatives and researchers are to disproportionalities. This likely relates to the Weber-Fechner law. A disproportionality measure with improved sensitivity for human judgement is 10 √Sin[θ]. This puts an emphasis on the first digits of a scale of 10, which can be seen as an inverse (Bart Simpson) report card. The suggested measure has a sound basis in the theory of voting and statistics. The measure of 10 √Sin[θ] satisfies the properties of a metric and may be called the slope-diagonal deviation (SDD) metric. The cosine is the geometric mean of the slopes of the regressions through the origin of z given w and w given z. The sine uses the deviation of this mean from the diagonal. The paper provides (i) theoretical foundations, (ii) evaluation of the relevant literature in voting theory and statistics, (iii) example outcomes of both theoretical cases and the 2017 elections in Holland, France and the UK, and (iv) comparison to other disproportionality measures and scores on criteria. Using criteria that are accepted in the voting literature, SDD appears to be better than currently available measures. It is rather amazing that the measure has not been developed a long time ago and been used for long. My search in the textbooks and literature has its limits however. A confusing element is that voting theorists speak about "proportionality" only for the diagonal while in mathematics and statistics any line through the origin is proportional

Comparing votes and seats with a diagonal (dis-) proportionality measure, using the slope-diagonal deviation (SDD) with cosine, sine and sign

Let v be a vector of votes for parties and s a vector of their seats gained in the House of Commons or the House of Representatives - with a single zero for the lumped category of "Other", of the wasted vote for parties that got votes but no seats. Let V = 1'v be total turnout and S = 1's the total number of seats, and w = v / V and z = s / S. Then k = Cos[w, z] is a symmetric measure of similarity of the two vectors, θ = ArcCos[k] is the angle between the two vectors, and Sin[θ] = Sqrt[1 – b p] is a measure of disproportionality along the diagonal in {w, z} space. The geometry that uses Sin appears to be less sensitive than voters, representatives and researchers are to disproportionalities. This likely relates to the Weber-Fechner law. A disproportionality measure with improved sensitivity for human judgement is 10 √Sin[θ]. This puts an emphasis on the first digits of a scale of 10, which can be seen as an inverse (Bart Simpson) report card. The suggested measure has a sound basis in the theory of voting and statistics. The measure of 10 √Sin[θ] satisfies the properties of a metric and may be called the slope-diagonal deviation (SDD) metric. The cosine is the geometric mean of the slope b of the regression through the origin of z given w and slope p of w given z. The sine uses the deviation of this mean from the diagonal. The paper provides (i) theoretical foundations, (ii) evaluation of the relevant literature in voting theory and statistics, (iii) example outcomes of both theoretical cases and the 2017 elections in Holland, France and the UK, and (iv) comparison to other disproportionality measures and scores on criteria. Using criteria that are accepted in the voting literature, SDD appears to be better than currently available measures. It is rather amazing that the measure has not been developed a long time ago and been used for long. My search in the textbooks and literature has its limits however. A confusing aspect of variables in the unit simplex is that "proportionality" concerns only the diagonal in the {w, z} scatter plot while generally (e.g. in non-normalised space) any line through the origin is proportional

Disproportionality in Gifted Education

Gifted education has historically involved disproportionate rates of identification and enrollment for both students of color and students from lower socioeconomic backgrounds, depriving these groups of more challenging learning opportunities. Giftedness transcends subgroups, spanning all racial, socioeconomic, and disability categories; still, data indicate that Asian and White students are identified and enrolled in gifted programs at rates exceeding their respective proportions (overrepresented) within the general student population. Conversely, the rates for Black, Hispanic, and American Indian students enrolled in gifted education are smaller than their respective proportions (underrepresented) within the general student population. This dissertation aims to disrupt current educational practices by provoking a reevaluation of gifted policy, bringing multicultural considerations to the forefront, and applying inferential statistics to disproportionality data rather than relying solely on descriptive reports. These studies used a cross-sectional observational design with two research arms. Arm A was designed to determine disproportionality rates within a large and diverse school district; Arm B was constructed to examine national disproportionality rates. The two-pronged, micro/macro approach allowed for the determination of disproportionality rates present in extant databases across special student populations, such as race, gender, SES, disability status, and language proficiency status. The data of participants enrolled in gifted programs were then compared to the total number of students from those special populations, yielding a range of proportionality. In specific terms, Arm A looked at prevalence rates by race/ethnicity, gender, and SES for elementary students enrolled for gifted services, while also asking whether the relevant race/ethnicity, gender, and SES proportions differed from those within the general student population. For its part, Arm B examined Gifted and Talented Education (GATE) program enrollment and focused on whether disproportionalities exist in student populations according to race/ethnicity, IDEA status, and ELL status, while also exploring the ranking of U.S. states in terms of racial disproportionalities in GATE program enrollment. Both studies deployed descriptive and inferential analyses, including a one-sample z test of proportions. Across both arms, results indicated and confirmed statistically significant disproportionalities among all variables, in both local and national samples. Findings specifically showed that each state across the nation contained racial disproportionalities in enrollment data for underrepresented groups. The largest racial disproportionalities among states were almost all located in the south-east region of the U.S. Results indicate causes of disproportionalities lurk beneath assessment and identification procedures, which are the most common arguments made in the literature. The present study argues that disproportionalities in gifted education are rooted in a culture-bound construct that guides our society’s conceptualization of giftedness itself

Comparing votes and seats with cosine, sine and sign, with attention for the slope and enhanced sensitivity to disproportionality

Let v be a vector of votes for parties and s a vector of their seats gained in the House of Commons or the House of Representatives. We use a single zero for the lumped category of "Other", of the wasted vote, for parties that got votes but no seats. Let V = 1'v be total turnout and S = 1's the total number of seats, and w = v / V and z = s / S the perunages (often percentages). There are slopes b and p from the regressions through the origin (RTO) z = b w + e and w = p z + ε. Then k = Cos[v, s] = Cos[w, z] = Sqrt[b p]. The geometric mean slope is a symmetric measure of similarity of the two vectors. θ = ArcCos[k] is the angle between the vectors. Thus Sin[v, s] = Sin[w, z] = Sin[θ] = Sqrt[1 – b p] is metric and a measure of disproportionality in general. Geometry appears to be less sensitive to disproportionalities than voters, representatives and researchers tend to be. This likely relates to the Weber-Fechner law. Covariance gives a sign for majority switches. A disproportionality measure with enhanced sensitivity for human judgement is the sine diagonal disproportionality SDD = sign 10 √Sin[v, s]. This puts an emphasis on the first digits of a scale of 10, which can be seen as an inverse (Bart Simpson) report card. What does disproportionality measure ? The unit of account can be either the party or the individual representative. This distinguishes between the party average and the party marginal candidate. The difference z – w is often treated as a level, and Webster / Sainte-Laguë (WSL) uses the relative expression z / w – 1. For the party marginal candidate z – w already is relative, with the unit of account of the individual representative in the denominator. The Hamilton Largest Remainder (HLR) apportionment has the representative as the unit of account. The "Representative Largest Remainder" (RLR) uses a 0.5 natural quota. The paper provides (i) theoretical foundations, (ii) evaluation of the relevant literature in voting theory and statistics, (iii) example outcomes of both theoretical cases and the 2017 elections in Holland, France and the UK, and (iv) comparison to other disproportionality measures and scores on criteria. Using criteria that are accepted in the voting literature, SDD appears to be better than currently available measures