4.1 Simple PCA

First, I conduct a PCA to explore the structure of the party system dataset. This type of analysis is a popular statistical approach that can help with the analysis of complex datasets (Dennis Reference Dennis1988; Downes Reference Downes1988; Ben-Ur and Newman Reference Ben-Ur and Newman2002; Kestilä Reference Kestilä2006; Zoco Reference Zoco2006; Guzmán and Sierra Reference Guzmán and Sierra2009; Jakulin et al. Reference Jakulin, Buntine, La Pira and Brasher2009; Falcó-Gimeno and Vallbé Reference Falcó-Gimeno and Vallbé2013; Akkerman, Mudde, and Zaslove Reference Akkerman, Mudde and Zaslove2014; Savoy Reference Savoy2015). The goal of the PCA is to recover the minimum amount of information needed to reproduce the maximum information present in the data matrix. This method projects high-dimensional data onto a lower-dimensional space. The lower-dimensional space is determined by the directions in which the data vary most. As such, the least amount of information possible is lost during projection. The projection is linear, meaning that it can create an accurate summary of the data if the data are Gaussian-distributed.

In practice, PCA estimations begin with a calculation of the dimension in which the data have the most variation. It then finds n orthogonal dimensions that explain the most significant part of the remaining variation within the data.Footnote ⁶ Within the result, the eigenvalues are the scale, and the eigenvectors are the direction of the new (reduced) dimensions. The eigenvectors are the principal components. The eigenvector with the highest eigenvalue is the first principal component of the dataset (i.e., the dimension in which the data have the most variation or the direction in which the data are the most dissimilar). The second principal component is the second eigenvector, and it is orthogonal to the first dimension, and so on.

I analyze the matrix by row. I project the 675-dimensional party seat share data matrix to a 20-dimensional space (the number of variables). To do this, I first mean center the data. Then, I find the eigenvalues and the eigenvectors of the covariance matrix. I do not scale the variables, because all the variables are seat shares and such commensurable variables do not require scaling (Jolliffe Reference Jolliffe2002). Moreover, because the seat share data are compositional, scaling would change the crucial condition that the seat shares add up to 1. Thus, in this case, scaling may not be an optimal solution (Online Appendix B).

One of the benefits of PCA is that the lower dimensions of the data are more easily interpretable than they are in a complex dataset (Jolliffe Reference Jolliffe2002). The first four principal components illustrate what makes party systems most unalike. As Figure 1 shows, the first dimension (PC1) contrasts countries where the two biggest parties are much larger than the other parties with countries where the two biggest parties are not much larger than the other parties. I call this dimension “Size of the Biggest Two Parties.” The second dimension (PC2) contrasts countries where the two biggest parties are similar in size with countries where the biggest party is significantly larger than the second-biggest party. This dimension contrasts countries that have two-party competition with countries that have one dominant party. We can understand this dimension as the “Competition Between the Biggest Two Parties.” The third dimension (PC3) is most heavily influenced by the size of the third party, and so it contrasts countries that have a big third party with countries that have a small third party (we call this dimension “Third Party”). The fourth dimension (PC4) is somewhat unclear. This dimension is influenced by the third, fourth, and fifth parties; tentatively, I call it “Multipartism.” Based on the plot, this last dimension may separate countries that have balanced multiparty systems from countries in which the third to the fifth parties are nonexistent or very small. All dimensions (PC1, PC2, PC3, and PC4) are orthogonal to each other. The purpose of the PCA is to reduce the dimensionality of the matrix. Therefore, the next step is to evaluate the appropriate number of dimensions to use for the analysis.

Figure 1 Loadings, PCA. Notes: The plot shows the loadings, the weight of parties in determining the principal components in the PCA.

Previous studies suggest different rules for choosing which PCA dimensions to analyze (Jolliffe Reference Jolliffe2002; Peres-Neto, Jackson, and Somers Reference Peres-Neto, Jackson and Somers2005). According to Jolliffe (Reference Jolliffe2002), a good rule of thumb is to use PC dimensions that explain between 70% to 90% of the (cumulative) percentage of the total variation in the data. In the following sections, I will focus on the first two dimensions, because as Table 1 shows the first two eigenvalues explain 87.12% of the variation in the data. Figure 2 shows the scree plot of the PCA.Footnote ⁷

Table 1 Eigenvalues and the explained variance, PCA seat shares.

Notes: The table shows the eigen values, the explained variance, and the cumulative variance explained by the principal components.

Figure 2 Scree plot, PCA seat shares. Notes: The plot shows the scree plot of the PCA on seat shares. On the x-axis are the numbers of the eigenvalues, and on the y-axis are the unexplained variance.

Figure 3 shows how some selected countries’ legislative party systems have changed from 1970 to 2013 on the two first dimensions. The advantage of the PCA that it can handle the dynamics of party system changes. The PCA shows that neither competition nor the weights of the two biggest parties change in some countries, such as in Iceland and Finland between 1970 and 2013 (Online Appendix A). The same is true for Western Germany (Figure 3c). On the other hand, the United Kingdom has shifted noticeably in the second dimension (Figure 3d). This means that while the country stayed close to a two-party system, the relative sizes of the two big parties have changed.

Figure 3 Individual countries in the PCA two-dimensional plane. Notes: The plots show how the countries’ party systems change along the PC1 (size) and PC2 (competition) dimensions. The shading of the arrows shows the time progression, with darker colors signifying times closer to the present day.

Figure 4 Electoral versus legislative party system changes 1970–2013. The party system changes throughout the years are plotted in the two-dimensional plane determined by a PCA on both seat and vote shares. In black, I plot the outer polygon surrounding all the positions of a given country based on seat share PCA, whereas in dark grey, based on vote share PCA.

Some countries stay in the same place in Dimension 2 (i.e., the competition between the two biggest parties stay the same) but move within Dimension 1 (sometimes the two biggest parties have a comfortable advantage, while other times the other parties grow), such as in Austria, the Netherlands, Belgium, and Italy (Online Appendix A). Figure 3b shows how the 1993 electoral reform in Italy increased the fragmentation of the party system at first, but then by the 2010s, the size and the competitiveness of the party system returned to the prereform era.Footnote ⁸ In addition, some countries—notably France, Portugal, and Norway—move in both directions. Figure 3a shows the immense changes that the French party system went through over the past decade, starting from the dominance of the UDR (Union for the Defense of the Republic) in 1973 before increases took place in terms of party fragmentation and the overall competitiveness of the party system. Eventually, the country had an almost British-style balanced two-party system by the 2010s.

We can quantify these changes. Each country’s party system changes can be superimposed by a polygon (Kollár-Hunek et al. Reference Kollár-Hunek, Heszberger, Kókai, Láng-Lázi and Papp2008). The area of the polygon represents how much the country’s party system changed between 1970 and 2013 (Figure 4a–d). In addition, we can calculate the standard deviation of all the data points for each country on the PC1 and on the PC2 dimensions $(Sd_{diffxy}= Sd_{x}-Sd_{y})$ . This measure is positive when in a country the sizes of the two biggest parties relative to other parties changed more throughout the years, while it is negative when the competition between the two biggest parties changed more. It is close to 0 if the changes are roughly the same. I also calculate the area–perimeter ratio of the polygons A/P. This measure is high if the polygon is elongated in any direction and low if the party system of the countries changes on both dimensions roughly the same way over the years. According to Figure 4a, France has the biggest polygon area (0.038). The United Kingdom, Italy, and Germany have polygon s that are roughly one-third the size of France’s, which means they have experienced much less change in their party systems between 1970 and 2013. In the case of France, the $Sd_{diffxy}$ is close to 0 ( $-$ 0.0012) while the party systems of Italy and Germany change more in terms of the sizes of the biggest two parties (0.0824 and 0.0321), whereas the United Kingdom’s party system changes more in terms of the competitiveness of the biggest two parties ( $-$ 0.0638) in the same period.

4.2 Difference Between the Electoral and the Legislative Party Systems

While many parties can get (some) votes in elections, not all of these votes translate to seats in the legislature. How the vote shares translate to seat shares depends on the permissiveness of the electoral system, the threshold to enter the legislature, and the geographical concentration of voters who prefer the same party. I call the number and sizes of parties that get votes in elections: electoral party system. A legislative party system, on the other hand, is the number and sizes of the parties in the legislature.Footnote ⁹ Next, I use the PCA results to compare the changes in the electoral and in the legislative party systems of the counties throughout the years.

First, I conduct a PCA on the vote shares of the parties in the previous 17 European countries from 1970 to 2013. The first two principal components that I find through the analysis of the vote shares are highly correlated with the first two principal components that I find through the analysis of seat shares (both have a correlation of $cor=0.98$ ). To be able to compare the legislative and the electoral party systems, I conduct a joint PCA of the seat shares and the vote shares. I create a matrix with both the vote shares and the seat shares of the parties from 1970 to 2013 in the 17 European countries I have in my sample. The matrix consists of 1,350 rows and 20 columns. Each country-year is represented as two rows—one with the seat shares of all parties, and one with the vote shares of all parties. Similarly to the previous analysis, I center the variables, but I do not scale them, and I analyze the matrix by row.

Next, I plot the legislative and the electoral party system changes of each country on Dimensions 1 and 2 (Online Appendix F). Around the points that represent the positions of the countries in each year, I superimpose outer polygons. In most of the countries with PR electoral systems, the electoral and the legislative party system changes are almost identical between 1970 and 2013 (Austria, Belgium, Denmark, Finland, Iceland, the Netherlands, and Sweden). In Figure 4a–d, I present the same countries as in Figure 3. These countries either have single member district (SMD) electoral systems (the United Kingdom and France) or mixed-member electoral systems (Germany and Italy), which means that vote shares do not directly translate to seat shares. In the plots, the polygons represent the legislative party system change (black) and the electoral party system change (grey) of the country. Figure 4a–d shows that even though Italy and Germany have mixed-electoral systems, the countries experienced very similar changes in their electoral and in their legislative party systems.Footnote ¹⁰ In case of France and the United Kingdom (both with single-member districts), the changes in the legislative party systems are much more radical than the changes in the electoral party systems. I calculate the differences in the area between the electoral and legislative party system change polygons. I also calculate the area–perimeter ratio of the polygons A/P. In addition, again we can use the standard deviation differences of the x-axes and y-axes to quantify in which direction do electoral systems distort the party systems. France’s legislative party system change polygon is three times as big as its electoral party system change polygon. The area–perimeter ratio suggests that the competitiveness in the legislature changes almost the same amount as the sizes of the two biggest parties, whereas the electoral party system mainly changes in one dimension. In the United Kingdom, this measure is almost the same for the legislative and the electoral party systems. In France, the standard deviation difference on Dimension 2 is twice as big as on Dimension 1 (Figure 4a). The results are similar in case of the United Kingdom (Figure 4d) in which case the impact of the electoral system is almost limited to Dimension 2.

Overall, with the separate vote and seat share PCA, I find that the same features are the most important two dimensions separating the electoral and legislative party systems. A joint PCA and a comparison where the countries fall on the first two dimensions shows the effect of the different electoral systems: the legislative party systems change more (especially on the competition dimension) in countries with more majoritarian electoral systems, while the electoral and the legislative party system changes are almost identical in countries with PR electoral systems.

4.3 Issues with the Data Structure

The data are compositional and sparse. I address these issues with several further analyses. The party seat share data are a compositional dataset, as the party seat shares add up to one. Some technical issues arise when we conduct a PCA on a compositional dataset, as the data points in each row are not independent from each other (Aitchison Reference Aitchison1983). Therefore, the correlations between the variables might have a negative bias (Jolliffe Reference Jolliffe2002).Footnote ¹¹

I conduct a principal component factor analysis on the data to evaluate the common and the specific variance of the variables. Similarly to the PCA, a factor analysis reduces the original dataset into a smaller number of dimensions or factors. The assumption behind this method is that there are latent variables (factors) that drive variation in a set of observed variables. These factors are linear combinations of the latent variables. Thus, the variation in the variables can be decomposed into common variation (also known as communalities) and variable-specific variation. I anchor the factors to the principal component dimensions. The communalities of the two biggest parties (0.0118 and 0.0064) reveal that the biggest parties vary the most with the other parties. The third party however, has the highest specific variance.

In addition, a compositional dataset does not have subcompositional coherence (Aitchison Reference Aitchison1983). This means that if we have only a subset of the data, the PCA on the covariance of this subset will lead to a different result from an analysis on the entire dataset. To address these concerns, I conduct the PCA on the log-ratio transformed data (Aitchison Reference Aitchison1986 ; see the analysis in Online Appendix B).Footnote ¹² The PCA on the transformed variables sorts the countries in groups based on the number of parties in the legislature. Thus, the result is not very informative. This is because party systems have different sizes and there are a lot of zeros in the matrix. If there are a lot of zeros and small values in the dataset, performing the PCA on the original dataset can be more informative than any of the other approaches, as the absolute variation in the variables may be an important feature of the data (Baxter and Freestone Reference Baxter and Freestone2006).

I also relax the linearity assumption of the PCA. I use kernel principal component analysis (kPCA) and nonlinear component analysis (NLPCA) to analyze the data. The kPCA offers one solution to how to find the appropriate reduced dimensional space if the data are nonlinear. With this method, we first map the data to a higher-dimensional nonlinear feature space, and after that, we run a normal PCA on the transformed data (Schölkopf, Smola, and Müller Reference Schölkopf, Smola, Müller, Gerstner, Germond, Hasler and Nicoud1997). Even though we cannot extract the loadings from this estimation process directly, the biplots of the analyses show that that the first two principal components are very similar to the first two principal components I obtained from the simple PCA. The kPCA thus shows that the untransformed data are close to a Gaussian distribution (Online Appendix C).

I also conduct a NLPCA on the data. Contrary to the kPCA, the nonlinearity of the NLPCA does not come from the transformation of the space on which we project the data, but from the potentially nonlinear optimization of the data matrix. The traditional PCA minimizes the loss function over the eigenvectors and eigenvalues. The NLPCA also minimizes the loss over the admissible transformations of the data columns (de Leeuw Reference de Leeuw2005). The dimensions that the NLPCA recovers are also similar to the ones that the simple PCA finds. These analyses reveal that the result we got with the simple PCA does not depend on the compositional nature of the data (Online Appendix C).

5.1 Typologies

Throughout the years, several authors have developed new party system typologies. In theory, the typologies should be static, and countries could exit and enter a group. However, the groups themselves should not change. Despite this, the authors find slightly different ways of splitting the countries. Through the PCA, one can understand better why periodically new typologies emerged, as well as why few typologies have been developed in recent years. The PCA shows that at different times, different groups of countries can become similar to each other on the two-dimensional plane. This means that all typologies could have been correct at a certain point in time and may have lost their relevance a decade later.

It is probably fair to evaluate the previous party system typologies based on how the party systems were situated on the two-dimensional plane of the PCA vis-a-vis each other when the relevant studies were written. In this paper, I show this with the typology of Rokkan (Reference Rokkan1970) and compare his typology to the dimensions that the PCA finds in 1970.Footnote ¹³

Rokkan (Reference Rokkan1970) classifies countries based on whether the parties are roughly the same size or whether there are one or more dominant parties opposed by small parties. Rokkan constructs three categories: (1) the British–German system, in which one party faces another or another big party (and maybe a small one); (2) the Scandinavian system, in which a big party faces three to four small parties; and (3) even multiparty systems, in which there are two to five parties of about the same size. Rokkan splits the final group into two subcategories: (3a) Scandinavian split working-class systems, in which workers do not back the same parties, and (3b) segmented pluralisms, in which different identity groups create roughly equal-sized parties (Rokkan Reference Rokkan1970). I show the classification of Rokkan in Reference Rokkan1970 along with the countries on the two-dimensional plane in Reference Rokkan1970 in Figure 5. In Figure 5a, I designate different symbols to different categories. The PCA quite clearly separates the different groups.

Figure 5 Comparison of Rokkan’s (Reference Rokkan1970) typology to PCA results. Notes: Figure 5a shows the first two dimensions of the PCA and the classification of Rokkan. Figure 5b shows the demarcation lines between groups. Top-left: Category (1). Top-right: Category (3)—Top(3b) and Bottom(3a). Bottom-right: Category (2).

Figure 6 Individual countries in the two-dimensional plane. Notes: The figures show how the countries’ party systems change along the PC1 (size) and PC2 (competition) dimensions through time. The lines represent the demarcation lines of different categories that Rokkan (Reference Rokkan1970) established. Top-left: 1. “British–German” party systems. Top-right: 3. Even multiparty systems—3a (bottom): Scandinavian “split working class” systems; 3b (top): Segmented pluralism. Bottom-right: “Scandinavian” party systems.

In Figure 5b, I draw the demarcation lines that separate these groups. I estimate these lines through logistics models.Footnote ¹⁴ With the help of this plot, I can evaluate the original typology of Rokkan. In addition, the plot helps in sorting the countries in other years and/or countries that are not in the original typology into types. First, Figure 5b shows that Rokkan does not create one category that logically should exist. He does not include a category for countries that have few parties, one of which is dominant. In the plot, these countries should be placed in the bottom-left quadrant. It could be the case that no such country existed in Reference Rokkan1970, but the plot shows that there is a country that Rokkan does not classify—France—which could fall into this category. Indeed, in 1968, in the French legislative elections, the UDR (Union of Democrats for the Republic) won 354 of the 458 legislative seats of the French National Assembly, while the FGDS (Federation of the Democratic and Socialist Left) ended up a distant second with 57 seats. Rokkan also did not classify the Italian party system in Reference Rokkan1970. With the help of Figure 5b, we can infer that Rokkan probably would have classified Italy as a “Scandinavian split working class” party system along with Finland and Iceland (Category [3a]) at this time.

The classification of all the 17 countries between 1970 and 2013 can also be inferred based on the PCA. In Figure 6, I plot the paths through which the legislative and the electoral party systems evolved from 1970 to 2013 in France, Italy, Germany, and the United Kingdom superimposed with the demarcation lines that the analysis recovered. In this way, we can identify the times when individual countries crossed key thresholds in the typology of Rokkan (Reference Rokkan1970). The electoral party system changes are very similar to the legislative party system changes in most countries. Italy has an even multi(legislative and electoral)-party system in most of these years, but it is perhaps closer to a two-party regime in 1976 and 2002. Germany is a two-party regime until the unification of the country in 1990, at which point it becomes an even multiparty system. The trajectories of the legislative and the electoral party system changes are different in case of countries with majoritarian electoral systems, in case of the United Kingdom and France. As for the legislative party system, France is probably a dominant party regime after the elections of 1970, 2002, and 2007; a “Scandinavian-type” party system in 1973–1981, 1988, and 1997; an even multiparty system in 1986 (the only year it has PR elections); and close to a two-party system in 1993. In 1968, which is the only PR elections in the country, the legislative and electoral party systems of France are very close to each other. Based on the electoral party system, the country starts as a Scandinavian system (2) in the 1970s and 1980s and becomes Scandinavian “spit working class” system (3a) in the late 1990s and early 2000s, and it is a segmented pluralism in 1978, 1986, and 1993. The United Kingdom’s legislative party system is effectively a two-party regime in most years, with the Labour Party dominating in the late 1990s and early 2000s. The electoral party system also starts out as a two-party system, but the country moves toward multiparty system in latter years. The plot shows that in the late 1990s and early 2000s when there are more parties in the electoral party system, one of the parties gains a significant majority in the legislative party system. This shows the spoiler effect of the small(er) parties in a majoritarian electoral system.

5.2 Indices and PCA

We can also compare the PCA results to the quantitative party system size indices that previous scholarship has been identified. Based on the literature review, below I compare the PCA results to several measures. From indices that are more sensitive to the sizes of small parties to indices that are less sensitive to the sizes of small parties—HH, Fractionalization Index (Rae Reference Rae1967), Entropy (Kesselman Reference Kesselman1966)—ENP (Laakso and Taagepera 1979), Shapley ENP (ENP in which I replace the parties’ seat shares with their Shapley-Shubik indices; Grofman and Kline Reference Grofman and Kline2011), and the size of the Biggest Party in the legislature normalized with the number of legislative seats (Taagepera Reference Taagepera1999; Dunleavy and Boucek Reference Dunleavy and Boucek2003). In addition, I include the Number of Parties in the Government, as this measure became popular for finding political outcomes (Bawn and Rosenbluth Reference Bawn and Rosenbluth2006). In addition, I include the Number of Parties in the Government, as this measure became popular for finding political outcomes (Bawn and Rosenbluth Reference Bawn and Rosenbluth2006).

I represent the correlations between these measures and the first two dimensions of the PCA in the first seven columns and bottom two rows of Table 2. This correlation matrix shows the Pearson correlation coefficients between the variables and their significance levels. All of these measures are highly correlated (0.7–.98) with Dimension 1 (PC1) and with each other. Through this method, we can implicitly compare the typologies with the party system size measures. As we can see, these indices relate closely to Dimension 1 and, thus, to the typologies of Duverger (Reference Duverger1954) and Blondel (Reference Blondel1968). Overall, this analysis shows that ENP distinguishes multiparty countries from two-party systems. However, this variable may not be the best solution when our theory calls for the measurement or control of interparty competition.

Table 2 Correlation table, traditional measures of party system size and opposition structure and principal components.

Note: Computed correlation using Pearson method with pairwise deletion. Standard errors are in parentheses. Abbreviations: Fraction, Fractionalization Index; HH, Herfindahl–Hirschman Index; Entropy, entropies; ENP, effective number of parties; P.in Gov, number of parties in the government; Shap.ENP, effective number of parties (Shapley); Big Party, size of the biggest party over the size of the legislature; ENOP, effective number of opposition parties; OPOP, the difference between the first and the second biggest opposition parties over the size of the legislature; BOPP, size of the biggest opposition party over the size of the legislature; Competition, size difference between the biggest and the second biggest parties over the size of the legislature. PC1: Dimension 1, Sizes of the two biggest parties. PC2: Dimension 2, Competition between the biggest and the second biggest parties.

To measure the competition within the party system, we might calculate indices that measure the structure of the opposition. The concentration of the opposition could be measured similarly to the party system size. This notion has been discussed previously in a few studies (Maeda Reference Maeda2010, Reference Maeda2015). However, very few political science research have used these measures. The Effective Number of Opposition Parties (ENOP) is calculated the same way as the ENP suggested by Laakso and Taagepera (Reference Laakso and Taagepera1979), except I calculate the measure only for the opposition parties. The Difference Between the Biggest and the Second Biggest Opposition Parties (OPOP) measures the competition between the biggest and the second biggest opposition parties. I normalize the difference between the seat shares of the two biggest opposition parties by the number of total available legislative seats within a country. The Size of the Biggest Opposition Party (BOPP) measures the size of the biggest party in the opposition also normalized with the number of the legislative seats. Finally, I calculate the Competition, which is the seat share difference between the biggest and the second biggest parties normalized with the number of legislative seats.Footnote ¹⁵ The last two rows of Table 2 show that the opposition measures and the competition measure relate both to PC2 and to PC1 while the party system size measures are closely related to only PC1. The only exception is the size of the Biggest Party (Big Party), which seems to be somewhere in between the opposition measures and the party system size measures. This makes sense because it is possible that the biggest party in the party system is not a government party but an opposition party.

Figure 7 Measures on the PC dimensions. Notes: The plot shows how party system size measures relate to each other and the two first dimensions of the PCA. PC1: Dimension 1, Sizes of the two biggest parties. PC2: Dimension 2, Competition between the biggest and the second biggest parties. Abbreviations: HH, Herfindahl–Hirschman Index; Fract, Fractionalization Index; Entropy, entropies; ENP, effective number of parties; P.in Gov, number of parties in the government; Shap.ENP, effective number of parties (Shapley); Big Party, size of the biggest party over the size of the legislature; ENOP, effective number of opposition parties; OPOP, the difference between the first and the second biggest opposition parties over the size of the legislature; BOPP, size of the biggest opposition party over the size of the legislature; Competition, size difference between the biggest and the second biggest parties over the size of the legislature.

I also plot the indices on PC Dimensions 1 and 2 to show how the indices relate to these dimensions (Figure 7). First, I regress each index on the PC1 and PC2 dimensions and plot the coefficients.Footnote ¹⁶ Figure 7a also shows that most of the indices that the current scholarship uses are closer to Dimension 1 while opposition concentration measures are closer to Dimension 2. Figure 7a offers a way in which we can decide between measures with multicriteria decision-making process, as the curve can serve as the basis of a pareto-optimization. The more we would like to take into account competition between the parties with our measure and less the number of parties, the closer index we should choose to Dimension 2. Alternatively, if our theory calls for the size of the party system, we can choose one of the variables close to Dimension 1. Similarly, we can evaluate how the measures relate to the PC dimensions by using the Sum of Ranking Differences method. First, I calculate how the country-years are ranked on the PC1 and PC2 dimensions. Then, I calculate how the various country-years are ranked by the indices. Finally, I deduct the PC1 and PC2 rankings from the rankings of the measures. The sum of the differences is the Sum of Ranking Differences. The low values indicate that the index is close to the dimension, and higher values indicate that the index is far. Since the PC method is nondirectional, I repeat the process with inverse ranking as well to find the measure that is the closest to the PCA solution. I find that the HH followed by the Fractionalization and ENP (identical in their ranking) are the closest to the Dimension 1’s ranking of country-years. This result shows that neither transformation of the HH changes the measure in a meaningful way. At the same time, the measure Competition is the closest to Dimension 2 followed by OPOP, ENOP, and BOPP (Héberger and Kollár-Hunek Reference Héberger and Kollár-Hunek2011; Bajusz, Rácz, and Héberger Reference Bajusz, Rácz and Héberger2015). I plot the normalized results against the theoretical distribution of random rankings in Figure 7b.