Extending the Laakso-Taagepera Index to integrate both party and ethnicity

Abstract

We show how to extend the Laakso-Taagepera measure of the effective number of parties so as to incorporate both party and ethnicity in a way that allows both ethnic-specific, party-specific, and composite measures that we refer to as extended L-T indices. While the aim of this article is methodological, we also illustrate our approach with U.S. two-party data from presidential elections, using two-group, three-group, and five-group categorizations of ethnicities to show how party ethnicization in that country has grown since 1952. For comparison, and to show the generality of our approach to situations with more than two parties, we present data from the Federation of Bosnia and Herzegovina (Federacija Bosne i Hercegovine), where we have three ethnic groups but also more than two parties.

Keywords

Elections ethnic parties ethnic voting Laakso-Taagepera index

Introduction

In this article, we are interested in two issues in the intersection of ethnicity and party: (1) the extent to which members of a given ethnicity distribute their support across different parties, on the one hand, and (2) the extent to which any given party draws it support from one or more ethnicities, on the other. We can understand these linkages by looking at an m x n matrix, or more specifically an E x P matrix based on E (ethnic groups) and group P (political parties), where each is taken to be a logically exclusive and mutually exhaustive set of categories. If we look at the raw data, the E_iP_jth cell is the number of cases in the category (E_i and P_j).¹ But, like any table, the E x P matrix may be percentaged either by rows or by columns or by total, depending on whether we wish to examine parties, ethnic groups, or a composite of the two. But while such m x n matrix contains all the basic data we need to examine the two issues of interest, and any data analyst should be reluctant to throw away information, often it is useful to look at summary measures.

In both the context of political party support and the context of ethnic diversity, two mathematically linked summary measures have been most commonly used to specify a single number to capture the overall features of party support or of ethnic membership: the Laakso-Taagepera Index (Laakso and Taagepera, 1979), which we abbreviate as the L-T Index, and the Herfindahl-Hirschman Index (Hirschman, 1964), which we abbreviate as the H-H Index. Here we show how to extend the former to study the intersection of party and ethnicity in terms of summary measures.

The Herfindahl-Hirschman Index is very widely used in economics, sociology, and political science (see, e.g., Alesina and Ferrara, 2005; Fearon, 2003). In linguistics, it is better known as the Index of Ethno-Linguistic Diversity (Greenberg, 1956). If we partition a set into logically exhaustive and mutually exclusive subsets with the ith subset comprising a share p_i of the total, we can define the H-H Index as ∑(p_i)². To the extent that there is one subset that is preponderant, the H-H Index will tend toward 1. As fragmentation increases, the H-H Index will move toward 0.

Laakso and Taagepera (1979) were concerned with answering the question: “How many parties are there?” While this might seem a trivial question, some reflection shows that it is not. For example, it is clear that we have a very different situation in terms of party constellations if there is one party with 90% of the vote and one party with 10% of the vote, then we do if we have two parties each with 50% of the vote, even though we have exactly two parties in both situations. The Laakso-Taagepera Index of the Effective Number of Parties is defined as 1/∑(p_i)², where the p_i may be specified as the vote shares of the parties in an election (the effective number of electoral parties), or as the seat shares of the parties in parliament (the effective number of parliamentary parties). If, for example, there were n parties, all with the same share of votes or seats, then the L-T Index is simply n. The L-T Index, the inverse of the H-H Index, can be viewed as more easily interpretable than the H-H Index. But both the L-T Index and the H-H index can also be expressed in a perhaps even more familiar way, in terms of a means and variance framework (Feld and Grofman, 2007).²

But while the H-H Index and the L-T Index have each been applied to capture ethnicity or party, taken separately from one another, our interest in this paper is in the intersections of ethnicity and party. We show how to summarize such intersections in ways that directly extend the standard Laakso-Taagepera index. And we can gain useful information about the linkage between party systems and patterns of ethnic voting by assessing the calculated values of the extended indices in comparison to the simple L-T Index.

Our extended L-T indices will provide a single number for each party to represent its ethnic diversity, and a single number for each ethnicity to represent its support spread across the parties, and we will also provide a way to combine the party-specific (or ethny-specific) extended L-T values into one composite score that represents a weighted average of the individual values. The extended indices can be interpreted in the same way as the usual L-T Index with the value telling us the number of groups that create an equivalence to a situation where all the groups are of equal size. Using the extended Laakso-Taagepera indices has the same fundamental advantage over simply looking at string of numbers (or here, a table of numbers), namely that we can use a single number to represent a distribution of values, just as we use the L-T index to calculate an effective number of electoral or legislative parties. But, of course, we must always remember that we are using summary measures, and that different distributions can produce the same summary score.

If we percentage the E x P matrix by rows, then each row is an ethnic group, and the sum of the entries in that row is 100%. For convenience of notation, we will label the cell values in the rows in such a table as e_ij. If we percentage the E x P matrix by columns, then each column is a party, and the sum of the entries in that column is 100%. Again, for convenience of notation, we will label the cell values in the table percentaged by columns as p_ij. When we percentage an E x P matrix by rows we are creating conditional probabilities: p (e_i │p_j); when we percentage an E x P matrix by columns we are creating conditional probabilities: p (p_i │e_j).

Depending upon which percentaging we use, we can use an E x P matrix to simultaneously incorporate both parties and ethnic groups, either to examine the effective number of ethnic parties, or to examine the degree to which any given party is ethnically homogenous by generating for it an index of its effective number of ethnic components. In addition, we can offer a composite index to characterize the overall system of contestation with respect to both party and ethnicity.

While the focus of this paper is a methodological one, to offer a new—and we believe useful—extension of the L-T index, we also present two illustrative empirical applications of our new index: (a) U.S. two-party presidential election data since 1952 for two, or three or five ethnic groups; and, to show applicability beyond the two-party context, (b) an election in the Federation of Bosnia and Herzegovina (Federacija Bosne i Hercegovine) that involves three ethnic groups and five parties.

Of course, we recognize that to define the E x P matrix we need a categorization of ethnic groups, and this can be a controversial task (see, e.g., Chandra, 2012; Fearon, 2003). For example, with respect to language or religion, one could find Northern Ireland to lack ethnic parties, unless one differentiates “Christianity” into Catholic and Protestant. To be useful for classifying the link between party system and ethnicity any categorization of ethnic groups requires attentiveness to what ethnic/linguistic/religious cleavages are politically salient in a given country. Thus we recognize that no application of our extended L-T approach can be done without country-specific knowledge as to politically relevant categorization. A given ethnic categorization might be relevant in one country but not another, and that might depend upon the relative sizes of groups in the two countries (Posner, 2004), and might change over time.

More generally, the degree to which the categorization scheme is fine-grained will affect the results. We deal with these problems here by being explicit about our ethnic group categories in the illustrative data: using the three largest groups in Bosnia, ones whose relevance to the political system is constitutionally enshrined and, in the case of the United States, compare results done from a two-group black versus non-black perspective versus that from a three-group perspective that includes the two largest minority groups, African-American and Hispanics, and for 2016, where we have presidential exit poll data, for a five-group perspective that also include Asian-American, the next largest minority group in the U.S. that is one of the categories of racial and ethnic groups identified in the Voting Rights Act of 1965. We use the U.S. case to show changes in the extended L-T indices that demonstrate how ethnicity has come to matter more in differentiating the support bases of the two parties. The Bosnian case, which provides contrast to the U.S. (essentially) two-party system without ethnic parties, illustrates the general applicability of our approach

Then we discuss the interpretation of the extended L-T metrics we have defined and provide illustrations about how differences between them and the corresponding original L-T Index provide us useful information about why the original L-T values are as high (or as low) as they are. We end the paper with a brief discussion section.

Extending the Laakso-Taagepera Index to simultaneously cover both ethnic and party relationships

In this section, we offer an extension of the Laakso-Taagepera Index to a two factor E x P framework that will give us the mean effective number of parties with respect to ethnic groupings.

Begin with an E x P table and percentage the data in that table by rows. Each row is an ethnic group, and the sum of the entries in that row is 100%. Label the values in the row as e_ij. We can calculate the L-T index for any ethnic group, i, summed over parties, as

\frac{1}{\sum_{j} {(e_{ij})}^{2}}

(1)

Similarly, we can take the E x P table and percentage the data in that table by columns. Each column is a political party, and the sum of the entries in that column is 100%. Label the values in a column as p_ij. We can calculate the extended L-T index for any party, j, summed over ethnic groups, as

\frac{1}{\sum_{i} {(p_{ij})}^{2}}

(2)

These formulas give us m extended L-T values for the ethnic groups and n extended L-T values for the parties as equations (1) and (2), respectively.³To obtain a single overall value for the mean effective size of ethnic groups with respect to parties, we simply calculate the weighted average given by

\sum_{j} ([e_{ij}] (\frac{1}{{\sum_{j} (e_{ij})}^{2}}))

(3)

Similarly, to obtain a single overall value for the mean effective number of parties with respect to ethnic group composition we simply calculate the weighted average given by

\sum_{i} ([p_{ij}] (\frac{1}{{\sum_{i} (p_{ij})}^{2}}))

(4)

In other words, what we have done is to combine the results of equation (2) to put together a measure of party x ethnic effective size that averages the L-T values for each party weighted by the party size, and similarly to combine the values from equation (1) to create a measure of ethnic x party effective size that averages the L-T values for each ethnicity weighted by ethnicity size.

There is one last use of the L-T Index in the context of both parties and ethnicities, namely when we percentage the E x P matrix by total, using (ep)_ij to represent the cells in that matrix, to obtain the measure. We refer to this measure as composite effective ethnicity-party fragmentation size.

\sum_{i} \sum_{j} ((\frac{1}{{\sum_{i} (e p_{ij})}^{2}}))

(5)

Applying the extended L-T index to the U.S. and the Federation of Bosnia and Herzegovina

Using data from the United States and then, for comparison purposes, the Federation of Bosnia and Herzegovina (Federacija Bosne i Hercegovine), we show how the parties in each country, and the party system overall, fit within our categorization schemes.

United States

Using data from a CNN exit poll from the November 2016 election, we present data on support for the Democratic Party presidential candidate (Hillary Clinton) and the Republican Party presidential candidate (Donald Trump) in terms of a five-fold classification of ethnicity that is operationalized to be mutually exclusive and logically exhaustive, with groups listed in order of share of the 2016 electorate, {non-Hispanic White,⁴ African-American,⁵ Hispanic, Asian-American, other}.⁶ To simplify the analysis we only look at votes for the two major party candidates; this takes us from an initial survey of 24,558 down to 22,021 respondents. We first present the raw data from the exit poll, then this same data percentaged by total, then the data percentaged by rows, and then the data percentaged by columns.

Now we can use the data in Table 1 to calculate the usual Laakso-Taagepera Index and then the mean effective number of parties with respect to ethnicity and the mean effective number of parties with respect to ethnicity for the five ethnic groups with respect to the two parties. To calculate mean effective number of parties with respect to ethnicity, we begin with the LT value for each row in the E x P matrix percentaged by rows, Table 1(c). For example, the LT party value for the largest group, non-Hispanic whites is 1/(.39²+.61²) = 1.91, while the value for the next largest group, African-Americans, is 1/(.92²+.08²) = 1.17. Etc. To get the composite score we average those five values (one for each ethnic group) weighted by the share of the ethny in the electorate—a share that is shown in the last column in Table 1(b).

Number of parties = 2

L-T index of effective number of parties = 2.0

Mean effective number of parties with respect to ethnicity = 1.81⁷

Effective number of parties for Non-Hispanic Whites = 1.91

Effective number of parties for African-Americans = 1.17

Effective number of parties for Hispanics = 1.72

Effective number of parties for Asian-Americans = 1.70

Effective number of parties for others/decline to state = 1.91

Table 1.

2016 U.S. presidential exit poll for five ethnic groups and two parties.

(a) Raw data
CNN
Exit
Poll	n = 24558; two-party vote data for n = 22021
Group	Clinton	Trump	Sum (C + T)
NH White	6064	9342	15,407
Black	2544	229	2773
Hispanic	1676	711	2387
Asian	587	244	831
	380	244	624
Total	11,251	10,770	22,021
	0.51	0.49
(b) Data percentaged by total
CNN
Exit
Poll	n = 24558; two-party vote data for n = 22021
Group	Clinton	Trump	Sum (C + T)
NH White	0.28	0.42	0.70
Black	0.12	0.01	0.13
Hispanic	0.08	0.03	0.11
Asian	0.03	0.01	0.04
other/no answer	0.02	0.01	0.03
Total	0.51	0.49	1.00
(c) data percentaged by rows
CNN
Exit
Poll	n= 24558; two-party vote data for n =22021
Group	Clinton	Trump	Sum (C + T)
NH White	0.39	0.61	1.00
Black	0.92	0.08	1.00
Hispanic	0.70	0.30	1.00
Asian	0.71	0.29	1.00
other/no answer	0.61	0.39	1.00
(d) data percentaged by columns
CNN
Exit
Poll	n = 24558; two-party vote data for n = 22021
Group	Clinton	Trump
NH White	0.54	0.87
Black	0.23	0.02
Hispanic	0.15	0.07
Asian	0.05	0.02
other/no answer	0.03	0.02
Total	1.00	1.00

Similarly, we can use the data in Table 1 to calculate the usual Laakso-Taagepera Index and then the mean effective number of ethnic groups with respect to party support for the two parties with respect to the five ethnic groups. To calculate the latter, we begin with the L-T value for each column in the E x P matrix percentaged by columns, Table 1(d). For example, the LT party value for the Democrats is

1 / ({. 54}^{2} + {. 23}^{2} + {. 15}^{2} + {. 05}^{2} + {. 03}^{2}) = 2.72

while the value for the Republicans is 1.32. In other words, the Democrats are an ethnically diverse party while the Republicans are a party where one group, here non-Hispanic whites, makes up the overwhelming majority of the party’s supporters. To get the composite score we average those two values (one for each party) weighted by the share of the party in the electorate—that share is shown in the last row in Table 1(b). We summarize these results below.

Number of ethnic groups = 5

L-T index of effective number of ethnic groups = 1.91.

Mean effective number of ethnic groups with respect to party support = 2.03⁸

Effective number of ethnic groups with respect to Democratic party support = 2.72

Effective number of ethnic groups with respect to Republican party support = 1.32

The mean effective number of ethnic groups with respect to party support is higher than the L-T Index of effective number of ethnic groups because, even though one of the parties is heavily concentrated in its ethnic support, that is, non-Hispanic whites for the Republican party, the other is ethnically diverse.

Our last L-T related measure takes the E x P matrix and percentages it by total. Now we are essentially looking at coalitional segments consisting of the ith ethnic group and the jth party. We refer to this index as the Compound L-T Index for short. Because there are multiple segments the Compound L-T Index we get will be higher, probably considerably higher, than for either the original L-T Index or the other extensions of the L-T Index described above. For the five ethnic groupings in the U.S. shown in Table 1 and the two major U.S. parties; in 2016 this Index had a value of 3.6.

To provide some historical comparisons we look at data from Zingher (2014) from 1952 to 2008. For 1972–2008 his data identifies the level of group support for the Democrats for the categories of African-American and Latino, and we can also identify the share of the Democratic voters who come from the remaining, almost entirely non-Latino white group.⁹ For 1952–1968, we only are given the share of African-American voters, and so the residual category will also include Latinos as well as Asian-American and those who list themselves as other, but in this time period, among voters, the number who fall into any of the latter categories are quite small. Zingher does not report comparable data for Republican support, but we know that the Republican party was overwhelmingly non-Latino white over this time period, with support very close to what we found in 2016, so looking only at Democrats for this data is not too much of an issue. On the other hand, there are substantial changes in the racial and ethnic composition of Democratic party support over the period 1972–2008 that the extended L-T index captures.

Figure 1 reports the mean effective number of ethnic groups with respect to party support with respect to a three-group and a two-group breakdown of ethnic groups. What we see very clearly is that the combined forces of white desertions of the party and growing Latino voting strength acted to substantially increase our measure of mean effective number of ethnic groups with respect to party support whether we look at a three-group or a two-group model. But by comparing the two-group model to the three-group model that includes Hispanics, it is obvious how growth in Hispanic population drove the dynamic in the 21st century. On the other hand, by looking at the two bloc model for the period 1952–1968 we see how much of we now see that the portrait of the Democratic party as the party of minorities also required the gains in black registration and turnout that came in the 1960s with the passage of the Civil Rights Act of 1964 and the Voting Rights Act of 1965.

Figure 1.

Mean effective number of ethnic groups with respect to party support (1952–2008). Source: Zingher (2014) and the author’s calculations.

Of course, ceteris paribus, the more groups we consider the higher will be this metric, so we need to recalculate our 2016 data to make it comparable to the data we are using in the 1952–2008 period. Doing so we find that, in 2016, for the three-group model, we have a mean effective number of ethnic groups with respect to party support value of 2.18, intermediate between what we found in 2004 and what we found in 2008. That measure for the two-group model (black vs non-black) is 1.55, again intermediate between what we found from the Zingher (2014) data in 2004 and 2008. The 21st century is a period of intensifying “ethnification” of party conflict in the U.S.

Federation of Bosnia and Herzegovina (Federacija Bosne i Hercegovine)

The 1995 Dayton Peace Agreement (DPA) divided Bosnia-Herzegovina into two “entities,” the “Federation of Bosnia and Herzegovina” and the Republika Srpska, and established a weak over-arching state structure. While those accords established two entities,¹⁰ they recognized three “constituent peoples”: Bosniaks, Serbs and Croats. We analyze data from the 2001 World Values Survey in Bosnia and Herzegovina, which offers individual-level data on party support and ethnicity, language, and religion. While the 2001 World Values survey is not the only survey that allowed cross-tabulation of party and ethnicity, it is the only one that (a) covered the whole of Bosnia-Herzegovina,¹¹ with 800 respondents from the Federation of Bosnia and Herzegovina and 400 respondents from the Republika Srpska, and most importantly (b) directly asks a question about ethnicity that does not require one to use indirect proxies like “language” or “religion” that are not very good measures.¹²

Because we using this data only to illustrate the applications of our metrics and our classification,¹³ and we wish to keep the analysis manageable, we limit ourselves to one part of Bosnia-Herzegovina, the Federation of Bosnia and Herzegovina, to demonstrate how our methods can apply to a “region” within a country.¹⁴ We examine data for five parties that contested in the Federation of Bosnia and Herzegovina in the election held prior to the 2001 survey, and we limit our analyses to the three largest ethnic groups—Bosniaks, Serbs, and Croats. Excluding those who gave no answer to the party or ethnicity question reduced our sample size from 800 to 472.

The three wartime nationalist parties were the Party of Democratic Action (SDA), the Serb Democratic Party (SDS) and the Croatian Democratic Union (HDZ) based, respectively, in the Bosniak, Serb and Croat communities. These three parties were the key victors of the pre-war 1990 election and also the initial post-war 1996 and 1998 state-wide elections. Another relevant party is the Social Democratic Party (SDP), the former Communists, which advances a liberal democratic and multi-ethnic agenda, but acquires most of its support from the Bosniak electorate. The last of our five parties is BOSS, a small party that, as we will see, draws its support from Bosniaks.

Table 2 shows the useable raw data for Federation of Bosnia and Herzegovina from the 2001 World Values Survey, then this same data percentaged by total, then the data percentaged by rows, and then the data percentaged by columns.

Table 2.

2001 Federation of Bosnia and Herzegovina survey data for three ethnic groups and five parties.

(a) Raw data
Bosnia 2001—Federation of Bosnia and Herzegovina only—raw numbers
Groups/Parties	SDP	BOSS	HDZ	SDA	SB&H	Total with data
Bosniak	131	19	0	126	118	394	0.83
Croatian	23	2	38	0	3	66	0.14
Serbian	11	0	0	0	1	12	0.03
Total with data	165	21	38	126	122	472
	0.35	0.04	0.08	0.27	0.26		1.00
(b) Data percentaged by total
	SDP	BOSS	HDZ	SDA	SB&H
Bosniak	0.28	0.04	0.00	0.27	0.25
Croatian	0.05	0.00	0.08	0.00	0.01
Serbian	0.02	0.00	0.00	0.00	0.00
Total	0.35	0.04	0.08	0.27	0.26	1.00
(c) Data percentaged by columns
	SDP	BOSS	HDZ	SDA	SB&H	Total
Bosniak	0.79	0.90	0.00	1.00	0.97	0.83
Croatian	0.14	0.10	1.00	0.00	0.02	0.14
Serbian	0.07	0.00	0.00	0.00	0.01	0.03
Total	1.00	1.00	1.00	1.00	1.00	1.00
(d) Data percentaged by rows
	SDP	BOSS	HDZ	SDA	SB&H	Total
Bosniak	0.33	0.05	0.00	0.32	0.30	1.00
Croatian	0.35	0.03	0.58	0.00	0.05	1.00
Serbian	0.92	0.00	0.00	0.00	0.08	1.00
Total	0.35	0.04	0.08	0.27	0.26	1.00

The distribution of party support in the Republika Sprska at that same time was very different, largely reflecting the differences in ethnic composition in the two parts of Bosnia.

We can use the data in Table 2 to calculate the usual Laakso-Taagepera index and then the mean effective number of parties with respect to ethnicity and the mean effective number of parties with respect to ethnicity for the three ethnic groups with respect to the five parties. The usual L-T index is 3.78.

To calculate mean effective number of parties with respect to ethnicity, we begin with the L-T value for each row in the E x P matrix percentaged by rows, Table 2(c). For example, the L-T party value for the largest group, Bosniaks is

1 / ({. 33}^{2} + {. 05}^{2} + {.0}^{2} + {. 32}^{2} + {. 30}^{2}) = 3.28 .

For the next largest group, Croatians, it is 1/(.35² +.03² +.58² +.0² +.05²) = 2.19, while the value for the last and smallest group, Serbians, is 1/(.92² +.0² + 0² +.0² +.08²) = 1.18.

To get the composite score we average those three values (one for each ethnic group) weighted by the share of the ethny in the electorate—that share is shown in the last column in Table 2(a). This gives us a value for the mean effective number of parties with respect to ethnicity of 3.08 since Bosniaks are the overwhelmingly preponderant group in this data set and so we get a weighted average not that far from the value for this group alone.

Analogously we may calculate the mean effective number of ethnic groups with respect to party support and compare it to the usual calculation of the L-T effective number of ethnic groups, which in this case equals 1.39. The values for the five components needed for the first calculation, the effective number of ethnic groups with respect to party j, are as follows.

SDP	1.53
BOSS	1.21
HDZ	1.00
SDA	1.00
SBiH	1.21

Weighting these values by the party’s share of the electorate we find mean effective number of ethnic groups with respect to party =1.25. The mean effective number of ethnic groups with respect to party is lower than the L-T index of effective number of ethnic groups because there are some parties that highly homogeneous in their ethnic support.

We summarize the main L-T related results for Bosnia below.

Number of parties = 5

L-T index of effective number of parties = 3.78

Mean effective number of parties with respect to ethnicity = 3.08

Number of ethnic groups = 3

L-T index of effective number of ethnic groups = 1.39

Mean effective number of ethnic groups with respect to party = 1.25

We can use the data in Table 2 to calculate the usual Laakso-Taagepera index and then the mean effective number of parties with respect to ethnicity and the mean effective number of parties with respect to ethnicity for the three ethnic groups with respect to the five parties. The usual L-T index is 3.78.

To calculate mean effective number of parties with respect to ethnicity, we begin with the LT value for each row in the E x P matrix percentaged by rows, Table 2(c). For example, the LT party value for the largest group, Bosniaks is

1/(.33²+.05²+.0²+.32²+.30²) = 3.28.

For the next largest group, Croatians, it is 1/(.35² +.03² +.58² +.0² +.05²) = 2.19, while the value for the last and smallest group, Serbians, is 1/(.92² +.0² + 0² +.0² +.08²) = 1.18

SDP	1.53
BOSS	1.21
HDZ	1.00
SDA	1.00
SBiH	1.20

Weighting these values by the party’s share of the electorate we find mean effective number of ethnic groups with respect to party = 1.25. The mean effective number of ethnic groups with respect to party is lower than the L-T index of effective number of ethnic groups because there are some parties that highly homogeneous in their ethnic support.

We summarize the main L-T related results for Bosnia below.

Number of parties = 5

L-T index of effective number of parties = 3.78

Mean effective number of parties with respect to ethnicity = 3.08

Number of ethnic groups = 3

L-T index of effective number of ethnic groups = 1.39

Mean effective number of ethnic groups with respect to party = 1.25

Our last L-T related measure takes the E x P matrix and percentages it by total. For the three ethnic groupings in the Federation of Bosnia and Herzegovina shown in Table 2 and the three major ethnic groups, this Compound L-T Index had a value of 4.45. As we would expect this segmentalized aspect of ethnicity-party linkage yields the highest value of any of our L-T related indices.

Comparing the extended L-T indices to the original L-T index

While above we have provided empirical results for both the original L-T Index and our extended indices for the U.S. and a portion of Bosnia, in this section we turn to a more analytic approach to comparisons between the various indices.

Proposition 1a

Equation (4), the mean effective number of parties with respect to ethnicity can be either higher or lower than standard L-T value for effective number of parties.

We can illustrate this point with 3 × 3 E × P matrices. For example, to obtain a value for equation (4) that is lower than the usual L-T value, imagine that we have three ethnic groups of sizes ½, 1/3 and 1/6, respectively, and that each ethnic group has its own ethnic political party with all members of the group in that party, so the parties are also of sizes ½, 1/3 and 1/6 respectively. Now ∑_je_ij = ∑_ip_ij, so equation (4) = 1/(1/2)*1 + (1/3)*1 + (1/6)*1 = 1, while the usual L-T index gives us a value of 1/((1/2)² + (1/3)² + (1/6)²) =36/23 = 1.57. This is a pure ethnic party system.

To get a higher value from equation (4) than the usual L-T value, let the three ethnic parties again have a support base of ½, 1/3 and 1/6, respectively, and let the ethnic groups also be of sizes ½, 1/3 and 1/6, respectively but let party 1 consist of equal parts of ethnic groups E₁, E₂, and E₃, that is, it has one-third of its voters from each of the three ethnic groups (1/6 + 1/6 +1/6 = 1/2). Since party 1 has half the total party support, while E₁ also has half of the voters that leaves 2/3^rd of the voters in group E₁ (2/6^th of all the voters) to be distributed to the remaining two parties. Similarly, E₂, which has one-third of the voters, now has half of its members (1/6^th of all the voters) that can be distributed to the remaining two parties. However, at this point, all of the voters in E₃ are in party one so there are none left for the other two parties. Assume now that half the voters in party 2 are from E₁ and half from E₂. That uses up of all the voters in E₂, so that party 3 is an ethnic party with 1/6th of the population supporting it, all of whom come from E₁. Now the value of equation (4) = ½*3 +1/3*2+1/6*1= 2.33. Here, for two of the parties, we have a mixed constellation with what in forthcoming work by the present authors we label as ethny incorporating parties, that is, ones where a given (small) ethnic group gives the bulk of its support to a party the bulk of whose membership comes from one or more other ethnic groups. The third party is what we would call, based on its ethnic support, an ethnic party.

To get the same value from equation (4) as from the usual L-T value, let the size of the parties be ½, 1/3 and 1/6 respectively, but now let each political party have a support base that perfectly mirrors the ethnic composition of the whole population, that is, equation (4) = (1/2)*1.57 + (1/3)*1.57 + (1/6)*1.57 = 1.57. This party constellation has neither ethnic parties nor ethnic-incorporating parties.

We have a parallel result.

Proposition 1b

Equation (3), the mean effective number of ethnic groups with respect to parties can be either higher or lower than standard L-T value for effective number of parties.

But we can go beyond simply saying that extended L-T values may differ from one another in either direction.

Proposition 2a

Ceteris paribus, when we have all or mostly ethnic parties, that is, parties drawing virtually all of their support from a single ethnic group, then we expect the mean effective number of parties with respect to ethnicity be lower than the usual L-T index.

Proposition 2b

Ceteris paribus, when we have all or mostly ethnic-incorporating parties, that is, parties with support from multiple ethnic groups with no ethnic minority group majority of the party’s supporters, then we expect the mean effective number of parties with respect to ethnicity to be higher than the usual L-T index. The two will have the same value only the very special circumstance that each ethnic group divides its support equally over each of the parties.

Equation (5) can be taken as a measure of overall fragmentation with respect to both party and ethnic grouping. Because we are dividing by the total, it automatically takes into account the relative sizes of ethnic groups and party support bases.

Proposition 3

Equation (5) gives a value that coincides with the usual L-T Index only when the number of parties and the number of ethnic groups are the same and there are only ethnic parties, or when the ethnic groups divide their votes equally among all parties. In general, we expect the mean effective number of parties with respect to ethnic group composition to be (considerably) higher than mean effective number of parties with respect to ethnicity because there will be more segments to be squared and summed.

Taking into account these propositions, we can also use comparisons of values for the extended L-T indices with that of the corresponding usual L-T index to gain insight into the way in which ethnic fragmentation affects party fragmentation. We now briefly revisit our main findings for the U.S. and Federation of Bosnia and Herzegovina cases:

For the U.S., both our generalized versions of the L-T index and the original versions of that index yield very similar findings to one another, with values close to 2 (1.81, 2; 2.03, 1.91) due to the fact that the dominant ethnic grouping, non-Hispanic whites, tends toward support of the Republican party, but still with both Republicans and Democrats receiving the bulk of their support from this group. The Democrats are much more ethnically diverse, which gives them support comparable to the Republicans, despite less support from non-Hispanic whites. This creates a two-party system in terms of effective number of parties, and very competitive parties at the national level, as well as something close to two effective ethnic groups.

For the Federation of Bosnia and Herzegovina, our two generalized versions of the L-T index and the original version of that index yield somewhat different results from the simple L-T index. The original L-T index for parties is 3.78 but, when we look by ethnicity, the equivalent extended index is lower at 3.08. We can see that one key reason for the higher value of the original L-T index is the fragmentation of Bosniak support into multiple parties. The original L-T index for ethnic groups is 1.39 but, when we look by party, the equivalent extended index is lower at 1.25. We can see that one key reason for the high value of the original L-T index is the existence of ethnic parties in Bosnia.

These interpretations of the cross-tabular data for the two countries will, of course, come as no surprise to political scientists knowledgeable about these cases, but the extended L-T indices provide a compact way to summarize the information in E x P tables. Moreover, for effective party size, comparison of the original L-T index value to that of the comparable extended index helps us interpret how the ethnic distribution of party support affects the size distribution of the parties and thus the effective number of parties.¹⁵

Discussion

There are already many articles that use the H-H Index, to measure ethnicity,¹⁶ and of course, the L-T Index has been used in hundreds of articles for measuring party fragmentation at the electoral and/or legislative level.¹⁷ Even if we limit ourselves to articles that integrate both ethnic and party considerations there is a large literature on ethnic parties,¹⁸ as well at work on voting behavior that looks more generally at how ethnic groups vote.¹⁹ And there is other work that applies a conditional probability approach to the party and ethnicity context (Chandra, 2012; Huber, 2012). As we see it, the distinctive contribution of this essay is the simplicity and directness of our approaches for any party system for which we have data on ethnic voting patterns.

We offer measurement tools for assessing the linkages between ethnic groupings and party systems that are readily interpretable because they directly extend the well-known L-T Index to simultaneously cover both parties and groups in terms of effective size of components.²⁰ There is no other work of which we are aware that does this. And, by looking at comparisons between our measures and the usual L-T Index we can gain insight into how ethnic voting patterns, for example, the existence of ethnic parties, may affect the usual L-T index calculations.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Bernard Grofman

Notes

Author biographies

Bernard Grofman is Jack W. Peltason Chair of Democracy Studies and Distinguished Professor of Political Science at the University of California, Irvine. His research deals primarily with issues of representation, including minority voting rights and party competition. A Fellow of the American Academy of Arts and Sciences, he has an honorary PhD from the University of Copenhagen for his work on the study of elections. He is co-author of five books with major university presses, and co-editor of 26 other books, with over 300 research articles and book chapters.

Jon Fraenkel is Professor of Comparative Politics, School of History, Philosophy, Political Science and International Relations, Victoria University, Wellington, New Zealand.

References

Alesina

La Ferrara

(2005) Ethnic diversity and economic performance. Journal of Economic Literature 43(3): 762–800.

Alvarez

Nagler

(2004) Party system compactness measurement and consequences. Political Analysis 12: 46–62.

Bernauer

Bochsler

(2011) Electoral entry and success of ethnic minority parties in central and eastern europe: a hierarchical selection model. Electoral Studies 30(4): 738–755.

Birnir

(2007) Ethnicity and Electoral Politics. Cambridge: Cambridge University Press.

Bochsler

(2010) Territory and Electoral Rules in Post-Communist Democracies. London: Palgrave-McMillan.

Bochsler

(2011) It is not how many votes you get, but also where you get them. territorial determinants and institutional hurdles for the success of ethnic minority parties in post-communist countries. Acta Politica 46(3): 217–238.

Chandra

(2004) Why Ethnic Parties Succeed. New York, NY: Cambridge University Press.

Chandra

(2005) Ethnic parties and democratic stability. Perspectives on Politics 3(2): 235–252.

Chandra

(ed) (2012). Constructivist Theories of Ethnic Politics. Oxford: Oxford University Press.

10.

Clark

Golder

(2006) Testing the mechanical and strategic modifying effects of electoral laws rehabilitating duverger’s theory. Comparative Political Studies 39: 679–708.

11.

Dalton

(2008) The quantity and the quality of party systems: party system polarization, its measurement and its consequences. Comparative Political Studies 41: 899–920.

12.

Dumont

Caulier

J-F

(2003) The ‘Effective Number of Relevant Parties’: How Voting Power Improves Laakso-Taagepera’s Index. Belgium: Catholic University of Louvain. Unpublished manuscript, Comparative Politics Center.

13.

Fearon

(2003) Ethnic and cultural diversity by country. Journal of Economic Growth 8(2): 195–222.

14.

Feld

Grofman

(2007) The Laakso-Taagepera Index in a means and variance framework. Journal of Theoretical Politics 19(1): 101–106.

15.

Ferree

(2006) Explaining South Africa’s racial census. Journal of Politics 68(4): 803–815.

16.

Gaines

Taagepera

(2013) How to operationalize two-partyness. Journal of Elections, Public Opinion and Parties 23(4): 387–404.

17.

Greenberg

(1956) The measurement of Linguistic diversity. Language 32: 109–115.

18.

Golosov

(2010) The effective number of parties: a new approach. Party Politics 16(2): 171–192.

19.

Grofman

Kline

(2012) How many political parties are there, really? A new measure of the ideologically cognizable number of parties/party groupings. Party Politics 18: 523–544.

20.

Grofman

Selb

(2009) A fully general index of political competition. Electoral Studies 28(2): 291–296.

21.

Hirschman

(1964) The paternity of an index. The American Economic Review 54(5): 761–762.

22.

Horowitz

(1985) Ethnic Groups in Conflict. Berkeley, CA: University of California Press.

23.

Huber

(2012) Measuring ethnic voting: do proportional electoral laws politicize ethnicity? American Journal of Political Science 56(4): 986–1001.

24.

Hulsey

Keil

(2019) Ideology and party system change in consociational systems: the case of non-nationalist parties in bosnia and herzegovina. Nationalism and Ethnic Politics 25(4): 400–419.

25.

Ishiyama

(2009) Do Ethnic Parties Promote Minority Ethnic Conflict? Nationalism and Ethnic Politics 15(1): 56–83.

26.

Kline

(2009) How we count counts: the empirical effects of using coalitional potential to measure the effective number of parties. Electoral Studies 28: 261–269.

27.

Laakso

Taagepera

(1979) Effective number of parties: a measure with application to West Europe. Comparative Political Studies 12: 3–27.

28.

Lijphart

(1994) Electoral Systems and Party Systems. A Study of Twenty-Seven Democracies 1945-1990. Oxford: Oxford University Press.

29.

Lublin

(2014) Minority Rules. Electoral Systems, Decentralization, and Ethnoregional Party Success. Oxford: Oxford University Press.

30.

Lublin

(2017) Electoral systems, ethnic diversity and party systems in developing democracies. Electoral Studies 47: 84–93.

31.

Mattes

Taylor

Africa

(1999) Judgment and choice in the 1999 South African election. Politikon 26(2): 235–247.

32.

Posner

(2004) The political salience of cultural difference: why Chewas and Tumbukas are Allies in Zambia and Adversaries in Malawi. American Political Science Review 98: 529–545.

33.

Reilly

(2007) Democracy and Diversity: Political Engineering in the Asia-Pacific. Oxford: Oxford University Press.

34.

Reynolds

(ed) (2001). The Architecture of Democracy: Constitutional Design, Conflict Management, and Democracy. Oxford University Press.

35.

Reynolds

(2010) Designing Democracy in a Dangerous World. Oxford University Press.

36.

Ruiz-Rufino

(2013) Satisfaction with democracy in multi-ethnic countries: the effect of representative political institutions on ethnic minorities. Political Studies 61: 101–118.

37.

Sartori

(1966) European political parties: the case of polarized pluralism. In: LaPolambara

Neiner (eds) Political Parties and Political Development. Princeton, NJ: Princeton University Press, 137–176.

38.

Selway

(2010) Cross-Cuttingness, Cleavage Structures and Civil War Onset. British Journal of Political Science 41: 111–138. DOI: 10.1017/S0007123410000311

39.

Taagepera

(2007) Predicting Party Sizes: The Logic of Simple Electoral Systems. Oxford and New York, NY: Oxford University Press.

40.

Van Cott

(2003) Institutional change and ethnic parties in South America. Latin American Politics and Society 45(2): 1–39.

41.

Van Cott

(2005) From Movements to Parties in Latin America: The Evolution of Ethnic Politics. New York, NY: Cambridge University Press.

42.

Zingher

(2014) An analysis of the changing social bases of America’s political parties: 1952–2008. Electoral Studies 35: 272–282.