Statistical Characteristics of the 2021 State Duma Election

Lyubarev A.E.

Abstract

This paper examines vote returns in the 2021 State Duma election using statistical methods. The author estimates the effective number of parties (candidates) for the federal constituency and single-seat constituencies, as well as the disproportionality index. Vote splitting is studied in the context of federal and single-seat constituency. It is revealed that many parties are subject to the so-called "contamination effect," which means an improved result of the party in the federal constituency given that it has a candidate in a single-seat constituency. Furthermore, being placed first on the ballot has a certain impact on the candidate's final performance. For all parties participating in the elections, the author calculates indices of territorial homogeneity of voting and evaluates the differences in vote returns between the regional capital and the region as a whole. Calculations are made to establish number of TECs where both the turnout rate and the results of United Russia exceed 90%. Correlation coefficients were calculated between the results of the parties that participated in these elections, as well as between the results of the parties in the 2016 and 2021 elections.

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The 2021 State Duma election results are no secret. There is a paper by Alexander Kynev [5] that analyzes the results of political parties (including on a region-by-region basis) in said election and compares them to previous election's results. Our recent work [6] analyzed vote overflows from parties in the 2016 election to parties in the 2021 election.

Still, some issues have been left without attention. This paper aims to examine the vote returns in the 2021 State Duma election using methods that we applied to the previous such election. These are methods that make it possible to assess electoral competition, disproportionality of election results, vote splitting in voting for parties and candidates, geographical features of voting, and correlation between party results. Most of these methods have been described in [11]. The application of these methods to the 2016 State Duma election has been summarized in [4].

General characteristics, competition and disproportionality

Fourteen political parties competed in the 2021 State Duma election. Five parties passed the five percent threshold, receiving a total of 89.1 percent of votes. The leading party, United Russia, received 49.8%, while CPRF, which fell 30.9% behind, placed second as usual. For the first time since 2003, a fifth party joined the State Duma — New People, which placed fifth with 5.3%.

The other nine parties gained less than 3% each, totalling 8.8%. Invalid ballots amounted to 2.1% (a record for a federal election since 1995).

In single-seat constituencies, the elected candidates included 198 from United Russia, 9 from CPRF, 8 from A Just Russia, two from LDPR, one from Rodina, Civil Platform and the Party of Growth each, plus five self-nominated candidates.

To characterize party competition / fragmentation, we use the effective number of parties. To calculate this number, we propose several different indices. The Laakso-Taagepera index is the most praclical; it is calculated by \(1/\sum_i n_i^2\), where \(n_i\) is the share of votes received by the \(i\)-st/nd/th candidate (of the number of valid votes). For the systems with a dominant party, we propose the Golosov index, which is calculated by \(\sum_i{1/[1+(n_1^2/n_i)–n_i]}\), where \(n_i\) is the share of votes received by the \(i\)-st/nd/th candidate, and \(n_1\) is the share of votes received by the leading candidate (all shares are calculated out of the number of valid votes) [1; 3].

For the elections in the federal constituency, the Laakso-Taagepera and Golosov indices amounted to 3.20 and 2.44, respectively. Both indices are higher than the corresponding values for the 2007, 2011, and 2016 elections, but lower than the values for the 1993, 1995, 1999, and 2003 elections.

Competition indicators by single-seat constituencies are shown in Table 1. If we compare them to the results of the 2003 and 2016 elections, competition in 2021 was more intense than in 2016 yet less so than in 2003 for most of the average values. Between the 2003 and 2016 values, there was also a number of constituencies where the winner's gained over 50% (65 and 84, respectively), and a number of constituencies where the winner's advantage over the opponent was under 10% (72 and 16, respectively). Only the average value of the Laakso-Taagepera index was higher in 2021 than in 2003 (when it amounted to 3.52). On the other hand, the Golosov index also puts the result between the 2003 and 2016 values (they were 3.16 and 2.75, respectively).

Another important indicator is the degree of disproportionate seat allocation. We typically use two disproportionality indices: the Loosemore–Hanby index that is calculated by \(½ \sum_i|v_i ‑ s_i|\), and the Gallagher index tht is calculated by \(\sqrt{½ \sum_i(v_i ‑ s_i)^2}\). In both formulas, \(v_i\) is the share of votes given to the \(i\)-st/nd/th party (of the number of valid votes), and \(s_i\) is the number of seats the party gained. For the 2021 election, the Loosemore–Hanby index amounted to 22.3% and the Gallagher index to 16.2%. Both indices are slightly higher than in the 2016 election (they were at 21.2% and 15.8%, respectively), but the Loosemore–Hanby index is lower than in the 1993, 1995, 1999, and 2003 elections. The Gallagher index is higher than in 2003 (in 2007 and 2011, when elections were conducted under proportional representation, both indices were markedly lower).

Vote splitting

Most parties participating in the elections nominated a significant number of candidates in single-seat constituencies. On election day, LDPR candidates were on the ballot in 224 constituencies, CPRF in 223, United Russia and A Just Russia in 217, and New People in 200. In other words, representatives of all major parties participated in the elections in most single-seat constituencies. Other parties nominated fewer candidates, but still quite a lot for the most part: Russian Party of Pensioners for Social Justice (the Party of Pensioners) had candidates in 187 constituencied, Yabloko in 142, Rodina in 134, and Communists of Russia in 130. The outsider parties nominated fewer candidates: Party of Growth with 91, REP The Greens with 89, Russian Party of Freedom and Justice (RPFJ) with 65, Civic Platform with 56, and Green Alternative with 27. There were almost no self-nominated candidates (only 10), and none at all from parties that did not participate in the elections in the federal constituency.

As a result, in a single-seat constituency, most voters could vote for a representative of the same party whose list they had supported when voting in the federal constituency. At the same time, the previous State Duma election, as well as regional elections — all held under the mixed system — show that when it comes to single-seat constituencies, Russian voters tend to support representatives of a different party rather than the one for whose list they vote in the whole constituency. This phenomenon is not exclusive to Russia and is known as "vote splitting".

In a previous article, we proposed several parameters that assess the behavior of voters in a mixed system, including the "vote splitting" [10]. The following indicators can be estimated for each party that nominated both the list and a candidate:

─ average vote gap index (VGIavg) is the arithmetic average of the difference between a candidate's result and the a party list's result in all single-seat constituencies;

─ candidate outperformance index (COI) is the proportion of single-seat constituencies where a candidate's result is higher than that of party;

─ asymmetry of the distribution of the vote gap index by single-seat constituencies;

─ candidate influence index (CII) is the ratio of the average result of a party in constituencies where its candidates ran for office to the average result of a party in constituencies where it did not have candidates (the index can only be calculated for parties that did not have candidates in all constituencies);

─ the correlation coefficient between the results of candidates and the party in the context of single-seat constituencies.

Table 2 contains these indicators for all 14 parties that took part in the elections. Vote returns are presented as a percentage of the number of valid ballots. First of all, we should consider the correlation between the candidate and party results. For most parties, the correlation is quite high (over 0.7). LDPR has a slightly lower correlation coefficient (0.68). The coefficient is even lower (0.5-0.65) for Communists of Russia, REP The Greens, and RPFJ, and considerably low (under 0.3) for Green Alternative and Civic Platform.

The interrelated values of VGIavg and COI deserve special consideration. The negative VGIavg value and the COI value of <50% suggest that, on average, voting for candidates in constituencies is less intense than that for the party itself. The table shows that United Russia, CPRF and LDPR have these very values.

Results like that are quite consistent with those of previous elections. CPRF also had a COI value of less than 50% in 1995, 1999, 2003, and 2016, and its VGIavg was negative in 1995, 1999, and 2016. In 2003, United Russia also had a negative VGIavg value, and its COI amounted to 48%. In 2016, these values were almost the same as this time (-1.81% and 35%).

In LDPR's case, negative VGIavg and low COI are particularly typical. For example, LDPR's VGIavg reached -4.79% and ‑8.09% in 1995 and 2003, respectively, while COI reached 5% and 1%. However, the 2016 numbers were already less distinct (-3.66% and 16%). At this point, the party's values are even closer to the average. In other words, LDPR candidates began to achieve higher results, approaching those of the party.

New people also exhibit a negative VGIavg value, although that close to zero, and its COI value (44%) is close to the average. To put it another way, on average the party's candidates performed slightly worse than the party itself.

Positive VGIavg and COI>50% mean that on average, candidates got more votes in constituencies than their parties. A Just Russia (like in the 2016 regional elections) and all non-parliamentary parties seem to repeat the pattern. However, our analysis has shown that higher support for candidates compared to party support (up to 100% of the COI value) is characteristic of outsider parties regardless of their nature, and this is evident not only in Russia, but also in Germany and Ukraine [10]. A possible explanation for this phenomenon may lie in the fact that in the vast majority of cases, the number of single-seat candidates is less than the number of party lists participating in the elections. Voting for the lists of outsider parties and their candidates occurs largely at random.

A high positive asymmetry value means that some candidates get significant results against the background of low results of the majority of candidates from a given party. We can see that Civic Platform, Rodina and Party of Growth indicate the greatest asymmetry, as one candidate from each of these parties won in "arranged" single-seat constituencies (in these constituencies, the candidate outperformed the party by 46.9%, 43.3%, and 25.2%, respectively).

United Russia is the only party to have a negative asymmetry. This is presumably a result of the fact that in several "coordinated" constituencies the party deliberately nominated weak candidates whose results were substantially weaker than those of the party list (the gap was 15-18% in five constituencies, 21-24% in two, and 50.8% in the one where the leader of Civic Platform got elected).

As for the candidate influence index (CII), we have to first bear in mind that LDPR had no candidate in one constituency only, CPRF had none in two, and A Just Russia had none in eight. Therefore, the perplexing CII values of less than one (meaning that the result of the party is higher in constituencies with no candidates) for these three parties cannot be considered acceptable. Other parties have CII value above one, which is quite predictable and is evidence of the "contamination effect": the presence of candidates nominated by a party in single-seat constituencies improves the results of that party's list in that territory [2: 225].

The highest CII values were observed in liberal parties such as Yabloko (2.63), Party of Growth (2.16), and Green Alternative (2.30). Yabloko and Party of Growth indicated the same effect in 2016 [4: 1112–1115]. We believe that candidate influence is not the only factor at play in this case — these parties nominated many candidates in Moscow and St. Petersburg, where they have a lot of support. Removing the Moscow and St. Petersburg constituencies from the calculation reduced the CII values for these parties to 1.69 and 1.29, respectively, which shows that the influence of the candidates cannot be dismissed, either.

Communists of Russia had the lowest CII among the non-parliamentary parties (1.06). In 2016, the party's index stood at only 0.89, which we considered as proof of its spoiler nature: the presence of a candidate hindered rather than helped the party, as in this case it was more difficult to confuse it with CPRF [4: 1112–1115]. The situation straightened out in the last campaign. It is also intriguing that Communists of Russia had the highest VGIavg value and a 100% COI value (with party list result at 1.3%, the average of its single-seat candidates amounted to 5.6%), which was not the case in the 2016 election. It appears that Communists of Russia candidates were more active this time.

Still, we have one more assumption as to why Communists of Russia have a high VGIavg value. We noticed the relatively high results of this party's candidates who were placed at no. 1 on the ballot, which prompted us to check if this placement affected candidate results for all parties.

Table 3 summarizes the results of our calculations. As we hypothesized, for most parties, VGIavg values for candidates placed at no. 1 were higher than VGIavg values for all candidates, with the exception of Yabloko, Civic Platform, and REP The Greens. For the latter two parties the result is clearly not significant, given the small number of candidates placed at no. 1 (for Civic Platform, the high VGI value (46.9%) of R.G.Shaikhutdinov also played a major role).

Yabloko does not really fit into the overall picture, most likely because the party has a clear ideology and its voters are quite well-educated. Another point worth noting is the small difference values for United Russia, CPRF, and New People: it is likely that the share of random votes for candidates from these parties was small.

Communists of Russia, on the other hand, exhibited the largest difference value, as we predicted. We should bear in mind that in the federal constituency ballot, CPRF was was placed at no. 1. Therefore, some CPRF voters could mistakenly vote for the Communists of Russia candidate if they were also listed at no. 1 (or simply right above the CPRF candidate). This assumption also agrees with the results of the regional elections, which were held at the same time as the 2021 Duma election: Communists of Russia passed the five percent threshold in the three regions (Altai Krai, Amur Oblast, and Omsk Oblast) where they were listed at no. 1 on the ballot.

We subsequently proposed two integral indicators that characterize the election campaign as a whole. The first one is the general vote splitting index (GVSI), which estimates the total proportion of voters who voted for one party list and for a different party or nonpartisan candidate. This is how the indicator was calculated. In each single-seat constituency, we compared the vote shares for the party list and the party candidate, and then selected the lower of the two numbers. The selected numbers were summed up, resulting in the total share of unsplit votes in the constituency. Next, the resulting shares of unsplit votes for all constituencies were averaged, and the average value was subtracted from 100%.

To estimate the share of those voters who had the opportunity to vote for a candidate and a list of the same party, but for some reason did otherwise, we used the specific vote splitting index (SVSI). Its calculations went as follows. In each single-seat constituency, the positive and negative VGI values of all parties that had both a list and a candidate in that constituency were summed separately. Next, the modules of the obtained sums were compared, and the largest of the two numbers was selected. Finally, we averaged the values obtained for all constituencies.

For the 2016 State Duma elections, the GVSI value was 15.6% and the SVSI value was 14.4%. These values are close to those of 2016 (19.7% and 15.5%, respectively). The 2016 and 2021 GVSI values are much lower than in 1995-2003, because in those years the high values (about 60-70%) resulted from the large number of popular independent candidates. However, the SVSI values for the 2016 and 2021 elections were also lower than in previous campaigns (the lowest values were 17.8% in 1999, and 17.8% in 2003). SVSI amounted to 23.5%. These results support the point that there was more partisan voting in the 2016 and 2021 elections.

Certain electoral and geographic features

In a recent paper [7], we calculated indices of territorial homogeneity of voting (THVI) for the 2021 State Duma election for the Russian Federation as a whole and for each region. The formula for this index went as follows:

\(I = \sum\limits_{j=1}^m((1 - \frac{n - ((\sum\limits_{i=1}^n s_{ij})^2 / \sum\limits_{i=1}^n s_{ij}^2)}{n - 1})p_j)\) ,

where \(n\) is number of territorial units; \(m\) is number of parties participating in the elections; \(s_{ij}\) is the share of votes received by the \(j\)-st/nd/th party in the \(i\)-st/nd/th territorial unit; \(p_j\) is the share of votes received by the \(j\)-st/nd/th party in the country (or region) as a whole.

For the Russian Federation, the THVI by region amounted to 0.867, which is slightly below the 2011 and 2016 values, significantly below the 2007 value, about equal to the 2003 value and exceeds the 1993, 1995 and 1999 values.

This index (also called the nationalization index) can be calculated on a party basis. Data on all 14 parties that took part in the elections is presented in Table 4. We can see that United Russia, CPRF, and LDPR exhibit fairly high index values (0.9 or slightly lower), which is a sign of a high homogeneity of voting. The indices of A Just Russia, New People, Communists of Russia, and the Party of Pensioners are slightly lower. The indices of REP The Greens, RPFJ, Green Alternative and Civic Platform are between 0.7 and 0.8. And lastly come the rather low indices of Rodina, Party of Growth and Yabloko — parties that performed relatively well in certain regions.

Voronezh Oblast (0.868), Khanty-Mansi AO (0.876) and Samara Oblast (0.880) indicated the lowest index values (broken down by TECs). The republics of Karachay-Cherkessia (0.988), Chechnya (0.990) and Kabardino-Balkaria (0.991) indicated the highest indices.

In terms of intra-regional differences in voting, the difference in voting results between the regional center and the region as a whole (calculated for the 80 regions with a regional capitals) seems especially interesting. An overview of such differences for the 1995-2018 elections can be found in [8]. The 2021 election revealed the same patterns. Table 5 indicates that turnout, the proportion of invalid ballots, and the results of United Russia, LDPR, and Communists of Russia are all mostly higher in the regional periphery. The other 11 parties perform better in regional capitals.

Nine regions where turnout in the capital exceeded that in the region included the republics of Ingushetia, Karelia, Komi, North Ossetia, Stavropolsky and Khabarovsky Krais, Volgograd and Irkutsk Oblasts, and Khanty-Mansi Autonomous Okrug. Kabardino-Balkarian Republic was the only region where the result of Yabloko was lower in the capital.

The seven regions where United Russia's result was higher in the capital included Kabardino-Balkarian and Chechen Republics, Zabaikalsky and Stavropol Krail, Astrakhan and Kemerovo Oblast, and Khanty-Mansi Autonomous Okrug. The difference did not exceed 1% in five regions and amounted to 2.9% in Stavropol Krai. On the other hand, Khanty-Mansi Autonomous Okrug indicated a difference of 19.8% (in the regions where United Russia's result was lower in the capital, there was often a greater difference: 17.3% in Mordovia, 20.4% in Voronezh Oblast, 18.5% in Lipetsk Oblast, 16.5% in Penza Oblast, 21.6% in Tambov Oblast). Of these seven regions, Kabardino-Balkarian and Chechen Republics, as well as Kemerovo Oblast, usually exhibited abnormal results; results in Astrakhan and Stavropol were often abnormal as well. Protest sentiments seem to have been quite strong in Zabaikalsky Krai's periphery. As for Khanty-Mansiysk, 2018 was the first time when it exhibited such an anomaly.

We also monitor the number of TECs with extra-high turnout and extra-high monolithic voting for the "party of power" (that is, where both turnout and the result of the "party of power" exceed 90%). This indicator peaked between 2004 and 2008 (214 in 2004, 215 in 2008, 183 in 2007), then it started to go down. For example, there were 122 such TECs in 2011, and 115 in 2016 [11: 210–214]. This time the number of TECs with ">90x>90" indicators went down to 38. That said, United Russia got over 90% of votes in 39 TECs, while the number of TECs with turnout over 90% amounted to 145. It is likely that the federal center began to instruct the "electorally governed" regions not to "overdo it".

All TECs with ">90x>90" indicators were concentrated in five regions: the republics of Dagestan (2), Tatarstan (4), Tuva (11) and Chechnya (all 20), and one TEC (Diveyevskaya) in Nizhny Novgorod Oblast.

Correlation analysis

Knowing the results of voting for party lists in 225 single-seat constituencies, it is not too difficult to calculate the correlation between the results of the parties that ran in the election. This data helps to understand the connection between voting for different parties, although its interpretation is normally far from unambiguous.

Table 6 presents a correlation matrix broken down by all 225 constituencies for all 14 parties that participated in the elections. As can be seen from the table, the two parties that stand out are United Russia and Rodina. United Russia has a negative correlation with all the other parties (and the same occurred in 2016). Rodina has no significant correlations with any of the parties (in 2016, the same occurred with Patriots of Russia, while Rodina had significant correlations with several parties) [9].

The remaining 12 parties can be broken down into three clusters. The first includes three parliamentary opposition parties (CPRF, LDPR, A Just Russia) plus Communists of Russia. They exhibit significant positive correlations with each other, as well as with some of the parties in the third cluster.

The second cluster includes three parties that can be very tentatively described as "liberal" (although voters may very well consider them as such): Yabloko, Party of Growth and Civic Platform. These parties also have significant positive correlations with each other (although the correlation between Party of Growth and Civic Platform falls short of the 99.9% level, but is within the 99% level). They have no significant correlations with the parties in the first cluster, with the exception of a fairly weak correlation between Yabloko and Just Russia.

The parties in the third cluster (New People, Party of Pensioners, REP The Greens, Green Alternative and RPFJ) display significant positive correlations both between themselves and the parties of the second and third clusters. Green Alternative stands out in particular with significant positive correlations with 10 parties out of 13 at the 99.9% level (as well as positive correlations with 12 parties at the 95% level—with all but United Russia).

Most interesting are correlation coefficient values between the parties that are considered ideologically close. This time the situation is slightly different than it was in 2016. At that time, there was a high correlation between Yabloko and Party of Growth (0.77); now it is markedly lower (0.57). On the other hand, the correlation between CPRF and Communists of Russia has increased: the coefficient was 0.37 in 2016, and now it is 0.57. Nevertheless, the correlation of both communist parties with the Party of Pensioners is higher (0.62 and 0.60). The highest correlation this time (0.785) was between Green Alternative and RPFJ.

From our experience, the correlation coefficients reflect not so much the common ideological ground of parties, but more so the social proximity of their electorates. As we observed in the section "Certain electoral and geographic features", CPRF's results were better in the capitals than in the regional periphery, while the opposite was true for Communists of Russia. This fact manifested itself in the correlation.

For the 11 parties that ran in both the 2016 and 2021 elections, we can calculate correlation coefficients between the vote returns in those elections. The calculation results are presented in Table 7.

The results of United Russia, Yabloko and LDPR are the most constistent (0.85–0.91). Correlation coefficients of Party of Pensioners, CPRF and Party of Growth is slightly lower (0.72–0.79). A Just Russia and Rodina coefficients are even lower (0.67–0.69). A rather low, although significant correlation was also present for REP The Greens and Communists of Russia (0.50–0.58). And there is hardly any correlation in the results of the outsider Civic Platform, so it is likely that voting for this party was done at random.

Conclusion

The electoral situation in any country is undergoing constant change. Every new election makes adjustments and provides new material for analysis. And this is especially true in such important election as national parliamentary election. In this context, it is important to analyze these elections after they have taken place using all available scientific methods. The next stage could be the incorporation of this analysis into the overall picture of the country's political history.

There are many interesting points to the 2021 State Duma election. One of the most important consequences is the emergence of a fifth parliamentary party. A more thorough analysis of the vote returns gives us extra information both on the electoral behavior of Russian citizens and on the parties themselves.

While studying the official vote returns in Russian elections, we arrive at an inevitable question: are we studying the actual results of the expression of the will of citizens? Or does the occurring electoral fraud render such studies pointless?

Indeed, there are studies that measure the share of "anomalous" votes [11: 107–112; 12]. And these studies show that "anomalies" are not countrywide: there is a small number of regions where the official results appear nothing like the actual expression of the will, and there are also regions where distortions are not quite so strong, which means we can apply statistical analysis to vote returns there.

At the same time, there are indicators that are resistant to fraud. For example, disproportionality indices only describe the stage where votes are converted into seats, regardless of how these votes were received. In other cases, the indices reveal the big picture. Thus, the effective number of parties (candidates) allows for an overall assessment of competition in elections, so it takes into account any factors that affect competition — the dropout of strong candidates, campaign inequality, and fraud.

On the other hand, the very data obtained by statistical methods provides some information about how distorted vote returns are. If such data follows certain patterns that remain relatively stable over time and can be explained by established political theories, we can assume that the distortions are insignificant. Occasionally, there are quite acceptable explanations for temporal or geographical exceptions, too. The absence of such acceptable explanations may indicate fraud, however.

Received 21.01.2023, revision received 25.02.2023.

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References

1. Golosov G.V. Fragmentatsiya partiynykh sistem: noviy metod izmereniya i yego primeneniye k rezultatam vyborov rossiyskikh regionalnykh zakonodatelnykh sobraniy (2003–2008) [Party System Fragmentation: Applying a New Assessment Method to Regional Legislative Assembly Election Results (2003–2008)]. – Elektoralnoye prostranstvo sovremennoi Rossii. Politicheskaya nauka: Yezhegodnik 2008. Moscow: ROSSPEN, 2009. P. 9–26. (In Russ.)
2. Golosov G.V. Rossiyskaya partiynaya sistema i regionalnaya politika, 1993–2003 [Russia's Party System and Regional Politics, 1993–2003]. Saint Petersburg, 2006. 300 p. (In Russ.)
3. Golosov G.V. The effective number of parties: a new approach. – Party Politics. 2010. V. 16. No. 2. P. 171–192.
4. Kynev A., Lyubarev A., Maksimov A. Kak vybirala Rossiya – 2016. Rezultaty monitoringa izbiratelnogo protsessa [The 2016 Russian Election. Electoral Process Monitoring Results]. Moscow: Liberal Mission Foundation, 2017. 1142 p. (In Russ.)
5. Kynev A.V. Obshchiye itogi vyborov i izbiratelnoi kampanii-2021 [General election and election campaign-2021 results]. – Liberal Mission Foundation website, 6 December 2021. URL: https://liberal.ru/reports/obshhie-itogi-elektoralnyh-rezultatov-izbiratelnoj-kampanii-2021?download=32099 (accessed 21.01.2023). (In Russ.) - https://liberal.ru/reports/obshhie-itogi-elektoralnyh-rezultatov-izbiratelnoj-kampanii-2021?download=32099
6. Lyubarev A.E. 2016 and 2021 State Duma Elections: An Evaluation of Vote Overflows. – Electoral Politics. 2022. No. 1 (7). P. 3. - http://electoralpolitics.org/en/articles/vybory-v-gosudarstvennuiu-dumu-2016-i-2021-godov-otsenka-peretoka-golosov/
7. Lyubarev A.E. Assessing Territorial Homogeneity of Vote Returns in the Russian Federation and its Subjects. – Electoral Politics. 2021. No. 2 (6). P. 1. - http://electoralpolitics.org/en/articles/otsenka-territorialnoi-odnorodnosti-itogov-golosovaniia-v-rossiiskoi-federatsii-i-rossiiskikh-regionakh/
8. Lyubarev A.E. Intraregional Differences of Electoral Indices during the Elections in Russia in 1995-2018. – Electoral Politics. 2019. No. 1. P. 3. - http://electoralpolitics.org/en/articles/vnutriregionalnye-razlichiia-elektoralnykh-pokazatelei-na-rossiiskikh-vyborakh-1995-2018-gg/
9. Lyubarev A.E. Korrelyatsionnyi analiz itogov golosovaniya na rossiyskikh federalnykh i regionalnykh vyborakh 2011–2018 gg [Correlation Analysis of Voting Returns in the Russian Federal and Regional Elections of 2011]. – Political Science (RU). 2021. No. 1. P. 205–225. DOI: 10.31249/poln/2021.01.09. (In Russ.)
10. Lyubarev A.E., Shalayev N.E. Rasshchepleniye golosov v smeshannykh izbiratelnykh sistemakh: popytka kompleksnogo issledovaniya [Vote Splitting in Mixed Electoral Systems: A Comprehensive Study]. – Sotsiodinamika. 2015. No. 8. P. 125–286. (In Russ.) - http://e-notabene.ru/pr/article_16076.html
11. Lyubarev A.E. Zanimatelnaya elektoralnaya statistika [Entertaining Election Statistics]. Moscow: Golos Consulting, 2021. 304 p. (In Russ.)
12. Shpilkin S. Anomalnyye i normalnyye: statisticheskiy analiz itogov dumskikh vyborov [Normality and Abnormality: A Statistical Analysis of State Duma Election Results] // Novaya realnost: Kreml i Gollem. Chto govoryat itogi vyborov o sotsialno-politicheskoi situatsii v Rossii [New Reality: Kremlin and Gollem. What Election Results Say About the Social and Political Climate in Russia] / ed. by K. Rogov. Moscow: Liberal Mission Foundation, 2021. P. 42–54. (In Russ.)