The Potential and Limits of Digital Election Forensics
1. The Potential and Limits of Digital
Election Forensics
Jozef Janovsk´y
Keble College
University of Oxford
A dissertation submitted in partial fulfilment of the requirements for the
degree of Master of Science in Applied Statistics
13 September 2013
2. This thesis is dedicated to all of my close friends and family
with whom I did not spend enough time this summer.
3. Acknowledgements
I would like to thank Professor Brian D. Ripley for his supervision, as well
as the Department of Statistics and Keble College for providing me with the
ideal conditions for dissertation writing. I would also like to thank Princeton
University for their election data.
I would not have been able to write this thesis without the financial support
of Tatra banka Foundation, SPP Foundation and Vlado Gallo, for which I
am most grateful. I must also thank my parents for their continuous and
unconditional support.
Last but not least, special thanks go to Niko and Daisy, who helped me get
back on track when I needed it the most.
4. Abstract
This dissertation focuses on statistical electoral fraud detection. Primarily,
it aims to answer the question of whether fraudulent electoral data can be
separated from fraud-free electoral data by analysing only the distributions
of specific digits in election results.
A large dataset of polling-station level election results was compiled and anal-
ysed. It can be said that the hypothesised digital patterns related to the so-
called Benford’s law have only limited empirical validity. The distributions of
the significant digits in vote counts tend to be more positively skewed than
in Benford’s law. On the contrary, the last digit in vote counts of large con-
testants is distributed uniformly. Unlike previous research, this thesis also
analysed digital distributions in vote shares, the patterns of which are no less
present in the data as compared to vote count patterns.
Solid evidence was found that fraud-free vote shares can be approximated by
a normal distribution on the simplex. This distribution served as the basis for
two models of fraud-free vote counts which are compared. The model with
the better fit was selected, and using this model, large numbers of artificial
electoral contests were simulated from each fraud-free election contest. Fraud
was then artificially imputed into a subset of the simulated election contests
and the synthetic data were used to train a logistic classifier. The information
contained in digital distributions was sufficient to allow for a good separation
of the election contests according to different fraud levels.
All in all, digital patterns seem to provide a substantial amount of information
on election result distributions. Nevertheless, the focus of future research
should shift from Benford-like patterns, which were merely adopted from other
fields, to patterns actually present in election results.
7. List of Tables
1.1 Expected Frequencies of the First (FSD), Second (SSD), Third (TSD) and
Fourth (FoSD) Significant Digit According to Benford’s Law . . . . . . . 7
2.1 Descriptives for First-Past-The-Post Elections . . . . . . . . . . . . . . . 15
2.2 Descriptives for Qualified Majority Elections . . . . . . . . . . . . . . . . 17
2.3 Descriptives for Proportional Representation Elections . . . . . . . . . . 19
3.1 Means and Standard Deviations of the Distributions of Predicted Fraud
Level Percentages by True Fraud Levels Over the 5,620 Test Sets . . . . . 49
3.2 Means and Standard Deviations of the Distributions of Predicted Fraud
Type Percentages by True Fraud Levels Over the 5,620 Test Sets . . . . . 53
iii
8. List of Figures
2.1 First Significant Digits in Fraud-Free Vote Count Distributions . . . . . . 20
2.2 Examination of the Compliance of Vote Count Distributions with the Con-
ditions for 1BL Occurrence Stated in [Scott and Fasli, 2001] . . . . . . . 21
2.3 p-Values of Pearson’s χ2
Tests of Compliance of Fraud-Free Vote Count
Distributions with 1BL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Second Significant Digits in Fraud-Free Vote Count Distributions of Con-
testants Competing in At Least 500 Polling Stations . . . . . . . . . . . . 23
2.5 p-Values of Pearson’s χ2
Tests of Compliance of Fraud-Free Vote Count
Distributions with 2BL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Last Digits in Fraud-Free Vote Count Distributions of Contestants Com-
peting in At Least 500 Polling Stations . . . . . . . . . . . . . . . . . . . 25
2.7 p-Values of Pearson’s χ2
Tests of Compliance of Fraud-Free Vote Count
Distributions with LDU . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 First Significant Digits in Fraud-Free Vote Shares of Contestants Compet-
ing in At Least 500 Polling Stations . . . . . . . . . . . . . . . . . . . . . 27
2.9 Second Significant Digits in Fraud-Free Vote Share Distributions of Con-
testants Competing in At Least 500 Polling Stations . . . . . . . . . . . . 28
2.10 Differences in Digital Distributions of Fraud-Free and Fraudulent Election
Results for Contestants Competing in At Least 500 Polling Stations . . . 29
3.1 Illustration of the Fit of the Normal Distribution on the Simplex to the
Empirical Vote Shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 First Significant Digits in Vote Counts of Small and Large Contestants
Competing in At Least 500 Polling Stations Simulated from the Multino-
mial and Na¨ıve Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 First Significant Digits in Vote Shares Simulated from the Multinomial
and Na¨ıve Model for Small and Large Contestants Competing in At Least
500 Polling Stations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
iv
9. 3.4 Image Plots of ROC Curves from Test Set Evaluation of Binary Logistic
Regressions for Different Values of Fraud Parameters . . . . . . . . . . . 47
3.5 Image plots of ROC Curves from Test Set Evaluation of Binary Logistic
Regressions With Two Different Types of Fraud: Prevalent Ballot Stuffing
on the Left and Prevalent Vote Transferring on the Right . . . . . . . . . 48
3.6 Violin Plots of the Distributions of Predicted Fraud Levels Percentages by
True Fraud Levels Over the 5,620 Test Sets . . . . . . . . . . . . . . . . . 50
3.7 Comparison of Importance of the Five Digital Patterns for Classification
of Different Fraud Levels Using the Difference In Deviances . . . . . . . . 52
3.8 Violin Plots of the Distributions of Predicted Fraud Level Percentages by
True Fraud Types Over the 5,620 Test Sets . . . . . . . . . . . . . . . . . 54
3.9 Comparison of Importance of the Five Digital Patterns for Classification
of Different Fraud Types Using the Difference In Deviances . . . . . . . . 55
3.10 Second Significant Digits in Vote Counts for Small and Large Contestants
Competing in At Least 500 Polling Stations Simulated from the Multino-
mial and Na¨ıve Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.11 Second Significant Digits in Vote Shares for Small and Large Contestants
Competing in At Least 500 Polling Stations Simulated from the Multino-
mial and Na¨ıve Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.12 Last Digits in Vote Counts for Small and Large Contestants Competing in
At Least 500 Polling Stations Simulated from the Multinomial and Na¨ıve
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
v
10. Introduction
“Electoral fraud is clearly the gravest form of electoral malpractice, and should be combated
overtly and publicly by all those with a stake in democratic development.”
[L´opez-Pintor, 2011, p. 3]
Without a doubt, elections constitute the very cornerstone of representative democ-
racy. Ensuring that a particular election is conducted democratically is, however, a
non-trivial task. The traditional approach, based on election observation [see Bjornlund,
2004, Hyde, 2008], has its limitations: observers monitor only a small number of polling
stations and their accounts can be questioned as partial. As Mebane writes, ‘election
monitoring is usually more focused on the conditions under which elections are con-
ducted – on whether they are free and fair – than whether they are accurate’ [Mebane,
2010c, p. 1; emphasis added].
In search of a better assessment of election accuracy, that is, the degree to which
official election results correspond to the true results, various methods of fraud detection
have been proposed. These are statistical techniques, attempting to identify patterns
in the large quantities of data produced in elections and use these patterns to distin-
guish between accurate and inaccurate electoral results. Although the techniques differ
substantially in their assumptions, they can all be considered tools of the emerging dis-
cipline called election forensics [Mebane, 2006]. Among the most widely applied as well
as controversial are methods of the so-called digital election forensics. Their proponents
claim that in fraud-free electoral data, distributions of digits at certain positions cor-
respond to theoretical distributions. Deviations from these theoretical distributions are
then considered to indicate electoral inaccuracies.
1
11. Given the high relevance of digital election forensics in the current academic and
non-academic debate, this dissertation will concentrate almost entirely on it; only the
literature review in the first chapter will briefly describe non-digital methods. From the
second chapter onwards, the applicability of different digital methods will be evaluated
using both empirical and simulated data. The second chapter will introduce the largest-
ever cross-national electoral dataset compiled at the polling-station level, collected almost
entirely by the author. The dataset will then be used to assess the occurrence of theo-
retical digital patterns in real-life elections. The third chapter will be simulation-based.
Many electoral contests will be simulated and electoral fraud will be artificially applied
to a subset. Logistic regression, using information on digital patterns, will be utilised to
separate the fraudulent and fraud-free electoral contests. The overall assessment will be
provided in the conclusion, together with an assessment of the limitations of the study
and implications for future research.
2
12. Chapter 1
Methods of Election Forensics
This chapter aims to provide an introduction to the context and methods of election
forensics. Throughout the whole chapter, the main driving question is: “How can we use
statistics to differentiate between fraud-free and fraudulent electoral data?”
The chapter starts with a section (1.1) introducing basic definitions that will be used
throughout the text. The second section (1.2) provides a short overview of general election
forensic techniques that have been used in the past. The following two sections then
concentrate on digital methods. Section 1.3 explores one of the most widely discussed
statistical ‘laws’, the so-called Benford’s law, which has constituted the main and the
most controversial digital forensic tool applied by election researchers. Section 1.4 is
dedicated to other digital methods.
3
13. 1.1 Terminology
The following lists define core terms used in this dissertation. The first relates to the
organisation of elections:
Constituency An electoral unit in which seats are contested.
Election contest A competition for representation within a constituency.
Electoral contestant A political party, movement or candidate participating in an elec-
tion contest.
Election A set of election contests from all constituencies.
Polling station An electoral unit on the level where votes are collected.
The second list is focused on electoral outcomes:
Vote counts The number of votes received by an electoral contestant in a polling station.
Vote shares Vote counts divided by their sum in a given polling station.
Election results Vote counts, vote shares and voter turnout.
Election-level results Election results for all polling stations in a given election.
Constituency-level results Election results for all polling stations by constituencies.
Size of vote count/share distribution The number of polling stations from which
election results are available for a given contestant.
The remaining terms relate to fraud and election forensics:
Election inaccuracy Discrepancy between the official and true election results.
Election fraud Election inaccuracy caused by intentional manipulation of true election
results. Unintentional errors are not considered fraud.
Ballot stuffing Fraud by filling ballot boxes with ballots for a specific contestant.
Vote transferring Fraud by transferring votes from a contestant to another contestant.
Digital forensics Statistical methods aimed at identifying election fraud by examina-
tion of distributions of digits in election results.
Significant digit For any non-zero real number, first significant digit (FSD) is defined
as its left-most non-zero digit. Second significant digit (SSD) is the digit to the
right of the FSD. The FSD attains values from {1, 2, . . . , 9} and the SSD from
{0, 1, . . . , 9}. Note that single-digit integers have no SSD.
Last digit For a non-negative integer, the last digit (LD) is defined as its right-most
digit ({0, 1, . . . , 9}).
4
14. 1.2 Non-Digital Election Forensics
This section briefly describes scholarship on general election forensics. It introduces
various ideas that have been used to detect electoral fraud.
The first line of reasoning compares election results in monitored and non-monitored
polling stations. Using the logic of field experiments, systematic differences in election
results between these polling stations are to be related to fraud [see Hanlon and Fox,
2006, Callen and Long, 2011, Enikolopov et al., 2012].
Another approach is to regress vote counts on relevant covariates and point to outliers
as being susceptible to fraud [see Wand et al., 2001a,b, Mebane and Sekhon, 2004]. This
method is designed to detect small-scale fraud occurring in a limited number of polling
stations. Large-scale systematic fraud is unlikely to be spotted by such a method.
Ecological regression has been employed to study the so-called flows of votes. Contes-
tants’ vote shares are regressed on their vote shares in a previous election. Homogeneity
of regression coefficients across all polling stations is assumed to avoid the ecological fal-
lacy. Their unusual values are used to make claims about the presence and magnitude of
fraud [see Myagkov et al., 2005, 2007, 2008, 2009, Park, 2008, Levin et al., 2009].
Based on the assumption that electoral fraud is in practice mostly implemented by
ballot stuffing, several studies have looked at the relationship between turnout and con-
testants’ vote shares using parametric or non-parametric regression [see Myagkov et al.,
2005, 2007, Vorobyev, 2011]. Most recently, Klimek et al. [2012] developed a parametric
model with parameters directly related to the number of fraudulent votes.
Unfortunately, the application of most non-digital methods to more than a few elec-
tions is problematic because specific information is required. Not always, for example, are
election monitors allowed to observe the election in randomly selected polling stations,
not always is fraud small-scale, and not always are previous election results for the same
contestants available. Because of these practical problems, it would be of great value
to have forensic methods which would require as little input as possible at our disposal.
With this in mind, I now move to the discussion of digital forensic methods.
5
15. 1.3 Digital Forensics Using Benford’s Law
Digital forensics aims to validate electoral results based on election results only. It claims
that fraud-free data exhibit certain digital patterns and systematic deviations from these
patterns signal fraud. By far the most popular line of reasoning has been associated with
the so-called Benford’s law. This section starts with different explanations of why Ben-
ford’s law emerges in many empirical datasets. Next, its applications to fraud detection
are described, focusing on election forensics.
1.3.1 The Mathematics of Benford’s Law
In 1881, Simon Newcomb published a two-page note on the frequency of significant digits
in what he called ‘natural numbers’ [Newcomb, 1881]. After his observation that the first
pages of logarithmic tables are worn out much faster that the last ones, he followed his
intuition that numbers occurring in nature’ should be approached as ratios, and derived
the formulas for the expected frequencies of the first significant digit (FSD):
F(FSD = d) = log10
d + 1
d
, for d = 1, 2, . . . , 9,
and the second significant digit (SSD):
F(SSD = d) =
9
i=1
log10
10i + d + 1
10i + d
, for d = 0, 1, . . . , 9.
Newcomb also noted that the differences between expected frequencies of the third and
latter significant digits are minuscule; indeed the distribution of the j-th significant digit
approaches uniform distribution exponentially in j (Hill [1998]). Expected frequencies of
the first four significant digits are reported in Table 1.1. Newcomb’s findings remained
unnoticed for a long time, maybe due to the vagueness of the explanation (based on the
concept of ‘natural numbers’) he proposed for the phenomenon.
Almost 60 years later the law was rediscovered by Benford [1938] who published a
more rigorous analysis. He compiled 20 datasets with more than 20,000 observations in
total and showed that several of the datasets followed the law to a large degree. Benford
6
16. Table 1.1: Expected Frequencies of the First (FSD), Second (SSD), Third (TSD) and
Fourth (FoSD) Significant Digit According to Benford’s Law
Digit FSD SSD TSD FoSD
0 . 0.1197 0.1018 0.1002
1 0.3010 0.1139 0.1014 0.1001
2 0.1761 0.1088 0.1010 0.1001
3 0.1249 0.1043 0.1006 0.1001
4 0.0969 0.1003 0.1002 0.1000
5 0.0792 0.0967 0.0998 0.1000
6 0.0669 0.0934 0.0994 0.0999
7 0.0580 0.0904 0.0990 0.0999
8 0.0512 0.0876 0.0986 0.0999
9 0.0458 0.0850 0.0983 0.0998
Expected frequencies of first four significant digits using formulas from [Newcomb, 1881].
found the best fit when all 20 different datasets were merged into a single table.
Benford’s explanation of the phenomenon was very similar to that of Newcomb; he
believed that natural as well as human phenomena fall into a geometric series which
yields the observed digit patterns. He went as far as stating that “Nature counts
e0
, ex
, e2x
, e3x
, . . . and builds and functions accordingly.” [Benford, 1938, p. 563]
On this basis he formulated the ‘Law of Anomalous Numbers’, which is a generalisation
of Newcomb’s formulas for integers of limited length. Instead of ‘length’ he speaks of
‘orders’, with the order equal to one for numbers 1-10, two for 10-100, three for 100-1000
and so on. The Law of Anomalous Numbers for the FSDs states:
Fr
1 = log
10(2 · 10r−1
− 1)
10r − 1
+
8
10r
1
log 10
,
Fr
a
a=1
= log
(a + 1)10r−1
− 1
a10r − 1
+
1
10r
1
log 10
,
where a stands for all digits except 1 and r is the digital order.
Over the course of the 20th century, plenty of explanations for the wide occurrence of
Benford’s law in real-life datasets were proposed. In his comprehensive overview of the
then scholarship Raimi [1976] concluded that none of the pure mathematical explanations
(e.g. those based on number theory) proved satisfactory and urged for a statistical inter-
pretation of the law. He cited several statistical results describing satisfactory conditions
7
17. for statistical models under which Benford’s law emerges (see below).
Hill [1998] elaborated upon the idea that it may be the process of mixing different
distribution that leads to better compliance with Benford’s law. He introduced a proper
probabilistic framework and derived ‘the log-limit law for significant digits’. It states that
if we select probability distributions at random and then sample each of them in a way
that is scale neutral, then the digital distribution of the combined sample converges to
Benford’s law. Hence Hill explained Benford’s surprising result that the union of all his
tables fit the law best [also see Janvresse and de la Rue, 2004, Rodriguez, 2004].
Another line of statistical reasoning was associated with the notion of multiplicative
processes. Furry and Hurwitz [1945] looked at the logarithm of a product log Yn =
log Πn
i=1Xi = n
i=1 log Xi of n independent and identically distributed random variables
Xi. Since under very weak conditions the central limit theorem applies to the latter sum,
then with increasing n the distribution of Yn approximates log-normal distribution. The
authors proved that log Yn(mod 1) approximates uniform distribution as n increases [also
see Adhikari and Sarkar, 1968, Adhikari, 1969, Boyle, 1994].
Which distributions satisfy Benford’s law? Scott and Fasli [2001] reported simulation
results showing that positively skewed non-zero unimodal distributions defined on a set
of positive numbers do follow it. The skew must be substantial with the mean at least
twice as high as the median. They found that the law is approximately followed by log-
normal distributions with the value of scale parameter no smaller than 1.2. Using signal
processing, Smith [1997] reached a similar conclusion, stating a good fit for distributions
wide in comparison to the unit distance on a logarithmic scale, e.g. wide log-normal
distributions.
Morrow [2010] proved that the compliance with Benford’s law improves as a random
variable is raised to higher powers. Looking at exponential-scale families of distribu-
tions closed under power transformations, sufficiently high values of scale parameter shall
therefore yield a good fit of log-normal distribution to Benford’s law. Results on other
distributions and distributional families can be found in [Leemis et al., 2000, Pietronero
et al., 2001, Engel and Leuenberger, 2003, Grendar et al., 2007].
8
18. 1.3.2 Applications to Fraud Detection
The applicability of Benford’s law to a wide range of datasets gave rise to the idea of
using it to distinguish between manipulated and non-manipulated datasets. It has been
most popular for the examination of financial statements in financial fraud detection.
Busta and Sundheim [1992] used it to examine tax returns yet in 1992, but it was only
after the publication of Nigrini’s accounting-related PhD thesis [see Nigrini, 2000] that
digital forensics gained popularity. Different methods of separating fraudulent from fraud-
free data using Benford’s law have been used, ranging from simple tests [see Wallace,
2002] to neural networks [see Busta and Weinberg, 1998, Bhattacharya et al., 2011] and
unsupervised procedures [see Lu and Boritz, 2005, Lu et al., 2006].
One of the first applications of Benford’s law outside accounting is related to Carslaw’s
research on cognitive perceptions [see Carslaw, 1988]. Recently, Diekmann [2007] has
studied the digital distribution of unstandardised OLS regression coefficients published
in academic journals.
Following the wide use of Benford’s law in accounting and other fields, its variations
have also been applied in electoral research by examining digital distributions of vote
counts. Although the law was originally derived for continuous distributions, it could
well be applicable to discrete distributions. It has been hypothesised that fraud-free vote
counts are Benford-distributed, and if a deviation is found then it may be attributed to
election fraud. However, since polling stations are typically rather similar in size, the
first significant digit law should often not be expected. That is why the focus shifted
from testing Benford’s first digit law (1BL) to Benford’s second digit law (2BL). Mebane
has been the main proponent of this fraud detection strategy [see Mebane, 2006, 2007,
2008, Mebane and Kalinin, 2009, Mebane, 2011] but other authors have used it as well
[see Pericchi and Torres, 2011, Breunig and Goerres, 2011].
This approach has been criticised for the lack of a convincing theoretical explanation
as to why we should expect to observe 2BL in fraud-free electoral data [see Carter Center,
2005, Deckert et al., 2011]. Mebane [2010b] came up with two mechanisms that may lead
to data satisfying 2BL but not 1BL. The first one assumes that three types of voters
exist: those who favour the incumbent, those favouring opposition and those who make
9
19. their decisions at random. All polling stations are assumed to be of the same total
size and proportions of the voter types across polling stations vary according to uniform
distribution. Voters’ choices are, in this model, also subject to a small probability of
mistake.
The second mechanism features the same three types of voters. For each voter type,
the probabilities of voting for either of the two alternatives are the same in all polling
stations. Voter type proportions in each polling station vary according to normal distri-
butions. Polling station sizes are distributed uniformly. Relying on simulations, Mebane
[2010b] claimed that both the second and the first mechanism led to the distribution obey-
ing 2BL but not 1BL. Nevertheless, due to the specific nature of Mebane’s mechanisms,
their applicability to real-life elections remains questionable.
The overall lack of support for the occurrence of Benford’s law in electoral results did
not stop political scientists from assuming it. Several studies [see Mebane, 2010a,b, Cant´u
and Saiegh, 2011] simulated electoral results based on the assumption that Benford’s law
holds (either 1BL or 2BL). The most sophisticated of the simulation analyses is the one
by Cant´u and Saiegh [2011]. They artificially introduced fraud to the simulated data by a
simple mechanism of moving a proportion of one contestants’ votes to another contestant
and adding some extra ballot-stuffed votes. They proceeded to train a supervised machine
learning classifier (na¨ıve Bayes) to distinguish between the fraudulent and fraud-free
simulated electoral contests; independent variables having been related to vote count
digital distributions.
In order to tackle the low validity of the Benford’s law assumption, Cant´u and Saiegh
[2011] calibrated the synthetic data with real-world electoral data. Their ad hoc cali-
bration, however, does not help to answer the question of the applicability of Benford’s
law to fraud-free electoral data in general. This dissertation aims to improve on their
methodology by both assessing the validity of the Benford’s law assumption on a large
empirical dataset and using empirical data for synthetic data generation.
10
20. 1.4 Other Digital Election Forensics Methods
It can be shown that under weak theoretical conditions, last digits of large-enough vote
counts are expected to occur with equal frequency. Proofs for certain continuous distri-
butions were provided by [Mosimann and Ratnaparkhi, 1996, Dlugosz and M¨uller-Funk,
2009] but these are not well-suited for inherently discrete electoral returns. Beber and
Scacco [2012] extended the previous work and used simulations to illustrate the behaviour
of several distributions. These showed that uniformity cannot be expected [Beber and
Scacco, 2012, p. 5]:
1. If a distribution has a standard deviation too small (about less than 10) because
draws from such distributions cluster within a very narrow range of numbers.
2. If a distribution has a fixed upper bound and draws that cluster at this bound.
However, even minor variations in polling station size (in tens of votes) will restore
last digit uniformity.
3. If a distribution has a mean relatively small compared to its standard deviation
because such a distribution generates a large number of very small counts.
When the numbers on electoral sheets are artificially modified by electoral commis-
sioners to favour a given party, they are likely to deviate from uniformity. The reason is
that people are rather bad at generating random numbers and so they introduce biases
into the data [see Mosimann et al., 1995]. The focus on the last digits of a sufficiently long
number is equivalent to focusing on inconsequential noise. This approach complements
the focus on ballot-stuffing which is typical for significant-digit analysis.
Apart from last-digit uniformity (LDU), Beber and Scacco [2012] discussed other
digital patterns that humans (even with incentives to randomise) tend to introduce into
data. For example, based on some experimental research they claimed that humans select
lower digits more often then higher digits, they avoid repetitions of digits and that they
tend to select pairs of distant numbers infrequently. While these constitute interesting
hypotheses, the focus of this dissertation will remain on the validity of 1BL, 2BL and
LDU for fraud-free election data as these are the three main open questions in the current
election fraud discussion.
11
21. Chapter 2
Empirical Data Analysis
It is striking how little empirical evaluation of the validity of digital election forensics
assumptions has been performed. Despite having direct political implications and thus
high social relevance, empirical studies applying Benford’s law to fraud detection have
either assumed the law’s validity or tried to ‘support’ it by illustrating its fit in one or
two elections only. Mebane [2006] looked at two elections (from the U.S. and Mexico),
Mebane [2007] at a single Mexican election, Mebane [2008] at one U.S. election, Mebane
and Kalinin [2009] at four Russian elections, Breunig and Goerres [2011] at five German
elections and Pericchi and Torres [2011] at 5 elections and a referendum from 3 countries.
Clearly, no compelling evidence has yet been used to support the use of Benford’s law.
On the other hand, critics of the applicability of Benford’s law to election results have
not provided comprehensive empirical evidence either. The Carter Center [2005] cited an
analysis showing a bad fit 2BL in a single election, and the most influential 2BL critique
by Deckert et al. [2011] only analysed two elections at the polling-station level. Any
analysis of electoral returns from a handful of elections can hardly provide satisfactory
evidence to reject the existence of Benford-like patterns in election results.
The hypothesis of last-digit uniformity has also not been thoroughly empirically stud-
ied; only a single article has been published on the topic in the election context. Since
the article demonstrates the phenomenon in only 4 elections, more empirical validation
is needed.
Having said all of the above, the natural next step would be to evaluate the digital
patterns on a substantial number of real-life elections. Large amounts of low-level cross-
12
22. national electoral data have been collected by the author for this purpose. To the best
of my knowledge, not only have cross-national polling-station data never been used to
a comparable extent in election forensics, they have not even been comparably used in
political science generally.
This chapter continues with a brief description of the dataset. The focus then shifts
to an evaluation of 1BL, 2BL and LDU on the dataset.
13
23. 2.1 Description of the Dataset
To assess the validity of Benford’s law validity, online availability of election results (at
the polling-station level, as defined in Section 1.1) was checked for all countries in the
world. The process of data collection, data cleaning and data manipulation was very
time-consuming and tedious as the format and quality of posted election results varies
greatly from country to country. The final dataset contains vote counts from 24 coun-
tries gathered either from primary online sources (typically national election commission
websites) or from reliable secondary sources (data used in peer-reviewed journal articles).
It is essential to determine the appropriate level of analysis. First, polling-station
electoral data must be analysed, as stressed by Mebane [2011]. Polling stations constitute
the level at which manipulation with ballot boxes occurs, and no further information is
lost as compared to working with more aggregated data.
Second, elections are organised in constituencies with separate election contests. Since
voters in different constituencies vote for different contestants, it is often not sensible
to combine election results across constituencies. Even if cross-constituency election
results could be sensibly combined, for example by looking at political parties rather
than individual candidates in British general elections, their distributions are likely to
be substantially different and merging them could result in mixtures that are hard to
analyse. For example, a regional party may be very successful in a few constituencies
only and not even run candidates in other constituencies. This is why the primary focus
of this dissertation rests on election contests as opposed to elections.
In order to make the dataset description as clear as possible, this section will be
organised according to the type of electoral system at use in a given election. The
importance of constituencies in this analysis requires an understanding of how they differ
across electoral systems. It has also been established that different electoral rules induce
different types of strategic behaviour of voters [see Duverger, 1959, Cox, 1997] and the
effect of election rules on digital distributions has been analysed by Mebane [2010a,b].
The three most widely employed electoral systems in the world are: first-past-the-post
(FPTP), qualified majority (QM) and proportional representation (PR). FPTP is applied
in single-seat constituencies with each voter casting a single vote and with the candidate
14
24. Table 2.1: Descriptives for First-Past-The-Post Elections
Country Type Year Fraud PS Const PS per C PS Size Cand
Canada LH 1997 No 59169 301 48-281 1-1147 3-11
LH 2000 No 61329 301 48-299 1-1933 3-10
LH 2006 No 62411 308 26-281 2-1972 4-11
LH 2008 No 65209 308 44-344 3-796 4-10
LH 2011 No 66449 308 44-395 3-799 3-9
Germany LH 1983 No 58214 248 124-478 7-4078 7-10
LH 1987 No 59169 248 132-494 10-2480 8-12
LH 1990 No 81489 328 132-496 12-2445 9-15
LH 1994 No 80053 328 129-496 8-2476 10-17
LH 1998 No 79134 328 104-496 4-2221 11-24
LH 2002 No 77353 299 142-492 6-2257 8-17
LH 2005 No 75978 299 141-494 6-2176 8-15
LH 2009 No 75059 299 125-493 6-1897 9-19
Jamaica LH 2011 No 6629 63 70-155 2-607 2-4
Mexico LH 2009 No 132201 300 323-764 1-1077 1-11
LH 2012 No 136766 300 323-763 1-2545 1-12
UH 2012 No 136908 300 327-757 1-1654 1-12
P 2012 No 138741 1 - 1-2196 12
Romania LH 2012 No 18456 311 25-134 5-1501 21-25
UH 2012 No 18456 135 63-278 5-1518 4-8
UK (LDN) SH 2004 No 624 14 35-55 857-4894 7-8
SH 2008 No 624 14 35-55 266-6038 8-12
SH 2012 No 625 14 35-55 1227-4640 5-9
US (CHI) P 1924 May 2233 1 - 95-893 3
P 1928 May 2922 1 - 112-1273 3
‘LH’ and ‘UH’ stand for elections to lower and upper houses of national legislatures, ‘P’ for presidential
elections and ‘SH’ for elections to sub-national legislative bodies. ‘May’ in column ‘Fraud’ represents
uncertainty as to whether election fraud was present. ‘PS’ denotes the total number of polling stations
form the given election included in the analysis, while ‘Const’ is the number of constituencies. ‘PS per
C’ shows the range of the number of polling stations per constituency. ‘PS Size’ reports the range of the
number of valid votes cast at the polling-station level. ‘Cand’ shows the number of candidates on the
constituency level.
obtaining the most votes taking the seat. This system is also known as ‘plurality voting’
and is used to elect UK MPs, for example.
As Table 2.1 shows, the dataset contains election results from 5,357 election contests
from 25 FPTP elections in 7 countries. In Canada, Germany and Jamaica the system is
employed in elections to the lower house of their national legislature while in Mexico and
Romania it is used for both lower and upper house elections.1
Mexico and the U.S. employ
1
The seats allocated to the parties in Germany and Romania are actually proportional to the vote
counts in multi-member constituencies (mixed-member proportional electoral system). Simply put,
FPTP votes are only used to determine who the deputies are (but not their total number).
15
25. FPTP variants to elect the president. In Mexico the whole country constitutes a single
constituency, while in the U.S. it could be argued that states represent the constituencies
better. However, since the collected data only comprise of results from Chicago, all of
them fall into a single constituency. Last, ward-level data for the 14 FPTP seats in the
London Assembly elections are also included.
Table 2.1 reports the total number of polling stations included in the analysis for each
election in the ‘PS’ column. For several elections, a small number of polling stations had
to be excluded in order to avoid mixing standard polling stations with ‘quasi-stations’ such
as those for postal voting from abroad. The remaining columns of Table 2.1 refer to the
number of constituencies in column ‘Const’, the range of the number of polling stations
per constituency (‘PS per C’), the range of the number of valid votes cast in polling
stations (‘PS Size’) and the range of the number of candidates on the constituency level
(‘Cand’). The columns of Table 2.2 and Table 2.3 are constructed and labelled similarly.
The only distinction between FPTP and qualified majority (QM) is that the latter
requires the winner to obtain a certain percentage of the vote, otherwise a second round
of voting is held. Typically, the pool of candidates is restrained in the second round as
compared to the first. The most common variant of QM is called ‘majority runoff’ (MR),
with at least 50% of the vote required to win in the first round. If no candidate gets 50%,
the two most successful candidates from the first round compete in the second round, and
the one with more votes gets the seat. This system is often employed to elect presidents,
e.g. in France and Ukraine.
In comparison with Table 2.1, Table 2.2 contains one new column (‘Rnd’) denoting
election round. Out of the 22 elections (from 12 countries) included, 16 are first round
and 6 second round. Given the popularity of MR for presidential elections, it is hardly
surprising that 17 out of the 22 elections included are presidential. Therefore, they only
use a single constituency. The remaining are Czech senatorial elections conducted in
27 constituencies and London Mayoral elections with a single London-wide constituency.
The London Mayor is elected using the so-called ‘instant MR’.2
2
Voters are asked to express two preferences: first preferences acting as first-round MR votes and
second preferences acting as potential second-round MR votes. If no candidate gets over 50% based on the
first preferences then the second preferences are redistributed to the top two candidates (according to the
first preferences) from the remaining candidates. The candidate with a majority after the redistribution
is declared the winner.
16
26. Table 2.2: Descriptives for Qualified Majority Elections
Country Type Year Rnd Fraud PS Const PS per C PS Size Cand
Afghanistan P 2009 1 May 22858 1 - 1-990 5
Armenia P 2013 1 No 1988 1 - 14-1736 7
Cyprus P 1998 1 No 1018 1 - 47-607 7
P 1998 2 No 1018 1 - 49-673 2
Czech Rep UH 2012 1 No 4812 27 101-289 3-688 5-13
UH 2012 2 No 4811 27 101-289 1-620 2
P 2013 1 No 14903 1 - 4-1923 9
P 2013 2 No 14903 1 - 5-1847 2
Montenegro P 2008 1 No 1141 1 - 7-829 4
P 2013 1 No 1169 1 - 4-910 2
Nigeria P 2003 1 May 2576 1 - 15-1177 30
Romania P 2009 1 No 18053 1 - 6-2340 12
P 2009 2 No 18053 1 - 2-3747 2
Russia P 2012 1 May 95193 1 - 2-4791 5
Sierra Leone P 2012 1 May 9386 1 - 24-714 9
Uganda P 2011 1 May 23827 1 - 1-1094 8
UK (LDN) M 2004 1 No 624 1 - 929-4918 9
M 2008 1 No 624 1 - 1632-6058 10
M 2012 1 No 625 1 - 1248-4625 7
Ukraine P 2004 2 No 33044 1 - 1-3527 2
P 2010 1 No 33554 1 - 1-2775 18
P 2010 2 No 33551 1 - 2-2856 2
‘UH’ stands for elections to upper houses of a national legislature, ‘P’ for presidential elections and
‘M’ for mayoral elections. ‘Rnd’ stands for the election round. ‘May’ in column ‘Fraud’ represents
uncertainty as to whether election fraud was present. ‘PS’ denotes the total number of polling stations
form the given election included in the analysis, while ‘Const’ is the number of constituencies. ‘PS per
C’ shows the range of the number of polling stations per constituency. ‘PS Size’ reports the range of the
number of valid votes cast at the polling-station level. ‘Cand’ shows the number of candidates on the
constituency level.
Proportional representation is used in multi-member constituencies in which, typically,
candidate lists of different political parties compete. Seats are awarded to political parties
in a manner that is ‘proportional’ to their vote counts. PR is a very popular system for
electing lower houses of national parliaments, e.g. in Sweden and Russia.
Table 2.3 contains descriptive statistics on 199 electoral contests from 32 PR elec-
tions in 14 countries. While most of them (22) are elections to lower houses of national
legislatures, sub-national legislative elections from the Czech Republic, Hong Kong and
London are also included as well as supranational European Parliamentary elections from
Bulgaria, Romania and London. The electoral system in South Africa is unique in having
two parallel proportional layers: a set of representatives for the national parliament is
17
27. elected proportionally in a nation-wide constituency and another set is elected propor-
tionally in each of the nine South African provinces. Constituency identifiers for Swedish
2006 and 2010 elections are missing and therefore these elections will be studied on the
election level only.
Proportional electoral systems have many parameters that can be varied (number and
size of districts, threshold, allocation formula, rigidity of candidate lists) and therefore
may differ a lot. Some research suggests these parameters can influence the distribution
of votes (Chatterjee et al. [2013]). For the purposes of this thesis, however, no further
distinctions between PR systems will be made.
As a last note, the elections in Afghanistan (2009, presidential), Finland (2011, lower
house), Mexico (2009 lower house; 2012 lower house, upper house and presidential) and
Sweden (2002, lower house) include a category ‘Others’ which aggregates votes for the
least successful candidates. Although this category is herein treated as a unique candi-
date, the distortions caused by this simplification should be minimal.
18
28. Table 2.3: Descriptives for Proportional Representation Elections
Country Type Year Fraud PS Const PS per C PS Size Cand
Armenia LH 2012 No 1982 41 34-79 6-1605 9
Aruba LH 2009 No 59 1 - 729-1096 8
Bulgaria EP 2009 No 11639 1 - 4-878 14
LH 2009 No 11872 1 - 5-2285 18
China (HK) SH 2008 No 519 5 65-156 318-7309 6-14
SH 2010 No 504 5 65-153 43-2889 2-8
SH 2012 No 1077 6 67-539 236-7589 7-19
Curacao LH 1998 No 105 1 - 170-1098 14
LH 2006 No 106 1 - 196-1262 14
LH 2010 No 106 1 - 153-1378 8
LH 2012 No 105 1 - 246-1905 8
Czech Rep SH 2012 No 13670 13 348-2055 1-885 23-30
Finland LH 2011 No 2326 14 92-361 68-6786 18
Germany LH 2002 No 77353 16 415-13336 6-2245 7-19
LH 2005 No 75978 16 406-13127 6-2183 7-16
LH 2009 No 75059 16 405-13322 6-1900 8-18
Montenegro LH 2009 No 1152 1 - 7-819 16
Romania EP 2009 No 18127 1 - 15-1692 9
Russia LH 2003 May 95181 1 - 2-4861 23
LH 2007 May 96182 1 - 1-8720 11
LH 2011 May 94678 1 - 1-3470 7
South Africa LH 2004 No 16963 1 - 17-5750 21
LH 2004 No 16962 9 347-4114 17-5592 13-21
LH 2009 No 19725 1 - 2-5535 26
LH 2009 No 19725 9 625-4482 8-6187 16-25
Sweden LH 2002 No 5976 29 39-621 69-1890 8
LH 2006 No 5783 - - 89-2056 14
LH 2010 No 5668 - - 101-2052 12
UK (LDN) EP 2004 No 624 1 - 956-4996 10
SH 2004 No 624 1 - 937-4950 9
SH 2008 No 624 1 - 264-6049 14
SH 2012 No 625 1 - 1238-4660 13
‘LH’ stands for elections to lower houses of national legislatures, ‘EP’ for elections to European Par-
liament and ‘SH’ for elections to sub-national legislative bodies. ‘May’ in column ‘Fraud’ represents
uncertainty as to whether election fraud was present. ‘PS’ denotes the total number of polling stations
from the given election included in the analysis, while ‘Const’ is the number of constituencies. ‘PS per
C’ shows the range of the number of polling stations per constituency. ‘PS Size’ reports the range of the
number of valid votes cast at the polling-station level. ‘Cand’ shows the number of candidates on the
constituency level.
19
30. It can be seen that vote counts exhibit a pattern of decreasing FSD frequencies. The
red circles in the left-hand plot, representing FSD frequencies in the combined table of all
16,546,457 vote counts, come close to 1BL, although their distribution is more positively
skewed. For each digit, the orange crosses are the means of frequencies of the given
digit across all vote count distributions. Unlike the red combined data frequencies, they
take into account how vote count distributions are nested within election contests. These
show even more positive skew than is present in 1BL. On average, the FSDs in vote count
distributions follow a distribution more skewed than 1BL, with a noisy fit.
Figure 2.2: Examination of the Compliance of Vote Count Distributions with the Con-
ditions for 1BL Occurrence Stated in [Scott and Fasli, 2001]
1 2 5 10 20
0
1
2
3
4
Mean/Median in Vote Count Distributions
Adjusted Mean/Median Ratio (Log Scale)
(54,809 Contestants)
RelativeFrequencyDensity
The Boundary for
Compliance with 1BL
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
12
p Values from Unimodality Tests
p Value
(54,198 Contestants)
RelativeFrequencyDensity
The left panel plots the ratio of the mean and the median for all vote count distributions. Before
taking the ratio, 1 is added to both the mean and the median to avoid zero counts. The red line shows
the approximate boundary for 1BL compliance as stated by [Scott and Fasli, 2001]. The right panel
summarises the p-values obtained from testing the unimodality of vote count distributions by the dip
test.
Scott and Fasli [2001] reported a good fit of 1BL to unimodal distributions with the
mean at least twice the size of the median. Do these conditions hold for empirical vote
count distributions? Vote count distributions are almost always positively skewed with
about 92% of them having a mean larger than the median (also see the left panel of
Figure 2.2). However, the skew is typically not as strong as required by the conditions of
[Scott and Fasli, 2001], which only hold for 0.27% of the distributions.
21
31. The right panel of Figure 2.2 looks at the unimodality of vote count distributions
by the dip test [Hartigan and Hartigan, 1985]. The dip test tests the null hypothesis of
distribution unimodality using the maximal difference between the empirical distribution
function and the unimodal distribution function that minimises this maximal difference.
The histogram of p-values shows that most vote count distributions do not satisfy uni-
modality.
Figure 2.3: p-Values of Pearson’s χ2
Tests of Compliance of Fraud-Free Vote Count
Distributions with 1BL
0.0 0.2 0.4 0.6 0.8 1.0
0
20
40
60
80
All Vote Count Distributions
p Value
(54809 Contestants)
RelativeFrequencyDensity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
[Scott and Fasli, 2001] Distributions Only
p Value
(144 Contestants)
RelativeFrequencyDensity
The histogram on the left summarises the p-values obtained from testing the fit of all vote count
distributions to 1BL using Pearson’s χ2
test. The histogram on the right only reports the p-values for
distributions with the adjusted ratio (mean+1)/(median+1) ≥ 2, as well as the dip test p-value ≥ 0.01
(the plot on the right).
To complement the visual assessment of fit, Pearson’s χ2
test was applied to all
54,809 vote count distributions to test their compliance with 1BL. If 1BL held, we would
expect the p-values to be approximately uniformly distributed on the unit interval. Fig-
ure 2.3 shows that this is not the case; for example about 82% of the p-values are smaller
than 0.01. Even looking at the distributions that satisfy the conditions from [Scott and
Fasli, 2001], the p-values remain strongly skewed. These conclusions do not change when
controlling for distribution size, contestants’ strength or electoral system (although PR
elections fit relatively best and FPTP elections relatively worst).
22
32. 2.2.2 Benford’s Law for the Second Significant Digit
Similarly to the first significant digit, the digital distribution of the second significant
digit is on average more positively skewed than 2BL. The fit of vote count distributions
to 2BL is illustrated in Figure 2.4. In order to make the goodness of fit clearer, Figure 2.4
only plots vote count distributions of a size larger or equal to 500 polling stations. Last,
Figure 2.4 also separates large and small contestants by the criterion of having a median
vote count larger or not larger than 10.
Figure 2.4: Second Significant Digits in Fraud-Free Vote Count Distributions of Contes-
tants Competing in At Least 500 Polling Stations
q
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Vote Counts of Small Contestants
Second Significant Digit
(426,196 Vote Counts of 144 Contestants)
RelativeFrequency(in%)
0 1 2 3 4 5 6 7 8 9
0
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10
15
20
25
30
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2BL Frequencies
Average Frequencies
Combined Data Frequencies
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Second Significant Digit
(2,854,344 Vote Counts of 840 Contestants)
RelativeFrequency(in%)
0 1 2 3 4 5 6 7 8 9
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15
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25
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2BL Frequencies
Average Frequencies
Combined Data Frequencies
The plots summarise the observed digital frequencies for each of the digits 0-9. The right-hand (left-
hand) panel plot distributions for contestants with the median vote count of more (equal to or less)
than 10 votes. The red circles denote digital frequencies in the combined data of all vote counts, the
orange crosses show mean frequencies across all distributions for each digit and the blue line connects
2BL frequencies.
Looking at vote count distributions of small contestants and large contestants sep-
arately, distinct patterns are observed (Figure 2.4). The fit for small contestants is
unsatisfactory and it exhibits the above mentioned pattern of a strong positive skew. A
different story is visible in the right-hand panel of Figure 2.4. Large contestants have
vote count distributions that tend to obey 2BL rather closely. Only a slight systematic
deviation from the law is visible and the fit is substantially better than for 1BL.
23
33. To assess the goodness of fit quantitatively, I tested all 37,571 constituency-level vote
count distributions against 2BL using Pearson’s χ2
test. The p-values for both cases are
plotted in Figure 2.5. Although both histograms exhibit a positive skew, the agreement
with the 0-1 uniform distribution is much better than for 1BL; approximately 18.8% of
the p-values fall under 0.01 when considering all contestants. As suggested by Figure 2.4
the fit is even better for large contestants (about 3.8% of the p-values fall under 0.01).
Figure 2.5: p-Values of Pearson’s χ2
Tests of Compliance of Fraud-Free Vote Count
Distributions with 2BL
0.0 0.2 0.4 0.6 0.8 1.0
0
1
2
3
4
5
All Contestants
p Value
(37571 Contestants)
RelativeFrequencyDensity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
Very Large Contestants Only
p Value
(3089 Contestants)
RelativeFrequencyDensity
The histograms summarise the p-values from testing the fit of vote count distributions of all contestants
(the left-hand plot) and the contestants with the median vote count at least 100 (the right-hand plot)
to 2BL using Pearson’s χ2
test.
Dividing the distributions according to the electoral system at use, those from PR
elections tend to obey 2BL best and those form FPTP elections worst. Also, vote count
distributions of contestants competing in many polling stations tend to have lower p-
values as the test is then able to detect even small departures from 2BL.
All in all, it seems that with a noisy fit, the SSDs tend to obey a slightly more
positively skewed digital distribution than 2BL. Combined with the results of previous
subsection, this finding puts into question previous research in electoral forensics that
na¨ıvely assumed that Benford’s law holds for fraud-free vote counts.
24
34. 2.2.3 Last-Digit Uniformity
As Beber and Scacco [2012] pointed out, distributions with a large number of small
counts are unlikely to have the last digit distributed uniformly. This is exactly the case
for vote counts of small contestants (median vote count less than approximately 20). LD
frequencies of small contestants competing in at least 500 polling stations are plotted in
the left-hand panel of Figure 2.6. Clearly, the lower digits are significantly more frequent
than higher digits. This pattern can be simply explained since the last digits of vote
counts of small contestants often constitute their first significant digits as well.
Figure 2.6: Last Digits in Fraud-Free Vote Count Distributions of Contestants Competing
in At Least 500 Polling Stations
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Vote Counts of Small Contestants
Last Digit
(397,171 Vote Counts of 912 Contestants)
RelativeFrequency(in%)
0 1 2 3 4 5 6 7 8 9
0
20
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60
80
100
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Average Frequencies
Combined Data Frequencies
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Last Digit
(2,697,506 Vote Counts of 836 Contestants)
RelativeFrequency(in%)
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15
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LDU Frequencies
Average Frequencies
Combined Data Frequencies
The plots summarise observed digital frequencies for each digit 0-9. The right-hand (left-hand) panel
plot distributions for contestants with the median vote count of more (equal to or less) than 20 votes.
The red circles denote digital frequencies in the combined data of all vote counts, the orange crosses
show mean frequencies across all distributions for each digit and the blue line connects LDU frequencies.
The distinction between the digital distributions of small and large contestants is
very clear. The right-hand panel shows a very good fit of the LDs for large contestants
to uniformity. Generally, the higher the median of a vote count distribution, the better
its fit to LDU.
Unsurprisingly, testing the fit of all constituency-level vote count distributions yields
25
35. a non-uniform distribution (as shown in the left-hand panel of Figure 2.7), with 51% of
the p-values falling below 0.01. Focusing on the large contestants only (the right-hand
panel of Figure 2.7), the p-values become close to uniform. With 1.3% of the p-values
smaller than 0.01, it can be assumed that vote counts obey LDU for large contestants.
Figure 2.7: p-Values of Pearson’s χ2
Tests of Compliance of Fraud-Free Vote Count
Distributions with LDU
0.0 0.2 0.4 0.6 0.8 1.0
0
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4
6
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All Contestants
p Value
(54809 Contestants)
RelativeFrequencyDensity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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0.8
1.0
Large Contestants Only
p Value
(19152 Contestants)
RelativeFrequencyDensity
The histograms summarise the p-values obtained from testing the fit of all fraud-free constituency-level
vote count distributions (the left-hand plot) and fraud-free vote count distributions of large contestants
(with the median vote count above 20, the right-hand plot) to LDU using Pearson’s χ2
test.
As described on in Section 1.4, Beber and Scacco [2012] reported three criteria that
typically constrain distributions from achieving uniformity. Most importantly, a vote
count distribution needs to have a large enough standard deviation (at least 10), but one
that is smaller than the mean, in order to follow the LDU closely. This rule of thumb
works well on this empirical dataset; about 1.5% of the p-values for such distributions are
below 0.01 and the distribution of the p-values is close to uniformity. Overall, support for
the hypothesis that vote counts of large contestants satisfy LDU appears to be strong.
26
36. 2.3 Digital Patterns in Fraud-Free Vote Shares
Interestingly, although Benford’s law was conveniently defined for continuous distribu-
tions, no research effort has been made to evaluate its fit regarding vote shares. This
section briefly explores this possibility.
2.3.1 Benford’s Law for the First Significant Digit
Figure 2.8 shows the digital fit of vote shares for distributions at least 500 in size, with
the large and the small contestants separated by the boundary of 20% for a median vote
share. While vote share distributions of large contestants exhibit a poor fit to 1BL (the
right-hand panel), the fit is much better for small contestants (the left-hand panel). The
bad fit for large contestants is not surprising, since their vote shares generally fall in one
order of magnitude only (between 10-100%) and do not exhibit a strong positive skew.
Figure 2.8: First Significant Digits in Fraud-Free Vote Shares of Contestants Competing
in At Least 500 Polling Stations
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Vote Shares of Small Contestants
First Significant Digit
(5,237,537 Vote Shares of 1,626 Contestants)
Frequency(in%)
1 2 3 4 5 6 7 8 9
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Average Frequencies
Combined Data Frequencies
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Vote Shares of Large Contestants
First Significant Digit
(1,403,513 Vote Shares of 460 Contestants)
Frequency(in%)
1 2 3 4 5 6 7 8 9
0
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40
60
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1BL Frequencies
Average Frequencies
Combined Data Frequencies
The left-hand (right-hand) panel reports FSD frequencies of vote shares for contestants who competed
in at least 500 polling stations with a median vote share of less (more) than 20%. The red circles denote
digital frequencies in the combined data of all vote counts, the orange crosses show means of digital
frequencies of all distributions and the blue line connects 1BL frequencies.
27
37. The good fit for vote share distributions of small contestants is more surprising. This
pattern is no less present in the data than any of the vote count patterns described in the
previous sections. Judging the fit by Pearson’s χ2
tests even leads to a slightly better,
although still unsatisfactory, fit, as compared to the case of 1BL for vote counts. For
instance, about 73% of vote share distributions yield p-values below 0.01. The plot is
very similar to the left-hand panel of Figure 2.3 and is not reported here.
2.3.2 Benford’s Law for the Second Significant Digit
Figure 2.9 shows the distributions of the SSD in vote share distributions at least 500
polling stations in size; the large and small contestants are separated by the criterion of
having a median vote share larger or smaller than 20%. Just as with the FSDs for vote
shares, the fit for small contestants is good, but the fit for large contestants is much worse,
with the SSDs distributed almost uniformly. Pearson’s χ2
tests yield results analogous
to those for vote counts, with a slightly better overall fit.
Figure 2.9: Second Significant Digits in Fraud-Free Vote Share Distributions of Contes-
tants Competing in At Least 500 Polling Stations
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Vote Shares of Small Contestants
Second Significant Digit
(5,148,041 Vote Shares of 1,576 Contestants)
Frequency(in%)
0 1 2 3 4 5 6 7 8 9
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Second Significant Digit
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Frequency(in%)
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Average Frequencies
Combined Data Frequencies
The left-hand (right-hand) panel reports SSD frequencies of vote shares for contestants who competed
in at least 500 polling stations with the median vote share of less (more) than 20%.The red circles denote
digital frequencies in the combined data of all vote counts, the orange crosses show means of digital
frequencies of all distributions and the blue line connects 2BL frequencies.
28
38. 2.4 Digital Patterns in Potentially Fraudulent Elec-
tion Results
Last, digital distributions from potentially fraudulent elections shall be explored. Most
of the patterns described above hold for these election contests as well. To avoid repeat-
ing the same material, only patterns showing substantial differences are reported here.
They relate to the distribution of the LD in vote counts of large contestants and to the
distribution of the FSD in vote shares of large contestants.
The two identified patterns are plotted in Figure 2.10. First, in potentially fraudulent
elections, large contestants (median vote count of at least 20 votes) do not tend to have
vote counts with uniform last digits as the distribution is positively skewed. Especially
interesting is the large variance of the frequency for digit 0 as compared to the other digits.
This phenomenon goes in line with the reasoning of [Beber and Scacco, 2012], who noted
that manipulation of election sheets by election officers may introduce a non-uniform
Figure 2.10: Differences in Digital Distributions of Fraud-Free and Fraudulent Election
Results for Contestants Competing in At Least 500 Polling Stations
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Last Digit
(836 Contestants in Fraud−Free and 29 in Fraudulent Contests)
Frequency
Potentially Fraudulent
Fraud−Free
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Fraud−Free
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The left (right) panel compares vote count (vote share) LD (FSD) distributions of large contestants
competing in at least 500 polling stations. Results from fraud-free elections are coloured in green and
those from potentially fraudulent elections are in red. The same distinction between small and large
contestants is used as in the previous sections: smaller or larger than a median vote count of 20 and
smaller or larger than a median vote share 20% for the left and right panel respectively.
29
39. pattern into the LD distribution.
Second, large contestants (median vote share of at least 20%) tend to have a vote share
FSD digit distribution much flatter than that of contestants in fraud-free elections. This
pattern may be related to the fact that the vote shares of large contestants in fraudulent
elections are artificially increased and therefore tend to be higher than vote shares in
fraud-free elections.
All in all, some distinctions in the digital patterns have been identified. Interestingly,
the most widely adopted digital patterns in election forensics (1BL and 2BL for vote
counts) do not yield substantial differences. However, it must be stressed that the number
of potentially fraudulent contests included is small, and the conclusions of this subsection
should by no means be regarded as definitive.
30
40. Chapter 3
Synthetic Data Analysis
This chapter aims to assess the usefulness of digital patterns for separating fraud-free
and fraudulent electoral contests. Empirical data cannot be used for this purpose for two
main reasons. First, election results (as defined in Subsection 1.1) are rarely available
for fraudulent elections. Second, the degree or even the very presence of election fraud is
inherently unobservable.
Due to these two reasons, simulations are more suitable for the assessment of the
potential and limits of digital election forensics. If data mimicking election contests can
be simulated, then election fraud can be artificially introduced into their subsets and
supervised machine learning procedures can be used to classify the simulated contests
according to their type. In the following, Section 3.1 describes how fraud-free election
results can be modelled, Section 3.2 reports the design implemented for data simulation
and evaluates the goodness of digital fit of the synthetic data to the empirical data.
Finally, Section 3.3 reports the results from applying a logistic learner to the simulated
data.
31
41. 3.1 Models for Election Results
As in all contests, election contests feature more than one contestant. Therefore, election
contest modelling constitutes a compositional problem, i.e. the election results of the
contestants represent interrelated portions of a whole. Surprisingly, previous simulational
studies in digital election forensics did not take the compositional nature of election results
into account [see Myagkov et al., 2009, Mebane, 2010a,b, Cant´u and Saiegh, 2011]. This
dissertation adopts a compositional approach to election results modelling.
Two main approaches to compositionally approximate election results exist: either
vote counts or vote shares are modelled. A standard way of simulating vote counts is
by multinomial distribution [see Wand et al., 2001a,b, Mebane and Sekhon, 2004], and
a standard way of modelling vote shares is by a compositional framework introduced in
[Aitchison, 1986], namely the recently refined concept of additive logistic normal distri-
butions [see Katz and King, 1999]. These two approaches are explained in more detail
below. For alternatives see [Jackson, 2002] and [Linzer, 2012].
3.1.1 Theoretical Framework
This subsection defines the compositional terms needed for model description. It is
predominantly based on [van den Boogaart and Tolosana-Delgado, 2013].
As mentioned earlier, by a composition or a D-composition x = (x1, x2, . . . , xD), I
mean a data point of D portions of the total. Individual values xj, with j ∈ {1, . . . , D},
of a D-composition are denoted as amounts and each of them is associated with a single
element of the composition. Summing the amounts of all the elements in a composition
gives the total amount or the total t. Finally, amounts xj divided by the total amount
are called portions and denoted by pj.
It is obvious that taking both the D vote counts or vote shares from a single polling
station yields a composition with D elements (contestants). Vote counts cj constitute
the amounts, and the total number of valid votes cast in the given polling station is the
total t. Vote shares sj = cj/t represent portions of the composition.
More generally, the transformation of amounts into portions is called the closure of a
32
42. composition and is defined by operation: C(x) = 1
t
(x1, x2, . . . , xD). A composition x is
called a closed composition if a composition y exists such that C(y) = x. The set of all
possible closed D-compositions, i.e. the following:
SD
= x = (x1, x2, . . . , xD)i=1,...,D : xi ≥ 0,
D
i=1
xi = 1
is called the D-part simplex. The D-part simplex therefore constitutes the set of all
possible vote share compositions of D contestants.
Three operations on the D-part simplex will be needed. Perturbation x ⊕ y of com-
positions x and y is the closure of their component-wise product:
x ⊕ y = C(x1 · y1, . . . , xD · yD),
powering λ x of composition x by scalar λ is the closure of its component-wise powers
to the λ:
λ x = C(xλ
1 , . . . , xλ
D),
and the Aitchison scalar product x, y A of compositions x and y is defined as:
x, y A =
1
D
D
i>j
log
xi
xj
log
yi
yj
.
It can be shown that the D-part simplex together with perturbation, powering and the
Aitchison scalar product defines a (D − 1)-dimensional Euclidean space structure on the
simplex [Pawlowsky-Glahn, 2003, van den Boogaart and Tolosana-Delgado, 2013, p. 37-
41]. Statistical modelling on the simplex using these operations is therefore equivalent
to statistical modelling in RD−1
. Using isometric1
transformations, the vote shares of
D contestants can be transformed into RD−1
, standard multivariate techniques can be
applied there and the results can be re-transformed into the original simplex.
One standard isometric linear mapping is called isometric log-ratio transformation. If
V is a D × (D − 1) matrix with its columns constituted by D − 1 normalised linearly
1
Transformations preserving angles and distances as defined in [van den Boogaart and Tolosana-
Delgado, 2013, p. 40].
33
43. independent vectors orthogonal to 1 = (1, . . . , 1), then we define ilr: SD
→ RD−1
as:
y = ilr(x) := log(x) · VT
with the inverse transformation x = C [exp(y · V)] . ilr() induces the Aitchison measure
λS = λ ({ilr(x) : x ∈ A}) for the simplex, analogous to the Lebesgue-measure λ [van den
Boogaart and Tolosana-Delgado, 2013, p. 43].
3.1.2 A Model for Vote Shares
Using the above methodology, we can define a model for vote share compositions [see
van den Boogaart and Tolosana-Delgado, 2013, p. 51-53]. A random vote share com-
position S has a normal distribution on the simplex NS(m, Σ) with mean vector m and
variance matrix Σ if projecting it onto any arbitrary direction of the simplex u with the
Aitchison scalar product leads to a random variable with univariate normal distribution,
of mean vector m, u A and variance clr(u) · Σ · [clr(u)]T
. Taking V as the basis of the
simplex, the coordinates ilr(s) of random vote share composition S have the following
joint density with respect to the Aitchison measure λS:
f(s; µV , ΣV ) =
1
(2π)D−1 · |ΣV |
exp −
1
2
(ilr(s) − µV ) · Σ−1
V · (ilr(s) − µV )T
(3.1)
which is a multivariate normal distribution with mean vector µV and variance matrix
ΣV . Normal distribution on the simplex (NDS) was first defined by [Pawlowsky-Glahn,
2003] and it is probabilistically equivalent to the additive logistic normal distribution
introduced by [Aitchison, 1986].
A practical problem arises with fitting the NDS, since computing ilr(s) = log(s) · VT
requires non-zero vote shares. However, zero vote counts are very common. As is usually
done, in order to overcome this technical obstacle, one vote is added to all observed vote
counts, and adjusted vote shares s are computed based on the adjusted vote counts c
(summing up to the adjusted total t ).
One more adjustment has to be performed before fitting the NDS. Vote shares in
empirical datasets are often not independent of vote totals (e.g. some contestants are
34
44. more successful in towns where polling stations tend to be larger and vice versa). The
most straightforward way to account for this effect is by using a simple linear model
for random vote share composition Si with the logarithm of the vote total log(ti) as a
predictor:
Si = a ⊕ log(ti) b + εi (3.2)
where a and b are to-be-estimated compositional constants and εi ∼ ND
S (1, Σ) is random
compositional noise. The logarithm of the vote total is used instead of simple vote totals
in order to decrease the leverage of huge polling stations, since the distribution of vote
totals is virtually always positively skewed. Interpretation of parameters a and b is of no
importance to us as the model serves for data generation only.
Following [van den Boogaart and Tolosana-Delgado, 2013, p. 129-131], Equation 3.2
can be rewritten as:
ilr(Si) = ilr(a) + ti · ilr(b) + ilr(εi) (3.3)
with ilr(εi) ∼ N(0D−1, Σilr). This is a standard linear model that can be fitted by
maximum likelihood using R packages stats and compositions.
3.1.3 A Multinomial Model for Vote Counts
The multinomial distribution provides a simple and intuitive way of modelling discrete
compositions [van den Boogaart and Tolosana-Delgado, 2013, p. 62-63]. Therefore it can
be conveniently used to model vote counts conditional on the total number of valid votes
cast. The probability of observing vote count composition c = (c1, c2, . . . , cD) in a polling
station with D contestants and the vote total of t is:
f(c; p, t) = t!
D
j=1
p
cj
j
cj!
(3.4)
where p = (p1, p2, . . . , pD) are the probabilities of any vote being cast for contestants
1, . . . , D. It is assumed that all votes within a polling station are independent with the
same p (expected counts are then t · p). Since p can be interpreted as expected vote
share composition then it can be estimated using predicted vote share compositions ˆs
35
45. from Equation 3.2. A potential problem with this distribution is that since the variance
matrix t · (diag(p) − pT
p) is fully determined by p, it is often not flexible enough for
modelling complex covariance structures.2
2
diag(p) stands for a matrix with p on the main diagonal and zeros elsewhere.
36
46. 3.2 Synthetic Data Generation
This section introduces the methodology of the simulational part of the analysis. Subsec-
tion 3.2.1 describes two ways fraud-free data are herein simulated and Subsection 3.2.2
compares the goodness of digital fit of the two models. Subsection 3.2.3 introduces the
simple model of fraud imputation into the simulated fraud-free election results and Sub-
section 3.2.4 reports how logistic regression models for discrimination between fraud-free
and fraudulent results are set up.
3.2.1 Fraud-Free Data Simulation
Based on the previous section, two ways of simulating election results are considered
here. Both approaches start with fitting the NDS model defined by Equation 3.2 to the
observed vote shares.
The first approach simulates vote counts in polling station i as draws from multinomial
distribution with the total given by vote total ti and the probability vector pi given by ˆsi
(as defined in Equation 3.4). This can be done using function rmultinom.ccomp() from
R package compositions.
The second approach rests on a simple model that will hereinafter be denoted as the
na¨ıve model. It simulates vote counts cij of contestant j in polling station i using a
two-step procedure. First, for each polling station i, value s ∗
i is computed by sampling
ilr(s ∗
i ) from the model given by Equation 3.3, that is, from N (ilr(a) + ti · ilr(b) , Σilr),
and applying inverse ilr(). This step is implemented in rnorm.acomp() function from
R package compositions. Second, vote counts are ‘na¨ıvely’ approximated as cij := ti ·
ˆs ∗
ij − 1, where stands for ceiling. Using ceiling assures that after subtraction of
the vote previously added to compute vote shares, all vote counts remain non-negative
integers. A slight inconsistency is induced by this procedure as the computed vote counts
do not need to sum exactly to the vote total. From a practical point of view, however,
these deviations are minimal.
37
47. 3.2.2 Goodness of Fit of the Synthetic Data
Both of the models rest on the assumption that empirical vote shares can be reasonably
well modelled by the normal distribution on the simplex. This subsection starts by
validating this assumption and then moves on to comparing the digital fit for the two
models outlined above. The decision will be made as to which of the models fits the
empirical patterns better and should therefore be used for simulations.
3.2.2.1 Fit of the Normal Model for Vote Shares
The fact that vote share distributions are often unimodal and rather bell shaped made
some researchers believe that they follow a normal distribution [see Myagkov et al., 2007,
2009]. This is of course impossible as the support of vote share distributions is bounded
by 0 and 1. If normality is to be expected in vote shares, then it would be normality on
the simplex.
The goodness of fit of the multivariate normal distribution on the simplex to em-
pirical vote share compositions can be assessed in at least two ways: either statistically
tested or visually explored. Complete compositional normality can be tested by apply-
ing a multivariate normality test to the isometric log-ratio transformations of vote share
compositions. A multivariate normality test introduced by [Szekely and Rizzo, 2008] is
implemented in command acompNormalGOF.test() of package compositions and was
applied to the election contests contained in this dataset. Overall, the test almost always
rejected the null hypothesis of multivariate normality. However, more exploration is
needed as non-normality in a single direction is sufficient to reject multivariate normality.
A visual assessment of compositional multivariate normality can be done using QQ-
plots. Multivariate normality on the simplex induces univariate normality of the loga-
rithm of a ratio of any two of its elements. Plotted values of the log-ratio transforma-
tion can then be compared with the standard normal distribution. Although bivariate
marginal normality does not necessarily imply joint normality, for most practical prob-
lems this assessment is good enough [van den Boogaart and Tolosana-Delgado, 2013].
However, a thorough visual examination is particularly difficult as the number of
election contests to examine is very high. For this reason, all election contests in elec-
38
