Developments in Positive Empirical Models of Election

Frauds∗

Diogo Ferrari† Walter R. Mebane, Jr.‡

July 19, 2016

∗Prepared for presentation at the 2016 Summer Meeting of the Political Methodology Society, RiceUniversity, Houston TX, July 21–23. Work supported by NSF award SES 1523355. Thanks to KevinMcAlister for suggestions.†Department of Political Science, University of Michigan, Haven Hall, Ann Arbor, MI 48109-

1045 (E-mail: dferrari@umich.edu).‡Professor, Department of Political Science and Department of Statistics, University of Michi-

gan, Haven Hall, Ann Arbor, MI 48109-1045 (E-mail: wmebane@umich.edu).

Abstract

We present new developments sparked by the positive empirical model of election frauds

introduced by Klimek, et al. (PNAS 2012). Taking a censored-Normal finite mixture likelihood

implementation of Klimek et al.’s idea (Mebane, MPSA 2016) as a point of departure, we develop

a variation of the model for which we estimate parameters using a Bayesian MCMC algorithm.

The variation uses functional forms somewhat more familiar to political scientists than those used

in the Klimek conception: e.g., logistic-binomial models for turnout and votes instead of censored

Normal distributions. We consider the effect that introducing covariates to condition the means of

turnout and of leading party vote share has on the performance and interpretability of the models.

We assess the performance of the logistic model using simulated data in Monte Carlo sampling

experiments. We apply both the Normal and logistic models to data from presidential elections in

Brazil, the United States (California) and Kenya. The logistic model’s performance is promising,

except for a problem with label switching. In the applications to real data the results from the

logistic model are similar to results from the Normal likelihood model.

1 Introduction

Election forensics is the field devoted to using statistical methods to determine whether the results

of an election accurately reflect the intentions of the electors. Election forensics techniques apply

statistical methods to low-level aggregates of votes such as polling station or ballot box counts of

eligible voters and votes recorded for parties. Many statistical methods for trying to detect

election frauds have been proposed (e.g. Myagkov, Ordeshook and Shaikin 2009; Levin, Cohn,

Ordeshook and Alvarez 2009; Shikano and Mack 2009; Mebane 2010; Breunig and Goerres

2011; Pericchi and Torres 2011; Cantu and Saiegh 2011; Deckert, Myagkov and Ordeshook 2011;

Beber and Scacco 2012; Hicken and Mebane 2015; Montgomery, Olivella, Potter and Crisp

2015). Multimodal distributions of turnout and vote choices are directly or indirectly the basis for

several approaches (Myagkov, Ordeshook and Shaikin 2008, 2009; Levin et al. 2009). Mebane

(2016) presents a finite mixture likelihood model that implements the sharpest contribution

motivated by multimodality, which is a model proposed by Klimek, Yegorov, Hanel and Thurner

(2012). All methods for election forensics may be affected by the fundamental ambiguity that it is

difficult to distinguish results caused by election frauds from results produced by strategic

behavior, but in combination the various methods can estimate, characterize and locate features of

low-level aggregates of votes that may indicate frauds (Hicken and Mebane 2015; Mebane 2015).

Klimek et al. (2012) specify functional forms that define two mechanisms by which votes are

added to a leading party: manufacturing votes from genuine nonvoters; and stealing votes from

the other parties. The “leading party” is the party that the model allows to gain votes from

frauds.1 These mechanisms operate either in an “extreme” manner, so election data have turnout

near 100 percent with nearly all votes going to the leader, or in an “incremental” manner, where a

substantial number but not almost all votes are reallocated to the leader. The model includes

parameters that express whether vote manufacturing or vote stealing is the predominant form of

fraud that occurs. If frauds as described by their model occur, then turnout and vote proportion

distributions are bimodal or trimodal.1A limitation of the model is that only one party may be represented as a party that gains votes from frauds.

1

As Mebane (2016) discusses, the Klimek et al. (2012) idea that multimodalities characterize

frauds is motivated by theory developed by Borghesi and Bouchaud (2010). Mebane (2016)

argues that the same theory implies that multimodalites are also created by strategic behavior. In

both cases the core mechanism is that imitative similarities are induced among electors. Whether

frauds and strategic behavior trigger empirical models that focus on multimodality in distinctive

ways is a question that Mebane (2016) preliminarily address, but more empirical and theoretical

research is needed to understand the distinctions.

The model of Mebane (2016) is based on a conception that closely follows the specification

defined by Klimek et al. (2012). We briefly review that specification in section 2.1. The

specification uses functional forms that are unusual for political science. Here we describe an

alternative specification that may be more familiar to political scientists, in that it includes

familiar logistic forms for turnout and vote choice proportions. In fact Borghesi and Bouchaud

(2010) also refer to logits of these proportions, so the alternative model is in that respect more

directly in line with their theorizing.2 Using the logistic functional forms means that covariates

can be incorporated in the model in a straightforward way to affect the distribution of turnout and

vote choice means.3

We adopt a Bayesian approach to formulating the alternative model, which we estimate using

Markov Chain Monte Carlo (MCMC) methods (Plummer 2003; Plummer, Stukalov and

Denwood 2016). At least with small datasets, estimation using the alternative model is fast

enough that we can perform Monte Carlo simulation exercises in which we estimate the model

using simulated data with known characteristics. We report some of those results. We also

compare results from using the two models with several real datasets.

2When vote counts are large, as they usually are, there is not any practical difference between the Normal dis-tributions used in the specification taken from Klimek et al. (2012) and the binomial distributions that go with ouralternative model that uses logistic functions.

3While the model specification in section 2.2 includes covariates, we will not illustrate estimation using covariateswith real data in this version of the paper. Meanwhile see Ferrari (2016).

2

2 Model

In the Klimek et al. (2012) formulation the baseline assumption is that votes in an election with

no fraud are produced through processes that can be summarized by two Normal distributions:

one distribution for turnout proportions and another, independent distribution for the proportion

of votes going to the leader. The model conditions on the number of eligible voters. Some votes

are transferred to the leader from the opposition, and some are taken from nonvoters. Two kinds

of election fraud refer to how many of the opposition and nonvoters votes are shifted: with

“incremental fraud” moderate proportions of the votes are shifted; with “extreme fraud” almost

all of the votes are shifted. Mebane (2016) and Klimek et al. (2012) have parameters that specify

the probability that each unit experiences each type of election fraud: fi is the probability of

incremental fraud and fe is the probability of extreme fraud. Other parameters fully describe

bimodal and trimodal distributions that the model characterizes as being consequences of election

frauds.

In our alternative specification the conception of frauds is essentially the same. Instead of

using Normal distributions as the fundamental distributions to model turnout and votes for the

leader, the alternative model uses binomial distributions and logistic models for those

distributions.

2.1 Original Specification

The original Klimek et al. (2012) conception includes three kinds of votes: votes without fraud;

votes with “incremental fraud”; and votes with “extreme fraud.” With no fraud the distribution of

votes, given the number of eligible voters, is a product Normal distribution. Fraud corresponds to

differing proportions of votes going to the leading party that should have gone to other parties or

should not have been counted as votes at all. With incremental fraud a small proportion xi of

nonvotes in electoral unit i are counted for the leading party while a proportion xαi , α > 0, of votes

for opposition instead go to the leading party. With extreme fraud a large proportion 1 − yi of the

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nonvotes are counted for the leading party and a proportion (1 − yi)α of genuine opposition votes

instead go to the leading party.

Following are definitions for the original empirical frauds model adapted from Klimek et al.

(2012) and implemented in Mebane (2016). Because of its reliance on censored and folded

Normal distributions we refer to this specification as the Normal model.

Observed data come from n electoral units (e.g., polling stations), and the observed number of

eligible voters in each unit is Ni, i = 1, . . . , n. Votes for parties are observed as the count of votes

for the leading party, denoted Wi, and the sum of votes cast for all other parties (the “opposition”),

denoted Oi. The number of observed nonvotes (“abstentions”) is Ai = Ni −Wi − Oi. The observed

number of valid votes is Vi = Ni − Ai.

Using N(µ, σ) to denote a normally distributed random variable with mean µ and standard

deviation σ, the conceptual specification of the model is as follows. For each electoral unit

i = 1, . . . , n, for some α > 0 define:

1. true turnout, τi ∼ N(τ, στ), 0 ≤ τi ≤ 1;

2. the leading candidate’s true vote proportion, νi ∼ N (ν, σν), 0 ≤ νi ≤ 1;

3. no fraud (occurs with probability f0 = 1 − fi − fe): the number of votes for the leading

candidate is W∗i = Niτiνi and the number of nonvoters is A∗i = Ni (1 − τi);

4. incremental fraud (occurs with probability fi): xi ∼ |N (0, θ) |, 0 < xi < 1, is the proportion

of genuine nonvotes counted as votes for the leading candidate and xαi is the proportion of

votes genuinely cast for others but instead counted as votes for the leading candidate, so the

number of votes for the leading candidate is W∗i = Ni

(τiνi + xi (1 − τi) + xαi (1 − νi) τi

), and

the number of nonvoters is A∗i = Ni(1 − xi) (1 − τi);

5. extreme fraud (occurs with probability fe): using yi ∼ |N(0, σx)|, σx = 0.075, 0 < yi < 1,

1 − yi is the proportion of genuine nonvotes counted as votes for the leading candidate and

(1 − yi)α is the proportion of votes genuinely cast for others but instead counted as votes for

4

the leading candidate, so the number of votes for the leading candidate is

W∗i = Ni (τiνi + (1 − yi) (1 − τi) + (1 − yi)α (1 − νi) τi), and the number of nonvoters is

A∗i = Niyi (1 − τi).

A finite mixture likelihood is defined in Mebane (2016).

2.2 Alternative Specification

The alternative specification we introduce here uses logistic functional forms and binomial

distributions. We use the logistic functional forms to allow turnout and vote choice means to

depend on covariates. We refer to this specification as the logistic model. Unlike the Normal

model which is estimated using likelihood methods, for the logistic model we adopt a Bayesian

estimation approach.

2.2.1 Data Model

Let i, i = 1, . . . , n, denote the electoral unit that generates the observation, for instance polling

station or ballot box. As in the Normal model, Ni is the observed number of eligible voters in

each unit, Wi denotes the observed count of votes for the leading party, and Ai denotes the

observed number of nonvotes.

LetWi be the true proportion of supporters of the leader andAi the true proportion of

abstentions at i. Let Zi ∈ {O, I, E} be an indicator variable such that Zi = O means no fraud has

occurred at i, Zi = I means incremental fraud has ocurred, and Zi = E means extreme fraud has

occurred. Wi,Ai and Zi are all unobserved. Also unobserved are S li and El

i, the incremental and

extreme proportions of votes stolen from absentees (l = a) or opposition (l = o) at i. We use ai,

wi, sli and el

i to denote the realizations ofAi,Wi, S li and El

i.

We observe Wi and Ai but we do not observe the labels Zi that indicate the distribution that

generated that observation. That is, we do not know if it comes from a no fraud, incremental

fraud, or extreme fraud distribution of votes and turnout. If no fraud has occurred, for instance,

the observed proportion represents the true realization of the random variableAi. If election

5

results have been manipulated and votes were manufactured from nonvoters, then the total

abstention is just a fraction of the true realization of the random variableAi. We have

Ai =

NiAi , if Zi = O

NiAi(1 − S ai ) , if Zi = I

NiAi(1 − Eai ) , if Zi = E

(1)

and

Wi =

NiWi(1 −Ai) , if Zi = O

Ni

(Wi(1 −Ai) + S a

iAi + S oi (1 −Ai)(1 −Wi)

), if Zi = I

Ni

(Wi(1 −Ai) + Ea

iAi + Eoi (1 −Ai)(1 −Wi)

), if Zi = E

(2)

Suppose incremental fraud has occurred (Zi = I) at i and votes have been manufactured from

nonvoters and stolen from the opposition. Then the true total abstention is Niai, but the observed

abstention is just a fraction of the true value. If a proportion sai ∈ [0, 1] of nonvoters was counted

as if they had voted to the leader, then the observed total abstention is Niai(1 − sai ). Similar

reasoning applies to the number of votes for the leader. The observed votes for the leader, Wi, is

the true number of votes for the leader, Niwi(1 − ai), plus the added votes from nonvoters (Nisai ai)

and from opposition (Nisoi (1 − ai)(1 − wi)).

We assume independence across observations and between random variables such that, given

observed covariates xi and coefficient vectors γ and β, the conditional distributions of Wi and Ai

6

are:

H(Ai | Ni, xi; z, s, e, θ) =

(Ni

Ai

)ηAi

a (1 − ηa)Ni−Ai , if Zi = O(NiAi

)ϕAi

a (1 − ϕa)ni−Ai , if Zi = I(NiAi

)ϑAi

a (1 − ϑa)ni−Ai , if Zi = E

(3)

ηa =

(1

1 + exp(−x′iγ)

)ϕa =

(1

1 + exp(−x′iγ)

)(1 − sa

i )

ϑa =

(1

1 + exp(−x′iγ)

)(1 − ea

i )

and

F (Wi | Ni, xi; Ai, z, s, e, θ)

(Ni

Wi

)ηWi

w (1 − ηw)ni−Wi , if Zi = O(NiWi

)ϕWi

w (1 − ϕw)ni−Wi , if Zi = I(NiWi

)ϑWi

w (1 − ϑw)ni−Wi , if Zi = E

(4)

ηw =

(1

1 + exp(−x′iβ)

) (1 −

Ai

Ni

)ϕw =

(1

1 + exp(−x′iβ)

) (1 − so

i

1 − sai

) (1 − sa

i −Ai

Ni

)+

Ai

Ni

(sa

i − soi

1 − sai

)+ so

i

ϑw =

(1

1 + exp(−x′iβ)

) (1 − eo

i

1 − eai

) (1 − ea

i −Ai

Ni

)+

Ai

Ni

(ea

i − eoi

1 − eai

)+ eo

i

Justification for the functional form of ηc, φc, ϑc, c ∈ {a,w} is in the appendix.

Using

αi = (1 + exp(−x′iγ))−1 (5a)

ωi = (1 + exp(−x′iβ))−1 , (5b)

7

we can express these distributions less formally as sets of binomial distributions:

Ai | (Zi, S ai , E

ai ) ∼

Binomial(αi,Ni) , if Zi = O

Binomial(αi(1 − S ai ),Ni) , if Zi = I

Binomial(αi(1 − Eai ),Ni) , if Zi = E

Wi | (Zi, S ai , E

ai ,Ai) ∼

Binomial(ωi(1 −Ai),Ni) , if Zi = O

Binomial(ωi(1 −Ai) + S aiAi + S o

i (1 −Ai)(1 − ωi),Ni) , if Zi = I

Binomial(ωi(1 −Ai) + EaiAi + Eo

i (1 −Ai)(1 − ωi),Ni) , if Zi = E

Given the dataD = (Wi, Ai)ni=1, N = (Ni)n

i=1 and X = (xi)ni=1, the likelihood function can be

written as

L(D | N,X; z, s, e, θ) =

n∏i=1

P(Wi, Ai | Ni, xi; z, s, e, θ)

=

n∏i=1

F (Wi | Ai,Ni, xi; z, s, e, θ)H(Ai | Ni, xi; z, s, e, θ) (6)

2.2.2 Prior Distribution

We consider the collection of random variables

Y = (W,A,Z,Sa,So,Ea,Eo) ∈ Wn×An×Zn×Sn×En ≡ [0, 1]n× [0, 1]n×{I, E,O}n× [0, 1]4n

8

with distributions given by

Zi ∼ Cat(φ), φ = (φO, φI , φE) ∈ ∆2

NiWi ∼ Binomial(ωi,Ni)

NiAi ∼ Binomial(αi,Ni) (7)

NiS li ∼ Binomial(ζ l

i ,Ni), ζi < k ∈ [0, 1], l ∈ {o, a}

NiEli ∼ Binomial(χl

i,Ni), χi ≥ k ∈ [0, 1], l ∈ {o, a}

The constant k captures an arbitrary cutpoint above which the fraud is considered extreme.

The set of parameters that index those distributions are θ = (φ, ω,α, ζo, ζa, ,χo,χa). The

parameter φ = (φO, φI , φE)T ∈ ∆2 is the vector capturing the expected proportion of units i that

experienced no fraud (φO), incremental fraud (φI), and extreme fraud (φE). If i are polling station

observations, for instance, φI is the expected proporiton of polling stations that experienced fraud.

The parameters ζ li and χl

i are fraud parameters at the unit level of observation. They capture,

respectively, the probability of incremental fraud and extreme fraud at the unit i. They also

capture the probability of incremental fraud and extreme fraud associated with stealing votes from

nonvoters (l = a) and from opposition (l = o).

For ζ li , χ

li, β, γ and φ we use the hyperprior

ζ li ∼ Unif(0, k) β ∼ Nd(µβ, τId×d)

χli ∼ Unif(k, 1) γ ∼ Nd(µγ, τId×d) τ ∈ R+

φ ∼ Dirichlet(σ1, σ2, σ3), σ1, σ2, σ3 ∈ R+

ωi captures the expected proportion of votes for the leader candidate at i, and αi represents the

expected turnout proportion at i. As specified in (5), these proportions depend on observed

covariates Xi = (X1i, ..., Xdi)′.

The assumptions on the dependence structure of the random variables in the model are,

9

∀i, j = 1, ..., n and i , j, (A1)(Wi yW j) | X, (A2)(Ai y A j) | X, (A3)(Wi y A j) | X,

(A4)lloS li y El

i y Zi. The assumptions (A1) to (A3) mean that, after taking into account the

covariates and their effect on the expected turnout and on the expected support for a specific party

at i, elements at unit j and unit i are independent.

For now, the model also assumes that the probability of incremental fraud is independent of

the probability of extreme fraud. This assumption can be relaxed in extended versions of the

model if there is information available about such dependence. It is reasonable to assume it in the

extent that, for instance, each fraud depends exclusively on the probability of getting caught in

each case. The model keeps this assumption for simplicity. Also, (A4) says that incremental and

extreme fraud at i are independent of the expected fraud in the election. In other words, it says

that if there is a positive probability of incremental fraud in the election, that probability do not

affect the chance of fraud in a specific unit i. Suppose that the probability of fraud at i is .5.

Suppose also that the probability of fraud in general is very low. Then it is necessary to consider

both probabilities together. That is, the joint probability of fraud at i and of incremental fraud in

the election as a whole gives the final fraud probability at i. Another way to put it is that the

probability of fraud in the election is weighted by the probability of fraud in unit i and vice-versa.

The prior distribution for this model is given by

p(Y, θ) =

n∏i=1

[p(NiWi | β)p(β)

]× [(p(NiAi | γ)p(γ)] × [p(NiS a

i | ζa)p(ζa)] × [p(NiS o

i | ζo)p(ζo)]

× [p(NiEai | χ

a)p(χa)] × [p(NiEoi | χ

o)p(χo)] × [p(zi | φ)p(φ)] (8)

2.2.3 Posterior Distribution

Combining equations (6) and (8), the posterior distribution is given by:

p(Y, θ | D) ∝ L(D | N,X; z, s, e, θ)p(Y, θ)

=

n∏i=1

∑j∈{O,I,E}

I(zi = j)F (Wi | Ai,N,X; z, s, e, θ)H(Ai | N,X; z, s, e, θ)

p(θ,Y)

10

2.2.4 Restricted models

This section presents a restricted version of the model developed in the previous sections

2.2.1–2.2.3. There are three justifications for having a restricted version of the model. The first is

substantive. If there is no reason to believe that each polling place are on average different in

terms of the probability of ocurrence of fraud, that is, ζ li = ζ l and χl

i = χl for all i in (7), then the

number of parameters can be reduced substantially. Once triggered with some probability, the

result is a proportion S ai of votes manufactured from nonvoters at i and so on. Therefore, we can

simplify the model and use 4 parameters (ζa, χa, ζo, χo) instead of 4 × n parameters.

The second justification is computational. If it is believed that ζ li and χl

i are the same for all i,

using a simplified version can reduce the computatitonal resources and the time needed to

perform the estimation. Obtaining well behaved posterior distributions for 4 × n parameters ζ li and

χli may be inordinately difficult.4

Another simplification is excluding the covariates in the model. Most often, in practice, we

lack meaningful covariates for each observation i. Even if observation-level covariates are

available, we can treat the covariates as having a common mean across all observations and

assume the covariates are independent of any frauds and do not themselves create

multimodalities. In these cases, parameters γ and β in (3), (4) and (5) reduce to the intercept

terms. The true vote proportions for the leader and the proportions of abstentions for different

observations are drawn from distributions with common mean.

The logistic model with no covariates and with common ζ l and χl parameters across

observations has a structure that is directly comparable with the mixture of restricted normals as

used by Mebane (2016) and Klimek et al. (2012), that is, with the Normal model. We compare the

estimation produced by the restricted version of the model developed here using MCMC

procedure againt the model with restricted normals estimated using EM-algorithm.

In practice, it is common that one does not know with certainty to what extent covariates

4So far in simulated data we have found multimodal posterior distributions for the parameters when attempting toestimate all 4 × n of them.

11

affect behavior, whether local fraud probabilities are similar everywhere or votes are stolen in a

centralized manner, or even whether there are huge differences on average in the probability of

turnout or vote for the leader from one polling place to another.5 So it is important to compare

results under different assumptions to see whether and how inclusion of covariates and less

restricted sets of random variables affect the estimation when they are available. If the models

produce different results, one can use a model selection criterion to help decide which one has the

best fit to the data. Domain expertise and further analysis should inform any substantive

conclusion about whether an election is fraudulent (Hicken and Mebane 2015; Mebane 2016).

Table 1 describes qualitatively the assumptions for each of the two logistic models considered.

The modelM1 is the full model described in the previous section. The modelM2, imposes

restriction on the value of the parameters. On average withM2 the chance of fraud in each polling

place is the same, that is χli and ζ l

i are the same for every i, and there is no covariate effect. The

same is true for the average parameter for true abstention and true support for the leading party.

*** Table 1 about here ***

3 Data and Estimation Results

3.1 Simulated Data

This section uses simulated data sets to evaluate the performance of the logistic model.

Descriptions of the simulated data sets and of the parameters used to generate them are in the

appendix. Because we know the values of the parameters that generated the data, particularly the

fraud probabilities, we can compare the results produced by the model when the data comes from

different contexts. The choices of the parameter values that generate the data are intended to

capture the main points discussed in the previous sections and to discriminate the models’

capacity to provide the correct estimates in each case. We use a two-step procedure.5Another challenge when thinking about using covariates is whether it is justified to assume they affect only the

distribution of true votes and abstentions or also affect the incidence of frauds. Do covariates affect only means, as inthe current logistic model, or perhaps they modify variances as well?

12

In the first step, we use four different combinations of parameter values to generate simulated

data. Table 2 presents a qualitative description of the simulated data sets. In Table 2, a checkmark

indicates that the corresponding feature is present in the simulated election, and a crossmark

indicates that it is absent. The column “ID” is just an auxiliary label that guided the

implementation. In Data Set 1.1, the data correspond to an election with no fraud of the types

considered here (φO ≈ 1, φI ≈ 0, φE ≈ 0), covariates do not affect turnout and number of votes for

a specific party6 (β0 = γ0 = 1, β1 =, ...,= βd = γ1, ..., γd = 0), and the local fraud probabilities

(ζ li , χ

li) are the same for all polling places i. One single sample with n = 4, 900 observations was

generated for each one of the features described in the table 2, using the value of the parameters

described in the appendix. Using these data sets, a MCMC procedure was implemented using

JAGS (Plummer 2003; Plummer, Stukalov and Denwood 2016) to estimate the value of the

parameters using the logistic model for each one of the data sets. Only the full modelM1 was

used.

*** Table 2 about here ***

In the second step, we select two data sets among the ten described in table 2 based on the

performance of the modelM1. We selected the data sets for which the interval with the Highest

Posterior Density (HPD) covered the true parameter’s value for all or almost all parameters, and

the one with the worst performance in the estimation, that is, the data set for which the HPD

didn’t cover the true value of the parameter. The parameter vector φ was used for the selection

because it is arguably the most important parameter of the model. Then, 150 different data sets

were generated using the same underlying features. The MCMC procedure was repeated in each

one of them, using four chains in each case, and the results—averages and HPD limits—were

averaged across data sets and chains. For this second step, a smaller sample was used for each

data set due to computational time constraints. Each one of the 150 data sets for the two different

electoral contexts has n = 1, 444 observations. In the second step, the same procedure was

6Alternatively, the absense of effect of the covariates can be seen as a random distribution of those covariates inthe space such that there is no spatial concentration of them in the polling places.

13

repeated for both of the models described in the previous section. Together, the first and second

steps reduce both the too-optimistic and the too-pessimistic evaluation of the performance of

modelM1 that can be attributed to a particular sample generated in the first step, and it also

allows us to compare the performance of different models in each case.

The main quantities of interest to be estimated using samples generated by the MCMC are the

posterior averages of the fraud probabilities E[φ j | ·], and the posterior average of local fraud

probabilities E[ζ li | ·] and E[χl

i | ·] for some i, particularly those places with high and low averages.

Also, it is interesting to compute the marginal classification probability, which is the probability

that a particular observation i comes from the no fraud, incremental fraud, or extreme fraud

distribution. It is given by

p(Zi = j | D,N,X) ∝∫

φ jL(D | N,X; z, s, e, θ j)p(θ | D)dθ (9)

Table 3 shows the result of the MC estimation for the ten simulated data sets described in

Table 2 that is, for the step 1. We use four indepenent parallel Markov Chains, each with a burn-in

period of 10,000 iterations. Standard convergence diagnostics for multiple (Potential Scale

Reduction Factor (PSRD), trace plot) and single (Geweke test, running averages) chains were

performed in all the cases (Cowles and Carlin 1996). As it can be seen in the table, the 95% HPD

intervals are small and quite close to the true parameter value in 8 out of the 10 estimations of the

posterior distribution of φ. The MC averages across chains also are very close to the true values in

those cases.

*** Table 3 about here ***

The best performance occurred for the data set I-A-a and I-C-a, that is, those similated

elections with no fraud. The data set I-C-a was selected as the case of best performance for the

second step procedure, particularly because it includes covariates.

The poorest performance of the model occurred in the estimation of the parameters of the data

sets II-A-a and IV-C-d. In the latter case, the model actually captured the values close to true

14

three values of φ, but the labels were switched between no fraud and incremental fraud. In this

case, the model indicates, nevertheless, that frauds occur. In the case of data set II-A-a, however,

the estimation was not able to capture the incremental fraud probability. Therefore, features of the

data set II-A-a were selected to generate the samples for the second step procedure.

Table 4 shows the results of the second step for the data set II-A-a for φ using the modelM2.

These data sets correspond to an election with only incremental fraud, no effect of the covariates,

and same distribution of local fraud probability for every polling place. Therefore, the appropriate

model is the minimal model, that is,M2 in Table 1. Table 4 shows that the the φ parameters have

the right values but there is label switching: φ1 and φ2 are inverted.7

*** Table 4 about here ***

Table 5 shows the results of the second step for the data set I-C-a for φ using modelM1. These

data sets correspond to an election with tiny probabilities of fraud but with covariate effects. Table

5 reports only the φ estimates. The HPD intervals include the true values of φ, and the posterior

means are close to the truth: E(φ2) = .0015 and E(φ3) = .0008. There is no label switching.

*** Table 5 about here ***

3.2 Real Data

We estimate the Normal and logistic models using data from elections in three countries. The

elections are all presidential elections: Brazil 2014; United States (California only) 2008; and

Kenya 2013, but the electoral systems in the various elections are diverse. Two elections have

plurality winner-take-all rules (Brazil 2014 round 2, California 2008) and one has plurality rules

with a regional dispersion requirement (Kenya 2013). One election carries the two parties with

the most votes forward into a runoff election (Brazil 2014 round 1). For the Brazil and California

elections we have polling station or precinct observations. For Kenya we have ward observations.8

7Computing issues prevented estimation of all model parameters. We have samples for φ for all data sets.8Data source information: Brazil 2014 (Tribunal Superior Eleitoral 2014); California 2008 (University of Califor-

nia Berkeley 2009); Kenya 2013 (Maron 2012). Each “polling station” in Brazil corresponds to a unique combination

15

Figures 1–4 show empirical density plots that display features of the data that these

mode-sensitive models should respond to.9 For both rounds of the Brazil 2014 election (Figures 1

and 2) bimodality is visually apparent in the one-dimensional densities for the proportions of

votes counted for the party with the most votes but not for turnout proportions. For both rounds

the two-dimensional densities show a single mode with strong skew more than clear bimodality.

For presidential voting in California 2008 (Figure 3) the one-dimensional densities for the vote

proportions for the candidate with the most votes and for turnout proportions are each visibly

multimodal, but the two-dimensional density shows that the modes for the two variables do not

align so that overall there is a single mode with strong skew more than clear bimodality or

trimodality. For the Kenya 2013 election (Figure 4) multimodality is visibly apparent in both

one-dimensional and two-dimensional density plots.

*** Figures 1–4 about here ***

3.2.1 Normal Model

Likelihood estimates of the Normal model’s parameters (Mebane 2016) are reported in Table 6.

We report fi, fe, α, θ, τ and ν. Table 6 also includes reports of the likelihood ratio test statistic for

the hypothesis that there are no frauds (i.e., that fi = fe = 0), along with the number of polling

station, precinct or ward observations for each election.

*** Table 6 about here ***

In most cases allowing for frauds improves model performance significantly. The likelihood

ratio test statistics reported in the ‘LR’ column in Table 6 show that in all but one instance the

hypothesis that there are no frauds can be rejected. Only for round 1 of the election in Brazil is

of the SIGLA UE, CODIGO MUNICIPIO, NUM ZONA and NUM SECAO (state, city, zone and section) variables. For Cali-fornia, absentee precincts (which lack eligible voter counts) are merged with the corresponding in-person precincts.

9We use the sm package (Bowman and Azzalini 1997) of R (R Development Core Team 2011) to estimate thetwo-dimensional empirical densities.

16

the hypothesis not rejected.10

Parameter estimates allow us to diagnose the types of frauds occurring in the elections,

including diagnosing when the “fraud” estimates do not point to genuine frauds—maleficent

actions—but instead to consequences of strategic behavior (see Mebane 2016). The estimates

show a very high probability of incremental fraud and a substantial probability of extreme fraud

in Kenya. These estimates probably correspond to maleficent acts instead of strategic behavior.

The incremental fraud probabilities ( fi) are small and roughly the same size for the Brazil 2014

round 2 and California 2008 presidential elections, which suggests the “frauds” actually result

from strategic behavior. But other parameters suggest the origins of the fraudulent (or strategic)

votes differ in the two cases. In Brazil, α > 1 so that xαi < xi and vote manufacturing (from

nonvoters) is more important, while in California α < 1 so that xαi > xi and vote stealing (from

opposition) is more important.

The magnitude of frauds in terms of the number of votes estimated to be shifted to the leading

party is a way to assess frauds. A point estimate for the number of votes produced by frauds can

be determined by computing Mi =∑n

i=1 Ni fiiξi and Me =∑n

i=1 Ni feiξe, using expected proportions

ξi and ξe.11 pi = Mi/∑n

i=1 Vi and pe = Me/∑n

i=1 Vi express these fraudulent vote counts as

proportions of the valid votes. Table 7 reports these values.

*** Table 7 about here ***

The estimated numbers of votes moved by frauds range from negligible (the Brazil and

California elections) to large. In raw numbers the largest count is Mi = 748, 330 for the Kenya

2013 presidential vote, a count that is orders of magnitude larger than the negligible number of

votes moved by frauds (probably actually by strategic behavior, see Mebane 2016) in Brazil or

10For the estimates for Brazil reported in Table 6 only votes for a party are treated as valid votes, so a “Blank”ballot is not. Estimates using voting machine data show significant incremental frauds for both Brazil rounds: round 1(n = 424, 902), fi = .0027, fe = 0, α = 1.23; round 2 (n = 425, 136), fi = .000052, fe = 0, α = 14.0.

11 fii and fei are estimates of the probability that observation i is a case of, respectively, incremental fraud andextreme fraud. ξi = E[xi (1 − τi) + xαi (1 − νi) τi | Ψ = Ψ] and ξe = E[(1 − yi) (1 − τi) + (1 − yi)α (1 − νi) τi | Ψ = Ψ],where Ψ is the vector of parameters and Ψ is the vector of parameter estimates (Mebane 2016).

17

California. In proportional terms the largest proportion of fraudulent votes in Table 7 also occurs

in Kenya (pi + pe = .148).

3.2.2 Logistic Model

Estimates using logistic modelM1 finds signs of frauds in all the elections. Table 8 reports

posterior means from modelM1 averaged across all four MCMC chains and the 95% HPD

intervals for each chain.

*** Table 8 about here ***

For all four elections the results broadly agree with the results from the Normal model, except

the logistic model estimates differ due to apparent label switching. The φ2 parameter nominally

measures the probability of incremental fraud, but in each case its posterior mean is the largest of

the three probabilities in φ and in fact φ2 seems to measure the probability of no fraud. For

example, for California E(φ2) = .9967 and E(φ2) = .0032 while fi = .00053 and fe = 0. φ2 is most

likely the probability of incremental fraud not of extreme fraud.

If we acknowledge label switching and treat all the φ2 parameters as estimating the probability

of no fraud, then for Kenya it seems reasonable to interpret the φ1 parameter as estimating the

probability of incremental fraud. The posterior mean of E(φ1) = .375 for Kenya is very close to

the Normal model’s estimate of fi = .35. If for Kenya φ3 estimates the probability of extreme

fraud, then the Normal and logistic models’ implications agree: fe = .0015 and E(φ3) = .0010.

For Brazil, even if we accept that label switching is occurring and read φ2 as measuring the

probability of no fraud, it is unclear which kind of frauds φ1 and φ3 measure. In round 1 of the

election, E(φ1) = .0745 and E(φ3) = .0005, but fi = fe = 0. The Normal model finds that no

frauds occur but the logistic model finds frauds. The Normal model provides no guidance about

how to interpret the nonzero logistic model parameters. In round 2 of the election E(φ1) = .0050

and E(φ3) = .0271 while fi = .00040 and fe = 0. Both E(φ1) and E(φ3) are orders of magnitude

larger than fi, and it is not obvious merely from the estimates of φ alone how to interpret the

frauds to which they may correspond.

18

In Table 9 we report for Kenya posterior summaries for modelM1 abstention and vote choice

intercepts γ and ω and probability parameters ω and α as well as for selected fraud-generating

probabilities ζoi , ζa

i , χoi and χa

i (recall (7)). ζai (min) is the smallest of the E(ζa

i ) values and ζai (max)

is the largest. Plainly E(ω) is describing a mean for the lower mode that is apparent in the

two-dimensional density in Figure 4—in fact E(ω) approximates the smallest observed vote

proportion. E(α) corresponds to the same observation’s abstention proportion. Each of the E(ζoi ),

E(ζai ), E(χo

i ) and E(χai ) values, when worked through equations (3) and (4), matches the observed

data values. The values shown in the Table for ζa, ζ0, χa and χo are the means of the respective

underlying types of parameters, but evidently the model is overparameterized. E(ω) and E(α) do

not estimate true average vote and abstention proportions. The intended meanings of the

parameters are lost.

*** Table 9 about here ***

These results for Kenya raise profound doubts about whether modelM1 is suitable as a model

for frauds. The frauds probability results reported in Table 8 are in this light questionable.

For Kenya we have estimates from modelM2. Diagnostics regarding the MCMC chains for

this model are shown in the appendix. In Table 10 we report for Kenya posterior summaries for

the φ parameters, for the modelM2 abstention and vote choice probability parameters ω and α as

well as for the fraud-generating probabilities ζo, ζa, χo and χa. E(φ2) = .1419 is less than half as

big as fi = .35 but E(φ3) = .0010 and fe = .0015 are similar. This appears not to be a case of label

switching but merely of slightly differing estimates. Comparing the other parameters to the

comparable parameters in the Normal model may illuminate what is going on with the logistic

model. E(ω) = .22 should match ν = .35 and E(α) = .71 should match 1 − τ = .75, and to a rough

approximation they do. E(ω) = .22 is perhaps a bit low, or ν = .35 is a bit high, but otherwise the

two models suggest comparable values for the true proportion of votes for the leading party and

for the proportion abstaining. So it appears the models imply similar structures for the frauds that

may have occurred.

19

*** Tables 10 about here ***

The Normal model estimate α = 2.3 for Kenya means xαi < xi and (1 − yi)α < 1 − yi, which

suggests we should see E(ζo) < E(ζa) and E(χo) < E(χa), but instead with modelM2 we have

E(ζo) > E(ζa) and E(χo) > E(χa). The plots in appendix Figures 6 and 7 suggests the posterior

means E(χa) and E(χo) are poor summaries for the posterior distributions for χa and χo: the

distributions are visibly multimodal. For two of the three chains E(χo) < E(χa). Parameters ζo, ζa,

χo and χa do not have direct counterparts in the Normal model, but we can compare the fraud

magnitudes implied by the two models. Table 11 shows the fraud magnitude computed using

equation (9). The incremental fraud magnitude Mi was computed by first classifying each

observation i as coming from the distribution with highest estimated posterior probability for Zi

(equation (9)). Then a proportion of those observations classified as fraudulent were added,

following the procedure mentioned above to get the fraud magnitude. The estimated fraud

magnitude is smaller than with the Normal model, although the order of magnitude is the same,

and the source of the fraud is also different. The logistic model classifies all the fraud as

incremental while the Normal model shows votes being shifted by both types of frauds.12

*** Table 11 about here ***

4 Discussion

As a practical and statistical matter, it is good to shift the empirical frauds modeling framework

from a likelihood implementation to a fully Bayesian implementation. Among the advantages a

Bayesian implementation should confer is the ability to state well-motivated measures of the

uncertainty in quantities such as the estimated magnitude of frauds. Currently we have an initial

Bayesian formulation using logistic functions and binomial distributions. The model and

estimator need more work to fix a label-switching problem (cf. Grun and Leisch 2009), but the

results of this initial effort are promising. It may be that the apparent label switching is an artifact12Fraud magnitude estimation for the other countries is not ready because of computing issues.

20

of using the overparameterized model formM1, so that label switching is not a genuine problem.

Instead the point may be simply to use model formM2—perhaps however also including

covariates.

With luck soon we will have Bayesian versions of the Normal model as well as various

extended models. The model extensions we envision will work more smoothly and simply if label

switching is not a genuine concern.

5 Appendices

5.1 Appendix A - Likelihood Function

The justification of the functional form of ηc, φc, ϑc, c ∈ {a,w} is the following.

Note that

Zi = I =⇒ Ai = NiAi(1 − S ai )

Therefore

E[Ai | Zi = I, S ai ] = E[NiAi(1 − S a

i ) | Zi = I, S ai ] = Niαi(1 − S a

i )

= Ni

(1

1 + exp(−x′iγ)

)(1 − sa

i ) = Niϕa

The case of Zi = E is analogous. It justifies the functional form of ϕa and ϑa

Note that:

Zi = O =⇒ Ai = NiAi ⇐⇒ Ai =Ai

Ni

21

Therefore, replacingAi by AiNi

in (4)

E[Wi | (Zi = O, S ai , E

ai ,Ai)] = Niωi

(1 −

Ai

Ni

)

Similarly, note that:

Zi = I =⇒ Ai = NiAi(1 − S ai ) ⇐⇒ Ai =

Ai/Ni

1 − S ai

Therefore, replacingAi by Ai/Ni1−S a

iin (4)

E[Wi | (Zi = I, S ai , E

ai ,Ai)] = Ni

[ωi(1 −Ai) + S a

iAi + S oi (1 −Ai)(1 − ωi)

]= Ni

[ωi

(1 −

Ai/Ni

1 − S ai

)+ S a

i

(Ai/Ni

1 − S ai

)+ S o

i

(1 −

Ai/Ni

1 − S ai

)(1 − ωi)

]= Ni

[(ωi + S o

i (1 − ω))(1 −

Ai/Ni

1 − S ai

)+ S a

i

(Ai/Ni

1 − S ai

)]= Ni

[ωi(1 − S o

i )(1 −

Ai/Ni

1 − S ai

)+ S o

i

(1 −

Ai/Ni

1 − S ai

)+ S a

i

(Ai/Ni

1 − S ai

)]= Ni

[ωi(1 − S o

i )(1 −

Ai/Ni

1 − S ai

)+ S o

i − S oi

(Ai/Ni

1 − S ai

)+ S a

i

(Ai/Ni

1 − S ai

)]= Ni

[ωi

(1 − S a

i −Ai

Ni

) (1 − S o

i

1 − S ai

)+

Ai

Ni

(S a

i − S oi

1 − S ai

)+ S o

i

]= Niϕw

The case of Zi = E is analogous. It justifies the functional form of ηw, ϕw and ϑw

5.2 Appendix B - Simulated Data

The model was tested in a subset of simulated data sets generated by different combinations of

values of the parameters described in Table 12. The table provides a qualitative description of the

characteristic of the data sets. The fist column of the table describes the set of parameters that

were permuted, the second column provides a code to identify the parameter values, and the third

describes the meaning of the code in the second column. The data set I-A-a, qualitatively,

22

represents an election with very low probability of incremental and/or extreme fraud, no

association between covariates and turnout or chances of voting for specific parties, and every

polling place has the same local probability of fraud of each type—that is, stealing votes from

opposition or manufacturing voes from nonvoters. Data set IV-C-a, on the other hand, represents

an election that has incremental and extreme fraud, there are spatial concentration of covariates

and they affect turnout and chances of voting for the winner, and the local fraud probability is the

same in every polling place.

There are 48 qualitative combinations. For each data set, two covariates were randomly

generated.

X1 ∼ Cat(π), X1 ∈ {1, 2, 3, 4}

X2 ∼ N(µ, σ2)

Using a treatment contrast design matrix for X1, we can write the effect of the covariates in the

simulation as

lnωi

1 − ωi= βT Xi, β = (β1, β2, β3, β4, β5)T , Xi = (1, X12i, X13i, X14i, X2i)

lnαi

1 − αi= γT Xi, γ = (γ1, γ2, γ3, γ4, γ5)T

Spatial distribution of covariates, turnout, and proportion of votes for the winner were

organized in a 70x70 grid representing two spatial coordinates. Some parameter values used to

generate the data sets are described in Tables 13 to 16.

5.3 Appendix B - Logistic Model Diagnostics

Diagnostics regarding the MCMC chains for modelM2 for Kenya are displayed in Figures 5–19.

*** Figures 5–19 about here ***

23

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26

Table 1: Different assumptions for each modelModel αi ωi β γ χl

i ζ li

M1 i−specific i−specific included included i−specific i−specificM2 same for all i same for all i excluded excluded same for all i same for all i

Table 2: Qualitative Description of the Simulated Data SetsData Set ID Incremental Fraud Extreme fraud Covariate effect ζ l

i,χli same for all i

1.1 I-A-a x x x X1.2 II-A-a X x x x1.3 III-A-d x X x X1.4 IV-A-a X X x X1.5 IV-A-d X X x x2.1 I-C-a x x X X2.2 II-C-a X x X X2.3 III-C-d x X X x2.4 IV-C-a X X X X2.5 IV-C-d X X X x

Table 3: Results of MCMC estimation for φData.Set Parameter True MC.aver HDP.chain.1 HDP.chain.2 HDP.chain.3 HDP.chain.4I-A-a phi[1] 1.00 1.00 (0.998, 1)** (0.998, 1)** (0.998, 1)** (0.998, 1)**

phi[2] 0.00 0.00 (0, 0.001)** (0, 0.001)** (0, 0.001)** (0, 0.001)**phi[3] 0.00 0.00 (0, 0.001)** (0, 0.001)** (0, 0.001)** (0, 0.001)**

II-A-a phi[1] 0.70 1.00 (0.997, 0.999) (0.997, 0.999) (0.997, 0.999) (0.997, 0.999)phi[2] 0.30 0.00 (0, 0.001) (0, 0.001) (0, 0.001) (0, 0.001)phi[3] 0.00 0.00 (0, 0.002)** (0, 0.003)** (0.001, 0.003)** (0, 0.003)**

III-A-d phi[1] 0.70 0.71 (0.695, 0.72)** (0.695, 0.72)** (0.695, 0.72)** (0.694, 0.72)**phi[2] 0.00 0.06 (0.054, 0.067) (0.053, 0.067) (0.053, 0.068) (0.054, 0.068)phi[3] 0.30 0.23 (0.22, 0.244) (0.221, 0.244) (0.22, 0.243) (0.22, 0.244)

IV-A-a phi[1] 0.60 0.65 (0.788, 0.81) (0.789, 0.811) (0.788, 0.81) (0.185, 0.208)phi[2] 0.20 0.15 (0, 0.001) (0, 0.001) (0, 0.001) (0.59, 0.618)phi[3] 0.20 0.20 (0.189, 0.211)** (0.189, 0.211)** (0.19, 0.212)** (0.19, 0.212)**

IV-A-d phi[1] 0.60 0.66 (0.649, 0.679) (0.656, 0.684) (0.65, 0.678) (0.641, 0.67)phi[2] 0.20 0.17 (0.158, 0.183) (0.153, 0.177) (0.16, 0.183) (0.166, 0.19)phi[3] 0.20 0.17 (0.155, 0.176) (0.155, 0.176) (0.155, 0.176) (0.156, 0.177)

I-C-a phi[1] 1.00 1.00 (0.998, 1)** (0.998, 1)** (0.998, 1)** (0.998, 1)**phi[2] 0.00 0.00 (0, 0.002)** (0, 0.001)** (0, 0.002)** (0, 0.001)**phi[3] 0.00 0.00 (0, 0.001)** (0, 0.001)** (0, 0.001)** (0, 0.001)**

II-C-a phi[1] 0.70 0.60 (0.582, 0.611) (0.585, 0.613) (0.587, 0.615) (0.585, 0.613)phi[2] 0.30 0.40 (0.389, 0.417) (0.386, 0.414) (0.384, 0.412) (0.386, 0.413)phi[3] 0.00 0.00 (0, 0.003)** (0, 0.003)** (0, 0.003)** (0.001, 0.002)**

III-C-d phi[1] 0.70 0.71 (0.693, 0.719)** (0.694, 0.719)** (0.694, 0.719)** (0.694, 0.72)**phi[2] 0.00 0.07 (0.063, 0.078) (0.063, 0.078) (0.064, 0.079) (0.062, 0.077)phi[3] 0.30 0.22 (0.211, 0.235) (0.212, 0.235) (0.21, 0.234) (0.212, 0.236)

IV-C-a phi[1] 0.60 0.47 (0.455, 0.483) (0.454, 0.482) (0.455, 0.483) (0.452, 0.481)phi[2] 0.20 0.33 (0.318, 0.344) (0.318, 0.345) (0.318, 0.344) (0.32, 0.346)phi[3] 0.20 0.20 (0.189, 0.211)** (0.189, 0.211)** (0.189, 0.212)** (0.188, 0.211)**

IV-C-d phi[1] 0.60 0.18 (0.169, 0.195) (0.167, 0.192) (0.164, 0.193) (0.173, 0.198)phi[2] 0.20 0.62 (0.601, 0.631) (0.603, 0.632) (0.602, 0.636) (0.597, 0.627)phi[3] 0.20 0.20 (0.19, 0.213)** (0.192, 0.214)** (0.191, 0.213)** (0.191, 0.214)**

Table 4: Estimation of data set II-A-a . Average across 150 replicationstrue MC.exp MC.sd HPD.lower.1 HPD.upper.1 HPD.lower.2 HPD.upper.2

φ1 0.7000 0.2966 0.0177 0.2727 0.3207 0.2727 0.3207φ2 0.2990 0.7016 0.0177 0.6774 0.7255 0.6775 0.7255φ3 0.0010 0.0018 0.0014 0.0002 0.0039 0.0002 0.0039

Table 5: Estimation of data set I-C-a . Average across 150 replicationstrue MC.exp MC.sd HPD.lower.1 HPD.upper.1 HPD.lower.2 HPD.upper.2

φ1 0.999 0.9977 0.0015 0.9952 0.9997 0.9952 0.9997φ2 0.0005 0.0015 0.0013 0.0001 0.0035 0.0001 0.0035φ3 0.0005 0.0008 0.0008 0.0000 0.0023 0.0000 0.0023

Table 6: Normal Model Parameter Estimates

Election fi fe α θ τ ν LR nBrazil 2014 President:

round 1 0 0 — — .95 .47 0 6,265round 2 .00040 0 19 .76 .95 .56 113.38 6,265

California 2008 U.S. President .00053 0 .1 .27 .78 .59 612.6 22,691Kenya 2013 President .35 .0015 2.3 .51 .25 .35 2147.62 1,231

Note: LR is the likelihood ratio test statistic for the hypothesis that there are no frauds (i.e., thatfi = fe = 0). n is the number of polling station, precinct or ward observations.

Table 7: Normal Model Estimated Fraudulent Vote Counts and Proportions

Election Mi Me pi pe pi + pe

Brazil 2014 President:round 1 0 0 0 0 0round 2 2,094 0 .0000199 0 .0000199

California 2008 U.S. President 2,528 0 .000199 0 .000199Kenya 2013 President 748,330 21,282 .144 .00411 .148

Table 8: MC estimation for φ using modelM1

Parameter MC.exp MC.sd HPD.chain.1 HPD.chain.2 HPD.chain.3Brazil 1st round φ1 .0745 .0055 (0.065, 0.085) (0.066, 0.088) (0.062, 0.083)

φ2 .9250 .0057 (0.914, 0.934) (0.911, 0.933) (0.917, 0.938)φ3 .0005 .0004 (0, 0.001) (0, 0.001) (0, 0.001)

Brazil 2st round φ1 .0050 .0011 (0.003, 0.007) (0.004, 0.009) (0.002, 0.006)φ2 .9679 .0043 (0.96, 0.976) (0.961, 0.977) (0.957, 0.973)φ3 .0271 .0037 (0.02, 0.035) (0.017, 0.031) (0.024, 0.038)

Kenya 2013 φ1 .3750 .0347 (0.329, 0.457) (0.311, 0.426) (0.312, 0.447)φ2 .6240 .0347 (0.543, 0.671) (0.574, 0.688) (0.553, 0.688)φ3 .0010 .0010 (0, 0.003) (0, 0.003) (0, 0.003)

California 2008 φ1 .0000 .0000 (0, 0) (0, 0) (0, 0)φ2 .9967 .0004 (0.996, 0.998) (0.996, 0.998) (0.996, 0.998)φ3 .0032 .0004 (0.002, 0.004) (0.002, 0.004) (0.002, 0.004)

Table 9: MCMC estimation for parameters for Kenya 2013, modelM1

parameter MC.exp MC.sd HPD.chain.1 HPD.chain.2 HPD.chain.3β −5.2579 0.0854 (−5.3426, − 4.9712) (−5.4070, − 5.1614) (−5.3952, − 5.2211)γ 1.1206 0.0013 (1.1182, 1.1232) (1.1183, 1.1237) (1.1176, 1.1227)ω 0.0069 0.0005 (0.0069, 0.0091) (0.0057, 0.0074) (0.0052, 0.0070)α 0.7526 0.0004 (0.7513, 0.7532) (0.7522, 0.7534) (0.7521, 0.7533)ζa

85 (min) 0.0001 0.0001 (0.0000, 0.0003) (0.0000, 0.0003) (0.0000, 0.0003)ζa

819 (max) 0.5424 0.2018 (0.2191, 0.8000) (0.1826, 0.8000) (0.0804, 0.8000)ζa 0.1601 0.1601 (0.1601, 0.1601) (0.1601, 0.1601) (0.1601, 0.1601)ζo

3 (min) 0.0002 0.0002 (0.0000, 0.0006) (0.0000, 0.0007) (0.0000, 0.0007)ζo

205 (max) 0.7980 0.0020 (0.7940, 0.8000) (0.7940, 0.8000) (0.7941, 0.8000)ζo 0.3107 0.3107 (0.3107, 0.3107) (0.3107, 0.3107) (0.3107, 0.3107)χa

826 (min) 0.8764 0.0523 (0.8000, 0.9466) (0.8000, 0.9730) (0.8089, 1.0000)χa

1020 (max) 0.9274 0.0517 (0.8176, 0.9896) (0.8241, 0.9992) (0.8435, 1.0000)χa 0.9000 0.9000 (0.9000, 0.9000) (0.9000, 0.9000) (0.9000, 0.9000)χo

528 (min) 0.8686 0.0501 (0.8000, 0.9432) (0.8000, 0.9619) (0.8000, 0.9841)χo

148 (max) 0.9238 0.0538 (0.8123, 1.0000) (0.8195, 1.0000) (0.8258, 1.0000)χo 0.9003 0.9003 (0.9003, 0.9003) (0.9003, 0.9003) (0.9003, 0.9003)

Table 10: MC estimation for parameters for Kenya 2013, modelM2

MC.exp MC.sd HPD.chain.1 HPD.chain.2 HPD.chain.3φ1 .8571 .0109 (0.8357, 0.8786) (0.8355, 0.8781) (0.8355, 0.8782)φ2 .1419 .0109 (0.1204, 0.1632) (0.1204, 0.1629) (0.1209, 0.1633)φ3 .0010 .0010 (0, 0.0029) (0, 0.0029) (0, 0.0029)ω 0.22 0.00 (0.224, 0.225) (0.224, 0.225) (0.224, 0.225)α 0.71 0.00 (0.713, 0.714) (0.713, 0.714) (0.713, 0.714)ζo 0.52 0.00 (0.515, 0.524) (0.513, 0.523) (0.515, 0.524)ζa 0.25 0.00 (0.244, 0.247) (0.244, 0.247) (0.243, 0.246)χo 0.88 0.01 (0.816, 0.85) (0.973, 0.997) (0.8, 0.825)χa 0.86 0.01 (0.877, 0.912) (0.8, 0.833) (0.838, 0.883)

Table 11: Logistic Model Estimated Fraudulent Vote Counts and ProportionsElection Model Mi Me pi pe pi + pe

Kenya-2013-President M2 538677.5 0 0.15598 0 0.15598

Table 12: Data SetsSet of Parameters Type DescriptionType of Fraud in the I No fraudElection II Only Incremental Fraud

III Only Extreme FraudIV Both Exreme and Inremental Fraud

Effect and Spatial Distri A No effect of covariatesbution of covariates B Covariates affect turnout and probability of vot-

ing to a party but they are randomly distributedin the space

C Covariates affect turnout and probability of vot-ing to a party and they are spatially concentrated

Local Fraud Probability a Same everywhere, and probability of stealingfrom opposition is the same as stealing fromnon-voters

b Same everywhere, and probability of stealingfrom opposition higher than stealing from non-voters

c Fraud probability can be different in eachpolling place, but probability of stealing votesfrom opposition is the same as stealing fromnon-voters

d Fraud probability can be different in eachpolling place, and probability of stealing votesfrom opposition is higher than stealing fromnon-voters

Table 13: Parameters used to Generate Data Sets (Data Set I-A-a)1 2 3 4 5 6

Ni 752.60 500.00 627.00 754.00 878.00 1000.00ζa = 0.3, ζo = 0.3, χa = 0.8, χo = 0.8

ω = 0.5744, α = 0.3543β = (0.3, 0, 0, 0, 0)

γ = (−0.6, 0, 0, 0, 0)φ = (0.999, 5e − 04, 5e − 04)

n = 4900

Table 14: Parameters used to Generate Data Sets (Data Set IV-A-d)ζa ζo χa χo ω α Ni

Mean 0.30 0.41 0.87 0.91 0.57 0.35 752.60Std.dev 0.20 0.23 0.04 0.05 0.00 0.00 143.76

Min. 0.00 0.01 0.80 0.81 0.57 0.35 500.001st Qu. 0.13 0.21 0.84 0.86 0.57 0.35 627.00Median 0.26 0.41 0.87 0.91 0.57 0.35 754.003rd Qu. 0.45 0.60 0.90 0.95 0.57 0.35 878.00

Max. 0.79 0.80 1.00 1.00 0.57 0.35 1000.00β = (0.3, 0, 0, 0, 0)

γ = (−0.6, 0, 0, 0, 0)φ = (0.6, 0.2, 0.2)

n = 4900

Table 15: Parameters used to Generate Data Sets (Data Set I-C-a)Mean Min. 1st Qu. Median 3rd Qu. Max.

ω 0.57 0.23 0.38 0.57 0.76 0.87α 0.54 0.09 0.29 0.56 0.79 0.94

Ni 752.60 500.00 627.00 754.00 878.00 1000.00ζa = 0.3, ζo = 0.3, χa = 0.8, χo = 0.8

β = (0.3, 0.05, 0.1, 0.03, 0.3)γ = (−0.3, 1,−0.04, 1,−0.4)φ = (0.999, 5e − 04, 5e − 04)

n = 4900

Table 16: Parameters used to Generate Data Sets (Data Set III-C-d)ζa ζo χa χo ω α Ni

Mean 0.30 0.41 0.87 0.91 0.57 0.54 752.60Std.dev 0.20 0.23 0.04 0.05 0.20 0.26 143.76

Min. 0.00 0.01 0.80 0.81 0.23 0.09 500.001st Qu. 0.13 0.21 0.84 0.86 0.38 0.29 627.00Median 0.26 0.41 0.87 0.91 0.57 0.56 754.003rd Qu. 0.45 0.60 0.90 0.95 0.76 0.79 878.00

Max. 0.79 0.80 1.00 1.00 0.87 0.94 1000.00β = (0.3, 0.05, 0.1, 0.03, 0.3)γ = (−0.3, 1,−0.04, 1,−0.4)

φ = (0.7, 0.001, 0.299)n = 4900

Figure 1: Brazil 2014, Round 1, Empirical Densities

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

round 1 leader vote proportions

N = 6265 Bandwidth = 0.0296

Den

sity

0.80 0.85 0.90 0.95

05

1015

round 1 turnout proportions

N = 6265 Bandwidth = 0.003811D

ensi

ty

0.2 0.4 0.6 0.8

0.80

0.85

0.90

0.95

vote

turn

out

round 1 leader vote by turnout density

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Note: n = 6, 265 city-zone-section observations.

Figure 2: Brazil 2014, round 2, Empirical Densities

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

round 2 leader vote proportions

N = 6265 Bandwidth = 0.02842

Den

sity

0.80 0.85 0.90 0.95 1.00

05

1015

2025

round 2 turnout proportions

N = 6265 Bandwidth = 0.002587D

ensi

ty

0.2 0.4 0.6 0.8

0.85

0.90

0.95

vote

turn

out

round 2 leader vote by turnout density

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Note: n = 6, 265 city-zone-section observations.

Figure 3: California 2008, President, Empirical Densities

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

round 1 leader vote proportions

N = 22691 Bandwidth = 0.02338

Den

sity

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

3.5

round 1 turnout proportions

N = 22691 Bandwidth = 0.01548D

ensi

ty

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

vote

turn

out

round 1 leader vote by turnout density

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Figure 4: Kenya 2013, Empirical Densities

0.0 0.2 0.4 0.6 0.8 1.0

0.4

0.6

0.8

1.0

1.2

1.4

round 1 leader vote proportions

N = 1231 Bandwidth = 0.08419

Den

sity

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

round 1 turnout proportions

N = 1231 Bandwidth = 0.05335D

ensi

ty

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

vote

turn

out

round 1 leader vote by turnout density

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Figure 5: Kenya 2013, MCMC Diagnostics φ Posterior Distribution by Chain

phi[1]

phi[2]

phi[3]

0

10

20

30

0

10

20

30

0

250

500

750

0.825 0.850 0.875 0.900

0.100 0.125 0.150 0.175

0.000 0.005 0.010value

dens

ity

Chain

1

2

3

Figure 6: Kenya 2013, MCMC Diagnostics χa Posterior Distribution by Chain

chi.a

0

20

40

60

0.800 0.825 0.850 0.875 0.900value

dens

ity

Chain

1

2

3

Figure 7: Kenya 2013, MCMC Diagnostics χo Posterior Distribution by Chain

chi.o

0

20

40

60

0.80 0.85 0.90 0.95 1.00value

dens

ity

Chain

1

2

3

Figure 8: Kenya 2013, MCMC Diagnostics ζa Posterior Distribution by Chain

zeta.a

0

200

400

0.242 0.244 0.246 0.248value

dens

ity

Chain

1

2

3

Figure 9: Kenya 2013, MCMC Diagnostics ζo Posterior Distribution by Chain

zeta.o

0

50

100

150

0.505 0.510 0.515 0.520 0.525value

dens

ity

Chain

1

2

3

Figure 10: Kenya 2013, MCMC Diagnostics φ Traceplots

6e+04 8e+04 1e+05

0.82

0.86

0.90

Iterations

Trace of phi[1]

0.82 0.84 0.86 0.88 0.90

010

30

Density of phi[1]

N = 50000 Bandwidth = 0.00107

6e+04 8e+04 1e+05

0.10

0.14

0.18

Iterations

Trace of phi[2]

0.10 0.12 0.14 0.16 0.180

1030

Density of phi[2]

N = 50000 Bandwidth = 0.001067

6e+04 8e+04 1e+05

0.00

00.

008

Iterations

Trace of phi[3]

0.000 0.004 0.008 0.012

040

080

0

Density of phi[3]

N = 50000 Bandwidth = 7.798e−05

Figure 11: Kenya 2013, MCMC Diagnostics χa Traceplots

360000 390000

0.80

0.82

0.84

0.86

0.88

0.90

0.92

Iterations

Trace of chi.a

0.80 0.84 0.88 0.920

510

15

Density of chi.a

N = 50000 Bandwidth = 0.003236

Figure 12: Kenya 2013, MCMC Diagnostics χo Traceplots

410000 440000

0.80

0.85

0.90

0.95

1.00

Iterations

Trace of chi.o

0.80 0.90 1.00

05

1015

Density of chi.o

N = 50000 Bandwidth = 0.00764

Figure 13: Kenya 2013, MCMC Diagnostics ζa Traceplots

310000 340000

0.24

20.

244

0.24

60.

248

Iterations

Trace of zeta.a

0.242 0.2460

100

200

300

400

Density of zeta.a

N = 50000 Bandwidth = 9.5e−05

Figure 14: Kenya 2013, MCMC Diagnostics ζo Traceplots

260000 290000

0.50

50.

510

0.51

50.

520

0.52

5

Iterations

Trace of zeta.o

0.505 0.515 0.525

050

100

150

Density of zeta.o

N = 50000 Bandwidth = 0.0002564

Figure 15: Kenya 2013, MCMC Diagnostics φ Autocorrelations

●

● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 10 20 30 40

−0.

50.

00.

51.

0

Lag

Aut

ocor

rela

tion

●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●

● ● ● ● ● ● ● ●●

● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ●

ESS = 50773

Figure 16: Kenya 2013, MCMC Diagnostics χa Autocorrelations

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 10 20 30 40

−0.

50.

00.

51.

0

Lag

Aut

ocor

rela

tion

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●ESS = 6.4

Figure 17: Kenya 2013, MCMC Diagnostics χo Autocorrelations

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 10 20 30 40

−0.

50.

00.

51.

0

Lag

Aut

ocor

rela

tion

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●ESS = 10.5

Figure 18: Kenya 2013, MCMC Diagnostics ζa Autocorrelations

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●●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 10 20 30 40

−0.

50.

00.

51.

0

Lag

Aut

ocor

rela

tion

●●

●●

●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●●

●●

●●

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●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

ESS = 827.2

Figure 19: Kenya 2013, MCMC Diagnostics ζo Autocorrelations

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 10 20 30 40

−0.

50.

00.

51.

0

Lag

Aut

ocor

rela

tion

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

ESS = 228.8