arXiv:1411.4411v5  [stat.ME]  24 Sep 2018
Quality&Quantity manuscript No.
(will be inserted by the editor)
Estimation of voter transitions and the Ecological fallacy
Antonio Forcina · Davide Pellegrino
00.00.0000
Abstract This paper attempts an investigation into the features of ecological fallacy in the
context of estimation of voter transitions between two elections. After reviewing some theo-
retical ﬁndings not always well understood, we discuss two tools for checking whether bias
is present: (i) ﬁtting models with covariates; (ii) comparing the standard errors of transition
probabilities computed under ideal conditions against those based on bootstrap methods.
Concerning the effect of covariates, we provide theoretical arguments and empirical evi-
dence to show that, under certain conditions, modelling the effect of covariates may fail to
correct ecological bias. Our investigation relies on the analysis of real and artiﬁcial data sets:
the latter are obtained by a computer software which mimics voting behaviour. An applica-
tion to a recent election in the city of Turin is also used to illustrate our methodology and
ﬁndings.
Keywords Ecological bias, effects of covariates, voting transition, bootstrap.
1 Introduction
When trying to interpret the results of an election, substantial additional insights are avail-
able if estimates of voter transitions relative to a previous election are available. For example
we may measure the proportion of faithful voters within different parties, the size of protest
voters moving to minor parties or to abstention or the size and direction of strategic voting
(Herrmann and Pappi, 2008) when the supporters of candidates not admitted to the runoffs,
have to choose whether to abstain or support one of the competing candidates.
In order to compute voter transitions we need an estimate of the joint distribution of
voters’ decisions in the two elections under consideration. If available, these data could be
A. Forcina
Dipartimento di Economia
via Pascoli, 06100 Perugia, Italy
E-mail: forcinarosara@gmail.com
D. Pellegrino
Dipartimento interateneo di Scienze, Progetto e Politiche del Territorio,
Universit`a e Politecnico di Torino,
viale Mattioli 39, 10125 Torino, Italy

2
Antonio Forcina, Davide Pellegrino
summarised in an R×C contingency table with the voting options in the previous election
by row and those of the new election by column. Unfortunately, due to the special nature of
electoral data, the only information available is the distribution of voters, separately, in each
election.
Following the usual terminology, we call aggregate data the frequency distributions of
voters by local unit and voting options, separately, in each election. We also call individual
data the frequency distribution of voters according to the pair of voting options selected
in the two elections under examination. For the purpose of our analysis it is important to
distinguish between the full and the condensed versions of individual data, according to
whether the joint distributions are available within each local unit or only for the whole area
under investigation.
Condensed individual data may be available in the context of simultaneous elections like
in the case examined by Plescia and De Sio (2018). Estimates could be obtained from exit
polls, however, as discussed, for instance, in Russo (2014), these estimates may be seriously
biased due to non response, voters not telling the truth or not remembering their earlier
decision. There are three additional limitations with exit polls:
– it may be difﬁcult to sample from abstainers as they do not show up at the polls;
– while a sample size of a few thousands may provide accurate estimates of the proportion
of voters for each party in a given election, a much larger sample size is necessary to
estimate how the voters of a small party split among the options available in the new
election;
– often, voter transitions have strong local features which would cancel out when sampled
on a whole region or State.
Based on a comparison with estimates provided from individual data, Liu (2007) noted that,
with his data, estimates from exit polls were even worse that those provided by ecological
regression.
Due to these limitations, in most situations ecological inference is the only feasible ap-
proach for the estimation of voter transitions. Thus it may be useful to study in depth, in
the context of electoral data, under which conditions these estimates may be biased, a phe-
nomenon known as ecological fallacy. A brief history of the subject starts with the seminal
paper by Robinson (1950); substantial contributions were also provided by Firebaugh (1978)
and, more recently, Wakeﬁeld (2004). By elaborating on the last two papers, Gnaldi et al.
(2018) provide a more explicit characterization of the conditions for having ecological fal-
lacy; these results will be the starting point of the present paper which relies on the analysis
of real and artiﬁcial electoral data generated according to a well deﬁned model of voting
behaviour presented in section 4.
We also discuss how to detect the presence of ecological fallacy when individual data are
not available and concentrate on two approaches in particular: the ﬁtting of models where
transition probabilities are allowed to depend on covariates and bootstrap methods, see for
instance Efron and Tibshirani (1994).
The paper is organized as follows. In section 2 we deﬁne the setting and summarize
the basic results concerning ecological fallacy and diagnostic tools to detect its presence. In
section 3, after reconsidering the work of Liu (2007), we investigate the possibility of cor-
recting bias by modelling the effect of covariates; we also describe data generating mech-
anisms where modelling the effect of covariates is unlikely to make any improvement. In
section 4 we describe the model of voting behaviour implicit in the modiﬁed version of the
Brown and Payne (1986) method of ecological inference and analyse two sets of artiﬁcial
electoral data affected by ecological fallacy. In section 5 we present an application to the

Ecological Bias
3
election for Mayor in the city of Turin in 2016 by combining a bootstrap simulation and an
attempt to correct for ecological bias by ﬁtting models with covariates.
2 The ecological fallacy revisited
Suppose that we are interested in voting transitions within a relatively small area like a town
made of K polling stations. Let Y denote the choice of a voter among the C options available
in a given election and X the corresponding choice among the R options that were available
in a previous election. Let pui j denote the proportion of voters who chose Y = j among
those who voted X = i in the previous election in polling station u. Let also πi j denote the
probability that an eligible voter chosen at random among those who voted X = i will chose
Y = j; these quantities are the main object of ecological inference. They will be called voter
transitions or transition probabilities though, when i = j, they measure the faithfulness of
voters of party i; note that they sum to 1 by row.
Let also yu j and xui denote, respectively, the aggregate proportions of voters for party j
in the new election and for party i in the previous election in polling station u. By simple
algebra, as for instance in Gnaldi et al. (2018), it can be shown that the usual ecological
regression may be written as
yu j =
R
∑
1
xuiπi j +εu j, where εu j =
R
∑
1
xui(pui j −πi j);
a closer investigation of the nature of the error term εu j indicates that the condition for
unbiased estimates by any methods based on linear or non linear regression on the marginal
proportions xui is that these are uncorrelated with the pui j, the proportion of voter transitions
within each polling stations. The effects of these correlations on bias depends on their signs
and strengths; as explained by Wakeﬁeld (2004), section 3.3, in the 2 × 2 case the largest
biases are to be expected when correlations are all in the same direction. Predicting size and
direction of bias in general is very complicated, some insights are provided in the example
presented in section 4.
Basically, there will be ecological bias whenever the variations across polling stations
of any of the proportions pui j is correlated with any of the marginal proportions xui. This is
the x-bar rule of Firebaugh (1978) where x-bar stands for averages which, in the context of
voter transitions, are just the marginal proportions. Association between voting transitions
and marginal proportions may be intrinsic, like when, for instance, the proportion of faithful
voters of a given party is greater in those polling stations where the party was stronger in
the previous election. Association may also be induced when certain entries of the table of
voter transitions are correlated with an external variable which, in turn, is correlated with
the marginal proportions; an example of this phenomenon is outlined in section 4.2.
2.1 Robinson’s data
A rather complex set of multilevel models were ﬁtted by Subramanian et al. (2009) to a
slightly extended version of the data used by Robinson (1950); though these data are not
about voter transitions, they may be useful for clarifying the nature of the ecological fallacy.
To test the condition for ecological fallacy in this context, note that the proportion of resi-
dents belonging to the different ethnic groups (Whites born in US, Whites not born in US

4
Antonio Forcina, Davide Pellegrino
proportion Whites
0.5
0.6
0.7
0.8
0.9
1
Illiterate
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Whiles
not US born
Blaks
proportion Blaks
0
0.1
0.2
0.3
0.4
0.5
Illiterate
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Whites
not US born
Blaks
Fig. 1 Quantile regression of the proportion of illiterate by race against the proportion of Whites US born
(left panel) and the proportion of Blaks (right panel).
and Blacks) are the analog of our xui, the proportion of votes obtained by each party in the
previous election, while the proportion of illiterate within each ethnic group, are the analog
of our pui j; because C = 2 and pui2 = 1 −pui1, we restrict attention to pui1. The issue is
whether the variations by State of pui1, the proportions of illiterate, are correlated with those
of the xui.
An informal assessment may be obtained by examining the quantile regression plots of
Figure 1; these were obtained by classifying States in 10 groups according to the deciles of
the explanatory variables (for instance the proportion of Black residents). Then, within each
group of States, we plot the averages proportion of illiterates by race against the average
proportion of Blacks (or Whites).
It emerges clearly that the proportion of illiterate decreases even among Blacks when the
proportion of Whites increases; the relation is reversed, though with a different shape, when
the proportion of Blacks increases. Robinson’s data were also reanalysed by (Firebaugh,
1978, Section IV) who argued that region (South / rest of the Country) was the very deter-
minant of illiteracy. In this example the proportion of Blacks is positively correlated with
the proportion of illiterates within Blacks and Whites US born..
2.2 Tools for detecting the ecological fallacy
In the unlikely event that the full individual data were available, one could simply compute
the pui j proportions and plot these as functions of the marginal proportions xui; for instance,
for a give i we might chose a set of js of interest. A quantile regression as in Figure 1 may
be used for a visual inspection of the overall direction of correlations.
When only the condensed individual data are available as in Plescia and De Sio (2018),
for each cell in the table of voter transitions, we can compute the errors in the estimates from
ecological inference relative to the true values. However these errors, as in any statistical
context, will be the sum of bias and random variation, unless we have substantial reasons to
assume that ecological bias is negligible. in the given context

Ecological Bias
5
When only aggregated data are available, we cannot even know which estimates are
closer to the truth. A set of indirect diagnostic tools for checking whether conditions for un-
biased estimates are satisﬁed were discussed by Loewen and Grofman (1989) but criticized
by Klein et al. (1991). In this paper we propose two diagnostic tools more directly related to
the nature of ecological fallacy: (i) ﬁtting models with covariates, (ii) comparison of stan-
dard errors estimated from theory with the corresponding bootstraps estimates. Point (i) will
be the subject of next section; point (ii) is brieﬂy described below and an application will be
presented in section 5.
To detect ecological bias by bootstrap, a statistical method for obtaining non parametric
estimates of standard errors (Efron and Tibshirani, 1994), the chosen method of ecological
inference has to be applied several times to different subsets of polling stations selected at
random with replacement. The idea is that, if transition probabilities are not constant across
polling stations, the estimates obtained from different samples will exhibit an amount of
variability larger than expected from theory. The method can be computationally demanding
because the chosen method of ecological inference has to be applied, say, 300 times. In the
end, 300 different estimates of each entry of the table of voter transitions will be available so
that uncertainty in each estimate can be assessed without the need to make any assumption.
An illustration will be provided in section 5.
3 Ecological fallacy and covariates
Several methods of ecological inference allow to model the effect of covariates on transition
probabilities, for instance Brown and Payne (1986) and Rosen et al. (2001); this possibility
is also mentioned as a possible extension by Greiner and Quinn (2009). However, as far
as we know, nobody has noted that, only covariates which are correlated with the xui, the
proportion of voters for the different options available in the previous election, are relevant
for removing bias.
The assumption that the logits of transition probabilities can be a linear function of
covariates measured at the level of polling station is stated by Rosen et al. (2001) in their
equation (2); an identical assumption is formulated by Brown and Payne (1986) in their
equation (2.3). Both formulations can be rewritten as
log πui j
πuiC
= αi j +β ′
i jzui j,
(1)
where αi j is an intercept parameter, πui j is the transition probability from i to j in local unit
u, zui j is a vector of covariates affecting cell (i j) and β i j is a vector of regression coefﬁcients
acting on the logit scale.
Covariates measured at the level of polling stations which might affect voting decisions
are, for instance, the proportion of people aged over 65, the proportion of unemployed or the
proportion of immigrants. The idea behind the method is that, if voter transitions are not con-
stant and their variation across polling stations are determined by certain covariates, which
are correlated with the marginal proportions as in equation 1, then the induced ecological
bias will be removed.
Individual data from the 2002 New Orleans mayoral election were used by Liu (2007)
to show that, by modelling the dependence of transition probabilities on appropriate covari-
ates, one can reduce substantially the ecological bias. From a Statistical point of view this
is not surprising because, generally, a correctly speciﬁed statistical model leads to consis-
tent estimates. An example based on computer generated electoral data that conﬁrms this
expectation will be provided below.

6
Antonio Forcina, Davide Pellegrino
A detailed discussion of social mechanisms by which individual behaviours may be
affected by average features of a local unit (polling station) are discussed in depth by
(Firebaugh, 1978, pp 564–568). A speciﬁc examination concerning the way that the envi-
ronment voters live in can affect their decisions is contained in Johnston and Pattie (2006),
in particular the chapter: ”Talking together, voting together”.
3.1 Individual level covariates
According to Firebaugh (1978), in addition to covariates which, by social mechanisms, may
affect groups of voters in a given polling station, covariates may also act as micro properties,
when voters decide mainly on the basis of their personal condition (like unemployment,
party afﬁliation or other), rather than the value of the same variable averaged at the level of
their polling station.
A numerical example with artiﬁcial electoral data where voting decisions depend on in-
dividual covariates will be presented in section 4 and is based on the following assumptions.
Suppose that the probability that an individual is unemployed varies across polling stations
and that employed and unemployed had different probabilities of choosing among the vot-
ing options available in the previous election. Assume also that the probabilities of choosing
among the options available in the new election depend both on the party voted previously
and on the personal condition (employed or not).
In this context it is likely that the entries in the table of voter transitions will be corre-
lated with both the marginal proportions in the previous election and with the proportion
of unemployed. As we shall see, modelling the effect of covariates in this context may fail
to correct ecological bias. Intuitively, there are two reasons why this may happen: (i) as
explained by Gnaldi et al. (2018), when decisions are affected by the personal condition
(employed or not) transition probabilities are linear (not logistic) functions of the proportion
of unemployed and (ii) to apply a proper ecological inference in this context we should know
the proportion of unemployed within voters of different parties in the previous election, not
just the average proportion of unemployed.
4 Ecological fallacy in artiﬁcial electoral data
In this section we describe a stochastic data generating mechanism for producing artiﬁcial
electoral data having speciﬁc features; the analysis of these data will allow an empirical in-
vestigation into the ecological fallacy and on models with covariates. The idea is to generate
electoral data in a way that mimics voters’ decisions in a context with a very large number of
polling stations; this has two important advantages: (i) when the number of polling stations
is very large, the random variation in the ecological estimates will be very small, so that
bias, if present, is easily detected, (ii) for the same reason, the experiments we present can
be easily replicated with the software that will be made available. The data generating mech-
anism is derived from the modiﬁcation of the Brown and Payne (1986) model proposed by
Forcina et al. (2012): in our opinion, the original paper and its modiﬁcation, describe in de-
tail a quite realistic stochastic data generating mechanism which can be implemented on a
computer.
The artiﬁcial data will be analysed with three methods of ecological inference: the
Goodman linear regression, the frequentist version of the Multinomial-Dirichelet method

Ecological Bias
7
by Rosen et al. (2001) and the modiﬁed Brown and Payne (1986) method itself. In princi-
ple, the last method might be favoured by the fact that the data are generated according to
the model underlying the method; however, due to the size of the data, random variation
will be negligible relative to bias and here the objective is not to compare the three methods,
but to show that, under certain conditions, ecological fallacy is going to affect any method,
approximately in the same way.
4.1 The revised Brown and Payne model of voting behavior
The approach to ecological inference proposed by Brown and Payne (1986) is based on a
stochastic model of voting behaviour which may be seen as a simpliﬁed, but also realistic,
approximation of reality. The model assumes the presence of two random components: (i)
at the individual level, the decision about which party to vote in the new election is assumed
to be similar to the drawing of balls from an urn with a given composition, determined by
the party voted in the previous election and the polling station where the individual lives;
(ii) at the level of polling station, it is assumed that transition probabilities for voters of the
same party may vary at random across polling stations due to local speciﬁcities which are
too complex to be known and are treated as random effects.
The revised version of this model due to Forcina et al. (2012) assumes that voters of the
same party and living in the same polling station are not homogeneous and that they tend
to split into small clusters of people connected by personal relationships. To be speciﬁc,
the model assumes that the size of clusters is determined at random and only people within
the same cluster share the same set of transition probabilities. In turn, these cluster speciﬁc
transition probabilities vary at random as in a Dirichlet distribution, a model of random
variation of sets of probabilities which is also part of the Rosen et al. (2001) model. In the
ﬁnal step, the parameters of the Dirichlet distributions, one for each voting option available
in the ﬁrst election, are allowed to depend on covariates measured at the level of polling
stations.
An interesting feature of this model is that it can be used to design a computer algo-
rithm that generates at random electoral data with preassigned characteristics. These data
can be analysed both as full, individual level data and as aggregated data to gain a better
understanding of the ecological fallacy by computer experiments. The objective of these ex-
periments is not to compare the performance of different methods of ecological inference,
but to show that, when the basic conditions are violated in a given way, biased estimates are
to be expected no matter which method of estimation is used. As an illustration, we concen-
trate on three method of ecological inference: Goodman (1953) linear regression model, the
frequentist version of Rosen et al. (2001) Multinomial-Dirichlet and the revised Brown and
Payne model.
Our approach will consist in generating artiﬁcial data sets with 20,000 polling stations
each. The data generating mechanism depends on the following set of parameters which
have to be set in advance:
1. a set of transition probabilities and a model that speciﬁes which transition probability
depend on which covariate; recall that, because covariates which are uncorrelated with
the marginal proportions xui are irrelevant, we may restrict the range of relevant models;
2. the amount of random variation in the Dirichlet distribution which determines the amount
of heterogeneity about the behaviour of voters belonging to different clusters;
3. the average size of clusters, a parameters that has to be speciﬁed but has little effect on
the functioning of the model.

8
Antonio Forcina, Davide Pellegrino
4.2 Examples
Below we consider two examples, in the ﬁrst we assume that covariates affect the logits
of voter transitions as expected by both the Rosen et al. (2001) and the Brown and Payne
(1986) models, in the second, instead, covariates are assumed to act as micro properties.
In the ﬁrst setting it emerges clearly that, when covariates are ignored, all three methods
of inference provide estimates heavily biased; however the bias disappear when covariates
are inserted properly into the model. On the contrary, in the second example modelling the
effect of covariates does not remove bias, as explained above.
In the ﬁrst example we suppose that the same three voting options are available in both
elections and that voters split among these in equal proportions in the ﬁrst election. We also
assume that the logits of the transition probabilities within each row are linear functions of
the corresponding marginal proportion in such a way that faithfulness to the option voted
in the ﬁrst election is greater in the polling stations where the same party was stronger in
the ﬁrst election. As we shall see, this leads regression based methods to overestimate the
amount of faithful voters. On the other hand, errors in the ecological estimates with co-
Table 1 True and ecological estimates of voter transitions in example 1: T=true, K=King, B=Brown-Payne
and G=Goodman; C stands for model with covariates; U, V, Z denote the three voting options available in
each election
Method
T
KC
BC
U
V
Z
U
V
Z
U
V
Z
U
0.714
0.143
0.143
0.715
0.145
0.140
0.712
0.145
0.142
V
0.142
0.716
0.142
0.141
0.714
0.144
0.142
0.713
0.144
Z
0.142
0.142
0.715
0.142
0.141
0.716
0.144
0.142
0.714
Method
G
K
B
U
V
Z
U
V
Z
U
V
Z
U
1.000
0.000
0.000
1.000
0.000
0.000
0.936
0.033
0.031
V
0.000
1.000
0.000
0.000
1.000
0.000
0.031
0.937
0.031
Z
0.000
0.001
0.999
0.000
0.000
1.000
0.033
0.031
0.936
variates do not exceed 0.0024, a value compatible with random variation; estimates without
covariates are all heavily biased in the same direction: the values along the main diagonal
(proportions of faithful voters) are much larger than the truth while transitions towards other
parties are underestimated.
In the second example, for simplicity, we assume that there are only two voting options
in both elections, leading to a 2 × 2 table and assume that there is a binary variable, for
instance unemployment, which affects the probability that a voter chooses one of the two
available options in the ﬁrst election. This induces correlation between the party chosen in
the ﬁrst election and the proportion of unemployed. In addition, in the second election, we
assume that voters’ decisions depend both on the party voted in the previous election and on
whether they are employed or not. Here estimates without covariates with the three methods
are all heavily biased in a similar way. As covariates we used both the proportion of voters
for party V in the ﬁrst election and the proportion of unemployed and it turns out that the
estimates are again far away from truth, though in a different direction; in addition both
methods behave similarly.

Ecological Bias
9
Table 2 True and ecological estimates of voter transitions in example 2: T=true, K=King, B=Brown-Payne
and G=Goodman; C stands for model with covariates
Method
T
KC
BC
U
V
U
V
U
V
U
0.2583
0.7417
0.0461
0.9539
0.1399
0.8601
V
0.1462
0.8538
0.3588
0.6412
0.2648
0.7352
Method
G
K
B
U
V
U
V
U
V
U
0.4042
0.5958
0.4104
0.5896
0.4043
0.5957
V
0.0001
0.9999
0.0000
1.0000
0.0000
1.0000
5 An application to real data
This study concerns the second ballot in the 2016 Municipal elections in Turin, one of the
largest urban area in Northern Italy, Though in the Italian system, since 1993, Municipal
elections are based on a two-round system, during the two most recent elections in Turin,
held in 2006 and 2011, the candidate of the centre-left coalition was elected in the ﬁrst
round. Considering the traditional strength of the centre-left coalition in Turin and that,
due to personal rivalries, the parties on the centre-right were running with three different
candidates, most political analysts had predicted that candidate Piero Fassino, incumbent
mayor and one of the Democratic Party (PD) leader at the national level, would probably
win in the ﬁrst ballot.
Since 2013 the Italian political scenario is considered to be a tripolar party system, with
the Movimento 5 Stelle (M5S) having gained a strength comparable to that of the more
traditional centre-left and centre-right coalitions, a feature which has, gradually, extended
to several local political systems, see for example Emanuele and Chiaramonte (2013) and
Regalia (2018). Though the M5S was active in Turin since the 2011 Municipal election,
only after the 2016 Municipal election M5S became the third-largest political group in the
local scenario; the M5S candidate was Chiara Appendino, former party chief councillor. On
the whole there were six main candidates running at the ﬁrst ballot; the remaining ones were
grouped into two residual categories:
1. Airaudo, (LEFT), heading a left-wing post communist coalition;
2. Fassino, (PD), heading a centre-left candidate;
3. Appendino, (M5S), candidate of the Five-Stars Movement;
4. Napoli, (FI), running for Forza Italia, Berlusconi’s party;
5. Morano, (LN), Lega Nord’s candidate;
6. Rosso (UDC), another centre-right party;
7. (ODX), other candidates on the right;
8. (OSX), other candidates on the left;
9. (NOV), in both ballots this category denotes abstainer, blank votes and other unclassiﬁed
votes
The 2016 ﬁrst-round election took place on June 5th 2016 and, surprisingly, the centre-
left leader did not win as expected and Fassino (PD) and Appendino (M5S) entered into a
runoff election which was held two weeks later. Though the expected winner was again the
centre-left coalition leader, he was clearly defeated, a result that represents a true turning
point in the Turin local political history. The objective of the estimation and analysis of
voter transitions from the ﬁrst to the second ballot tries to throw some new light on what
has happened. One conjecture that we aim to verify is that a substantial amount of voters

10
Antonio Forcina, Davide Pellegrino
1
2
3
4
6
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8
9
10
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14
15
16
18
19
20
20
21
23
24
5
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#
#
ROME
TURIN
1 - Centro
10 - Lingotto
11 - Santa Rita
12 - Mirafiori Nord
13 - Pozzo Strada
14 - Parella
15 - Vallette - Lucento
16 - Lanzo - Madonna di Campagna
17 - Borgata Vittoria
18 - Barriera di Milano
19 - Falchera - Villaretto
2 - San Salvario
20 - Regio Parco - Barca Bertolla
21 - Madonna del Pilone
22 - Borgo Po
23 - Mirafiori Sud
24 - Cit Turin
25 - Cavoretto
3 - Crocetta
4 - San Paolo
5 - Cenisia
6 - San Donato
7 - Aurora
8 - Vanchiglia
9 - Nizza - Millefonti
Income levels
Macro-zones
Low
Middle
High
Fig. 2 Map of zones in Turin borough, darker area correspond to higher average income.
of centre-right candidates excluded from the ballot voted M5S rather than abstaining, which
would be an instance of strategic voting (Herrmann and Pappi, 2008), when voters make the
effort to go to the ballots to prevent the victory of the candidate they disliked most.
Turin has 909 valid polling stations, after removing hospitals, prisons and the like, with
an average of 786 voters each. To account for possible heterogeneity in voting behaviour,
we used a geographic information system software (GIS) to determine the position of the
909 polling stations and link to the most relevant social-economic variables from Census
data. By matching polling stations with census data, the following ﬁve covariates, measured
at the level of polling stations and expressed as ratios with respect to the number of eligi-
ble voters, were made available: aged 65 and over, registered immigrants, unemployed, low
qualiﬁed workers and people with low education. We started by ﬁtting a modiﬁed Brown
and Payne model without covariates and then tried several alternative models which allow
the most relevant cells of the table of voter transitions to depend on the available covariates.
An informal model selection procedure was used by comparing the estimates of the various
regression coefﬁcients on the logit scale with the corresponding standard errors and remov-
ing non-signiﬁcant covariates. Evidence that the ﬁnal model with covariates is a substantial
improvement is provided both by the substantial increase in the log-likelihood from about
-11800 to -5680 and the fact that all estimated regression coefﬁcients are signiﬁcant at the
5% level. Estimates of transitions provided by the model without covariates and by the ﬁnal
model are displayed in Table 3.
Estimates of the regression coefﬁcients and the corresponding ratios z between estimates
and standard errors for the model with covariates are displayed in Table 4. The absolute value
of the z ratio gives a measure of signiﬁcance in the sense that, the greater the value, the more
unlikely it is that the corresponding association is a random artifact; usually values greater
that 1.96 are taken as signiﬁcant. To interpret the results one should consider the sign of the
regression coefﬁcients together with the cell of the table of voter transitions and the nature
of the corresponding covariates.

Ecological Bias
11
city zones
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
% of eligible voters
0
5
10
15
20
25
30
35
40
45
65 and 0ver
immigrants
unemployed
low qualified
low educated
Fig. 3 Average covariate values within the 25 macro areas composing the Municipality of Turin.
Table 3
Estimates of voter transitionsbetween the ﬁrst and the second ballot in the city of Turin with and
without covariates
Second Ballot
First Ballot
without covariates
with covariates
M5S
PD
NOV
M5S
PD
NOV
LEFT
0.000
0.894
0.106
0.093
0.841
0.066
M5S
1.000
0.000
0.000
1.000
0.000
0.000
UDC
1.000
0.000
0.000
0.947
0.000
0.053
FI
0.828
0.096
0.076
0.797
0.203
0.000
PD
0.033
0.944
0.023
0.036
0.902
0.061
LN
0.808
0.096
0.096
0.754
0.153
0.092
ODX
1.000
0.000
0.000
0.479
0.000
0.521
OSX
0.496
0.000
0.504
0.600
0.189
0.211
NOV
0.005
0.000
0.995
0.025
0.006
0.969
Additional evidence that the estimates of voter transitions with covariates provide a
closer approximation to the truth is provided by the comparison of the estimates of the
standard errors of voter transitions according to theory (assuming no ecological bias) and
by bootstrap, for both the models without and with covariates. The results are displayed in
Table 5 where the ﬁrst row is the sum by columns of the differences between the bootstraps
standard error (Bse) and the theoretical standard error (Tse) for the cells where Bse is greater
than Tse; the second row contains again the sum by columns of the differences (Tse - Bee)
this time for the cells where Tse is greater.
Note that, for the model without covariates, the values in the ﬁrst row are substantially
larger than those in the second row which are close to 0; this indicates that Tse underestimate
substantially the true standard errors probably because transition probabilities vary across
polling stations due to ecological fallacy. On the other hand, the corresponding results for
the model with covariates indicate that Tse is approximately equal to Bse and that their

12
Antonio Forcina, Davide Pellegrino
Table 4 Estimated regression coefﬁcient for the model with covariates: An = preoportion of voters ages 65
and over; Im = proportion of immigrants; Bs = proportion of residents with low schooling, Bq = proportion
of low qualiﬁed workers
Cov
row
col
est.
z
Cov
row
col
est.
z
An
1
2
0.59
3.20
Bq
4
1
-0.85
-2.98
Im
1
2
0.44
3.21
An
5
2
-0.01
-2.08
Bs
1
2
-0.96
-3.19
Bs
6
1
0.16
4.63
An
4
1
-0.14
-4.61
Di
6
1
-0.50
-3.92
Im
4
1
-0.17
-3.09
Im
7
1
-0.21
-4.13
Bs
4
1
0.92
5.27
Bq
7
1
0.42
2.37
differences are small and go in both directions, as if they were caused by random variations.
Table 5 Differences between standard errors expected from theory (Tse) and estimated by bootstrap (Bse),
columns totals.
Without covariates
With covariates
M5S
PD
No vote
M5S
PD
No vote
Bse−Tse, Bse > Tse
0.329
0.073
0.338
0.043
0.049
0.040
Tse−Bse, Tse > Bse
0.008
0.001
0.000
0.034
0.023
0.071
In other words, Table 5 tries to provide a summary of the relevant differences between
standard errors expected from theory and resulting in reality as estimated by bootstrap. The
fact that the values of the ﬁrst row are substantially larger relative to the second indicates
that the estimates provided by the corresponding model are likely to suffer from ecological
bias.
6 Concluding remarks
One of the main conclusions of this paper is that, whenever the proportion of voters tran-
sitions are not constant across polling stations and their variations are correlated with the
strength of certain parties in the previous election, estimates from aggregated data will be
biased, no matter which method of ecological inference is used. Though methods of eco-
logical inference differ in the way they model the randomness intrinsic to voters’ decisions,
when the number of available polling stations is relatively large, this component is likely
to be almost negligible relative to potential ecological biases. Only methods that allow to
model the effect of covariates measured at the level of polling stations have a chance to
correct or reduce the bias.
By modelling the effect of covariates on transition probability, we are implicitly assum-
ing that covariates act as macro properties, to use the terminology of Firebaugh (1978),
meaning that certain average features of polling stations affect the decisions of those who
voted in the same way in the previous election. A detailed study of situations where this
may happen are discussed by Johnston and Pattie (2006). However, there may be situations
where voters’ decisions depend on their personal condition (e.g. young, unemployed) rather
than on the corresponding proportions within the polling station: that is we are faced with

Ecological Bias
13
micro properties. When this is the case, it is no longer true that modelling covariates is go-
ing to reduce ecological bias. We describe a speciﬁc data generating mechanism and give a
numerical example where modelling the effect of covariates fails to correct the bias.
We also argue that, unless data at the individual level are available, it will ba almost
impossible to ﬁnd objective evidence of ecological fallacy. However, when covariates mea-
sured at the level of polling stations are available, indirect evidence of bias may be achieved
by ﬁtting models with covariates. To be speciﬁc, when ﬁtting a model which allows transi-
tion probabilities to depend on a given covariates, we may check how much better does this
model ﬁt relative to the one without covariates and test whether the estimated regression
coefﬁcients are all highly signiﬁcant. When both these conditions are satisﬁed, it is very
unlikely that the detected dependence of transitions from covariates is a numerical artifact.
An additional technique which we consider for detecting ecological bias without hav-
ing access to individual data is based on comparisons between the standard errors in the
estimated transitions derived form theory with the corresponding standard errors estimated
by bootstrap. Essentially, the bootstrap consists in computing estimates of voter transitions
from several random samples of polling stations. When the conditions for no ecological fal-
lacy are violated, by selecting at random different subsets of polling stations, we are likely to
obtain estimates which differ more than what would be expected by the randomness intrinsic
in voters’ choices in a homogeneous context.
Finally, we would like to comment on the linear regression method of ecological infer-
ence, also known as the Goodman method, which, because of its computational simplicity,
is widely used in certain contexts. Because this method cannot account for the effect of
covariates, it should be applied only within an area which is sufﬁciently small to believe
that transition probabilities do not differ in a systematic way across polling stations. For in-
stance, the results presented in the previous section suggest that, though Goodman’s method
should not be applied to the city of Turin as a whole, its application should probably be valid
within suitably chosen sub-areas of the Municipality. In doing so, we need to balance two
conﬂicting requirement: (i) the number of polling stations should not be too small, other-
wise estimates will be affected by large random ﬂuctuations; (ii) if the area is too large, it is
likely that estimates are affected by a certain amount of ecological bias. An advantage of the
Goodman method which, as far as we know, has not yet been exploited, is that a bootstrap
estimate of standard errors of voting transitions can be computed very easily and compared
with those expected in an ideal situation.
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