Cho and Gaines have recently criticized work by Burden and Kimball on split-ticket voting in the USA, suggesting that their estimates of the volume of such voting (derived using King�s EI method) across Congressional Districts and States are unreliable. Using part of the Burden-Kimball data set, we report on a parallel set of estimates generated by a different procedure (EMax), which employs three rather than two sets of bounds. The results are extremely similar to Burden and Kimball�s, providing strong circumstantial evidence for their conclusions regarding the impact of campaign spending and other influences on the volume of split-ticket voting

Gschwend, Thomas

Johnston, Ron

Pattie, Charles

English

Research Papers in Economics

SONDERFORSCHUNGSBEREICH 504
Rationalit¨ atskonzepte,
Entscheidungsverhalten und
¨ okonomische Modellierung
Universit¨ at Mannheim
L 13,15
68131 Mannheim
No. 04-40
On Estimates of Split-Ticket Voting: EI and EMax
Ron Johnston
and Thomas Gschwend
and Charles Pattie
November 2004
Financial support from the Deutsche Forschungsgemeinschaft, SFB 504, at the University of
Mannheim, is gratefully acknowledged.
University of Bristol, email: r.johnston@bristol.ac.uk
Sonderforschungsbereich 504, email: Thomas.Gschwend@mzes.uni-mannheim.de
University of Shefﬁeld, email: C.Pattie@shefﬁeld.ac.uk 
On Estimates of Split-Ticket Voting: EI and EMax 
 
 
ABSTRACT: Cho and Gaines have recently criticized work by Burden and 
Kimball on split-ticket voting in the USA, suggesting that their estimates of the 
volume of such voting (derived using King’s EI method) across Congressional 
Districts and States are unreliable. Using part of the Burden-Kimball data set, 
we report on a parallel set of estimates generated by a different procedure 
(EMax), which employs three rather than two sets of bounds. The results are 
extremely similar to Burden and Kimball’s, providing strong circumstantial 
evidence for their conclusions regarding the impact of campaign spending and 
other influences on the volume of split-ticket voting. 
  2 
 
The use of King’s (1997) method of ecological inference to generate valid estimates 
of various statistics of interest has not gone uncriticized. Cho and Gaines (2004a)  
recently condemned Burden and Kimball’s (1998) paper which used EI to estimate 
split-ticket voting and, in a review of Burden and Kimball’s (2002) book on the same 
subject, questioned the validity of the estimates because of the ‘shaky statistical 
foundations of EI’ (Cho and Gaines, 2004b, p.243). EI does not meet their criteria for 
an acceptable procedure for decomposing aggregate data. 
 
We present alternative estimates of the same patterns which sustain Burden and 
Kimball’s, partly countering Cho and Gaines’ (2004b, p. 242) claim that those 
estimates are ‘fortuitously, roughly correct’. While accepting the latter’s statement 
that ‘an example is never tantamount to proof’, our additional evidence – using an 
alternative estimating procedure that has been available for some two decades, 
including a paper in Political Analysis (Johnston and Pattie, 2000)
1 – provides strong 
support for Burden and Kimball’s estimates. The volume of ticket-splitting estimated 
by the two procedures is highly correlated.  
 
EI and EMax 
 
EI addresses a long-established problem of ecological inference, generating estimates 
of the unknown cell values in a 2 x 2 x n matrix. On the assumption that no variables 
other than the row and column sums of each separate 2 x 2 sub-matrix influence the 
cell sizes, EI produces what King shows, with numerous examples, are accurate 
estimates of the unknown values for both the sub-matrices and the summed matrix. 
 
EMax, also based on the method of bounds, estimates the sub-matrix cell values, 
given that cell values for the total matrix (i.e. summed across the n sub-matrices) are 
also known. (The first papers, with applications to electoral data, were Johnston and 
Hay, 1982, 1983.) It deploys three sets of constraints rather than two – the row and 
column totals for each sub-matrix (as in EI) plus the cell values for the total matrix, 
with the latter usually estimated from a national survey. As with EI, EMax assumes 
that there are no influences on the estimated cell values other than the three sets of 
constraints; it generates the maximum likelihood estimates of the sub-matrix cell 
values. The outcome is consistent with fitting a log-linear model without any three-
way interaction (Johnston, 1995). 
 
EMax is a more restrictive method for addressing ecological inference problems than 
EI because three sets of constraints require more prior information than two. 
Moreover, EI has the added bonus that every estimate comes with a measure of 
uncertainty.
2 Although a third set of constraints allowing EMax to be deployed might 
not be available for many applications, in the case of ticket-splitting patterns in the 
US an estimate of the national split-ticket matrix can be derived from surveys 
containing reports of retrospective voting behavior. Why not use more information for 
the estimation problem if it is readily available, therefore? As is generally accepted, 
better data always trump better methods. But if the better data – in this case, the third 
set of constraints – are not available, how good are the estimates derived from EI?  
 
  3EMax Estimates of Split-Ticket Voting in the USA: 1988 
 
We report a comparison of the EMax and EI estimates of split-ticket voting at the 
1988 US elections, using the same data set as Burden and Kimball.
3 If the two 
procedures result in very similar estimates, then some of the force of Cho and Gaines’ 
criticism is removed. If two sets of estimates – derived from different methodologies 
– result in the same pattern of spatial variation, then the results gained from analyses 
based on one of the outcomes are not peculiar to that estimating procedure. 
 
The third set of constraints – the national split-ticket voting matrices – were obtained 
from the 1988 American National Election Studies (NES). For the contests for 
President and Representative we used data for all respondents living in the 354 
Congressional Districts contested by both the Republican and the Democrat parties; 
for split-ticket voting in the Presidential and Senate races we used the parallel Senate 
Election Study. The relevant matrices – using data for the two main parties only, as in 
Burden and Kimball’s paper – are in Table 1.
4  
 
Figures 1-4 plot Burden and Kimball’s EI estimates for selected cells in each of the 
two matrices against the EMax estimates, indicating very considerable agreement. 
Figure 1 shows that – save for a small number of outliers – there was a very close fit 
between the EI and EMax estimates for Democrat loyalty (DD: those who voted for 
Dukakis for President and for a Democrat candidate for the House). Figure 2 shows a 
similar very close fit (with only two substantial outliers) between the two sets of 
estimates for Democrat splitters (DR – those who voted for Dukakis for President and 
a Republican candidate for the House). 
 
There were only 33 cases in the contests for the Presidency and the Senate. For these, 
too, Figures 3 and 4 show close agreements between the EI and EMax estimates for 
Republican loyalists and splitters (RR and RD, respectively).  
 
Regression analyses between the two sets of estimates are in Table 2. All four 
comparisons of the EI (the independent variable) and EMax (dependent variable) 
estimates for Presidential and House voting have correlations very close to 1.00 – the 
first block in the table – as do the first two of the four for Presidential and Senate 
voting. The smaller r
2 values in the last two regressions in the latter group relate to 
Democrat loyalists and splitters: fitting curvilinear rather than linear relationships 
increased the r
2 values to 0.83 and 0.85 respectively. (None of the straight-line 
regressions gave a 45 degree slope with a constant of zero and regression coefficient 
of 1.0, however. For this data set, in general EMax produces somewhat higher levels 
of split-ticket voting at the lower end of the range than EI, with the two converging at 
the higher end of the range.) 
 
Conclusion 
 
The EI and EMax estimation procedures produced very similar outcomes for the two 
sets of matrices.
5 In relative terms, the pattern of ticket-splitting across the 
Congressional Districts and States varies only slightly between the two procedures. 
Burden and Kimball used their estimates to analyze the impact of campaigning on the 
volume of ticket-splitting: the greater the intensity of the campaign, the better the 
party’s performance. That their EI estimates of the volume of split-ticket voting by 
  4State and Congressional District correlate so highly with the EMax estimates offers 
strong circumstantial evidence sustaining their findings regarding such spatial 
variations and their causes 
 
Much quantitative social science involves estimating unknown variables of interest 
from known values. Burden and Kimball claimed (1998, 542) their use of EI 
produced ‘more accurate and efficient estimates of the direction and magnitude of 
ticket splitting in different states and congressional districts’, and were superior to 
those generated by previous research dependent on either aggregate data (which led to 
the classic ecological inference problem) or individual-level data that ‘could not be 
used to estimate district level parameters’. Using EMax, we have combined aggregate 
and individual level data and produced estimates very similar to Burden and 
Kimball’s. Given this stability of evidence, we suggest that Burden and Kimball’s 
second claim, regarding the forces which generate these variations across districts and 
states, should not be dismissed. 
 
Cho and Gaines (2004a, 169) state that ‘EI generally does not outperform OLS’. It is 
difficult to accept the logic of this statement, since EI incorporates information from 
the bounds; methods which do that – including EMax – are surely superior to any 
which do not. It is at the core of statistical inference that the more information you 
have, the better. EI deploys what is available in many cases – i.e. the bounds for the 2 
x 2 data matrix for each of the areal units studied. EMax uses a further set of 
information where that is also available – a third set of bounds – and our reworking of 
Burden and Kimball’s data shows very similar estimates generated by EI and EMax.  
 
Our results suggest a tempering of Cho and Gaines’ skepticism regarding the validity 
of EI-derived estimates: an alternative procedure has produced very similar estimates 
of the unknown variables, suggesting either that the Burden/Kimball estimates are 
better than Cho/Gaines argue, or that both of the procedures suffer from fatal flaws 
and yet produce the same outcomes. Only further methodological evaluations will 
determine which is the correct conclusion, but we have indicated sufficient doubt 
regarding Cho and Gaines’ outright condemnation to suggest at least a stay of 
execution on Burden and Kimball’s estimates let alone the wider criticism of King’s 
EI method implied by their stance 
 
  5 
 
Notes 
 
 
1 That technique was used in an early evaluation of spatial variations in ticket-splitting in the US – 
Johnston and Hay, 1984: Burden and Kimball did not develop ‘the first-ever accurate district-level 
estimates of vote-splitting’ (Cho and Gaines, 2004a, p. 169). 
2 As a statistical procedure, EI has standard errors for all of its estimates: developed as a mathematical 
procedure, EMax does not although we are currently addressing this issue. In addition, EI can use 
covariates which, as yet, EMax cannot. 
3 We are grateful to Profs Burden and Kimball for providing us with the relevant data files. 
4 These matrices are, of course, estimates from survey data only, and introduce some error to the 
entropy-maximizing procedure – an issue which we are currently addressing. 
5 Given Cho and Gaines argument of instability in the results from applying EI, it may be that Burden 
and Kimball serendipitously produced the same pattern of results as we did – an issue for further 
research on EI. 
  6 
 
References 
 
Burden, B. C. and Kimball, D. C. (1998) A New Approach to the Study of Ticket-
Splitting. American Political Science Review, 92: 533-544. 
 
Burden, B. C. and Kimball, D. C. (2002) Why Americans Split their Tickets:  
Campaigns, Competition and Divided Government. Ann Arbor MI: University 
of Michigan Press.  
 
Cho, W. K. T. and Gaines, B. J. (2004a) The Limits of Ecological Inference: The 
Case of Split-Ticket Voting. American Journal of Political Science, 48: 152-
171. 
 
Cho, W. K. T. and Gaines, B. J. (2004b) Review of Burden and Kimball (2002). Party 
Politics, 10:241-243. 
 
Johnston, R. J. (1995) The Geography of English Politics. London: Croom Helm. 
 
Johnston, R. J. and Hay, A. M. (1982) On the Parameters of Uniform Swing in 
Single-Member Constituency Electoral Systems. Environment and Planning A 
14: 61-74. 
 
Johnston, R. J. and Hay, A. M. (1983) Voter Transition Probability Estimates:  an 
Entropy-Maximizing Approach.  European Journal of Political Research 11: 
93-98. 
 
Johnston, R. J. and Hay, A. M. (1984) The Geography of Ticket-Splitting:  a 
Preliminary Study of the 1976 Elections Using Entropy-Maximizing Methods.  
The Professional Geographer 36: 291-296. 
 
Johnston, R. J. and Pattie, C. J. (2000) Ecological Inference and Entropy-Maximizing: 
an Alternative Estimation Procedure for Split-Ticket Voting. Political 
Analysis, 8: 333-345. 
 
King, G. (1997) A Solution to the Ecological Inference Problem – Reconstructing 
Individual Behavior from Aggregate Data. Princeton: Princeton University 
Press. 
  
 
TABLE 1. The estimated national pattern of split-ticket voting at the 1988 
Presidential, House and Senate elections (percentages of row totals) 
 
President            House                                                  Senate 
      R                     D        R                     D 
Bush   72.6   27.4    75.0   25.0 
Dukakis  19.9   80.1    33.9   66.1 
 
  8 
 
 
 
TABLE 2. Linear regressions of EMax on EI estimates of split-ticket voting, 
1998 (standard errors in parentheses) 
 
President House  a  b  r
2 
 
Bush   Republican  16.8  0.76  0.96 
       (0.008) 
Bush   Democrat  6.8  0.76  0.96 
       (0.008) 
Dukakis Democrat 12.0  0.84  0.97 
       (0.007) 
Dukakis Republican 4.1  0.84  0.97 
       (0.007) 
 
President Senate  a  b  r
2
 
Bush   Republican  15.2  0.71  0.95 
       (0.029) 
Bush   Democrat  13.9  0.71  0.95 
       (0.029) 
Dukakis Democrat  9.4  0.81  0.76 
       (0.080) 
Dukakis Republican 9.9  0.83  0.76 
       (0.080) 
 
 
 
 
 
  9 
DD - EI Estimate
100 90 80 70 60 50 40 30 20 10 0
D
D
 
-
 
E
M
 
E
s
t
i
m
a
t
e
100
90
80
70
60
50
40
30
20
10
0
 
 
 
 
 
Figure 1. EI and EMax estimates of Democrat loyalty – 1988 Presidential and House 
elections 
  10 
DR - EI Estimate
100 90 80 70 60 50 40 30 20 10 0
D
R
 
-
 
E
M
 
E
s
t
i
m
a
t
e
100
90
80
70
60
50
40
30
20
10
0
 
 
 
 
 
 
 
 
 
 
 
Figure 2. EI and EMax estimates of Democrat splitters – voted Democrat for 
President and Republican for House in 1988 
  11 
RR - EI Estimate
100 90 80 70 60 50 40 30 20 10 0
R
R
 
-
 
E
M
 
E
s
t
i
m
a
t
e
100
90
80
70
60
50
40
30
20
10
0
 
 
 
 
 
 
 
 
 
 
 
 
Figure 3. EI and EMax estimates of Republican loyalty – 1988 Presidential and 
Senate elections 
  12 
 
RD - EI Estimate
100 90 80 70 60 50 40 30 20 10 0
R
D
 
-
 
E
M
 
E
s
t
i
m
a
t
e
100
90
80
70
60
50
40
30
20
10
0
 
 
 
 
 
 
 
 
 
 
 
Figure 4. EI and EMax estimates of Republican splitters – voted Republican for 
President and Democrat for Senate in 1988 
 
 
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A New Approach to the Study of TicketSplitting.

A Solution to the Ecological Inference Problem – Reconstructing Individual Behavior from Aggregate Data.

Ecological Inference and Entropy-Maximizing: an Alternative Estimation Procedure for Split-Ticket Voting. Political Analysis,

On the Parameters of Uniform Swing in Single-Member Constituency Electoral Systems. Environment and Planning A

The estimated national pattern of split-ticket voting at the 1988 Presidential, House and Senate elections (percentages of row totals) President House Senate

The Geography of English Politics.

The Geography of Ticket-Splitting: a Preliminary Study of the 1976 Elections Using Entropy-Maximizing Methods.

The Limits of Ecological Inference: The Case of Split-Ticket Voting.

Voter Transition Probability Estimates: an Entropy-Maximizing Approach.

Why Americans Split their Tickets: Campaigns, Competition and Divided Government. Ann Arbor MI:

On Estimates of Split-Ticket Voting: EI and EMax

http://www.sfb504.uni-mannheim.de/publications/dp04-40.pdf

On Estimates of Split-Ticket Voting: EI and EMax

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