(Page 1 of 1) - Making Ecological Inference for R x C Tables easy - Standard Errors for EMax authored by Elff, Martin. and Gschwend, Thomas.

Making Ecological Inference for

R × C

Tables easy — Standard Errors for EMax

Martin Elff

Department of Political Science

University of Mannheim

68131 Mannheim, Germany

+49.621.181.2093 (voice)

+49.621.181.2099 (fax)

## email not listed ##

webrum.uni-mannheim.de/sowi/elff

www.sowi.uni-mannheim.de/lehrstuehle/lspwivs.htm

Thomas Gschwend

Mannheimer Zentrum fuer Europaeische Sozialforschung (MZES)

University of Mannheim

68131 Mannheim, Germany

+49.621.181.2809 (voice)

+49.621.181.2845 (fax)

Thomas.## email not listed ##

www.sowi.uni-mannheim.de/lehrstuehle/lspol1/gschwend.htm

For most substantive relevant ecological inference problems

scholars face a R × C table from which only the marginals can
be observed. We argue that most frequent used ecological in-
ference methods are computationally demanding and are inef-
ﬁcient since they do not make use of all the information that is
available. We take a fresh look at an estimator that was precisely
developed for these kinds of problems: EMax. This estimator
utilizes more available information at the estimation stage than
previous ecological inference estimators in the literature. As
methodological innovation that remedy the main disadvantage
of EMax in substantive applications, we create model-based
standard errors.

Requirements for ecological inference estimators

• Making e

ﬃcient use of all available data: less assumptions will be

necessary, results will be generalizable

• But: There is no best solution; only data- or application-speciﬁc solu-

tions are available

Limits of conventional ecological inference methods

• Computationally demanding
• Ine

ﬃcient because not all available information is used

• Stable results only for the 2 × 2-case

EMax as data- and application-speciﬁc Solution

• Suitable in situations where aggregate data are supplemented by sur-

vey data

• Allows for estimation of cell probabilities and cell counts for each of

the aggregates in an e

ﬃcient way

• Does not lead to out-of-bounds cell estimates
• Reliable results even in the R × C

Applications

• Useful for Estimation of Voter Transition or Split-Ticket Voting.
• Requires three bits of information: Besides district-level data also

national-level data.

• National-level information derived from cross-tabs of survey items.
• Examples:

–

Johnston, R. J. and Hay, A. M. 1983. “Voter Transition Proba-
bility Estimates: an Entropy-Maximizing Approach.” European
Journal of Political Research 11: 93-98.

–

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.

Entropy-Maximizing (EMax) principle

• Entropy: measure of amount of uncertainty that is contained in a set

of quantities

• Entropy in case of discrete probabilities without constraints:

= −

log

• Principle of maximal entropy:

–

A principle for constructing probability models — not an esti-
mation procedure!

–

Maximal entropy subject to constraints reﬂects our uncertainty
about unknown quantities

–

Example: distribution over real line that maximizes entropy is
the normal distribution

–

In case of ecological inference problem: a distribution of maxi-
mal entropy describes the likelihood of all possible cell counts for
given marginal constraints, cell probabilities describe the most
likely conﬁguration of cell proportions – How many possibilities
on individual-level are consistent with observed result given the
district level marginals and the national-level frequencies?

Our contribution: Standard errors for the EMax
approach to ecological inference

• Derivation of theoretical standard errors, that have not yet been avail-

able in EMax for ecological inference

• Standard errors, that are computable even for the case where one of the

margins is a survey sample

• Validation of theoretical standard errors by Monte Carlo study
• Validation of EMax application to real-world data

Derivation of an entropy maximizing probability
model for ecological inference

1. Complete data constitute a three-dimensional (I × J × K) contingency

table (n

i jk

2. Marginal tables (n

+jk

) and (n

) are known.

3. Marginal table (n

i j

) is estimated as ( ˆn

i j

) by a survey sample.

4. Cell entries n

i jk

for any combinations of indices i

, j, k are a realization

of a multinomial random variable N

i jk

with parameters

i jk

(note that

this implies

,j,k

i jk

= 1).

5. Parameters

i jk

maximize entropy subject to restrictions

i jk

= n

+jk

/n =: p

+jk

i jk

= n

/n =: p

i jk

= ˆn

i j

/ˆn =: p

i j

Where ˆn denotes the sample size and n denotes the size of the popu-
lation.

Maximizing entropy under these restrictions is equivalent to maximizing

the following Lagrangian

= −

,j,k

i jk

log

i jk

−

τ(1 −

,j,k

i jk

) −

i j

−

i jk

)

−

i jk

) −

+jk

−

i jk

)

with Lagrange-multipliers

i j

, β

, γ

. The

i jk

maximize this function if

and only if

∂E

∂π

i jk

= − log π

i jk

− 1

+ τ + α

i j

+ β

+ γ

= 0

⇔ log

i jk

= τ − 1 + α

i j

+ β

+ γ

that is, if and only if the

i jk

comply to a log-linear model with no three-way

interactions. The parameters of such a model can thus be computed using
a standard procedure like iterated proportional ﬁtting (IPF).

Derivation of the standard errors

Observational data is available only for the marginal tables. This is reﬂected
in the following log-likelihood of the marginal tables with given

i jk

i j

log

i j

log

ˆn

+jk

log

+jk

+ C

with

i j

i jt

isk

and

+jk

r jk

Second derivatives of are given, for example, by

∂

∂α

i j

∂α

i j

= − (n + n + ˆn)(δ

j j

i j

−

i j

)

+ δ

j j

i jt

i·t

−

i jt

ˆn

+jt

i jt

+jt

−

i jt

+jt

∂

∂α

i j

∂β

i k

= − (n + n + ˆn)(δ

i jk

−

i j

)

+ δ

ˆn

+jk

i jk

+jk

−

i jk

+jk

Application of the delta-method then leads to the standard error formula

SE( ˆ

i jk

)

= ˆπ

i jk

(1 − ˆ

i jk

) ˆ

i j

,ij

αα

+ ˆG

,ik

ββ

+ ˆG

,jk

γγ

+2 ˆG

i j

,ik

αβ

+ 2 ˆG

i j

,jk

αγ

+ 2 ˆG

,jk

βγ

where ˆ

i j

,ij

αα

, ˆ

,ik

ββ

, ˆ

,jk

γγ

, ˆ

i j

,ik

αβ

, ˆ

i j

,jk

αγ

and ˆ

,jk

βγ

are elements of the asymptotic

covariance matrix of

i j

, β

and

based on the second derivatives of the

above likelihood.

A Monte Carlo study

1. Artiﬁcial data: Population size 20,000,000; “Sample size” for one of

the margins: 1,000; cell probabilities given, conforming to no-three-
way-interaction model; array of probabilities size: 40×5×5

2. Compute theoretical standard errors
3. 200 Replicatations:

a) Generate random numbers (multinomial distribution), compute

marginal counts and “sample” of one marginal table

b) Compute IPF estimates of cell probabilities

4. Compare root mean square error from simulations with theoretical

standard errors

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Theoretical standard errors for each cell

Root mean squared errors for each cell

Figure 1: Comparison of simulated root mean square errors for cell
probabilities and theoretical standard errors (Colors denote a 2D ker-
nel density estimate)

Simulation shows that standard errors and RMSE are of the same order

of magnitude, yet numerical instabilities lead to some deviations of RMSE
for some of the cells.

Application of EMax to the analysis of split-ticket
voting in New Zealand

Johnston and Pattie (2000) use data from New Zealand on split-ticket vot-
ing to validate EMax estimates. In our application, we supply standard
errors that are not given by Johnston and Pattie.

The numbers of split-ticket voters in individual voting districts is esti-

mated on the following known quantities

1. the numbers n

i j

of votes for candidates of the parties j in districts i,

2. the numbers n

of votes of party lists k in districts i,

3. sample estimates ˆn

+jk

of numbers of combinations of candidate and

list votes on the national levels.

The following tables give examples of these data.

Table 1: List votes in New Zealand districts extract

District

Labour

National

Alliance

NZ First

· · ·

Albany

10271

13583

1967

1033

· · ·

Aoraki

14413

10393

2881

992

· · ·

Auckland Central

13647

7747

2321

671

· · ·

Banks Peninsula

14018

12643

2844

788

· · ·

Bay of Plenty

11342

11350

1769

3178

· · ·

Christchurch Central

13407

8887

3369

880

· · ·

Christchurch East

15084

7816

3665

719

· · ·

Clutha-Southland

9182

12882

1883

1043

· · ·

Coromandel

12390

10747

2241

2406

· · ·

Dunedin North

15052

6427

3902

401

· · ·

.
.
.

Table 2: Candidate votes in New Zealand districts – extract

District

Labour

National

Alliance

NZ First

· · ·

Albany

8753

13701

3775

751

· · ·

Aoraki

17415

10276

2031

705

· · ·

Auckland Central

12645

7360

6129

· · ·

Banks Peninsula

15475

14020

1474

510

· · ·

Bay of Plenty

8679

15781

1338

4185

· · ·

Christchurch Central

17229

7825

2690

641

· · ·

Christchurch East

18157

6995

2127

528

· · ·

Clutha-Southland

9218

15619

1049

1131

· · ·

Coromandel

3892

13432

1217

1237

· · ·

Dunedin North

18856

6161

1968

224

· · ·

.
.
.

Table 3: Survey data on New Zealand list and candidate votes – ex-
tract

Candidate votes

Labour

National

Alliance

NZ First

· · ·

List votes
National

197

· · ·

Labour

134

· · ·

NZ First

· · ·

Alliance

· · ·

.
.
.

On the base of these data, the application of the EMax procedure results

in the following estimates:

Table 4: Predicted list and candidate votes for the district

Albany

Candidate votes

Labour

National

Alliance

NZ First

· · ·

List votes
National

11949

1122

234

786

· · ·

Labour

403

3584

341

2476

· · ·

NZ First

219

486

1224

809

· · ·

Alliance

140

360

3088

· · ·

.
.
.

5000

10000

15000

20000

25000

30000

5000

10000

15000

20000

25000

30000

Real number of straight tickets

Predicted number of straight tickets

Figure 2: Comparison of predicted and observed straight-ticket votes
in New Zealand districts

Software for EMax with standard errors

• Implemented in R

• Implementation in Stata is planned