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Analysis Of Variance InA Descriptive Context: AGeographic ExampleAndrw Kirby aa Department of Geography , University ofReading , Reading, RG6 2ABPublished online: 28 Jul 2006.

To cite this article: Andrw Kirby (1982) Analysis Of Variance In A DescriptiveContext: A Geographic Example, Journal of Applied Statistics, 9:1, 5-15, DOI:10.1080/02664768200000002

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BIAS, V o l 9 , No 1, 1982 - 5 -

ANALYSIS OF VARIANCE IN A DESCRIPTIVE CONTEXT:

A GEOGRAPHIC EXAMPLE

Andrm Kirby Department o f Ceogmpl~y,

Ilniuersity o f Reading, Reading RC6 2AB

Abstract

This paper is designed to introduce students in the 16-19

group to Analysis of Variance. It takes as its subject a

political-geographic issue, namely electoral districting, or

gerrymandering. The topic of gerrymandering is outlined, and

the distribution of boundaries for the 1979 EEC elections in

London examined. Different gerrymanders are then explored,

using Analysis of Variance to describe various types of

solution, according to the distribution of the Labour vote

both within and between constituencies.

Introduction

This paper provides an unusual example of the use of Analysis

of Variance: the context used is that of a political-

geographic issue, namely gerrymandering. As I shall outline

below, there are numerous ways in which boundaries may be

drawn around areas in order to create political constituen-

cies, and different political results may obtain depending

upon the actual configuration chosen. These results can of

course be expressed simply (eg as numbers of 'winsf for a

particular party), but such an approach lacks precision. A

superior alternative is the application of an ~nalysis of

Variance strategy, which allows us to compare the voting

strekgth of both parties at once (fo= simplicity, only two-

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party contests are discussed here).

Gerrymander inq

The term 'gerrymandering' is indicative of its meaning. It

owes its development, not to a gerry called mander (as one

student answer has it) but to United States Governor Elbridge

Gerry. In 1812 the latter attempted to alter the political

boundaries of Massachusetts in order to limit the power of

his opponents, the Federalists. In this he succeeded

admirably: with a Republican/Democrat versus Federalist

distribution of votes of the order 51,766 to 50,164, the

tally of seats was a landslide 29 to 11, at the expense of

the Federalists. Of course, such manipulation is not easily

achieved, and one of Gerry's tortuous redistricting exercises

excited even local comment: a journalist compared its shape

to a salamander, and the name has stuck, "ensuring Elbridge

Gerry's political immortality" (Taylor and Johnston, 1979,

p.372).

In the intervening years, redistricting for political

advantage has evolved to the extent that a typology of

partisanship exists. This is outlined by Johnston as follows:

"(1) creating stacked districts, of unusual 'shapes, which seek

out relatively isolated pockets of the party's support

and amalgamating them to produce a seat in which the

party has a majority.

(2) creating packed districts, by concentrating the

opposition party's votes into a very few safe seats,

creating a large number of excess votes for them

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( 3 ) c r e a t i n g cracked d i s t r i c t s , by d i l u t i n g t h e oppos i t ion

vot ing s t r e n g t h s a s a minor i ty i n a l a r g e number of

s e a t s , producing many wasted vo tes f o r them"

(Johnston, 1979, p. 172) . I n simple terms then, gerrymanders c a n involve connecting

very s c a t t e r e d groups of v o t e r s ; o r concen t ra t ing your

opponents i n a smal l a r e a wi th enough vo tes t o win t h e s e a t s

many t i m e s over ; o r jo in ing each group of your opponents t o

j u s t enough of t h e suppor te r s of t h e p a r t y i n power t o make

s u r e they l o s e each time. I n each c a s e of course , d i f f e r e n t

vo t ing r a t i o s o b t a i n , and it is t h e s e t h a t a r e t o be measured

by Analysis of Variance.

An example from t h e European E lec t ions , 1979

This example is designed f o r sixth-form geographers, and

draws on vot ing d a t a r e l a t i n g t o t h e r e c e n t EEC e l e c t i o n s .

It takes a s i t s b a s i s t h e r e l a t i o n s h i p between-the s e a t s won

by t h e two main p a r t i e s i n Westminster E l e c t i o n s , and t h e

s e a t s won a t t h e Strasbourg e l e c t i o n s o f t h e same year.

London is used a s an example of how the c r e a t i o n of new

c o n s t i t u e n c i e s can a l t e r t h e underlying p o l i t i c a l support i n

an a rea .

Figure 1 d i s p l a y s t h e d i s t r i b u t i o n of Labour and Conservative

Par l iamentary 'wins ' and t h e boundaries of t h e l a r g e r EEC

c o n s t i t u e n c i e s . Although t h e r e is a wide d i s t r i b u t i o n of

both Labour and Conservative s e a t s , it is t o be noted t h a t

only two of t h e t e n EEC Const i tuencies were won by Labour,

and t h i s must be a t t r i b u t e d t o t h e ways i n which t h e

boundaries have been drawn. This i's i l l u s t r a t e d i n d e t a i l i n

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Figure 1 : The ten European constituencies i n Great London. including the Westminster seats according to t h e i r p o l i t i c a l choice i n 1979 (Conservative o r Labour), and the study area.

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r e l a t i o n t o t h e two s e a t s i n North-East London, which con-

t a i n t h e following Westminster c o n s t i t u e n c i e s (Table 1).

Table 1: Westminster Cons t i tuenc ies and Labour support (1979 e l e c t i o n )

Consti tuency Labour vo te a s proport ion of tu rnou t

. 1. Enf i e l d North 4 1 % 2. Edmonton 4 7% 3. Chingford 27% 4. Walthamstow 50% 5. Leyton 51% 6. Bethnal Green 50% 7. Newham North West 61% 8. Newham North Eas t 54% 9. Stepney and Poplar 62%

10. Newham South 64% 11. Wanstead and Woodford 19% 12. I l f o r d North 37% 13. Romford 32% 14. Upminster 35% 15. I l f o r d South 43% 16. Barking 52% 17. Dagenham 52% 18. Hornchurch 43%

A s we would expect , t h e r e is a good d e a l of v a r i a t i o n i n t h e

Labour vo te between t h e s e 18 c o n s t i t u e n c i e s . This means t h a t

it should be p o s s i b l e t o combine d i f f e r e n t a r e a s t o produce

d i f f e r e n t vot ing p a t t e r n s - s t r o n g Labour, s t r o n g Conserva-

t i v e , marginal Labour, and s o on, - i n t h e l a r g e r European

s e a t s . More important ly , a t l e a s t wi th in t h e p resen t c o n t e x t ,

w e can then compare t h e d i f f e r e n t conf igura t ions using an

Analysis of Variance approach; the summary s t a t i s t i c , t h e

F - t e s t , g i v e s us a simple y e t p r e c i s e account of t h e p o l i t i c a l

impl ica t ions of any s o l u t i o n .

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in Figure 1: see Table 2. Details of the calculations invol-

ved can be widely found: see for example Blalock (1960).

Table 2: Analysis of Variance Results: ~uropean Constituencies

Labour Seat conservative Seat

Total (2) =SO7 R = 50. N = 10

Total ss = 2527 Between ss = 595 Within ss. = 1932

ss DF Total ss 2527 17 Between ss 595 * 1=595 Within ss 1932 + 16=120.75 F-RATIO (Between Ss) = 4.92

Within. B B ~ - Total (E)=313 LL82O

7 R = 39.1 N = 8

Table 2 shows that the Between Sum of Squares is larger than

the Within Sum of Squares, although the latter is relatively

large. This solution can be taken as a base line, about which

we can experiment in order to achieve different political

results; more sinply, we can gerrymander, by aiming to increase

or decrease the F-ratio.

First, let us attempt to create a packed distribution, ie a

concentration of all one party's support into one constituency,

in order that they waste votes. Table 3 illustrates a

gerrymander that would benefit the Conservatives at the

expense of Labour; (as in all these examples, we are using

only two constituencies for convenience, and a more realistic

study would of course consider all ten seats). As expected,

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t h e Between SS h a s r i s e n , w h i l s t t h e Within SS h a s v i r t u a l l y

d i sappea red .

Tab le 3: Analys i s o f Var iance r e s u l t s : packed gerrymander

Labour S e a t Conse rva t ive S e a t

41 T o t a l ss 27 Between ss 19 Within ss 37 32 35 T o t a l ss 4 3 Between ss 4 3 Within ss

F-RATIO *

For o u r second a t t e m p t , l e t us aim f o r a gerrymander t h a t

w i l l b e n e f i t Gabour, by g i v i n g t h e l a t t e r bo th s e a t s . A t

t h i s s t a g e i t shou ld be p o s s i b l e t o produce a s o l u t i o n s imply

by t h i n k i n g i n terms of t h e F - r a t i o . Because t h e Conserva-

t i v e s must narrowly l o s e each s e a t , it fo l lows t h a t t h e

Within SS must be n e a r l y a s h igh a s t h f Between SS, and t h a t

consequent ly t h e d e s i r e d s o l u t i o n is one wi th a very sma l l

F - r a t i o ( s m a l l e r t han t h a t found i n Table 2 , f o r example) .

I n Johns ton ' s typology, such a s o l u t i o n is a l s o l i k e l y t o be

a s tacked d i s t r i b u t i o n . T h i s p a r t i c u l a r s o l u t i o n has reduced r

t h e Between SS d r a s t i c a l l y , w i t h t h e r e s u l t t h a t t h e F - r a t i o

i s a l s o d r a m a t i c a l l y reduced. It might however be u s e f u l f o r

s t u d e n t s t o rework t h e d a t a , i n an a t t empt t o f i n d a l t e r n a -

t i v e c o n f i g u r a t i o n s t h a t g i v e s i m i l a r r e s u l t s . Such an

e x e r c i s e would p rov ide f a m i l i a r i t y w i t h ' t h e t echn ique , w h i l s t

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underlining the number of possible solutions (eachofcourse

providing different political outcomes) that exist. In order

to underline this wide range of potential configurations,

the dissimilarity of the three partitions discussed here is

shown in Figure 2.

Table 4: Analysis of Variance results: stacked gerrymander

First Labour Seat Second Labour Seat

50 Total ss = 2527 41 Betweenss 13 47 Within ss 2514 2 7 50 SS DF 51 Total ss 2527 17 62 Between ss 13s 1 = 13 64 Within ss 2514+16 = 157.1 19 37 F-ratio = 0.08

E = 448 R = 44.8 N = 10

Discussion

The example has been designed simply to introduce the student

to the measurement of variance in different populations. It

is thus descriptive in intent, not analytic.' For this

reason, no consideration has been given to the assumptions

of the Analysis of Variance model, nor to the interpretation

of the F-statistic. It may be noted that the assumption of

normality within the data can be relaxed a good deal with

this model, and for this reason it has not been considered

here. Similarly, it is possible that criticism may be

levelled against the use of constituencies, which do not

strictly represent a sample. An argument which reconciles

these views is however presented in Blalock (1960, p.270).

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{ th in the study a"'* figurations * conrtityen~~es.

Three p s s \ b l e , European Figure 2: ,,c,ud,ng the ex~1tf109

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Consider ing t h e example i n a d i f f e r e n t c o n t e x t , t h e aim o f t h e

t o p i c i s two-fold. I t h a s been developed t o be p a r t of t h e

geograph ica l s t u d y o f r eg ions : i n t h i s c a s e , r e g i o n s dev i sed

by p a r t i c u l a r agenc ie s . Secondly, i t r e p r e s e n t s one more

v e h i c l e by which t h e l o c a l environment can be approached.

The d a t a employed h e r e can be r e a d i l y o b t a i n e d from any

( l o c a l ) newspapers, and can b e r e p e a t e d a t v a r i o u s s p a t i a l '

s c a l e s ; an obvious a l t e r n a t i v e would be a s t u d y of t h e ways

i n which l o c a l wards a r e aggrega ted t o produce Westminster

pa r l i amen ta ry c o n s t i t u e n c i e s . Taken t o g e t h e r , t h i s t y p e o f

p r a c t i c a l work can t h u s p rov ide u s e f u l i n s i g h t s on s t a t i s -

t i c a l , g e o g r a p h i c a l - p o l i t i c a l and l o c a l t o p i c s .

Notes

1. Although t h i s example i s pedagogic, i t should n o t be

i n f e r r e d t h a t t h e s t a t i s t i c a l a n a l y s i s o f s p a t i a l

phenomena i n g e n e r a l , o r e l e c t o r a l i s s u e s i n p a r t i c u l a r ,

i s poor ly developed. Some o f t h e problems o f s p a t i a l

a n a l y s i s were o u t l i n e d , by geographers , i n a s p e c i a l

i s s u e o f The S t a t i s t i c i a n , X X I I I ( 3 / 4 ) , 1974. A

d e t a i l e d view of t h e s t a t i s t i c a l bases of e l e c t o r a l

m a t t e r s is provided i n Gudgin, G and Tay lo r , P J, Seats ,

Votes and t h e S p a t i a l Organ i sa t ion of EZections,Pion,

1979.

REFERENCES

BLALOCK H M ( 1 9 6 0 ) , S o c i a l S t a t i s t i c s , ~ c ~ r a w - H i l l , New York.

JOHNSTON R J (1979) , P o l i t i c a l , E l e c t o r a l and S p a t i a l Systems, Oxford U n i v e r s i t y P r e s s , Oxford.

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TAYLOR P J and JOHNSTON R J (19791, Geography of Elections, Penguin Books, Harmondsworth.

Acknowledgements

My thanks go to an anonymous referee for hie helpful comments

on an earlier version of this paper.

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