An Outline for Racial Bloc Voting Analysis Paper

    School of Social Sciences
[Jniversity of California, Irvine



AN OUTLINE FOR RACIAL BLOC VOTING ANALYSIS*

Bernard Grofman
Professor of Polltical Sclence

N'lcholas Novlel lo
Doctoral Candldate

School of Soclal Sclences
Unlversity of Callfornla, Irvlne

Irvlne, Callfornla 92717

June I, .I983

Sectlon IX of thls paper
'ls not f or quotatlon 1n
1 ts present form

*Acknow1edgments: Thls research was part'la11y supported by NSF Grant
#SES 8l-07554, Program 1n Po'lltlcal Sclence. I'le are lndebted to Kathy
Albertl, Llll'lan !,lh1te, Helen !{1ldman, cheryl Larsson, Dee yox, and 0avld
l{esthof f of the llord Processlng Center of the School of Soc'lal Sclences,
Unlverslty of Ca1'lforn1a, Irvlne, for typlng a large number of rough
drafts of thls manuscrlpt. l{e would also llke to acknowledge the helpful
dlscusslons of related lssues the senlor author has had wlth Charles
Cotrell, Robert Br'lschetto, Fred Cervantes, and Jose Garza. After theflrst draft of thls artlcle was completed we became aware of the work of
James Loewen (.l982) and earl'ler references whlch he cltes and whlch we
have now l ncorporated and bu'l I t upon. Al1 J udgments expressed 1n th'ls
artlcle are, however, the so'le responslbll'lty of the authors.

An earller verslon of thts paper was presented at a pane'l on "Soclal
Sclence 1n the Courtroom" at the Southwestern Soclal Sc'lences Heetlng,
Houston, Texas, Harch 17-20, .l983



it.

TABLE OF CONTENTS

I. Introduction.

III.

Bas'ic Model for Single Member District Two Candidate General
Elect jons Involving a Bl ack Candidate and a l,Jh jte Candidate
Where Registration (or Population) Data by Race are Available
for Each Voting District.

Measuring the Extent of Racial BIoc Voting irt
Sing'le Member D'i stricts

0ther Regression Approaches.

20

l3

17

35

6l

23

?)

42

50

52

IV. Mod'ification to the Basic Model for the Single Member
District General Election Case Where There is More than
One Candidate of a Given Race

V. Modification to the Basic Model for the Single
Member District Case for Primary rather than General
El ecti ons

Vi. Extending the Bas'ic Model for the Single Member District
General Election Case to Allow for Differential Turnout
and Voter Choice by Race and by Party .

VI I . Coping with the Unavai l abi f ity
(or Populatjon) Data by Race.

Vi I I. Modification
D'i stricts or
Place System.

of Regi stration

to the Analysi s for the Case
At-1arge Elections Without a

of Mul timember
Desi gnated

IX. Estimating the
Possible Type

Proportion of Ballot Patterns of Each
Cast by Bl ack/hlh'ite Democrat/Non-Democrati c

Al low'ing For Different'ial Choice by Race

Voters.

Tables.

Vari abl e Li st

Appendix A:

Appendix B:
and by Party. 64



I. introduction

The problem with which this paper will deal is the measurement of

racial (or Iinguistic or ethnic) polarization'in voting at both the

primary and general election level jn both single-member djstrict

elections (or at-large elections with numbered places) and multimember

district elect'ions. Proving the existence of bloc voting along racial or

linguist'ic lines is often an'important component of cases a1'leging vote

dilution of. a racial or linguistic minority. (See e.9., McMillan v.

Escambia County of Florida, 688 F. 2d 960 at 966 n. .l2.) 
The techniques

we offer are intended to provide a standardized methodology through wh'ich

measures of the extent of racial or linguist'ic bloc voting can be

obtained, and include all the methods commonly used as well as a number

of modifications and extensions (€.9., to the case of multimember

d'istrjct electjons without a numbered place system) that we have
'l

developed.' We shall provide techniques of analysis both for the case

where data on the racia'l compos'ition of the electorate or potential

electorate in each voting prec'inct'is available and for the case where

the only d'irectly racially relevant data that are available are the vote

shares of the Black (tihite) cand'idate(s) in each voting precinct.

lFor simplicity, we shall henceforth discuss only Black minorit'ies,
but the results we offer are, of course, appl'icable to any identifiable
group. Again for simplicity we present our models'in terms of Black
candidate(s) vs. White candidate(s) contests, but the same kind of
analysis would apply if we were dealing with candidates who were the same
race but whose electoral support varied with the race of the voters.



2

II. Basic Model for Single-Member Djstrict Two-Candidate General

Elections Involving a Black Candidate and a White Cand'idate Where

Registration (or Populat'ion) Data by Race Are Ava'ilable for Each

Voting'Precinct

For simp'l'icity, I et us assume that regi strat'ion data by race by

voting precinct are ava'ilable. The analysis for the case where the

available data are on the racial compos'it'ion of the populat'ion in each

voting prec'inct (rather than on the racial composition of the registered

voters) is essentially identical to that we give below.

Let2

= the proportion of total registration which is

lih i te

I -x = the proport'ion of total regi strati on whi ch 'i s

Black

PWW = the proportion of White registered voters who

vote for the White candidate

Pgw = the proportion of Black registered voters who

vote for the !,lhite candidate

Pgg = the proportion of Black registered voters who

vote for the Black candidate

PWa = the proportion of White registered voters who

vote for the Black candidate

2R t ist of all variables used 'in th'is paper along w'ith their
def i n'itions i s prov'ided at the back to assi st the reader.



3

Pw = f,he proportion of total reg'istered voters

who vote for the White candidate

PB = lhe proport'ion of tota'l regi stered voter s

who vote for the Black candidate

f = the proportion of registered voters who vote

Tw = Pl,lw * P,,B = the proportion of registered white voters who vote

TB = PBB * PB,u = the proportion of registered Black voters who vote

(N0TEt PtllB + P*, S 1, PBB * PBw 5 I' and Pw * PB I l, since some white

(Black) registered vote s may not vote. )

It is true by defjn'ition that, for the electorate as a whole, we must

hav e

Py1 = X(fuur) + (l - x) (Psy);

i.e.,
P,,=(Pww-Pr*)x*PB,,,

Let P!{(j) = the value of Pw in the ith voting precinct, etc.; j.€.,

in general, we shall use the superscript (i) to denote the value of a

variable in the'ith voting precinct. Hence if

t,l') = *rX11) + bt

for al1 i, then

P,..,u = *l * bl

PBtl = bl ;

i.e., if we regress P,, on x, then the slope and intercept have a

straiqhtf orward "natura1" 'interpretation, 'in terms of the propo tion of



l^/hite (Bl ack ) reqi stered voters who vote f or the lrJhite candidate.
)

S'imi I arly, i f we regrest PB on x, we obtai n'

PUJB = nZ+ bZ

PBB = bz

Analogously, if we regress P,, on l-x, then we obta'in PWW as

the intercept of that regression equation; wh'ile if we regress PA

I - x, we obtajn' PWg as the'intercept of that regression equation

Let us specify a new notation such that

D

Dr = 
'|,Jl,,J

'wl^l Pww * PwB

0n

Dl =.BI,J

etc.

Bl^l

%.g*r&;

3hle have deliberately chosen to do those regressions in terms of
proportions rather than raw numbers. Virtually identical equations can
be defined for the case where we look at the number of Black (l,lhite)
reg'istrants, and at the number of voters tor ffiE-Wh jte (Bl ack ) cand'idate
inltead of at proportion{-6G the parameters (especial'ly the intercept)
do not have as direct and immed'iately meaningful interpretations as 'in
the regression equations based on proportjons defined above, since using
raw numbers confounds d'ifferences caused by differjng propens'ities of
White and Black voters to vote for Black candidates with effects caused
by differential voter turnout among White and Black voters, since tota'l
vote for Black candidates wjll be lowered in precjncts wjth high Black
population if Black turnout is lower than Wh'ite turnout. This effect
wi11, in general, lead to an underestimation of the maglitude of 

-Py1y1jf raw numbers rather than proportions are used. The differential
turnout issue ure consider separately below.

The methodology we use is based on that'in Goodman (.1953), a classic
art'icle whose techniques have become part of the standard statistical
tool kit (see e.g., Loewen, 

.l982; Langbein and Lichtman, 
.l978).



5

These primed variables give us the White (Black) vote for Wh'ite/Black

candidates as a proportjon of total White (Black) vote. It js PilW and

Pig in which we shall be most interested since these denote the proportion

of votes from t'Jhite (Black) voters which go to a candidate of their own race.

S'imi larly we shal I let

P.. D

Pil=T^fu, anci Pi=tr.
0f course, Pit =l-Pi.

These two primed variables give us the White (Black) share of the actual

vote .4

0f course, the value of ,2 for the regress'ion equations specified

on page three and four above (where r is Pearson'S correlation

coefficient) tells us how good our assumption is that a constant

proportion of White (81ack) registrants vote for the White (Black)

cand'idate(s) and thus whether the vote share of the White (Black)

candidate is 'in fact almost perfectly directly proportional to the

proportion of hlh'ite (Alack) registration 'in each voting precinct, 'i .8.,

directly proportional to x (l - x). It is important to emphas'ize that,

in analyses involving racial registration data and votes for cand'idates

by the race of the candidate, r (or rZ) is not a measure of the

extent of racial bloc voting. Rather, d h'igh value of r (or

equivalently of rZ) is a necessary prerequis'ite for us to have

4Often those dojnq racial bloc votinq will reqress P; on x
rather than P6 on f. Great care must-be taken-in inteirpreting the
parameters jn such a regression equation. tie d'iscuss th'is issue'in a
methodol ogical appendi x.



confjdence in estimates of parameters such as PWW and Pi' which

are the ones actua'l1y to be used in the measurement of the oegree of

raci al polarization.5'6

Tu rnout

Our est'imating procedure also tells us about (differential) turnout

among White and Black registered voters, since P,.B * PWhl = T,, = proport'ion

of t/hite registe ed voters who vote (i.e. White turnout as a proport'ion of

White registered voters) while PBB * PBW = TB = proportion of Black

registered voters who vote (i.e. Black turnout as a proport'ion of Black

regi stered voters ) .

We can directly estimate turnout, as a proportion of registered voters,

by making use of the identity

Total votesT,,* * Tr(1-x) = T = regi stered voters

and regressing T on x. The intercept of this regression will give us

Tg while the sum of the slope plus the'intercept will give us Tr.

5To those with statistical tra'ining, this may seem trivially
obvious, but jt'is a point wh'ich federal courts have fa'iled to fu11y
qrasp. (See espec'ially McMillan v. Escambia County, Florida 688 F.2d 960
it goo n. 12 ( .t 

982) , one-oT-tTiFl ead

6The.e is no clear professional consensus as to what value of r
is adequate. Indeed, this is a misleadjng question to ask. Art'icles
appear in the major professional pof itical scjence iournals containing
regressions with r values of.3 or even lower if the t-statistics for
particular regression parameters 4re statist'ically significant. Values
of Pearson's r above .5 are relatjvely rare in political science.



Analogously if we let

PD*l = proportion of registered voters who are either Democrats
or Independents

PR*I = proportion of reg'istered voters who are either Republicans
or Independents

PD = proportion of registered voters who are Democrats

PR = proportion of registered voters who are Republicans,

PI = proport'ion of registered voters who are Independents or
unaffi'liated,

TD = proport'ion of registered Democrats who vote

TR*I = proportjon of registered Republjcans and Independents who
vote

then TDPD * TR*lPR*l = T, etc. (see Section V).

Checking Realism of Parameter Estimates

(l ) Verifying aggregate relationships v'ia mathematical ident'it'ies:

As one check on the realism of our regress'ion-obtained estimates of

P,,,,, P,,B, PBB, and Pr, (and thence of the accuracy of thejr primed

equivalents), we must have certain 'identities satisfied:

,PWu,*(l-x)PBW=P,n,

*PWB*(l-x)PBB=PB

Analogous'ly, we must have

T*x+Tr(l-x):T

(2) Extreme case analysis: In order to establish further confidence

in our regression-based estimates of parameters such as PilW, Pig, TA, TW



6o

etc. jt is important to directly check the realism of these estimates. To

do this we may look at d'istricts which are nearly all Black (t'lhite) in

their voting registration. Clear'ly, in a nearly .l00 percent Black

d'istrict, the share of the total vote in that precinct garnered by Black

candidates'is essentially identical to ti['), while in a nearly .l00 percent

White district the share of the total vote in that precinct garnered by

l,lh jte candidates 'is essentially jdentjcal to ti,l'). The average values of

the values of Pig and P,i* we obtain from look'ing at extreme-case (all

White all Black) districts enable us to verify the reasonableness of our

regreSsion-based est'imates of these Same parameters. Sim'il arly the

average value of T we detain from lookjng at all White or Black

districts, respect'ive1y, enables us to verify the reasonableness of our

regress'ion est'imates of Tg and TW. Such a check should always be

performed if a sufficient number of extreme case districts of each race

are available. Be aware, however, that some differences between the

regression-based estimates and those obtained from the average values of

homogeneous districts (especial 1y for completely racially homogeneous

districts) are to be expected. In particular, I,Jhite voters in homogeneous

I^Jhite districts are very 1ike1y to be a few percentage points different in

turnout and in the proport'ion of their votes which goes to Whjte

candidates than is true for tJhite voters as a whole--due to both

contextual effects (i.e., homogene'ity reinforcing the prevailing tendency

'in a group) and also the effects of class d'ifferences, (and perhaps also



9

part'isanship) between districts which are almost entirely l.lhite and those

which contain some number of Blacks:7

(3) The method of overlapping percentages: In some cases we may not

have d'istricts wh'ich are nearly 100% black or .l00% whjte but there may

still exist d'istricts which are on average overwhelmingly (say 90% or

more) of one race. For such districts we may make use of the methods of

"overlapping-percentages" (Duncan and Davis, I953; Loewen, 1982) to

determine an est'imate of the minimal bloc voting which coulci exist and

stjll be compatible with the observed levels of support for the

t^lhite/Bl ack candidate(s). As before let

x(i)

t-r(i)
o (i)
'hl

proportion of district
proportion of district
proportion of the vote

the Whjte candidate(s)

voters

proport'ion of the vote

the Black candidate(s)

voters

i which is White

i which is Black

in district i which goes to

as a proportion of registered

di str i ct i wh i ch goes to

a proportion of registered

d (i).B 1n

AS

TThe regress'ion-based estimates will normally be slightly superior
to those obtained from extreme-case analysis, because the reg ession
est'iate is based on all the data and thus provides a better estimate for
the behavior of the average I,Jhite/Black voter. However, where racial
polarizat'ion voting is clearcut, extreme case analysis, or the method of
overlapping percentages d'iscussed below, frdJ be sufficjent to establish
the presence of rac'ial bloc voting, and regression techniques superfluous.



t0

. e*(i)

Since even if all

would still leave

wh'ich has to come

Simi I ar1y, 'if

rf+ . *(i) , th.n

P (j) - r-*(i)_ 'l^lrww5 Tjl-
Bl ack voters voted f or the tlhite cand'idate(s), that

a remaining vote proportion of er(i) - (l-x)(i)
from White voters.

*.tr(i)<t-*(i),th.n

D (i) _ ,(i)
^ 'B ^PaaS t -f.rf

while only (ry) 1oo percent or

(4) Constra'ined percentages: The nrethod of overlappino percentages

provides a conservative estjmate of the amount of racial bloc voting. In

most circumstances, we bel'ieve it to be unreasonable to posit (as we did

above) tnat a full .l00 percent of Black voters vote for l,,lhite cand'idates

te

)

e

hth i

(i)

il
ent

rac

at

o

e

S

a

el
fe
ari

rac

her

Whi te voters

/
that only (

\
idates, whilt

of voters ol

d, ceteris pt

he opposite r

ht fail 'if tl

Bl ack cand

proporti on

race shoul

oters of t
clause mig

vote for

PB(i) - x

candidates. It is simply unreasonable to posit

100 percent of Black voters vote for

of t,Jh'ite voters do so. Clear'ly, the

who vote for candidates of that same

least as great as the proportion of v

vote. 0f course the ceteris paribus

( 'l-x 
)

1bo per

a g'i ver

ri bus, b

ace who

ere was

)(

rc

n



ll

partisan contest between candjdates of djfferent races'in which each

candidate was from a party opposite to the party registrat'ion held by a

majority of the members of h'is/her race.) If, however, w€ I Safely

assume that

Pttw i Paw

D rD.BB :'t,lB '

then we can improve on the method of overlapping percentages and

calculate the m'in'imal poss'ible values of PgA (PWW) as follows:

If ). er(i) < I - *(j), th"n

Pss S PB('i )

To see th'is we need merely note that if Pgg = P,.,B then

p,,J(i) = r(i,rrr,,, + 1r-x(i)1rrr(i) = pBB = p,,B

Similarly, 'if l. t*(i) . x(i), *,.n

Pww S P!,l(i )

If | . ,(i) . p,.(i), or L, .,-x(i) . Rr('i ) , then analogous

reasoning appl ies.

Regardress of whether P!{(i ) i *(t ) o. PB('i) | r-*(i I , we must have

p,',(i) ,r,rtt' -illl-t , ,



12

since

o > (t-*(i)) (t*(') - l) ,

and

D (i) _,(i)
o (i) - 'B nrB' Tif '

since

o , ,(i) (rr(,) - t)

tJe shall refer to techniques such as those above as involving the

method of constrained percentages. The method of overlapping percentages

wil I always y'ie1d lower est'imates of the amount of rac'ial bloc voting

than wiII the "constrained percentages" assumpt'ion that each race js at

least as l'ike1y to vote for candidates of h'is/her own race as he/she is

to vote for cand'idates of the opposite race.

(5) Mixed-case analysis: Sometjmes we have extreme cases for one

value of a variable but not folits other values. For example, there may

be, say, some nearly all [,lhite precincts but no nearly all Black

precincts. To cope with that we can use what we shall call mixed case

analysis. Let q adummyvariable(inthiscase q=B or q=[,J). We

h ave the i dent'i ty

Pq = P!\tq (PRoP0RTIoN 0F hIHITES IN THE MIXED DISTRICTS) +

Psq (PRoPoRTIoN 0F BLACKS IN THE MIXED DISTRICTS)

OO 'is s'imp1y the population mean. tr' 'is obta'ined from looking

at the values of tO in the all-white districts. With tO and

O*O known, and the proportjon of Whites/Blacks jn the mixed districts
known, we merely solve the above equation for tr'.



t3

III. Measuring the Extent of Racia1 Bloc Voting in S'ing1e Member

Di stricts

I n I ook 'i ng at r aci al b I oc voti ng, we are concerned with what

proport'ion of the voters of each race votes for cand'idates of that same

race, i.e., with the extent to which voting patterns are polarized along

racial lines. One way to do thjs js simply to look at PWW and Pie. If
both are h'igh, then voting is clearly racially polarized. Indeed, in my

view, we may have bloc vot'ing if either Pilw or Pfig are high. If our

principal concern is with minority vote dilution, then the parameter of

greatest interest is PilW, since that tells us the extent to which Whites

are wjlling to vote for minority candidates. However, only if PU]W is

high relative to PiW do we actually have a polarjzed sjtuat'ion rather than

one in which both Whites and Blacks are voting predominantly for the White

candidate.

For single member districts it'is useful to develop a measure of

racial polarization wh'ich incorporates both PWW and Pgg. We may

represent raw data'in the form of a LxZ table whose entries sum to

one. (See Table l.)

Table I about here

In a situation which is perfectly polarized for both the majority and

the minority communities, the upper right-hand cell (customarily denoted

b) and the bottom left-hand cell (customarily denoted c) will be zero.

in this notation a is used to denote the upper left-hand cell and d



For such a

t4

Zx? table, a'is used to denote the lower right-hand cell.

useful measure of polarjzation is Yule's Q:

P D -P D'tJI^l' BB 'l.lB' Bt^l

ffiBt^r

n -ad-bcY ad+bc

For Table 1, the expression for Yule's Q simplifies to

Yule's Q = = I ndex of Rac i al Po'l ari zati on .

An exactly identical value for Yule's Q would have been obta'ined had

we used Figure 2'instead, where Pi,tW, Pi' etc. are used directly in a

table that 'is percentaged by rows ('i .e., in which each row sum 'is one).

Figure 2 about here

The index of racial polarization g'iven in the equation above

(Yule's Q) will normally range from 0 to l; the closer the index is to
one, the greater will be the degree of overall racial polarization (b1oc

voting). For single member districts, values above .6 would indicate

consi derable pol ari zati on.

An alternat'ive operationalization of rac'ial polarization is based on

R'ice's Index of Likeness (Benson, .l969). 
We shal I def ine, R, the Rice

Index of Racial Polarized Voting:

R = 'l - R'ice Index of L'ikeness

= Pt - Dl ='BB 'rlB Pilw - Priw'



t5

This measure will normally run from 0 to l, but wil I usually be

less than .5. For single member djstricts, values above .3 would

jndicate considerable po1 ari zation.S

There is no single cutoff value necessary of either R or Q to

establish s'ignificant racial b'loc voting. Racial bloc voting, in our

view, 'is signif icant when it leads to mjnority vote dilution, i.e ., if it
'is of sufficient magnitude that the success of minority supported

cand'idates js made difficult or imposs'ible because many White voters are

either unwilling to vote for Black (or Black supported) candidates, or

(for mmds) are willing to vote for no more than a "token" number of

m'inority canuidates. We should also emphasize that racial bloc voting

must be looked at in both general electjons and primaries. The absence

of rac'ial bloc voting'in a general election (especially a partisan

contest) does not'imp1y the absence of rac'ial bloc voting, since such

bloc voting may be present in the primary, and thus constrain the'

election chances of Black cand'idates. Even the success of Black

cand'idates in a given primary in no proof of the absence of racjal bloc

voting, since in a multicandidate plurality election,'if l,'lhite voters

divide their.votes among many candidates while Black voters do not, Black

candidates can be elected even though few l.lh'ites vote for Black

cand'idates; but th j s electoral success of Bl ack cand'idates may not be

8N.ith.r Q nor R

voting for the nmd case
appropriate measures of
below.

is appropriate an
without a numbered
rac'ial bloc voting

an'index of racial bloc
place system. We discuss
for th'is case in Section VIIi



l6

repeated in subsequent primary elections with fewer viable Whjte

candidates (cf. Blacksher and Menefee, 
.l982: 60 and note 346; McMillan

v. Escambia Cou@.ida 683 F 2d at 1241 n. 6).



17

IV. Mod'ifjcation to the Bas'ic Model for the Single-Member D'istrjct

General Elect'ion Case trJhere There'is More than One Candidate of a

G'iven Race

To deal w'ith sing'le-member-district situat'ions where there is more

than one cand'idate of a given race is not difficult. We s'imp1y redefine

the proportion of votes for the l,ihite (Black) candidate as the proportion

of votes for the White (B1ack) candidate. Since every voter has only one

vote which can be cast either for a Black or a tihite candidate, combining

votes for candidates of the same race seems the obvious thing to do and

offers, in our view, tl'c best procedure for estimat'ing the magnitude of

rac'ial bloc voting. However, it may be the case that the different

candidates of the same race have djfferent appeal to voters of that race

and/or to voters of the opposite race. A straightforward way to check

for this, which, as we shall see be1ow, also provides a methodology to be

used when voter registration (or population) data by race is not

available, 'is to regress the vote shares for candidate j on the vote

shares for candidate j, for each 'i,i pair (see below for further

details). In multicand'idate races a matrix of correlation coefficjents

f or each (hypothet'ical ) cand'idate pairing should be generated. in

addit'ion, for each candidate we can estimate ('in a f ash'ion d'irectly

analogous to that specified in Section II) the proportion of Black voters

and the proportion of [.lhite voters (and for partisan elect'ions, the

proportion of registered Democrats and of registered

Republicans/Independents) who voted for that candidate by regressing
TNV -CVS {CVSjand#withx(andw.ithPo).Suchregressionsenableus



l8

enable us to "fine-tune" our analysjs to allow for variations/overlap in

the support bases of different candidates of the same race. Such

inter-candidate variations/overl aps may occur, for example, if candidates

of the same/djfferent race are of the same/d'ifferent party.

We can verify the accuracy of these regressions estjmates by

comparing the mean number of ballots cast by voters'of each race (of each

party) obta'ined by summing up over all candidates the voter proportions

estimated as being given to each candidate from voters of a given race

(or a given party) with that given by our direct estjmates of ,l^J and

ng (no and nR*l).

If there is racially polarized voting, then (in a single member

d'istrict race) we should obta'in high negative correlat'ions between

cand'idates of different races, although some candidates may be associated

with more rac'ial po'larization than others. In a single-member d'istrict

race with more than two candidates'is always necessary to run a

regression of total vote share of White candidates versus total vote

share of Black candidates. The negative correlation this g'ives us w'ill

almost'invariably provide a more revealing ind'ication of probable racial

polarization than will the correlations resulting from any of the

two-candidate (Bl ack cand jdate vs. l^lhite candidate) regressions, since

candjdates of the same race are competing *ittl each other as well as with

the candidate(s) of the oppos'ite race, and candidates of the same



'ideology will be drawing some of their support from the same

Hence, the magnitude of the (negative) correlations between

Black and Wh'ite cand'idates'in a single member distrjct race

than two cand'idates wjll understate the seeming magnitude of

l9

voters .9

individual

wi th more

racial

po1 ari zation.

In multimember districts without a numbered place system, even if

racial bloc voting is high, the correlation between'indjvidual pafrs of

tih'ite and Black candidates may not even be negative, espec'ia11y if the

candidates are of the same party, since voters possess multiple votes and

it is therefore "easier" to give votes to a cand'idate or candjdates of

the opposite race.

9As for candidates of the same race in an smd race with more than
two candidates, we m'ight expect a positive correlation between their
votes to the extent that they're drawing votes from the same types of
voters; but, of course, they're also competing for voters, and this would
tend to g'ive a negative corielation between the'ir vote shares across
voting precincts. In general, in a multicandidate race, no clear
pred'iction for the nature of the correlat'ion between the vote share of
candidates of the same race can be given.



?0

V. Mod'ification to the Bas'ic Model for the S'ing1e-Member D'istrict Case

for Primary rather than General Elections

In a primary elect'ion (except jn the handful of states with open

primaries), the only elig'ib1e voters are those who are party

registrants. This means that unless this data is directly available, we

must estimate party registration by race, and then'for each party's

primaries substitute the (estimated) proport'ion of (Black/White)

party-reg'istered votes for the proportions of registered voters in our

earlier ana'lysis. Where we have data by voting precinct for both party

registration and registration by race, we can estimate party reg'istration

by race by running a regression of party registrat'ion by race, 'i .e., we

est'imate PD, PR and Pl by using regressions based on identit'ies

which must be true for the electorate as a whole. Using notation

analogous to our earlier notation

t.€.,

PD = X(P!{D) + (l - x)PgO ;

PD=(fu0-Pgg)x+PBD.

Hence if

pD(i) = r3*1i) + b: for att i,

we have

P,uD=*3*b3

PBD = b3



21

In like manner vle may estimate the proportion of l.Jhite registered voters

who are Republicans (i.e., PWR) and the proportion of Black

reg'istered voters who are Republicans ('i.e., PBR), etc. Analogous to

our earljer results PBD * Pt* 51, and P*, + Pr* 51, since

some registered voters may not have a party affiliation.

0nce we have obta'ined values for PWO and PgO (PWn and PBR, PWt

and Pgt) we use Bayes Theorem to specify the proportion of registered

Democrats who are White, PDW, etc. as

D=.DW

etc.

'Ptt D-5-
,D

then we use

x-P*,PO-PwtPI x-P**P*-PIWP,
I - LPRWPR * PIWPI * PRBPR + PIBPII I - PR - PI

'instead of x, and use

'l -x-PRgPR-PtgPl
I - IPRWPR * PIWPI * PnBPR + PlgPtl-

instead of I - x in the regression equat'ions

(Republ ican ) primaries.

If our estimates are real'istic, it must be

for the Democratic

the case that



22

PI.JD * Pt.lR * Pt.lI = I = PI,JD * PwlR*I1

PBD * PBR * PBI = I = PBD * Pg(R+t)'

If the proportion of independent or unaffil'iated voters is low, it
will be preferable to combine these voters w'ith the voters of the party

which they most resemble, a'lthough 'if the percentage of independents or

unaffil'iated is low it will not matter greatly which party we combine

them with and so we can use PWln*t1 and PalR*t) to obtain the

analogues to x and I - x for Democratic primaries and Pw(O+t;

and Pg(O+t) to obtain the analogues to x and I - x for Republican

primaries. In thjs case we estimate the values of, sdY, PBD,

PAln+11, PWO and PWlR*11, 'in a manner simi lar to that shown above.



23

VI. Extend'ing the Basjc Model for the Single-Member District General

Elect'ion Case to Al low for Different'ial Turnout and Voter Choice by

Race and by Party

In the previous section we showed how, if the appropriate

registration data broken separately by race and by party were available'

'it was possible to calculate PWD, PBD, PWlR*t1, and Paln+t;.

In the basic model'in Section II we have implicitly assumed that Black

(White) turnout is (approximately) constant across voting precjncts, and

so is PWW (Pgg). If these assumptions gjve rjse to regression

estimates with low values of r (rZ), it may be possible to improve

the reliability of our estimates of the proportion of Wh'ite/Black voters

who vote for Black cand'idates by using a multivariate regression model

which permits White Republicans and White Democrats (or perhaps Black

Republicans and Black Democrats) to behave differently from one another.

Alternatively,'if the bivariate regress'ion est'imates are good (or if we

use data derived from extreme case analysis) we can use this data to

obtain estimates of unknown values such as T*, and Trr. Clearly

there are elections jn wh'ich we might expect that, say, tlhite Republicans

and l,Jhite Democrats would behave somewhat different'ly, e.g., in an

election between a Black Denpcrat and a l.lhite Republ'ican.

As noted above, 'it w'ill often be the case that the number of

independent voters w'il I be low and/or varying in the'ir electoral behav'ior

across districts so that regression estimates for these voters as a

separate category w'ill be unreliable. In that case, we can combine

Independents with the party they most resemble. We present the analysis



24

for this case below. (A mult'ivariate regress'ion analysis of the case

where Democrats, Republ icans and Independents are best treated separately

is given in Appendix B.)

(1) Tu-rnout: An equational approach. To obtain turnout by race and

by party we may seek to solve the fol'low'ing set of equations 'in terms of

the unknown parameters TWO, TBD, TWlR*t I and TAlR+i;:

PorT*o*PDBTBD=TD

P1R*t)wrw1n+t1 * t(**t)aTg(R+I) = TR*l

oowT*o * o(**t)wTw(R+t) = Tt^t

PoaTgo * o(**t;aTa(R+t) = TB

The values of TD and TR*I, are known from regress'ing T on

PR*I bY using the identitY

TD(l - Pp+1) * TR*IPR+I = J

The values of Ta and T',,, can also be derived either from

extreme case analysis or from a regression equation (see Section II). By

Bayes Theorem

n*, (x)
D=
'Dt.l

e*r(x)-5-P,,,D(x) + PBD(l - x)

D ='l -D.DB .DhI

D = 
Pw(n*l)(')

'1R+t1w - -



25

P1n*l)B=l-P1R*t1w

The values of PI,lD, PWlR+I;, PBD, Pg(R*t) are estimated vja the regressions

described in Section V. 0f course x and I - x are known.

Wjth PD,.,, PDB, P1R+lyw, P1n+t1a and TlnI, TB, TD, and TR*I known, ,{
we can simpljfy by assuming that TBD = TBIR*I1, then we can directly

solve by substitutid* Ta for TgO in the first equation on page 24 to

obtai n

PowTwo+PrtTt=T,

Rewrjting this equation to solve for T*r, we obtain

Tr^iD=#
Analogouslylo

r -TR*t-P(R*i)gTg
't^t1R+I) -E

lOIf we attempt to solve the four equations on page 26 for TWD,

etc. in terms of the known parameters we obtain

PoeTao - P1R*l)wTw(n+l) = TD - Tw

PowTwo * PDgTBD * P(R*l)wTw(R+t) * P1R*t)aTa(R+l) = T

to*T*o - t(o*t)gTg(R+r) = TD - TB

giv'ing us only three equat'ions and four unknowns since our original
equation set 'is not independent (To*, * Tn = To * Tu = T). Thus we can
proceed no further unless we make a''sfmplify'ing"assutllpt'ion. As above, we
could reasonably simplify by letting Tp(o*r\ = Tnn. (I,Je could also assume
that To/o-rr = T,.,/nrr\, but aS lOng aS"thel6 are"few CaSeS Of BlaCk
RepubliElffi/inoep8rl5d*di which option we choose won't matter greatly. We

can decide between them by solving under both assumpt'ions and (continued)



26

(2) Voter choice: a multivariate regression approach. For

simplicity, we present our analysis with independents comb'ined with

Repubf icans. We have

wh'i I e

P1a = [eygP1wD)14 + P1a(R+I )Pw(n*r)w] x

+ [PsoP(BD)1^1 + Pg(n*t)Pe(n+t)w] (l-x) ;

P,,rD=P (+) Pso

.I-D

PwlR+r1= + -(+) Paln+r;

(continued) then checking to see wh'ich set of parameters gives us a
better fit.) With this first assumption, we have

PogTg - t(**, 
)wTw1n+t; 

= TD - TI.J

PowTwo + (PoB * P1R*I1a)Ta * P1R*I)wTw(r+t) = J

PowTwo - P1n*t)BTB = To - Tg '
wh'ich can be rewritten as:

(l - Pow)Ts - P1R*l)wTw1r+t1 = TD - Tht

(l - tD, * I - P(R*t1w)Ta * PDwTwD * P1R*I)wTw(n+r) = J

- (l - P1R*tyw)Ta = PDr^rTwD = TD - TB

After some (sicl) algebraic manipulation we obtain:

TI,ID =

Twln+r1

These estimates will be
text, and very probably not

only a marginaf improvement over
worth the bother to calculate.

+ 2(T, - TB)PDWI[arw - 2Ttrpow - 4To * 2TB * Z(T9 - Tw)p(R+t)w

(R+I )l.l/.. (R+I ) t.l

those i n the



?7

Hence, if we may simplify by assuming that the Black vote does not vary

with party registration, i.e. t(eO)14 = PB(R+l)14 = PB14, then

p, = [p, - (t-x)pup1 p(wo)w

+ [(l-Pe) - (l-x)Ps(R*l)] Pw(R+r)',,

+ Pgw(l-x) '

where P* and P1R*t) are estimated from earl'ier regressions, as'it PB,,.

I,le may rewrite the above equation urll

llAn ulternatjve is to use this equation to define two new variables,
using Z, as an arbitrary label:

,, = r(J) - (t-x(i))pso

z2 = (,-r(il)) - (r-r(i))pg(R+r),

then we can specify a regress'ion equation:

r,i, - psw{r-x(i), = p(wo)wZt * ptnl(n+r)wzz

* b2r

If the voting patterns of tlJhite Democrats/White Republicans and
Independents are roughly constant across districts, then we should have

bzl-o
and the coefficients of the regression give us the values for P(WO)W

md P11(R+I)H.

In this manner, if we do w'ish to try to distinguish among Black
Democrats and Black Republicans plus Independents, we can set

,(;) - pww(x(i))

= [pr-x(')rrr] p(so)w

+ [(l-Pr) - r(i"*,**r)] Ps(R+r),, . (continued)



?8

Pw = (l - x)[PsoP(t,,tD)hJ * Ps(R*t)Pw(R+I)tl * Pew]

+ Po[P(wo)w - Pw(n*i)w]

JO' 'tl(R+l)W

Hence, in a multivariate regression equation of PW vs I - x and

PD the intercept gives us an estimate of PWln+l1W and the value of

the coefficient of the PD term added to the'intercept g'ives us an

estimate of P(WO)W. Similarly in a mult'ivariate regression equation

of Pg vs. I - x and PD the intercept gives us an est'imate of

PWln+t;A and the value of the coefficient of the PO term added to

the 'intercept gives us an estimte of P(WO)g. These four values enable

us to obtain the'ir primed equivalents PiwO)w, Pw(R+I),,,, etc., wh'ich are

analogous to Pf^tw and Pwg.

The multivariate regress'ion model given above allows us to find

separate parameters for White Democrats, t,Jhite Republjcans and Blacks (or

we could have used equations which, if there were sufficient numbers of

Black Repubf icans and Independents to yield rel'iable est'imate, would have

enab'led us to find separate parameters for Black Democrats, Black

(Fontinued) However, this regression is likely to either have a low
rt because of the 1ow proport'ions of Black Republicans and the low
variance in this variable, or to yield_values for Pg(n+t)W and
PlgO)51 which are essentially ident'ica'1.

The first method given in the text, based as it'is on bas'ic rather
than derived variables, appears to give more reliable parameter estimates
than the method given in this footnote. However, even the first method
tends to have trouble partitioning variance if the basic variables
(i.e., PR+I and x) are highly correlated, and we have come to
strongly prefer the substitutjon technique based on bivariate equations
described in the numbered section immed'iately following th'is one as the
preferred method for estimating P(91D)W, etc.



29

Republicans, Independents, and Whites). We do not have sufficient

independent variables to d'istinguish by party and by race for both races

sjmultaneously in a single multivariate regression equation.l2

(3) Voter choice: an equational approach. We can make use of

equations most of whose parameters can be derived d'irectly or indjrectly

from bivariate regressions or extreme case analysis, and then solve for

unknown parameters of interest such as P(WO)W. This method seems to

give considerably more reliable parameter estimates of voter choices than

the multivariate techniques described above, which suffer from the

problem of multicol l'inearitY.

Pdvw

Povg

Pfn*t)vw

proportion of votes cast by Democratic voters who go

to the l,lhite cand'idate(s)

proportion of votes cast by Democratic voters wh'ich go to

the Bl ack candi date( s )

proportion of votes cast by Republjcan and Independent

voters which go to the t,lhite candidate(s)

proportion of votes cast by Republican and Independent

voters which go to the Black cand'idate(s)

P(n*l)vs

l2If our bivariate regressions do not yield a good enough fit, 'it
may be usefu'l to try to replace the assumption of the model in Section II
that the proport'ion of Black/White voters who are registered Democrats or
Repubf ican'flI@dents is constant over all voting precincts wjth a

more realist'ic model which permits precinct by precinct variation in the
proportions of voters of a g'iven race who are Republicans, Independents
or Democrats. The multivariate model 'in numbered Section (2) of Section
VI perm'its us to cons'ider precinct by prec'inct variations in the
proportions of voters of a given race who are Repub'lican/Independent or
Democrats, but does not permit us to look at all four possible
combinations simultaneously in the same estimating equation. This is a

rather technical point and we have relegated a full d'iscussion of it to
Appendix B.



30

BVP = proportion of voters who are Black

I,{VP = proport'ion of voters who are Wh'ite

I^IDVP = proportion of voters who are registered as White Democrats

WRM = proportion of voters who are registered hlhite Repubficans

or Independents

BDVP = proportion of voters who are registered as Black Democrats

BRIVP = proportion of voters who are registered as Black

Repub'licans or independents

DVP = proportion of voters who are registered as Independents

or Democrats

RIVP = proportion of ..'oters who are registered as Republ'icans or

I ndependent s

t,le have

Pbvw = SFPirr)r^r *HF Piao)w

Pdvs = Hi Pi*olB * Hi P(so)s

t^le know that

wDVP=ffi

BDVP=l-WDVP

But

WD = xP*,

BD=(1-x)p*



3l

while,

DVP=ffi
t.Je can obtain POV* and PfjVg by regressions of P!{ and PB with

P1n*t) in the manner shown in Sect'ion II. If we can simplify by letting

Dr = Dr - Dr
'(eo)t.l - 'B(R+I)l.l - rBhJ '

which is a value known from earlier regress'ions (see Section II), then

we have

(DVP)Pdv!{ - (l - WDVP)Pdw
P 

f wo)w =

and

Dl = I - Dl'(I.ID)B ' '(hrD)hr

In like manner we can calculate the proportions of White Democrats

who vote for a particular cand'idate, C, from the proportions of

Democrats and Blacks who vote for that candidate.



VI I.

32

Coping with the Unavailabil ity of Registration (or Population)

Data by Race

As noted in Sect'ion IV above, when registration (or population) data

by voting precinct by race'is unavailable, it is possible to check for

racial bloc voting by regressing the aggregate vote share for cand'idates

of one race on the vote share of candidates of the opposit. ru...l3

l.Jh'ile a high magnitude (and negative sign) of the correlatjon coefficient

between aggregate Black candidate vote share and aggregate Wh'ite

candidate vote share is often taken to be sufficient evidence of racial

polarization, this js jn error. For single member d'istricts, a h'igh

negative correlation between Black candidate vote share and t,Jhite

candidate vote share is a necessary, but not a sufficjent, condit'ion for

the presence of racjal polarjzation. It is possible (although

empirically unlikely) that we could obtain a high negative correlation

between aggregate Black candidate vote share and aggregate t,Jhite

cand'idate vote share even if (in terms of PWW and Pgg or in terms

of Yule's Q or the Rice Index) only very fimited racial polarization

existed. This could occur either if the Black candidate(s) and the White

candidate(s) were consistently appealing to different constituencies that

were differentiated along lines not (to any significant extent) related

to race, or if there was a consistent pattern of Black candidates

appeaf ing more strongly to Black voters and White candidates appealing

more strongly to I,Jhite voters that did not involve .ery large differences

I 3such a method shoul d be
methods are ruled out because

used, of course, only if other better
of data l'imitatioffi



33

in the proportions of the votes from voters of each race that went to

candidates of that same race, e.9., if hJhites consistently divided the'ir

votes 5l percent - 49 percent (Llhite candidates vs. Black candidates)

wh'ile Blacks in the various precincts consistently divided their votes 49

percent - 5'l percent. For l'inear relationships, the correlat'ion

coefficient provides a measure of cons'istency of relationship, not

magnitude of effect.l4

If we do obtain a high negative correlation between the aggregate

vote shares of Black and White candidates, we would like to rule out the

possibif ity that this high correlatjon could be caused by factors

extraneous to race and to assure ourselves that race is indeed a factor

of major importance. As suggested in Section III, we can achieve these

ends by lookjng at voting precincts which are overwhelm'ingly composed of

voters of one race. In such lopsided Black (White) prec'incts Black

(l,lhite) candidates should be rece'iving very high (low) vote shares if
indeed voting is po'larized along rac'ial l'ines. Normally, even if we do

not have hard data on rac'ial proportions in each voting precinct, we

st'il I wj I I have suff ic'ient "common-sense" knowledge of electoral

demography to be able to isol'ate enough almost all White and almost all

l4Nonetheless for single member districts if racial polarization
aggregate is high, then we must obtain a high negative correlation
betwebn the vote shares of ETack and White candjdates (Puf and
Ps). However, for mult'imember districts without a place system the
c6rrelation between Py and Pg need not even be negat'ive, even if
there js consjderable rac'ial bloc voting.

0f course the correlation between Pi and Pd must be -1, since
these variables are complements.



34

Black precincts to make th'is test as a double check on the reliability of

a judgment of the existence of a high degree of racial bloc voting

determjned on the basjs of regression(s) of candidate vote shares.



35

VIII. Modificat'ions to the Analysis for the Case of Mult'imember

Districts or At-Large Elections I'lithou! a Designated Place System

It is straightforward to mod'ify our analysis to deal with mmd

situations where each voter may cast from 0 to k votes. Let

nW = mean number of cand'idates voted for by White

registered voters

nB = mean number of cand'idates voted for by Black

, rBgistered voters
total votes castn = mean number of candidates voted for = *rffi

It is apparent that

i-=n*x+ng(l-x)

Hence, i' is for mmds analogous to voter turnout, T, for smds, and

if we regress n on. x we obtain

nB=b6

n,u=16*b6

The key equations for the mmd case are ident'ities of the form

Io* = n,^,(x)Pww + nr('l-x)Pr*

mean # of votes x . mean # of votes (l-x) . mean # of

for Hhite cast by Wh'ites for votes cast by Blacks

candidates White candidates for White cand'idates.



36

Rewriti ng, we obta'in

@-. ls p-:Pl,J = - n F 
PBwi

i.e., if we regress Py on x, we obtain

PB,.,= fo b7

P,,,,,= h (mr+br)

0f course, if n, = n*, then PBbl = b7 and P,,,. = *7 * b7.

In like manner, by using the identity

PB = n14(x)P1n1g * nB('l-x)Pgg,

by regressing P3 on x t{e can 6tai n

PBB= fo b8'

Pr^lB= h (mr+br)

Thus

Dr = '** = 
*' * o'

' Hl'l PWbt*PWg ,g * bg + m7 + b7

and

pils= ,rh =#E



37

The coeff icientr P'Wl^l and P'BB must be interpreted with great

care in mmd races without a numbered place system. If, for example,

there are 5 White and I Black cand'idates in a race in which voters may

cast up to four votes, 'if hJhite voters were "color-blind" in the way they

cast their ballots (j.e. voted w'ith equal (2/3) probability for each of

the six candidates), the Black candjdates would receive exactly 1/6 of

the ballots of White voters. Thus we would find a P',,,,, of .83

(5/6ths), wh'ich would on the surface appear to be evidence of highly

polarized voting. For mmd races without a numbered place system,

comparing the obtained PWW and Pig values with what would have been

obtained had all candidates been voted for with equiprobability may help

us to judge the actual degree of rac'ial polarization, but even the

assumption that all cand'idates of the same race are voted for with equal

probab'i'l'ity is suspect; and thus comparisons with the equiprobability

model can be quite misleading. For mmd races without a place system,

it'is preferable to focus on racial bloc voting as measured by the

relative proportjon of l,Jhites/Blacks who vote for any given Black/White

candidate.l5 For mmd elections without a place system this will

prov'ide us a better *.usrre of racial bloc votjng than PWW or Pig,

or the Q or R values derived from them.

For mmd elections without a place system we shall develop a general

technique for estjmating the proportion of White/Black voters who vote/do

l5Wfrile it is possible to apply standard stat'istjcal tests to
determine the'likelihood that the observed P,',,, and P:^ values could
have been obtained by chance from a racially HHutrat paffern of voting,
such a test will only determine the ex'istence of racial polarization and
not jts magnitude.



38

not vote for any given candidate. lle can then use th'is technique to

separately consider differences in the support bases of the Black and the

tlhite candidate(s). In particular for the case where there 'is only a

single Black candidate th'is method will yield us estimates of the

proportions of t.Jhites who vote for that candidate vs. the proportion of

Blacki who vote for the cand'idate, the d'ifference between whjch provides

a direct measure of racial bloc voting.



39

Estimat'ing the Proportion of Whites/Blacks

. |.lho Vote for a Given l'Jhite/Black Candidate

1,1e provide the analysis for the case of a Black candidate. The model

i s 'identical for a Wh'ite cand'idate.

As before, 1 et

C. = the jth iandidate, a Black candidate
J

T = proportion of registered voters who turn out to vote

TNV = number of registered voters who turn out to vote

TB = proport'ion of Black registered voters who turn out to vote

TW = proportion of l,Jh'ite registered voters who turn out to vote

As noted previously, if we regress T vs x, we obta'in

TB = intercept of regress'ion of T vs x

Simi lar1y,

Tll, = intercept of regression of T vs l-x

Let

BVP = proportion of voters who are Black

= (1-x) IA / [xT* + (l-x) TrJ

l^lVP = proportion of voters who are }lhite

= *Tw / [xTw + (1-x) TgJ

PilVC, = proportion of Hh'ite voters who vote for the Black candidate, Cj

PfiVC, = proport'ion of Black voters who vote for the Black candidate, Cj

CVS, = proport'ion of the voters who vote for the Black cand'idate, Cj



40

# of votes for the Black candidate, C,

TNV

For any candidate, Cj

"" 
= ;':'; -l"..'lo':,'lJl'
- 1P1lvc3-P ivc' )'uvP - Pivc'

Hence, 'if we regress CVS, vs. liVP, then the

PfiVCr. Sjmilarly,'if we regress CVS. vs. BVP,

g i ves ut PdVC ..
J

Thus, to find P[ur. (PilVC.), we first regress
JJ

to obta'in T, 1f, ) ; then substi tute those v al ues 'i n

formulae to obtain BVP (I,JVP), and use these values

regress'ion of CVS, vs. BVP (CVS, vs. WVe;.16

intercept gives us

then the i ntercept

T vs.x (Tvs. l-x)

the specjfied

of BVP (h/VP ) 'i n the

Estimati ng the Proporti on of tlh i telBl ack

Democrats/Republ'icans who Vote for a Given l.lhite/Black

Candidate in a Democratic/Republican Primary

For a primary, the analyses of turnout and the racial pattern of the

votes for a given cand'idate is s'imilar to the analyses for a general

l6Alternatively we can define candidate j's share as a proportion
of the registered voters, and then, by knowing turnout, we can specify
the proportion of the reg'istered voters who didn't vote for cand'idate i.
By regressing these proport'ions on x we caiTdT-cul ate PiVC. and

PfiVC. from the'ir unprimed equ'ivalents, analogously to how weJcalculated

Pfr* Jand Pfig from their unprimed equivalents in Section II.



4l

election we have given above. 0f course, we must substitute Black/hlhjte

Democrat/Republican registered voters for Black/tlhite registered voters.

This requ'ires us to define registered voters in terms of only those

voters eligible to vote in the given'primary (see Section V for details).



42

IX. Estimating the Proportion of Ballot Pattern of Each Possible Type

Cast by Bl ack/l,lh'ite Democrat/Non Democrat Voters.

The proportion of White (Black) voters who vote for no candidates of

the oppos'ite race is the parameter for mmds which most closely

corresponds to PWW and Pdg for smd elections. However, since

there may be more than one candidate of race opposite to the voter's own

contesting an election, we would also like to know to what extent voters

of a given race are unwilling to vote for more than j candidates of the

opposite race; since when k js large and there are several candidates

of each race in the contest, "tokenism" may be present, i.e., voters may

be wilfing to vote for one (or perhaps even two) candidates of the

oppos'ite race but not for more. To estimate the magn'itude of racial b'loc

voting in an mmd race without a numbered place system, we propose

estimating the proport'ion of the t,Jh'ite (Black) electorate who vote for

exactly j (0 < j S k) tJhite/Black candidates, i.e., we w'ish to

ascertain what proportion of the ballots of the voters of each race

include votes cast for exactly i candidates of the opposite race.

However, because we don't have indiv'idual ballots, but only aggregate

total s, i n an mmd race w'i thout a p1 ace system ascertai ni ng ba1 1 ot

patterns is very difficult, and usually impossible.

For example, 'if every voter has two votes and we observe ballot

proportions for candidates one through three of (1/2,1/2,1/2), we can't

readily establish the relative proportions of voters who voted for C.,

and Cr, C1 and Cr, C, and Cr, C., on1y, C, on1y, and C,



43

only, although we do know that, on average, voters voted for only 1'1/?

cand'idates. NonetheleSS, regreSsion and simultaneous equations

techniques may help us estimate Such unknown ballot proport'ions'

1^1e begin w'ith a model for reconstructing ballot patterns for exactly

two of the K candidates in a given mmd race. [lle then extend th'is to

reconstruct (for exactly 2 candidates) Uatlot patterns first by race and

then by race and by party. Then we show how to extend the analysis to

the K candidate case. The method we give can also be used to

reconstruct voter ballot pattern proportions for sets of

single-member-d'istrict elections taking pl ace s'imultaneously within a

given jurisdiction (e.g., the votes for President, congressman' and

governor within a congressional district).

Let

T = proportion of registered voters who vote

BRT = # of registered voters who are Black

lrJRT = # of registered voters who are White = I - BRT

RT = # of reg'istered voters

Vt = # Votes for candidate I

YZ = # Votes for candidate 2

etc.

WVP = proportion of voters who are Wh'ite

BVP = proportion of voters who are B'lack = I - blVP

Nl,lD = # of White Democrats

NW(R+I ) = # of l,lhite Non-Democrats (Republ ican and Independent )

TA = proportion of Black registered voters who vote

TW = proportion of t^lhite registered voters who vote



44

TNV = total number of voters

.l = proportion of voters who vote for nejther candidate I or

candidate 2

. rZ = proportion of voters who vote for candidate I but not

candidate 2

13 = proportion of voters who vote for candidate 2 but not

. cand'idate I

,4 = proportion of voters who vote for both candidate I and

= candidate 2

w'ith other variables as before. Then, we have the fol low'ing two

identities

Vt = (13 + r+ )tt'tV

YZ= (.2 *.q)Truv

Moreover,'if we can assume that virtually a1l voters vote for either

candidate I or candidate 2 or both, then

11 =l-T

F'inal ly,

11*r2*r3+14=l
V. V^

Let us denote # as ut and # as at.

Specifying values for the four equations 'immediately above, and then

solv'ing for the undesired unknowns we obtain



45

rl - I - T

rr=T-a.'
rr=T-a,
14=dl*uZ T

In like manner, if we use si to represent ballot patterns of

lrJh'ite voters and tj to represent ballot patterns of Black voters, we

obtai n

vt = (s3 + sq)wvp 'TNV + (ts + t4)(l - t,JvP) '(TNv)

V2 = (s, + sO)[,lVP . TNV + (tZ + tO)(t - t^lVP) . (TNV)

st - I - TW

tt:l-TB
tl*s2+s3+54=l
tl+tZ t3+t4=l

Using ,9 and b9 to represent the slope and intercept obtained by

regressing Vt on WVP . TNV and ml, and btO to represent the slope

and intercept obtajned by regressing UZ on t^lVP 'TNV, we obtain as a

sol ut'ion

,l-l-TW yl=l-TB

,2=TW-b9-*9 2=TB-b9
*3=T1^1-btO-mlg Y3=TB-btO

X4=ffig+b9 '.l0*blo*Tw J4=bg*blo-TB'
We can extend this analysis to'include a breakdown of vote patterns

by race and by party.

Let

*l = Proportion of lihite Democrats who don't vote for

either candidate I or candidate 2



46

x4 = Proportion of White Democrats who vote for candidate 2
I

but not candidate I

etc.

J1 = proport'ion of hlh ite Republ 'i cans/Unaff i I i ated who don 't
vote for either Candidate I or Candidate 2

etc.

If we represent ballot patterns for candidates I and 2 in vector

form with a I 'indicating a vote and a 0 ind'icatjng the absence of a

vote, we can represent the relevant set of regres'ion equations in matrix

form as below:

(oo) (ot ) (to) (il )

N[{D ^l ^Z *3 *4

Nhl(R+i) yr y? y3 y4

NBD ,) ,Z ,3 ,4

NB(R+I) qt qZ q aq

# of Voters

Thus,

Vt = [rltlDx, + NI.l(R+i)yg * NBDz, + NB(R+I)qg * NWDxO +

NI,I(R+I)yq * NBDzO + tlg(R+I)eq

(*3**4)ltWO + (V3+V4) Nt^l(R+I) + (*3*24)NBD +

( o3*oa) NB(R+l ) .

YZ = [rlWDx, + NW(R+I)yZ * NBDz, + NB(q+l)QZ + NWD14 +

Nl,{(R+I )yq * NBDz4 + NB(R+I)qq



47

= (xZ+xiNWD + (t2*vl NI^I(R+I) + (zr+zlNBD +

( or+04) NB(R+I )

If, virtually all eligible voters who vote in the election vote for either

Vt or YZ or both, then we have

xl=l-TWO

Y1=l-Tw(n*t)

'1=l-Tgo
qr =l -Ta1n*t;

where TWO etc. are estimated by one of the methods described in

Section III.

0f course,

,l*^Z x3+x4=l
Yl *Yz* Y3+v4='l
,1*r?*zr+zo=1
Ql*Q2*q3*Q4=l

Treating the first two of the above equations as regression equat'ions

and the latter eight as'ident'ities, we can solve for Xi,yi,Zi,

and qi ('i=l ,4) . Representing the f irst two equations as

,{ t ) =

,l') =

a.,.,NWD('i) + urilw(R+I)(') + at3NBD(') + atotla(n+t)(i)

ar.,ttwo(i) + arrNW(R*t)(i) *.zrNgo(i) + aroNB(n+11(i)



48

and f or notat'ional convenience, letting xl = d31r Y1 = u32, ,1 = u33,

e1 = u34, we can solve for the proport'ion of Black/l,Jh'ite Democrats/

Repubficans who cast ballot patterns of each type, obtaining

*l = u3l yl = u32 ,1 = u33 ql = u34

^z 
= 1-al l-a31 J z = 1'a1?-a32 'z = 1-ul 3-a33 Qz = l-ul4-u34

,3 = l-u2l-u3l y3 = l'u?Z. u3.- ,3 = l-uz3-u33 Q3 = 1-aro-aro

*4 = 011+Yr.,+ar.,-l Y4 = ul ,+arr+arr-1 ,4 = u1r+arr+arr-1 94 = ul4*u24*u34-l

The solut'ions specified above provide us the ballot pattern

proportions we are seek'ing. If we are dealing with a primary ather than

a general election we use registered Democrats/Republicans rather than

registered voters.

If we wish to deal with more than two candidates we simply repeat the

process. For examp'le for three candidates we solve for each of the

proportions Vl n V2, Vl 
^ 

V2, Vl 
^ 

V2, and V.t n v-r, among Black/tlhite

Democrats/NonDemocrats, and then use these proportions to determine

vtAV2n v3' vtn v2nv3

vtnT2nv3'vtn vz/\v3' vtn v2n v3

Tt^v2nV3 ' Vt^[2AV3' vt^v2n v3'

etc. ,

by solving

Vt n V2 = xONWD * J4N(R+I) + z4NBD + qO(R+I)

V3 = .INI.ID + r2N(R+I) rrNBD + r4NB(R+i)

and analogous equat'ion sets, for all the possible triples, where the

values of xi,yi,Zi and a1 (i = 
.l,4) are known from the earlier



49

regress'ion equations.. in other words, we solve a set of equations w'ith

V.1 A V2, Vt n Tr, Vt A V2, and Tt 
^ 

[r, resPect'ive]y, substituting for

Vl in our earlier equation set.

It is straightforward to generalize this technique to deal w'ith a k

candidate race (k > 3), or with k (k > 3) single member district

races'in the same juridiction which we w'ish to simultaneously analyses.



50

proportion proportion
of votes of votes
to White to Black
candidate(s) candidate(s)

(normal i zed ) proport'i on
of }lhi te regi stered
voters who voted for
( a) wfri telBl ack
candidate(s)
( normal i zed ) proportion
of Black registered
voters who voted for
(a) t'lhite/Black
candidate(s)

votes from t'lh'ite voters
total vote

votes from Black voters
total vote

Pil Pi

Tabie I

Parameters of Racial Bloc Voting in a

Single Member District Elect'ion Represented

in a ZxZ Matrix Form

Pww ,\
q-:;P--tx /

Pwg ,\
4=7rrxt

,%(r-x) ft(r-x)



5l

proport j on of tJhi te
voters who voted for
(a) Hn'ite/Black
candidate(s)
proport'ion of Black
voters who voted for
(a) t,Jh'ite/Black
candidate(s)

proportion of proPortion of
votes to l,lh'ite votes to Black
candidate(s) candidate(s)

Ptiw Pils

Priw Pris

Tabl e 2

Alternative Representat'ion of Parameters of

Raci al Bloc Voting i n a Sing'le Member District Elect'ion

in a ZxZ Matrix Form



52

Variable L'ist

BD = the proport'ion of registered voters who are Black

Democrats

BDVP = proportion of voters who are registered as Black Democrats

BOVp = proportjon of voters in a Democrat'ic primary who are

BD.PDBurdC:r=p
'BP ' DB 'tlD ' Dtl

BI

BR

= the proport'ion of registered voters who are Black

Independents or Black Unaffiliated.

= the proportion of reg'istered voters who are Black

Republ icans

= # of registered voters who are Black

= proportion of Black voters who are in a Republican Primary

Repubf icans or Black Independents

= proportion of voters who are Black

= cand i date j

= proportion of the voters who vote for cand'idate, C.
J

= proportion of the registered voters who vote for

candidate C=
J

= turnout of Democrats (# of Democratic voters)

= proport'ion of voters who are registered as Independents

or Democrats

BRT

BRvp

BVP

C.
J

CVP.
J

CVS.
J

DT

DVP

PnsTen

TgRf Rg 
* TwR'PRt.l

BRIVP = proportion of voters who are registered as Black



nB

53

= mean number of candidates voted for by Black registered

v oter s

nD = mean number of candidates voted for by voters who are

registered Democrats

nR+I = mean number of cand'idates voted for by voters who are

registered Republ icans or Independents

n,,, = mean number of candidates voted for by Wh'ite registered

voters

n = mean number of cand'idates voted for = i9!9]=Y9t9t tStt
regi stered voters

NBD = # of Black Democrats who are registered voters

NB(R+I) = # of Black Republicans plus Independents who are

registered voters

Nl.lD = # of White Democrat who are registered voters

Nt^l(R+I ) = # of White Repubf icans p'lus Independents who are

regi stered voters

Pg = the proportion of total registered voters who vote for

the Black candidate

Pgg = the p.roportion of Black registered voters who vote for

the Black candidate

PgO = proportion of Black registered voters who are Democrats

Pgn = proportion of Black registered voters who are Republicans

Pg(R*l) = proportion of B'lack reg'istered voters who are either

Republ icans or Independents

PgW = the proportion of Black registered voters who vote for

the t,{hite candidate



54

PD

D,DB

o
'Dt^l

PI

PR

o,RB

P1n+r1

P1n+r)a

PRw

Pw

ttrt

D
'I^lD

P(wo)s

P(wo)w

PwR

Pwln*t 
1

proportion of

proportion of

proportion of

Unaffiliated.

proportion of

= proportion of registered voters who are Democrats

regi stered Democrats who

registered Democrats who

reg'i stered voter s wh o are

Prr(l-x)
PBD(l-x)+Prr(x)are Black =

are tlhite = l-POg

Independents or

regi stered

= proport'ion of reg'i stered

= proport i on of reg'i stered

I ndependent

voters who are Republjcans

Republicans who are Black =

voters who are Republicans

nr,(1-x)
PBR(l-x)+PwR(x)

or

= proport'ion of registered Republicans and Independents who a

re Black

proportion of registered Republ icans who are !,Jhite = l-PRB.

the proportion of total registered voters who vote for

the l,lh'ite c and i date

the proportion of White registered voters who vote for

the Bl ack cand'idate

proportion of h/hite registered voters who are Democrats

proportion of !,lhite Democrats votes which go to Black

candi dates I as a rati o of reg i stered l^lhi te Democrats]

proportion of l,'lhite Democrats votes wh jch go to white

candidate(s) [as a ratio of registered white Democrats]

proportion of I'lhite registered voters who are Republ icans

proportion of l,lh'ite registered voters who are either

Repubf icans or Independents



55

Pwln*113 = proportion of white Republjcans and Independents

(comb'ined) votes wh'ich go to Black candidate(s) [as a

ratio of registered l,lhite Republicans and Independents]

P = proportion of l,lht'ie Republ icans and Independents'hJ(R+I )51 Pr vt,vr

(combined) votes whjch go to White candidate(s) [as a

ratio of registered Wh'ite Republicans and Independents]

PWW = the proportion of hlhite reg'istered voters who vote for

the tlhite candidate

Dr = 
O'

' B PB*PW

Dr = 
ou,

' BB Pgw*PgB

PfiOVC. = proportion of Black Democratic voters who vote for the
J

Black/White cand'idate, C,

pl-.^ = proportion of Black Repubfican voters w'o vote for the' BRVCJ

Bl ack/lJh'ite candi date, ,j
pl.. = proportion of Black voters who vote for the Black/tJhjte,BVCJ

canci'idate Cj

D

Dr = 'BW
' BtlI Pgw*PgB

PliVg 
' 

= proportion of votes cast by Democratic voters which go to

the Elack candidate(s)

PilVC. = proport'ion of Democratic voters who vote for the Black/

' *h'ite candidate, cj

PfjVW = proportion of votes cast by Democratic voters which go

to the White candidate(s)



P'tl

P 'l^JB

P 'tjtJ

Pilovc,

Pilvc,

56

Pin*l)VW = proportion of votes cast by Republican and Independent

voters. wh'ich go to the Wh i te c and i date ( s )

Pin+t1Va = proportion of votes cast by Repubf ican and Independent

o1
'RVCj

voters which go to the Black candidate(s)

= proportion of Repub'l'ican voters who vote for the Black/

Wh'ite candidate, C.l
J

D..l^/

= qlrB-
o, 

hiB
= -d--o-

' l.lI^J " hlB

D
'l,JhJ=-

Pww*Pt^t 
B

= proport'ion of tlhite Democratic voters who vote for the

Black/t',lhite candidate, C,

Pfif nVC. = proportion of tJhi te Republ ican voters who vote f or the
J

Black/White cand'idate, rj
= proport'ion of White voters who vote for the Black/White

candidate, C,

a

R

PwwPg g -PwgPgw

= Rice Index of Rac'ial Polarization = PBB-PWB

= D -D't.ll'/ ' Btl

RDV = # of registered Democrat voters

RIVP = proportion of voters who are registered as Republicans or

I ndependent s

= Yule's a Index of Racial Polarization = D.=:'tl!{%gffill



57

RRV = # of registered Repubf ican voters

RT = total number of registered voters

T = turnout as a proport'ion of registered voters

TB = PBB*PB,r,'= the proportion of registered Black voters who vote

TgO = proportion of Black Democrats who turn out to vote

r = proport'ion of Black Republicans and Independents who turn'B(R+I )

out to vote

TO = turnout proportion of registered Democrats

TNV = total number of voters

T0T = total number of voters

TR = turnout proportion of registered Republjcans

T1n*l) = turnout proportion of registered Republican and

Independents combined

T* = P**+P*t = the proport'ion of registered hlhite voters who vote

TWO = proportion of White Democrats who turn out to vote

TWR = proportion of t/hite Republicans who turn out to vote

TWlR*t ) = proport'ion of [,Jhite Republ icans and Independents who turn

out to vote

Vj = # of votes f or cand'idate j

[,lD = the proportion of registered voters who are both White

and Democrats, i.e., the proport'ion of registered voters

who are tJhite Democrats

WDVP = proportion of voters who are registered as tJhite Democrats

WOVp = proport'ion of voters in a Democratjc primary who are l,lhite

= l -BDVP



58

hll = the proportion of reg'istered voters who are White

Independents or !'lh'ite Unaff i I i ated

l,lR = the proportion of registered voters who are White

Republ icans

l.lRM = the proportion of voters who are registered as White

Republicans or Independents

t.lRT = # of registered voters who are White

WnVp = the proportion of voters in a Republican primary who are White

= 1-hvp

[.lVP = proportion of voters who are White

x = the proportion of total reg'istration which is }Jhite

l-x = the proportion of total registration which 'is Black



59

R EF EREN C ES

Benson, 0. Political Science Laboratory. Columbus, 0hio: Charles,

Merri11, .l969

Blacksher, James ['1. and Larry F. Menefee. "From Reynolds v. Sims to City

of Mobile v. Bol1pn." Hastings Law Journal, Vo1.34, No. I

(September .l982), pp. l-64.

Duncan, 0. and B. Davis. "An Alternat'ive to Ecological Correlation."

American Sociological Review, Vo1. l8 (.l953).

Goodman, Leo A. "Ecological Regression and the Behav'ior of

Ind'ividuals." American Soc'iological Review, Vol . .l8, No. 6 (December

.l953), 
FF. 663-664.

Goodman, Leo A. "Some Alternatives to Ecological Correlation." American

Journal of Sociology, Vol.64, No.6 (May .l959), pp.6l0-625.

Iversen, G. R. "Recovering Individual Differences in the Presence of

Group and Individuaj Effects. American Journal of Sociology, Vol. 7g

(1973), pp. 420-434.

Jones, E. "Ecological Inference and Electoral Analysis," Journal of

Interdi scipl i nary H'istory, Vol . 2 (1972) , pp . 249'269 .



60

Langbeifl, L. and A. Lichtman. Ecological Inference. Beverly Hi11s:

Sage Publ icati ons, 1 978.

Loewen, James. Social Science in the Courtroom. Lexington, Mass.:

Lexington Books,.l982.

Ranney, Austin. "The Utility and Limitations of Aggregate Data in the

Study of Electoral Behav'ior. " 'in A. Ranney (Ed.), Essays on the

Behavior Study of Pol'itics. Urbana: Univers'ity of Illinois Press,
.l962, pp. 9l-.l02.

Robjnson, W.S. "Ecological Correlations and the Behavior of

Ind'ividuals, " American Sociological Review, Vol. 15, No. 3 (June

1950), pF. 35.|-357.

Sh'ive1y, W. P. "Ecological Inference: The Use of Aggregate Data to

Study Ind'iv'iduals, " American Pol itical Science Review, Vol . 63

( 
.l969), pp. I 

.I83-l ,l96.

Shively, l,/.P. "Utilizing Exterior Evidence in Cross-Level Inference,"

Political Methodology, Vol. I (.l974) pp. 61-74.

Stokes, D.E. "Cross-Level Inference as a Game Against Nature," In J. L.

Bernd (Ed, ), Mathematical Appl'icat'ions in Po'lit jcal Science, Vol. 4

Charlottesv'ille: Univer^s'ity of V'irginia, .l969, pp. 62-83.



6l

Appendix A: Other Regression Approaches

It is often the case that'individuals seeking to do racial bloc

voting analysis w'il1 regress Pil (i.e., the proportion of the vote wh'ich

goes to the tJh'ite candi date) , on x rather than PW on x. If Pil (Pd )

is used rather than PW (Pg), great care must be taken interpreting the

resulting coefficients 'lest they be wrongly understood. Indeed, unless

l,lhite turnout and Bl ack turnout are 'identical (wh'ich they almost never

are), it is best not to use this form of regression. If, however, it has

been used, we show below how to properly interpret the result of

regressi ng Pil on x.

By definition

Dl =.hl
(Pww*Pws ) (x )Pilw * ( PBs*Psx) ( l-x) Pfl,

PI^J*PB

Pt^l*PB

(Pww*Pws) (x)Pils * (PBs*Psp) (l-x)P[,

P

(P,u!{+P,,,B) (- )Tffi * (PBs*PBy) (l -x )

_ xPhtw + (l-x)PBl{

Phl * PB

and

'i.e.

Pd =

Pgw

Pgg*PBw
Dl =.h, Pt,l*PB



62

But then,

O+DO
D. = _.!!''Blil,-.',, 'BhlPli = T; + P; txJ+ PP, 

;

hence

Pgw = bi (Pil+PB)

P,,,,,, = (mi+ui)(Pr.r*Ps)

Similarly, if we regress Pil on x, we obtain

Pu.B = (mi+ui )(P!J+PB)

PBB = (bi )(P*+PB)

To obtain P*W and P'BB, we let

p,i*=#k %U.',.i,.:i)'
D, = 'r, - 

(bl) (P!{+PB) 
= 

bliT

'BB PBg*PBW PBII,*PBB TB

If l.lhite turnout and Black turnout are identical, then of course

PW,.*PWB = PB*P,, and

Priw =ri *bi
Pris -- b'Z

If, however, Black turnout is less than bJhite turnout, i.e.,

P,,,,,*PWB , PBB*PBl,li oS w€ might expect to be true, then



63

P',,,, * PI.,B t PB * PW , PB,. * PBB ;

and, therefore,

Pnr.r "i * bi

Pis ,rl *bl

Hence, using the coeffic'ients in the equations based on regressing

Pi(PL) on x (1-x) willproducemisleadjngestimatesof Pif.r and

Pilg if we fail to correct for d'ifferential turnout by race and simply

take PfiW to be estimated by ni*bi and Pfig to be estimated by b).

Another problem with regressing Pil(Pd) on x (l-x) is that we can

often obtain slope and intercepts outside the [0,.l] range. This can

readily occur if b'loc voting is high and turnout'is low. For example,

'if PBB , PW+PB, we have ni * bi. > l.



64

Append'ix B: Allow'ing for D'ifferential Voter C,hoice

by Race and by Party with Republicans' Democrats,

and Independents Separately Treated

t.Je propose a general model of the form

Pt^t = [PwoP(wo)w * PwnP(rln1xJ x

* [PsoP(BD)t.t + tsnP(BR)vJ (l-x)

+ x(t - Pwo - PwR)P1wr)j1 * (t - x)(t - Pso - psn)P1er)w

Our regressi on techni que f or est'imati ng P,,D, P,.,R, PBD, etc. gi ven 'in

the text above yields

P.

o = D - l-x p
't.lD x x BD

o

D ='R _ l-x p'l.JR x x 'BR

D='l-D-P. t.ll ' l^lD 'I.JR

PBI=l-PgO-Ogn

Al so

Pr.rr = I + + Psn -3. + rr, -P

Hence, we may rewrite our equation for PW in terms of only five

parameters, PBD, PBR, PD, PR and x:



65

Pw = [e, - (l-x) PBD] P(1n1D)r,J * [PR - (l-x) Psn] P(wn)ur

+ (1-x) [PgoP(BD)r^t * PBRP(a*)*]

+ [x - Po * (l-x) PBD - PR * (l-x) PBR] P(1ar)1nt + (l-x)(l - Pso - PsR) Plar)w

trr(') ano tr*(') u.. assumed to be approximately constant for all i.
If we assume each is known from a regression run of Black registrat'ion by

Democratic registration, we may rewrite our equations above in terms of

four variables--each of which in turn is a function of the three bas'ic

variables x, PD, and PR.

er(i) - t-x('i) pso p(wo)w

* p*(i) - t-x(j) PsR P(wn)w

+ x(i)-er(i)+ r-*(i, rr,-e*(i)+ r-*(i, rr* p(wr)r

* 1t-r(i), [(t-pso - paR) t(rr)1n1 *'psop(BD)1^1 + psnp(an1wJ

Unfortunately, we have four subsidiary variables but only three

'independent variables x, PD, and P* and thuS our est'imates of the

parameters of our regress'ion must be unreliable. To cope with this, we

may simpt'ify by letting t(rr)* = P(aR;w = Pow. Then the last term

of the equation above simplifies to

{r-x(i), Paw



66

Transferring the last term to the left-hand side, we obtain an

equation with only three independent varjables on the right-hand sjde.

t(fr) - (t-*(i ))psw = r,yr * 
^1ztz 

* *r3y3 * b, ,

where the yi corresponds to the variables specified in the equation

direcly above, i.e., pD(i) - (t-x(i))pgo , pR(i) - (t-x(i))pgn ,

x('i) - er(i) + (r-x(i))peo - e*(i) + 1r-x(i))pgn . Then we have

t(*r)h/ = mt j

o(**)t^t = mt2

P(*r)r.,t = mr3

If the assumpt'ions used to generate these relat'ionships are reasonable,

then we should also obta'in

btt - o '

In ljke manner, we may readily establish equivalent equations for

PB. Hence, i t 'is stra'ightf orward to calcul ate the proport'ions of

White/Black Republ jcans/Democrats/lndependents who vote for a candidate

of their own race. By substituting P(wo)w and t(uo)g (or P(wl )w and

t(gr)gr or^ P(Wn)W and t(t*)r) for PWW and PgA in the formulae given

in Section IiI of this paper (see Tables I and 2), we may calculate racial

polarizat'ion separately for Democratic, Republican, and Independent voters.

Alternat'ive1y, we may calculate, as before' an overall value of P 
WW

(Pge, etc. ) by solving



67

Pwt,t = P(wo)wPt^tp * P(*R)wPwo * P(rt 
)wPttt

PBB = P(eo)BPBD * t(t*)BPBR * P(gt)sPst

etc.

Remember, however, that

Pl^lH * Pl,tB 5 l, PBB * PBt{ . l, and e* + e, : 1

as previously, since some registered voters may not vote. Thus, we would

normally w'ish to look at the normalized values PilW, P{t, etc.

The model given above allows precinct by precinct variation for

voters of one race, but not for voters of both races. We could al1ow for

precinct by precinct variat'ions in party membership proportions in the

model offered above for Black voters'instead of for tlh'ite voters, but we

do not have sufficient independent information to do so for both l'lhites

and Blacks simultaneously. in effect, as we show below, we have three

equations in six urknowns. Three of these unknowns can be estjmated at

the aggregate level and assumed to be constant in all voting prec'incts.

Let

|.JD = the proport'ion of reg'istered voters who are both White

and Democrats, i.e., the proportion of registered

voters who are White Democrats

l.JR = the proport'ion of reg'istered voters who are l'lhite

Republ jcans

t,JI = the proportion of registered voters who are tJhite

Independents or tihi te Unaff i l i ated



68

BD = the proport'ion of registered voters who are Black

Democrats

BR = the proportion of registered voters who are Black

Republ icans

BI = the proportion of registered voters who are Black

Independents or Black Unaffjliated

And as before, 'let

x = the proportion of registered voters who are l.Jh'ite

l-x = the proportion of registered voters who are Black

PO = the proportion of reg'istered voters who are

regi stered Democrats

PR = the proportion of registered voters who are

regi stered Republ icans

PI = the proportion of registered voters who are

regi stered as Independents or Unaff i l i ated

Cl ear 1y

WD=xP*,

WR=xP**

lrll=xP*r=x(1 -PWO-PWn)

39 = (1-x) PgO

3P = (1-x) PgR

BI = (1-x) Pr, = (1-x) (l - PgO - Pgn).



69

Since the party reg'istration of minority voters will generally be

lopsidedly Democratic in all vot'ing precincts in which minority voters

may be found, the party registrat'ion of m'inority voters wil I be relat'ive1y

insensitive to precinct by prec'inct variations. Let us estimate PgO

and Pgp from the regression technique outlined in Sect'ion V. 0nce we

obtain PBD and Pgq, we may obta'in BD (and BR) by mult'ip1y'ing

PgO (Pan) by l-x. We may then determine BI e'ither via the same

technique or by mak'ing use of the identity

BD + BR + BI = l-x.

Five other i dent'ities are usef ul :

l,lD+t.JR+[.lI=x

tlD+BD =PD

tlR+BR =PR

WI + BI = PI

PD*PR+PI=l

For each voting precinct, i, we may set up a system of three

equat'ions in three unknowns by using the first three of these last five

equations, treat'ing PgO and PgR (and Pgl) as known from

regression analyses and assumed to be constant across a'l 1 voting

prec i ncts :



70

t*r(') *(i) * tr*(i) x(i) * p,,,r('i) x(i) = r(i)

,*r(') *{i) * pso(t-*(i); = pr(i)

. r**(') x(i) * pBR(t-x(t)) = e*(i).

Solving this equation set we obtain

t*Jt) =

l,le may now set

x(i)- e*(i)- tr(i)+ (PBR+ PBD)1r-*,(i))

e*(i) = *(i) ,ro(i) e1yo1w

* ,.( 
j ) p11R(i ) p(wr 

)w

* *(i)ptlr(j) P(wr)w

+ 1't-x(i))(psoP(so)w * PgnP(sR)w * PsrP(ar1y)

Substituting in the equation set above the values previously obtained

for ,*r(' ), ,**(') und ,*, ( t ), we have the same result obtai ned

earlier, to wit:

x(i)



71

p,^,(i ) = [pD(t,- rr*(r-*(')] r(hrD)h.t

+ [pR(, )- rr*(r-*(1))] p(wn)w

*;*(i,- t*,',- to,i)* (t-r(i))(psn* eur)1

+ (t-x( i ))tpsoP(ao1w* PsRPsR(w)* PgrPlai;w1

And, as before, we set t(rr)1n1 = P(BR)1J = P(BI)14 = PBtt and move

the last term over to the left-hand side to obtain three independent

variables on the right-hand side. As can be seen from the analysis

above, the model given above does permit the proport'ion of White voters

who are Repub'l'icans/Democrats/Independents to /ary across voting precjncts.

In each precinct the three proportions ,*r('), t**(') und t*r(t) u..

obtained via the solut'ion to a set of simultaneous equations. The values

so obtained are then entered into the regression equation used to estjmate

P(wo)w, P(wR)w' and P(wt)w'

The regression equation with e*(l) - (l-x)(i) PgW as the dependent

variable does not, however, provide us with sufficient informat'ion to

disaggregate the parametert P(BD)1.,, P(gR)W, and O(tt)*t and so we may

e'ither as before, make the strongly simpf ifying assumption that

o -o - D -p'(BD) (BR)t^l - '(Br)t,l - 'BhJ'

or determine those three parameters by running the appropriate regression

on e*(i)- ,(')tr* rather than on Pw - (t-*(i ))or*, o. (for simplic'ity)

we may estimate these parameters from jnformation derjved from voting

patterns in districts which are overwhelmingly minority in population.