The Danish Institute for Educational Research, Copenhagen

1. Introduction

This article suggests one general and three specific models for the description of changes in the distribution of votes by political party for the geographic subareas within a given area. Data from Danish parliamentary elections from 1960 to 1968 are chosen to show how the models work.

The main topics discussed are: What is uniform change? and What is differential change? Uniform change occurs when the support for a particular political party changes by the same amount in the same direction in all subareas, while differential change refers to deviation from uniform change in support for that party.

In addition to the description and testing of models for uniform and differential
change, this article suggests a macro-sociological theory of voting behavior for
interpretation of the statistical models.

Georg Rasch1 has proposed a theory of measurement especially applicable to
the social sciences, and his work has been the main inspiration for the models presented
here.

2. A Model for Uniform Change

A tendency to uniform change, that is, approximately the same amount of change in all geographical subareas, has frequently been observed for parties in democratic countries. This tendency was observed in many Danish political districts 1960—1968, in four successive elections to the Danish parliament which have been analyzed by Borre and Stehouwer.²

The present question is: How can we measure the amount of change in the support
for a political party in a geographical subarea?

A very common method is to take the proportion received by the party, of all
potential votes in that subarea, as a measure of the 'support' for that party. Using this

approach to construct a probability model, we take the proportion of votes for party h in subarea i in election t as an estimate of p£°, the probability that one voter chosen at random from the subarea has cast his vote for party h. That is, the stochastic equality

(1)

where a^ is the number of people voting for party h, and Nit is the number of
voters in the subarea in election t (the category of non-voters, i.e. those eligible
to vote who did not, is regarded as a separate 'party').

From this formula, there are several ways of formulating a model for uniform
change.

Borre and Stehouwer³ compute the difference in the party's proportion of votes in two successive elections and use this difference as a measure of the amount of change. In probabilistic terms, the model for uniform change (between elections 1 and 2) is then

(2)

that is, the same difference, 000,o00, is observed in all k subareas.

This model, however, fits badly with the actual Danish election statistics. Small parties, in particular, show systematic deviations from this model — for these parties the difference tends to be proportional to the proportion of votes in election 1. This suggests another model for uniform change:

(3)

that is, the same ratio, cOO,c00, is observed in all subareas.

This model has been used by Tage Bild4 to analyze the (very small) vote for the
Danish Communist Party; ii; can only be applied to very small political parties
because of one obvious weakness: formula (3) does not imply the logical condition

(4)

That is, the model does not imply that the sum of the proportions received by all parties in one subarea in an election must equal one. With such a model one can get the following results: Suppose the votes for a party in one subarea increase from 10 percent to 20 percent of the total vote. Suppose that in another subarea the same party begins with 50 percent of all votes cast. Then, according to formula (3), in the second subarea this party's support must double to 100 percent

of the vote in order to show uniform change, without regard to what happens
to competing parties.

Therefore it seems more reasonable to use a model which describes the support for one party relative to the support for other parties. This support for each party will be described by a parameter X^ which is not known a priori for any of the parties. The model suggested is:

(5)

That is, the probability of voting for party h is the parameter X^0 divided by the
sum, yit, of the parameters for all parties receiving votes in that election.

Furthermore, let us define uniform change as the condition that

(f\
W

That is, the same ratio, c®*, is assumed to appear in all subareas for a particular
party.

Formula (6) can also be expressed by

(7)

where Xs° is separated into two parameters, one independent of elections, the other
independent of subareas.

That this formula implies (6) can be seen by the fact that

(8)

That is, the ratio between two support parameters is the same for all subareas.
The two assumptions (5) and (7) can now be gathered into one formula:

(9)

This model is a version of Rasch's general model of measurement.5 The model
can be tested, and the size of the parameters £ and e can be estimated from the data.

If desired, the model can be applied to only n out of all m parties. The formula
then becomes

(10)

where p£° can be estimated by the proportion voting for party h in relation to the
proportion voting for one of the parties 1,..., n.

For generalization below of the model, a purely formal logarithmic transformation
of the parameters can be made:

(11)

Thus (10) can be written

(12)

Statistical methods for testing 1 his model and estimating its parameters are described
by the present author.6

The only test shown here will be a preliminary graphic test of model (12) on data from the elections of 1960, 1964, 1966, and 1968 for 41 subareas within the municipality of Copenhagen. The analysis will be limited to those eight parties which participated in all four elections: A Social Democrats, B Radicals, C Conservatives, D Liberals, E Justice Party, F Socialist People's Party, K Communists, U Independent Party.

Following the procedure described in Appendix 1, Figure 1 plots, for each of the 41 subareas, the relative support each party received in 1960 (measured by l^^c) — l^; see Appendix 1) against the mean of the party's relative support in all 4 elections (measured by l^^h>) — l..^(h-)). If model (12) for uniform change fits the data, then the points should deviate only at random from a straight line with slope 1 for each party. (For convenience the graphs in Figure 1 are referred to different origins.)

The data in Figure 1 show a remarkably close fit with the model, with two minor exceptions: parties E and U show greater deviations from their respective lines than the other parties. But one would expect this statistically because parties E and U receive very few votes. More interesting are the systematic deviations for parties B and U, where the points indicate better fit with straight lines having slopes greater than 1. (These deviations suggest a rule for generalizing this model to a model for differential change — see section 3 below).

6[h) measures the relative support for party hin subarea i independent of time
(that is, independent of elections), while a[h) measures the relative support for
party h at election t independent of space (that is, independent of the subareas).

Values of a[^h) are indicated in Figure 2 for each election. Note for example that the relative support for party A (Social Democrats) appears fairly constant, support for B (Radicals) increased throughout the period, and support for U (Independent Party) decreased throughout the period.

3. Models for Differential Change

Examination of 6[^h)-values (relative support in a particular subarea) reveals two groups of positively correlated parties: A, F, K (worker-supported parties) and B, C, D, U (non-worker-supported parties), each group correlating negatively with the other. This suggests the assumption:

(13)

That is, the Ø^-values of party h are proportional to a parameter, ois for the subarea
oj, which is independent of the party, with a coefficient, 9(h), specific to the party.
Formula (13) inserted into (12) gives

(14)

This model (the distribution analysis model of Rasch) can be tested on each election separately by maximum-likelihood estimation procedure.⁷ A method for preliminary graphic testing of the model is given by Christiansen and Stene⁸ and is applied to the Copenhagen data for 1960 in Figure 3. For each party, the residue from equal support in each area, measured by r£° (see Appendix 2), is plotted against the sign-_weighted mean residue from equal support (measured by r£^}).

If the model fits the data, the points for each party should only deviate at random
from a straight line with slope proportional to qf®. (For convenience the
graphs in Figure 3 are referred to different origins.)

The data fit the model quite well, but not as closely as in the previous analysis.
The best fit is observed for parties C, F, K and the worst fit for parties B, E.

For the Copenhagen data it was found that 0j as estimated separately for each election is approximately constant throughout the period 1960—1968.⁹ Also it is related approximately linearly to a measure for over-representation of manual workers in each subarea. Therefore (see equation (13)), 9^(h) is interpreted as 'the degree of manual-worker orientation of party h'. Figure 4 shows values of yf® as estimated separately for each election.

Minor changes in 9^(h) are observed over time, and this indicates that 9^(h) is time-dependent. As expected, the parties traditionally in favor of more equal income (A, F, K) have positive values of 9^(h), while the more conservative parties (U, C, D) have negative values of 9^(h). Party E (Justice Party) is in be ween, and it appears that B (Radicals) is approaching zero at the end of the period investigated.

When, as in this example, 9(h) is dependent on time, formula (14) must be generalized
into a model for differential change:

05)

This differential change is described by 9^, the orientation of party h at election
t on the dimension measured by oj. In this example, the dimension 0j has been identified
with manual-worker orientation.

The model can be further generalized into a model for differential change with
more than one dimension:

(16)

In this formula 0^ (BU, Øi2 „.., Øir) arid O?°: (cpj?, <få\ ..., <$>) are vectors of
dimension r, and

d?)

As in factor analysis, tests of this model are not very conclusive when the number
of dimensions is not known.

A test can be made, however, in the special case where only one of the elements
of O^ is dependent on time, that is, differential change occurs in only one dimension.
By convenient renaming of parameters, we can obtain:

(18)

that is, only the first term is dependent on time. When this expression is inserted
into equation (16), the model becomes

(19)

If this model for differential change is compared with the model for uniform
change, formula (12), it turns out that (19) differs from (12) only by adding the
one-dimensional term a^^ to the exponent.

This model fits the Copenhagen data quite well. (A simple preliminary test follows the procedure of Appendix 2, in computing the residues from the model in equation (12)). The model also fits the regional distribution of data from the same elections, previously published by Stehouwer and Borre in Scandinavian Political Studies.^w

For use with these data, the analysis includes eight political parties, plus the 'party' of non-voters. The close fit of the model with these data can be illustrated by the small differences between the actual and the 'computed' proportions (computed on the basis of estimated parameters) voting for each party in each subarea (region) in each election. The differences are in no instance greater than 2 percent, and are greater than 1 percent in only 17 out of 396 instances.¹¹

Table I shows that the differential change dimension oq is strongly positively correlated with degree of urbanization. Therefore (see formula (19)), we will call $® 'the degree of urban orientation of party h at election t'. Figure 5 shows values of p[^h) for each election.

Party B (Radicals) shows a marked increase in urban orientation, and there is
a general tendency to either increase or decrease throughout the period for most
parties. Party D (Liberals) is a clear exception to this rule.

4. A Macro-Sociological Theory for Voting Behavior

It has surprised me that most of the electoral data I have worked with so far can be described with such simple models. It is characteristic of the models that they describe the distribution of votes in geographic populations, with parameters either independent of time (independent of elections) or independent of space (independent of subareas within the total area).

This simplicity of description indicates that voting populations in these subareas are useful macro-sociological entities for description of behavioral distributions. Furthermore, it indicates that these distributions can be explained by subarea characteristics which do not change over time and by characteristics of the election which are the same for all subareas in the total area.

At this point we introduce three assumptions consistent with the models developed
above.

Assumption L The more the image of a political party satisfies the goals of the population
in one subarea, the greater the support for that party in that subarea.

The weakness of assumption 1 is that there is no external definition of what is
to be understood by 'image' and 'goal'. This will be a subject for further research.

Assumption 2. The image of one political party at a particular election is the same
in all subareas.

I expect this assumption to hold only in very cohesive political regions where each subarea receives approximately the same political stimuli. It would not hold, for example, in situations where local political events are important (e.g. local municipal elections).

Assumption 3. The goals of the population in one subarea do not change from one
election to the next.

That is, the population in the subarea will be considered a macro-sociological entity with a certain distribution of individual goals. I expect this distribution to be dependent on basic socioeconomic factors that only change slowly, so that it can be regarded as constant over short periods of time. This assumption will be unrealistic during situations of violent political crisis or great socioeconomic change.

Note that assumption 3 does not apply to the individual at the micro-level. It is
only the distribution of individual goals that is assumed constant, not the goals
of the individual person.

5. Interpretations of Parameters

All models mentioned in this article are special cases of the general model, formula
(16):

The parameters of this model will now be interpreted in light of the theory in
the above section.

Statistically, ØjO^ describes the differentiation of the support for each party in time and space, while of⁰ deiscribes the general support (independent of subareas) for each party at each election. According to the theory, oi<D[^h) is interpreted as the degree to which the image of party h at election t satisfies goals specific to the population of subarea i, while d^ is interpreted as the degree to which the image of party h at election t satisfies goals common to all subarea populations. 0_{ is a vector describing the position of the specific goals of the subarea on different dimensions of orientation (e.g. manual-worker orientation, urban orientation), while O[^h) is a vector describing the position of each party image at each election on these same dimensions. To sum up, these symbols have been given the following interpretations:

ØP Orientation vector of i:he specific goals of subarea i.
Of0: Orientation vector of the image of party hat election t.
ryf°: Satisfaction of common goals by the image of party hat election t.

All models described can now be interpreted as special cases of the general model.
If the model for uniform change, formula (12), is compared with the general

model, formula (16), it is evident that in this special case

(20)

that is, the orientation vector specific to the particular party does not change, whereas the satisfaction of common goals o^ for each party might change. This model provides a close fit with most of the parties in the Copenhagen data (see section 2 above).

Similarly, when the one-dimensional model for differential change, formula (14),
is compared to the general model, we obtain:

(21)

that is, both orientation vectors can be described in one dimension. For the Copenhagen data, 6j was interpreted as manual-worker orientation on the basis of ecological correlation. Only minor changes were observed in 9^, the manual-worker orientation of the parties (see section 3 above).

For the special case of the multi-dimensional model for differential change,
formula (19), we have

(22)

that is, the parties can change in only one orientation dimension. This dimension
was identified as the rural-urban dimension in the national data previously published
in this yearbook.12

6. Conclusion

The limited Danish electoral data analyzed so far indicate that simple models of
measurement, with parameters independent of either time or space, describe a
surprisingly high proportion of the change in voting behavior.

I expect that models of this kind will find broad applicability in future electoral research. An important subject for further empirical and theoretical investigations will be the analysis and interpretation of the parameters estimated from the electoral data.

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NOTES

1. Georg Rasch, 'A Mathematical Theory of Objectivity and Consequences for Model Construction', European Meeting on Statistics, Econometrics and Management Science, Amsterdam, 2—7 September 1968.

2. Ole Borre and Jan Stehouwer, Fire folketingsvalg, 1960—68, Århus: Akademisk Boghandel, 1970.

3. Borre and Stehouwer, op. cit.

4. Tage N. Bild, Om anvendelse af poissonmålingsmodellen, København: Københavns Universitets Sociologiske Institut. 1966 (mimeo).

5. Rasch, op. cit.

6. Søren Risbjerg Thomsen, Målingsmodeller for korttids-forandringer i vælgeradfærd, Århus: Institut for Statskundskab, Aarhus Universitet, 1971 (mimeo).

7. Thomsen, op. cit.

8. Ulf Christiansen and Jon Stene, 2. bind af G. RascKs lærebog i teoretisk statistik, København: Teknisk Forlag, 1969, Vol. 2, pp. 160-277.

9. Thomsen, op. cit.

10. Jan Stehouwer and Ole Borre, 'Four Elections in Denmark, 1960—68', Scandinavian Political Studies, Vol. 4, Oslo: Universitetsforlaget, 1969.

11. Thomsen, op.cit., section 7.7. approximately constant through the whole period.)

12. Stehouwer and Borre, op. cit.

13. Christiansen and Stene, loc. cit.