But How Does Place Matter? Using Bayesian Networks to Explore
a Structural Definition of Place
 
Colin Flint
Mark Harrower
Robert Edsall

Department of Geography
Pennsylvania State University
University Park, PA 16802

Paper presented at "New Methodologies for the Social Sciences: The Development and Application of Spatial Analysis for Political Methodology." University of Colorado at Boulder, 10th-12th of March, 2000.

ABSTRACT

Previous attempts by electoral geographers to investigate the contextual nature of politics have focused upon space rather than place. To further the inquiry of contextual politics a structural definition of place is needed. A structural view of place requires a consideration of what constitutes a place, how these elements of place interact, and how they combine to mediate political behavior within places. A spatial analysis of the change in the Nazi party vote between 1928 and 1930 illustrates how the search for spatiality in aggregate data concentrates attention towards the manifestation of place-specific behavior in spaces rather than the operation of the elements of place. Bayesian networks are introduced as an alternative technique that may examine a structural definition of place. Bayesian networks allow for an examination of how the component parts of place interact to produce a political outcome. Again, the Nazi party vote between 1928 and 1930 is used to exemplify the argument. Bayesian networks show how the institutional and economic aspects of place interacted to produce support for the Nazi party. Spatial analyses and Bayesian networks are complementary techniques that show how space and place create contextual settings for political behavior.

Keywords: Bayesian networks, spatial analysis, Nazi party, electoral geography, place, space.

Introduction
Defining Place and how it matters
Spatiality in aggregate data
A spatial analysis of the Nazi party vote
An overview of Bayesian approaches
A Bayesian network approach to place and politics
Conclusion

1. Introduction

Geographers have developed structural notions of place, whereby the practices and institutions within places mediate political behavior to produce place-specific outcomes. Place and politics are both the product of group and individual actions, or in other words, they are socially constructed. In addition, there is a recursive relationship between place and politics as political acts create and recreate places and, in turn, the nature of a place frames future political activity. Electoral geographers have mapped the geographical variation in political activity and, more recently, have used spatial analytical techniques to uncover the place-specific interaction of economic and social processes with politics. However, spatial-analytical tools have concentrated, up to now, upon spatial patterns rather than place-specific behaviors. Hence, geographers have been left with the task of showing that there is a geography, or place does matter, without being able to investigate how or why.

The two purposes of this paper are to call for a concentration upon place rather than space in the contextual analysis of politics and to introduce Bayesian Networks (BNs) as a technique to explore a structural understanding of place and politics. In order to make our argument we first discuss differences between compositional and structural conceptions of place, and show the theoretical superiority of the latter. Second, we describe how spatial regressions have incorporated the spatiality in aggregate data to show the place-specific nature of political behavior. Third, spatial regression techniques are criticized for focusing upon space rather than place. Fourth, we describe BNs and show how they may be used to identify the complexity of the mediating role of different aspects of place upon political behavior. Analyses of the growth of the Nazi party vote between 1928 and 1930 are used to illustrate both the spatial statistical techniques and BNs.

 Bayesian Networks (BNs) provide a way to uncover probabilistic relationships between variables while also denoting key subsets of variables and the interactions between them (Heckerman et al., 1995). Hence, BNs allow for the evaluation of causal relationships between the dependent and independent elements of a regression equation, and also the complexity of relationships within the explanatory variables. In the quantitative analysis of political behavior, BNs can be used to explore the complexity of the relationships identified by a spatial-regression analysis. Such exploration also provides insights into the way that different aspects of place interact, or, in other words, how the mutually constituted elements of place combine to produce place-specific behavior. The end result is an examination of the relationships within places rather than across spaces.

 Despite the promise of BNs to the contextual analysis of politics, we stress that this is our first attempt at applying this technique to political behavior. Thus, the spirit of the paper is an exploratory foray into both the technique itself and the analysis of a mediated and structurally complex notion of place. The paper is organized in the following way. First, we outline definitions of place and how place is theorized to mediate politics. In this section we emphasize the difference between a structural and a compositional notion of place. Next, we note how such notions of place have been modeled using spatial-structural regression techniques to highlight the spatiality in the data, spatial dependence and spatial heterogeneity. The fourth section of the paper uses the example of a spatial regression analysis of the growth of the Nazi party vote between 1928 and 1930 to illustrate the focus upon space rather than place. Section five introduces an alternative focus and an alternative technique by describing the logic and construction of BN's. Our purpose in adopting BNs is to unpack the results of a spatial-structural regression to uncover the complex interactions between the different components of a place and how they combine to influence political behavior. In section six, we use the growth in the Nazi party vote between 1928 and 1930 as an example of how BNs can further our understanding of the contextual influences upon political behavior by evaluating the causal role of different aspects of place and how they are related to each other. In the concluding section, we discuss future steps in the use of BNs as a tool for contextual analysis to assess its potential as a technique that can provide a systematic analysis of place rather than space.
 

2. Defining Place and How It Matters

 Place matters because it structures the way we behave (Pred, 1990). Our everyday experiences and actions are a framed by the institutions, practices and people with whom we interact. The problems people face, the possible avenues towards solutions, and the interpretation of what needs to be done and why differ depending upon the economic trajectory of a place, how institutions within a place filter information, and the senses of identity that are developed and given meaning within places. The uniqueness of a place is not a matter of the variation in the size of particular socio-economic groups, i.e., the size of the black population or Muslim populations. Such a compositional view of place misses the complexity of a place and how its institutionalized practices and customs mediate behavior.

 When place is considered a structure rather than an entity composed of different attributes, it becomes the unit of analysis. Political behavior is place-specific because of “the intricacies of interaction, the specificity of particular times and spaces, the sense of living as meeting, the context” (Thrift, 1983, p. 39). An analysis of place-specific political behavior, at the very least, needs to capture the institutions that are interacting, the senses of identity, and the actions of different socio-economic groups. Political outcomes are place-specific because knowledge is interpreted and acted upon within the varying contexts of institutionalized memory, interpretation of contemporary events, and endorsement of political responses (Thrift, 1983, p. 45). A contextual analysis of political behavior considers the role of institutions and identity in mobilizing particular groups. In other words, a structural view of place considers the setting of political actors rather than just the attributes of those actors.

Thus, the analysis of place and politics is, at the outset, an ontological issue. The scale of analysis for a contextual approach to politics is the geographic setting that mediates the political outcomes of interest. The key questions are what are the components of place that determine political behavior, and how do these components interact? A number of geographers have established theoretical frameworks to guide us in an interrogation of these questions.

John Agnew’s Place and Politics (1987) established a theoretical basis for the analysis of place-specific behavior. Agnew identified three aspects of place; location, the role a place plays in the world-economy; locale, the institutional setting of a place; and sense of place, identities forged and given meaning within places. Of course, these three aspects are separated purely for heuristic reasons. Within places, the existence and vitality of institutions such as unions and chambers of commerce is partly a function of the type and health of local economic activity. In addition, a sense of what it means to be working class or a member of an ethnic minority will be developed within institutions, i.e. within a union, and in relation to the activities of other institutions, the police or religious congregations, for example. Thus, Agnew (1987) provides a view of places as being constituted by economic, institutional and socio-cultural processes.

Doreen Massey’s (1994) definition of place makes explicit some of the implications of Agnew’s (1987, 1996) work. For Massey (1994, p.120), places are “networks of social relations” that are dynamic over time. The current expression of social relations is, to some degree, a function of the legacy of previous social relations that have been altered. Places are continually changing as current political actors use the background of existing social relations to foster change. Also, for Massey the nature of a place is a product of its linkages with other places and not just a matter of its internal features. Trade, migration flows, and cultural exchanges are examples of how a place reaches out in ways that alter its economic, institutional and cultural make-up. Thus, Massey makes explicit the temporal dynamic of a place and the way that it is part of a broader network of places.

The importance of place in the study of political behavior demands an ontology that recognizes places as objects of study. Such a structural view of place promotes a holistic and relational view of place instead of a compositional perspective that counts the socio-economic make-up of places. This is where electoral geographers using aggregate data run into trouble when in dialogue with political science colleagues. “Holistic view” can translate into an argument that places are “complex” and “unique”. In other words, the idea that “place matters” is often asserted by reference to individual case studies, at best, or even anecdotal examples. What have been elusive are systematic studies showing how places matter.

To conclude this section, we will summarize the components of place that need to be identified to include a structural definition of place into a systematic analysis. An operationalization of Agnew and Massey’s definition produces the following measurable components of place; economic role, institutional setting, political-cultural identity, linkages with other places, and changes over time. The historical dynamism of political behavior within places has been illustrated by a number of studies (Agnew, 1987; Archer and Shelley, 1986; Flint, 1998a; Johnston and Pattie, 1998). Also, studies have shown the importance of linkages between places at either the local (Flint, 1998a) or extra-local scale (Cox, 1998). The empirical section of this paper will illustrate how BNs may be used to identify the relative importance of the other three aspects of place in mediating political behavior and how these aspects are related to each other. In this way it is hoped that place can be shown to matter while retaining the integrity of a structural notion of place.

Studies that attempt to uncover the role of place often end up looking at the spaces created by political behavior. For example, Flint’s (1998b) analysis of the Nazi party showed that its electoral support varied across regional spaces. Other studies of the regional nature of political behavior include O’Loughlin and Bell (1999), Johnston and Pattie (1988), and Archer and Taylor (1981). Also, the role of linkages between places in determining political behavior have been incorporated into contextual analyses by defining localized pockets, or spaces, of support (Flint, 1998a). In summary, space rather than place has dominated geographer’s contextual analyses of politics. Hence, the difficulty in substantiating the claim that place matters. The reason for the prioritization of space over place in contextual analyses lies in the search for spatiality in aggregate data that has driven many studies (O’Loughlin et al, 1994; Flint, forthcoming). The following section describes the two forms of spatiality in aggregate data and illustrates their role in focusing attention upon space rather than place. An alternative approach is to use spatial-regression analysis as an aid in constructing BNs that can explore the complexities of place.

3. Spatiality in Aggregate Data

 Electoral geographers commonly use aggregate data. This is an ontological decision as the purpose of geographic inquiry is to investigate the specificity of place and its influence upon political behavior. Aggregate data is used to measure attributes of places, or other geographic units, which, in combination, combine to form structures mediating behavior. The focus, therefore, is open the geographic setting rather than the individual. Two aspects of spatiality in aggregate data have been identified and incorporated into analyses by electoral geographers: spatial dependence and spatial heterogeneity.

Spatial dependence exists when the value of the dependent variable in one spatial unit of analysis is partially a function of the value of the same variable in neighboring units.  The existence of spatial dependence may be a manifestation of diffusion and can result in "Galton's problem", whereby "certain traits in an area are often caused not by the same factors operating independently in each area but by diffusion processes" (O'Loughlin and Anselin, 1992, p.17).  In other words, an increase in electoral support for a political party in one place may have been a function of increased support in neighboring places. The identification of spatial dependence and its incorporation into the analysis of political behavior operationalizes a key component of place, the importance of the linkages between places in defining context (Massey, 1994).

Spatial heterogeneity refers to a regional pattern in the data that results in instability of parameters across the whole study (Anselin, 1988, p.9).  In other words, the slope of any regression equation would not be constant when comparing regions with the complete data set. The identification of spatial heterogeneity within the data indicates the presence of geographical variation in political behavior and its incorporation into statistical models illuminates the place-specific behavior of voter and party. For example, white-collar employees may have supported a party in one region while blue-collar employees supported the same party in a different region.

The application of spatial dependence and spatial heterogeneity to electoral geography have been detailed elsewhere (O’Loughlin et al, 1994; Flint, 1998a; Flint, 1998b; Flint, 1999). In this paper, we identify the logic of spatial dependence and spatial heterogeneity in order to show how, by definition, they allow us to explore the role of space in electoral behavior rather than the role of place.

Spatial heterogeneity indicates the existence of regionally specific relationships across the data.  It may exist as a mere statistical nuisance, expressed as lack of constancy of the regression error variance, or it may be indicative of contextual variation in political behavior (O'Loughlin and Anselin, 1992, p.27).  Structurally significant heterogeneity is identified in OLS models by diagnostic tests for heteroskedasticity (Anselin, 1992). Heteroskedasticity is the presence of non-constant variance of the random regression error over all of the observations.  If heteroskedasticity is present, the OLS estimates are unbiased but inefficient and inference based upon the t and F statistics will be misleading and the measures of fit will be wrong (Anselin, 1988, p. 120).

If diagnostic tests for heteroskedasticity are significant, regional patterns of political behavior are indicated.  In other words, subsets of geographical units, or regional groupings of counties or census blocks, are identified in which different explanations for political behavior are found. After diagnostic tests have identified heterogeneity, previous studies, theoretical frameworks, and exploratory analysis can be used to identify regions that display voting behavior different from the remainder of the data.  The identification of these regions, called spatial regimes, is then used to estimate structural change models (Anselin, 1992, p. 32-1). To estimate a structural change model, cases within a particular region are identified by the use of a dummy variable and the structural change estimation reported separate regression coefficients for the two sets of cases, those in the region and those that are not.  The structural change model is represented by the equations

yi = ai + Xibij + ei for d = 0
yj = ai + Xjbij + ej for d = 1

where both the constant terms (ai(j)) and the slope terms (bi(j)) take on different terms (O'Loughlin & Anselin, 1992, p. 31). To diagnose whether the structural change estimation captures the heterogeneity within the region, tests for heteroskedasticity should be insignificant.

In addition, a Chow test was reported for the model as a whole and for each of the explanatory variables. The Chow statistic (Chow, 1960) is a test upon the stability of regression coefficients. The Chow statistic is distributed as an F variate with K, N-MK degrees of freedom (with M as the number of regimes). The test is a test of the null hypothesis

H0: g'b = 0

where b is a vector of all the regression coefficients (including the constant terms) and g' is a K by 2K matrix [Ik | -Ik], with Ik as a K by K identity matrix (Anselin, 1992, p. 32-2). The corresponding Wald test may be expressed as the equation

W = (g'b){g'[var(b)]-1g}-1(g'b)

where b are the estimates of the regression coefficients and var(b) is the corresponding (asymptotic) variance matrix (Anselin, 1992, p. 32-2). A significant value for the Chow test measuring the stability of the regression coefficients between the two spatial regimes indicates that heterogeneity existed at the regional scale. In other words, different political behavior existed within the regions, or contextual settings, defined by the spatial regimes.

The identification and incorporation of spatial heterogeneity into the analysis of political behavior identifies spaces of regionally specific behavior. These spaces are the product of the combination of relatively similar places and the political behavior that they mediate. The spatial regimes of a structural change model are the manifestation of how place matters in politics but they are not able to capture the processes that determine the role of place as a geographic structure. In other words, spatial heterogeneity and structural change models are a concept and technique that describes the product of place-specific behavior rather than its operation.

Spatial dependence is also incorporated into the spatial analysis of political behavior in order to identify a key component of place, linkages to other places (Massey, 1994). Spatial dependence may be incorporated into models of voting behavior in one of two ways depending upon the diagnostic tests reported in the initial models. The average value of the vote in neighboring geographic units in the first of a sequence of two elections, referred to hereafter as the temporal-spatial lag, was incorporated into the initial OLS model. If spatial dependence existed after the inclusion of this variable, then it was replaced by the spatially lagged dependent variable, the average value of the dependent variable in neighboring spatial units. In both of these cases, the definition of a neighbor may be calculated either by distance or contiguity. If the temporal-spatial lag is positive in sign and statistically significant, it indicates that the size of the vote in one unit was partially a function of the size in support in the neighboring units in the first of the two elections in that particular period of change. If the spatially lagged dependent variable was positive in sign and statistically significant, it indicates that the vote in a unit was partially a function of the vote in the same election in neighboring units. The inclusion of either of these variables models the role of the interlinkages between places in defining the contextual setting of the voter. Methodologically, the presence of spatial dependence produces biased and inconsistent regression coefficients (Anselin, 1988, p.59).

Spatial dependence can exist in two forms (Anselin, 1988, pp. 11 - 13). In its substantive form, spatial dependence is interpreted as spatial contagion, whereby the behavior in one spatial unit is partly explained by similar behavior in neighboring units. Methodologically, substantive spatial dependence is incorporated into the regression equation by adding the spatially lagged dependent variable. Formally this may be expressed by the equation

y = pWY + Xb + e

where y is a vector of observations on the dependent variable, X is a matrix of explanatory variables, including the temporal-spatial lag, bare the regression coefficients, e is an error term, p is a spatial autoregressive coefficient, and WY is the spatial lag, the average of the value of the dependent variable in neighboring units (Anselin, 1992, p. 27-1).

In addition to the substantive interpretation, spatial dependence may also have to be controlled for as a nuisance. This form of spatial dependence is known as spatial error dependence as it is associated with model specification errors that are not restricted to one unit but spill across the spatial units of observation. The usual assumptions of homoskedastic and uncorrelated errors no longer hold and so the spatial error model incorporates a spatial autoregressive process in the error term. To estimate regression coefficients in the presence of spatial error dependence, a spatial autoregressive model is estimated which may be stated in the following equations

y = Xb + e

e = We + x

where the notation is the same as above with We being a spatial lag of the errors and x is a "well-behaved" error term with mean of zero and variance matrix s2I (Anselin, 1992, p.29-1). The presence of both the spatial lag and the "well-behaved" error term creates a problem of simultaneity. Therefore, a maximum likelihood procedure that includes the estimation of a nonlinear likelihood function must be executed (Anselin, 1988, p. 59). If the spatial error dependence is ignored, the OLS estimates would be unbiased but could result in misleading inference if the variance estimates are not adjusted because the OLS variance expressions do not account for the dependence among the errors.

Substantive spatial dependence is a means of operationalizing the role linkages between places play in mediating political behavior. However, though the goal is to uncover the specificity of place the result is the inclusion of connections across space between places. Defining neighbors in terms of contiguity or distance is a spatial relationship that aims to uncover the nature of places. As with the consideration of spatial heterogeneity, incorporating spatial dependence into an analysis of contextual political behavior prioritizes the construction and mediating role of spaces rather than places. In other words, including the spatiality of aggregate data in the spatial analysis of political behavior shows the manifestations of place-specific behavior but not the mediation of politics and social processes within places, a mediation that produces the specificity of place.

There is a second implication of using spatial regression models to investigate the recursive interaction between place and politics. Regression analysis partitions the roles played by particular aspects of place. Interrelationships between the independent variables in a multiple regression analysis are controlled for rather than sought and incorporated as part of the model. Hence, a compositional view of place is promoted, whereby place-specificity is a product of the combination of different socio-economic attributes. These two implications of using spatial regression to explore politics and place are exemplified through an analysis of the Nazi party vote.
 

4. A Spatial Analysis of the Nazi Party Vote

Census data were used to create variables to measure different aspects of place.1 Location, or the economic role of a place, was investigated by using variables measuring the proportion of different classes and employment sectors within a county. Specifically, the following variables were used to measure location: BCTRADE, the percentage of the workforce who were blue collar workers in trade and transport; and TOTSELF, the percentage of the workforce who were self-employed. Locale, or institutional setting was engaged by including variables that measured religious affiliation and labor organization. Specifically, the following variables were used to measure locale: PROT, the percentage of the population who were Protestant; and MANIND, the percentage of the workforce who were manual industrial workers. Finally, sense of place was measured by including variables that identified alienation. Specifically, the following variables were used to measure sense of place; UNEMP, the percentage of the workforce who were unemployed; and TURNOUT, the electoral turnout as a percentage of eligible voters for the second of the two elections within the period of change in the Nazi vote. The dependent variable, NAZI30CH, was the percentage change in the Nazi party vote between the consecutive Reichstag (parliament) elections of May 1928 and September 1930.

It should be noted that these variables are also instruments to test theoretical frameworks that have been used to explain Nazism. Institutional setting is the place-specific manifestation of Burnham's (1972) theory of political confessionalism. Burnham argued that Catholics and the industrial proletariat would not have been attracted to the Nazi party because of their respective allegiances to the Center party and the Social Democrats and Communists. In other words, religious and labor institutions would have been an element of the particularity of places. Sense of place is the manifestation of local identities generated by alienation (Arendt, 1958; Kornhauser, 1959), and surrogately measured by unemployment and electoral turnout. Finally, the class theory (Lipset, 1960) argued that the economic policies of the Nazi party attracted the self-employed middle-class, while Flint (1998b) and Ault and Brustein (1998) found that artisans and skilled workers were also susceptible to the Nazi’s economic policies. The economic role of a place is investigated by the relative size of these economic groups, measured by the variables TOTSELF and BCTRADE.

Exploratory analysis is required before the estimation of spatial regression models. Multiple regression models were specified using the variables discussed above plus adding others by stepwise regression techniques. Though a theoretically informed model is preferred, additional variables were considered in order to counter the critique of electoral geography that contextual influences identified by geographers are a function of poorly specified models that do not include all the relevant explanatory variables (McAllister, 1987). The only other significant variable to be found was TRADJOBS, the percentage of the workforce employed in the trade and transport sector.

The spatial analysis of the Nazi party was conducted at the national scale in order to maximize the number of cases and, therefore, assist in facilitating the robustness of the Bayesian network created later. The results are displayed in Table One. The model provides evidence for a significant role for all three aspects of place but the direction of the relationships is surprising. The positive and significant value of the variable PROT illustrates, as expected, that places without strong Catholic institutions supported the Nazi party. The negative and significant value of the variable TURNOUT indicates that political alienation was not a factor in the Nazi party vote. Instead, the institutional setting was one of weakening political parties allowing the Nazis capture their support. The two variables measuring socio-economic status are harder to interpret. Both BCTRADE and TRADJOBS measure employment in the trade and transport sector, the former focusing upon a particular class of employee. BCTRADE displays a negative sign while TRADJOBS is positive. Other analyses have shown BCTRADE to be positively related to the Nazi party vote (O’Loughlin et al, 1994; Flint, 1998b). The positive sign of TRADJOBS may indicate support for the Nazi party in urban transport nodes, but this is just speculation.
Fig. 1.  click to enlarge.

In addition to significant socio-economic variables, regional dummy variables were included to capture heterogeneity across the national surface. Germany was divided into eight historical-cultural regions in order to capture the heterogeneity of German society and its possible influence upon voting behavior. The regions were Prussia, the Northwest, Rhineland-Westphalia, Silesia, Central Germany, Baden, Bavaria, and Württemberg (Figure 1, left). The regions were designed to capture cultural similarities and historical interactions and political organization. Also, the borders of these regions were related to the regions created by the Nazi party to organize their political campaigns. In sum, the regions are an attempt to capture a similarity in the message being disseminated by the Nazis and the similarity in the cultural setting within which it was received. The significant value of three of the regional dummy variables indicates the presence of regional heterogeneity. However, the significant value of the Breusch-Pagan statistic indicates heterogeneity within the regions.

Finally, the regression model is a spatial error estimation to control for spatial autocorrelation across the error terms. LAMBDA is the spatial autoregressive coefficient that controls for the spatial error autocorrelation and allows for the estimation of unbiased and efficient estimates (Anselin, 1992, p.29-2).

Spatial regression models are effective in illustrating spatial variation in political behavior as well as how linkages across space is a factor in defining place-specific politics. However, the emphasis upon space is to the detriment of the understanding of how different elements of place interact to mediate politics. Spatial regression does offer insights into the attributes of places that mediate political behavior. However, these attributes, as independent variables in a regression analysis, are treated separately. The alternative approach offered by BNs has the benefit of exploring the interaction between different aspects of place and how they combine to mediate political behavior. Spatial regressions promote a compositional view of place by separating out additive socio-economic attributes of place. On the other hand, BNs promote a structural view of place by showing the mutually constituted complexity of place and its mediation of politics.

5. An Overview of Bayesian Approaches


Bayesian networks (BNs) have recently gained popularity in the modeling of uncertain relationships among variables. For introductory and accessible texts see Pearl (1988), Charniak (1991), Heckerman et al (1995), and Jensen (1996). The term "Bayesian network" encompasses a variety of graphical models for representing knowledge and associations within a data set (including Bayesian belief networks, Bayesian inference networks, and graphical probability networks). However, the term Bayesian network (BN) is preferred as it is more neutral than including the term belief, causal, or inference (Charniak, 1991). Though the theory behind BNs has been in existence for over a century, only now have the difficult problems of computing probabilities given evidence and conditional probabilities become tractable using modern computing and breakthroughs in algorithms for "propagating" the uncertainty through the related variables. Indeed, less than ten years ago Charniak (1991) was lamenting the computation time needed to construct BNs.

A Bayesian network is a graph of relationships among variables in a data set. A network consists of a series of nodes, each representing a variable, and arcs (or edges), connections (with direction) between nodes representing a causal (but uncertain) relationship between the variables in the nodes (Charniak, 1991, p.50). The assignment of these relationships can be driven either by the data or by the analyst, but most often and most effectively, the relationships are derived from some combination of the two (Spiegelhalter et al, 1993). A connection between two nodes may be interpreted as either a causal path from one to the other or evidence that the nodes are correlated (Charniak, 1991, p. 54). Causal relationships are, of course, useful in making predictions given certain information, but learning causal associations is also important in exploratory analysis, in gaining insight into a data set (and its corresponding problem domain) (Heckerman, 1996a).

The expert input may be interpreted as the prior probability of a relationship that is then judged by the data to create a posterior probability (Mitchell, 1997, p. 157). A maximum likelihood approach is used to create the probabilities taking into account the observations and the assumed probabilistic distribution of the data (Mitchell, 1997, p. 157). Thus, Bayes theorem evaluates the probability of a hypothesis by considering its prior probability, the probability of particular observations given the hypothesis, as well as the actual observations (Mitchell, 1997, p.157). Bayes theorem may be stated

P(h|D) = P(D|h)P(h)
           P(D)

where P(h|D) is the posterior probability of our hypothesis (h), P(h) is the prior probability, P(D) is the probability of observing the data given assumptions of its distribution without reference to h, and P(D|h) is the probability of observing the data given that h is correct (Mitchell, 1997, p. 156).

The type of probabilistic approach of Bayesian methods is in contrast to the classical approach of statistics, which deals with confidence intervals and levels. Where classic deductive reasoning assumes that observation alone can be used to predict unobserved events, Bayesian methods incorporate beliefs about the probability of an outcome held prior to (and perhaps updated by) the observations. In this way, Bayesian networks allow inference without repeated trials by allowing direct construction by a domain expert of the probability tables associated with each node.

The arcs defined by a BN allow for the evaluation of the conditional independence of the variables, or nodes (Mitchell, 1997, p. 185).
Fig. 2:  click to enlarge.
Three varieties of conditional independence are defined (see Fig. 2, left, adapted from Henrion et al, 1991). First, marginal independence refers to source variables, those with no predecessors (Henrion et al, 1991, p. 74), variables U and W for example. Second, two variables are conditionally independent if they have “one or more common parent(s) but no arc between them” (Henrion et al, 1991, p. 74), variables X and V for example. Third, a variable is “conditionally independent of its indirect predecessors given all the variable’s immediate predecessors” (Henrion et al, 1991, p. 74), for example Y is conditionally independent of U given its immediate predecessor V.

The idea of conditional independence produces three types of path (see Fig. 3, right, adapted from Charniak, 1991).
Fig. 3.  click to enlarge
First, a linear path from node A to node B and on to node C, second, a converging path as both nodes A and C are predecessors to B, and third, a diverging path as node B is the predecessor to both nodes A and C (Charniak, 1991, p. 54). The linear path shows a causal path from A to B and then on to C. The converging path shows that both A and C are causes of B. The diverging path shows that B is a causal factor for both A and C.

A BN may include a variety of these paths. It is from the graphic visualization and probabilistic calculation of these relations that the structural nature of place can be explored. Variables representing different aspects of place can be included in a network. The existence of causal paths between these nodes and one representing political behavior graphically display the mediation of politics by elements of place. In addition, a converging path will show that different aspects of place play a role in mediating political behavior. Linear and diverging paths illustrate the complexity of place by showing how different aspects of place are related to each other and may have a less immediate impact upon political outcomes. Another avenue for inquiry, and one not pursued in this paper, is to reverse the direction of the acyclic arcs to explore the recursive interaction between politics and place.

Of course, the assumption of prior knowledge about the relationships between variables is a weakness as well as a strength. A drawback to the construction of Bayesian networks is the determination of its structure with respect to causality and dependence among the nodes. To determine the structure of a Bayesian network, two things are required: some order of the variables that indicates which variables are causes and which are effects, and an assessment of the subset of variables that are conditionally independent of one another. On the one hand, Charniak (1991, p. 61) dismisses the problems of defining prior probabilities. However, Heckerman (1996b, p. 13-14) has found that "the causal semantics of Bayesian networks are in large part responsible for the success of Bayesian networks as a representation of an expert system," and asserts that, rather than searching through n! different combinations of orders, people "can often readily assert causal relationships among variables, and causal relationships typically correspond to assertions of conditional dependence." The problems of defining prior probabilities are much more problematic for those considering social behavior than, say, medical diagnoses as the processes being investigated are much more contingent.

A computer, in an unsupervised network generation algorithm, explores all possible pairs of nodes for conditional dependency (though, in our experience, we have not encountered a network generation algorithm that explores all possible orderings of the variables). After the dependencies and causal relationships are in place in a Bayesian network, the local probability distributions for each node (given specific outcomes or values of its parents) is assessed and incorporated into the graph.

It is possible to level a criticism of BNs that often these prior probabilities seem arbitrary; it is not often easy, even with domain experts, to develop probabilities for every possible combination of elements in a set of variables. For this reason, the problem (and its associated computing complexity) is reduced somewhat by discretizing continuous variables into classes or rankings (Charniak, 1991, p. 51). Even so, the number of probabilities in a single node with three possible values (low, medium, and high) with p parent nodes, each having three possible values, is 3p+1. Thus, even the simplest of models become daunting to input all of the values in the node probability tables (Agena Ltd., 1999). However, the complexity of the network is reduced when the conditional independence of some nodes is found and hence the amount of connections and probabilistic outcomes is reduced (Charniak, 1991, p. 53).

In our application of Bayesian networks, the network structure -- that is, the conditional dependencies among the nodes of the networks -- was not provided in advance.  The construction of linkages establishing conditional dependencies among variables is a difficult and time-consuming task because it (most effectively) must be performed manually by an expert. The machine learning of the network structure is achieved by calculating the probabilities of possible network linkages and selects a model that maximizes the probability of the result.  A network structure is created that has a high probability given the data: the structure itself amounts to a hypothesis about the conditional dependencies of the variables in the data set.  The search, then, is for the maximum likelihood hypothesis given the data set.

The relative likelihood of each of the large number of possible network structures (Cooper and Herskovits, 1992). The number of possible structures increases exponentially with the number of nodes (variables) present, but a series of assumptions cut down on the number of tested solutions (and resulting complexity) without losing significant explanatory power (Cooper and Herskovits, 1992). A metric for the likelihood of the structure is determined and compared to a measure of the most likely structure found thus far in the search.  If the probability, given the data, of the network presently evaluated is greater than the previous "best" result, the present network is established as the most likely structure.

The maximum likelihood measure can be assessed both locally for each link (thus showing the most dominant dependencies in the data set), and globally (the measure for the comparison of one network to another in the search).  The log likelihood is given, since the probabilities (which can range from 0 to 1) are typically very small numbers, and it is easier to examine the exponents (usually large negative numbers) of the likelihood measures rather than the measures themselves.  Thus, the most likely hypotheses (networks) are those with the lowest negative (closest to zero) global log likelihoods, and those dependencies which are strongest are those with the lowest negative local log likelihoods.

Our purpose in using a BN is to identify how different aspects of place interact to produce place-specific political behavior. BNs are a tool for the systematic analysis of probabilistic relationships that geographers cite as being the key mediating factors of places.  The need for expert input into BNs gives them a role within the electoral geographer’s toolkit. Theory, case studies and complementary quantitative techniques can offer evidence for the expert in their construction of the network. In turn, the relationships found within a BN can be used to inform theory, case study and further quantitative analysis.

Following Spiegelhalter et al (1993), the BN of Nazi voting behavior may be thought of at three levels of representation. First, is the qualitative level to investigate the general relationships by creating arcs between the nodes (Spiegelhalter et al, 1993, p. 220). The second level, or the probabilistic domain, calculates the joint distribution of the nodes in terms of probabilities (Spiegelhalter et al, 1993, p. 222). The third level, or the quantitative domain, provides a numerical evaluation of the conditional distributions (Spiegelhalter et al, 1993, p. 223). A BN uses the conditional distributions of individual arcs to create a joint probability for the network as a whole. Hence, we end up with a graphic visualization of the structural relationships creating and place and mediating political behavior as well as a quantitative evaluation of the combined determination of that behavior.

6. A Bayesian Network Analysis of Place and Politics

A balance needs to be maintained between illustrating the complexity of place to such a degree that it becomes unclear or confusing and an over-simplification that prevents an analysis of the structural qualities of place. To negotiate such a tension, a network illustrating the complexity of place and how it mediated support for the Nazi party was constructed. From the relationships identified in that network two refined networks were constructed that focused upon the interactions of identity, class composition, and institutions in mediating Nazi electoral support. The choice of nodes in the refined networks was made by reference to relationships found in the complex network, the earlier regression analysis, as well as theories of place and theories of Nazi party support. Hence, we used expert knowledge derived from theory and prior data analysis to construct the refined network.

The variables used in the BNs are the same as the ones in the previous regression analysis. The construction of BNs requires discrete data. This is a drawback in the use of BNs as it requires expert intervention in deciding what breaks are used to recode continuous variables. For this analysis a binary classification was adopted, using the mean of the variable as the break. The binary classification produced more robust networks than those adopting three or four categories did.
Fig. 4.  click to enlarge.

The complex network (Fig. 4, left) illustrates, well, the complexity of place. The different components of place are related to each other and the nodes representing Protestant, manual industrial workers, and blue collar trade and transport workers are all parents of the Nazi node. The manual industrial workers node has a direct link to the Nazi node and an indirect link in terms of a linear path through the blue-collar trade and transport workers node. The same is true for the Protestant node as it displays a direct link to the Nazi node and a linear path via through the blue-collar trade and transport workers node. Nodes measuring alienation, electoral turnout and unemployment, play different roles. The electoral turnout node is the apex of a diverging path connecting to manual industrial workers and self employed. On the other hand, unemployment is an ending node conditionally dependent upon manual industrial workers and self employed. The class node measuring the self employed is at the end of arcs leading from manual industrial workers, electoral turnout, and protestant. In turn, the self employed node is a parent to unemployment, measuring economic alienation. Finally, the node measuring jobs in the trade sector is the end node of two arcs, one from manual industrial workers and the other, not surprisingly, from through the blue collar trade and transport workers. The global log likelihood score for the network was –3488, allowing for comparison of its explanatory value with subsequent networks.

The network is, perhaps, better thought of as a web. The web contains nodes measuring the institutional, identity, and class composition aspects of place. Sense of place, or identity within place, may be engaged by noting the level of alienation of the inhabitants. Economic alienation is measured via the node measuring unemployment and political alienation is identified with the electoral turnout node. The institutional setting of a place was measured by two nodes, religious institutions are measured via the protestant node and organized labor is measured via the manual industrial workers node. The economic role of a place was captured by nodes measuring class composition, self employed, and blue collar workers in trade and transport. The arcs between the various nodes show how the different components of place are mutually constituted. For example, political alienation is related to the institutions of organized labor as well as the size of the self employed group. As another example, religious institutions are related to the size of the self employed group and also to the presence of blue collar workers in trade and transport. A final example of the relationship between different aspects of place can be seen in the relationships of organized labor and self-employment to unemployment, or economic alienation.

When it comes to showing how these aspects of place structure political behavior, two institutional nodes, protestant and manual industrial workers, and one class node, blue collar trade and transport workers, explain Nazi party electoral support. In addition, the network also shows that the institutional aspects of place are translated through class composition to explain political behavior. Hence, the two linear paths manual industrial workers to blue collar trade and transport workers to Nazi party vote and also protestant to blue collar trade and transport workers to Nazi party vote. In other words, different aspects of place explain political behavior and these aspects are related to one another.

To try and make the explanation of Nazi party support clearer, two refined networks were created. The construction of these networks was based upon expert knowledge and the previous complex Bayesian network. Agnew’s (1987) theory of place suggests that measures of economic role, institutional setting, and sense of place should be included in the network. Complementing Agnew’s approach are theories of Nazi party support suggesting the role of institutionalized political competition (Burnham, 1972), political and economic alienation (Arendt, 1958; Kornhauser, 1959), and class (Lipset, 1960; Ault and Brustein, 1998). Hence, expert knowledge suggests that variables measuring identity/alienation, class or socio-economic status, and the role of institutions should be included in the refined network. The complex network identified protestant, manual industrial workers, and blue collar trade and transport workers as parent nodes of the Nazi vote. In addition, earlier regression analysis had identified electoral turnout as an explanatory variable. In combination, previous analyses and expert knowledge called for a network that included alienation (measured by electoral turnout), institutional setting (measured by manual industrial workers and protestant), and class (measured by blue collar trade and transport workers) as potential parents for the change in the Nazi party vote.

The choice of variables to be entered into the network was determined by Agnew’s (1987) theory of place. However, different interpretations of Agnew’s theory produced different ideas of how the aspects of place interacted. Seeing as the order that the variables are entered into the network may have an effect upon the relationships that are found (Heckerman, 1996b, p.13), two
Fig. 5.  click to enlarge.
networks were constructed. In the first refined network (Fig. 5, left), the initial node was alienation, measured by the size of electoral turnout, which was experienced within institutional settings, measured by the variables protestant and manual industrial workers, and also interpreted depending upon class position, measured by blue collar workers in trade and transport. Of course, the variable manual industrial workers also measures class position, and may be interpreted either way. Alienation is related to both blue collar trade and transport workers and manual industrial workers which, in turn, converge onto the node measuring change in the Nazi party vote. Thus the presence of alienation within a place is expressed through class position and labor institutions to produce political support for the Nazis. In a separate linear path, the institutional religious setting displays a relationship with the Nazi vote independent of the other nodes. In other words, the dominance of the religious institutional setting in determining political behavior in Weimar Germany is supported by these results.

Figure 6, right, shows the strength of the relationships in the first refined network. The figures are log likelihood scores, and the lower the score the greater the probability that the status of a parent node determines the status of its child. The log likelihood table shows that the strongest relationship (-419.39) is a product of considering the religious and labor institutional settings with class status (or Protestant, with manual industrial workers and blue-collar trade and transport workers). Further support for the role of institutional setting in mediating political behavior is offered by the next strongest relationship (-420.33), one that considers just the Protestant and manual industrial workers nodes. The third strongest relationship (-430.34) confirms the interaction between class standing and institutional setting in mediating political behavior, by quantifying the interaction of Protestant and blue collar trade and transport workers and their relationship to the Nazi party vote. The log likelihood table also indicates the relative unimportance of electoral turnout in mediating the Nazi party vote. The global log likelihood score for the refined network was –2427, illustrating its greater explanatory value compared to the complex network.

The relatively weak role played by electoral turnout in the first refined network suggested an alternative refined network. The second refined network (Fig. 7, left) places class position as the initial node, a position from which alienation is experienced and then interpreted through religious institutions.
Fig. 7. click to enlarge.
 Hence, the nodes measuring blue-collar workers in trade and transport and manual industrial workers (both deemed to be measuring class) were entered first, followed by electoral turnout and protestant.

Similar to the first refined network, religious institutional setting (measured by Protestant), displays relationship to the Nazi party vote separate from the other variables. Both the manual industrial workers and blue collar trade and transport workers nodes display direct relationships with political behavior. Entering electoral turnout into the network at a later stage confirms the findings of the first network that it has no relationship with Nazi party vote. However, there is a relationship from manual industrial workers to electoral turnout that suggests the role of class standing and/or labor organization in voter mobilization. Finally, the linkage from manual industrial workers and blue-collar trade and transport workers probably suggests an obvious correlation between two measures of class.

The log likelihood tables for the second refined network are, not surprisingly, the same as the previous ones (Fig. 6, above). The second refined network shows the dominant roles played by institutions and class standing in mediating the Nazi party vote. Religious institutions in particular, and the interaction between class standing and class organization, interacted to form settings that either nurtured or frustrated Nazi electoral success. The global log likelihood score for the second refined network was –2424, meaning that its explanatory value was almost identical to the first refined network.
Fig. 8.  click to enlarge.

A final, complex, network was created in order to test the robustness, or sensitivity, of the refined networks or, in other words, that the relationships found were not merely an artifact of the order in which the nodes were entered into the network (Fig. 8, right). A sensitivity network was created in which all the variables were entered in a random order. The acyclic graph illustrates that the same three nodes (protestant, manual industrial workers, and blue collar trade and transport workers) have direct pathways to the change in the Nazi vote. In addition, electoral turnout plays is located in a linear path, in this case between manual industrial workers and protestant. The role of these four variables in the complex and randomly generated network provides support for the claim that the relationships in the refined expert networks are not an artifact of the data.

7. Conclusion

In summary, the refined networks illustrate how different aspects of place are mutually constituted and, together, mediate political behavior. Sense of place, location and locale are seen to operate through variables measuring alienation, class position, and institutions. The arcs of the networks illustrate how these aspects of place interact to mediate politics within places. Using expert knowledge derived from theories of place and theories of political behavior, in conjunction with knowledge gained from previous analyses, BN’s were created which unpacked spatial regression models to show the complexity of place and its role in mediating actions.

With respect to the Nazi party vote, institutions and class position were found to be the most important aspects of place in determining political behavior. The dominant role of Protestantism in explaining the Nazi party vote has been a staple of previous analyses (for example, see Falter, 1991). However, the BN approach conceptualizes Protestantism as an institutional feature of place rather than an individual characteristic. The importance of institutional setting in explaining the Nazi party vote is confirmed by the role of the manual industrial workers node in the networks. Finally, the interrelationship between the manual industrial workers node and the blue-collar trade and transport worker illustrates how institutional setting constitutes class standing. In combination, different aspects of place interacted to form spatial settings conducive to Nazi party electoral success.

BN’s reorient the quantitative analysis of contextual politics from a focus upon space to place. Instead of looking at spatial linkages between places and variation in political behavior across regional spaces, BN’s provide a systematic way of analyzing the mechanisms by which place mediates political behavior. The next step is to incorporate space into the BN’s. Spatially lagged variables of political behavior may be included in the networks to capture local political activity. In addition, and once spatial analysis has identified regions of political behavior, different networks can be created for different regions to engage such spatial heterogeneity. Hence, the mutual construction of place and space can be included in the same analysis.

The purpose of this paper was to investigate how places mediate political behavior. Once the elements of place were identified through theory, the relationships amongst them were identified graphically and probabilistically. Hence the way that aspects of place are mutually constituted was explored. In addition, BN’s show how the complexity of place aligns to mediate political outcomes. Thus, BN’s offer a fruitful mechanism for unpacking the components of place to see how they interact to structure place-specific behavior. Using this technique, place is shown to be more than the sum of its parts, as it is the way that the elements of place combine that produces place-specific behavior.

notes
1. There is a wealth of aggregate data allowing for the spatial analysis of voting behavior in Weimar Germany. Census and election files were obtained from an archive "Wahl- und Sozialdaten der Kreise und Gemeinden des Deutschen Reiches, 1920-1933" at the Central Archive of the University of Cologne. The Cologne data were dissaggregated at a scale of over 6,000 geographic units that included Kreise (counties), villages, and neighborhoods within cities and then aggregated into 743 Kreise in order to match census and voting data. The explanatory variables were created from the 1925 census data. Areal boundaries of the Kreise were obtained from an OSS (Office of Strategic Services) Map, #6289 (1944) and digitally coded into a Geographic Information System (GIS) using the Arc/Info software package. Rusty Dodson, David Fogel and Steve Kirin (Department of Geography, University of California, Santa Barbara) coded the original map into a GIS. Subsequently, Michael Shin (Department of Geography and Regional Studies, University of Miami) and Colin Flint revised this map to incorporate the boundary changes and reduce the number of Kreisunits on the map to 743. The GIS was used to construct a contiguity or spatial weights matrix based upon first order contiguity to define the immediate neighbors of the Kreise.

The election file includes returns for the Reichstag elections between 1920 and 1933, and the census file contains socio-economic data collected from a variety of sources and at a variety of times.  Jürgen Falter and Wolf Gruner (1981) have described how this data set was revised during the 1970's to correct data errors, which were mainly a result of punching errors, and also to compensate for internal political boundary changes within Weimar Germany in order to make the territorial units within the data set as consistent and coherent as possible.  The changes in internal political borders in Weimar Germany were a product of major changes in administrative units between 1919 and 1933 that were partially a result of the incorporation of suburbs into urban areas and the reform of local government.  Falter and Gruner believe that the sources of the data set are the respective volumes of the Statistik des Deutschen Reiches (Statistics of the German Reich) issued by the Statistisches Reichsamt (State Statistical Office) in Berlin.

2. With the innovations in the computational algorithms for the propagation of evidence given uncertain relationships among variables, BNs demonstrate their utility as self-contained decision support systems. Once a network has been defined, and the probability distribution among the variables inputted (either by the expert, by the data, or some combination of both), BNs can be used to propagate known evidence, like states of certain variables, and predict the likelihood of states of other variables given that evidence. For example, a BN is an effective representation for the diagnosis of car problems: given the fact (evidence) that the car doesn't start, but that the radio works and the lights work, there is a 0.58 (of 1.00) chance that the likely cause for the failure is a bad distributor, and a 0.35 chance that the spark plugs need to be replaced. The assumption is that the certain variables are independent (like the status of the fuel pump and that of the power windows), but that others are dependent and interrelated (like the      status of the ignition and that of the alternator), and that these conditional dependencies can be modeled through some combination of prior (expert) knowledge and evidence. Such potential of the propagation of uncertainty through a network of relationships to reveal the conditional probability of a certain outcome makes a BN a very effective decision support tool. The user presents the network with a "what-if" scenario and the network delivers its prediction (with uncertainty and alternatives) for states of unknown variables.

 

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Table One. Spatial error estimation of the change in the Nazi party vote, May 1928-September 1930.

OBSERVATIONS = 743
VARIABLES = 12
DEGREES OF FREEDOM = 731
R2 = 0.47
LIK = -2425.92
 
Variable Coefficient Standard Deviation
CONSTANT 26.83* 4.35
PROT 0.14* 0.01
BCTRAD -0.31**  0.16
N309TURN -0.23*  0.05
TRADJOBS 1.02* 0.25
NORWEST 0.44 1.30
RHINE 0.60 1.30
CENTRAL -3.80* 1.44
SILESIA 2.13 1.64
BADEN 2.09 1.72
WURTBERG -10.37*  1.58
BAVARIA -3.02**  1.50
LAMBDA  0.43* 0.04
*  Statistically Significant at the 0.01 Level
** Statistically Significant at the 0.05 Level

Regression Diagnostics:
Diagnostics for Heteroskedasticity

Random Coefficients
Test: DF VALUE PROB
Breusch-Pagan test  11 120.42  0.000
Spatial B-P test 11 120.42  0.000

Diagnostics for Spatial Dependence
 
Test: DF VALUE PROB
Likelihood Ratio Test   1 91.50 0.000

Test on Common Factor Hypothesis
 
Test:  DF VALUE PROB
Likelihood Ratio Test 11 12.96 0.296
Wald Test 11 12.24 0.346