Missing:
Progress in Human Geography 28,6 (2004) pp. 807 – 814
Quantitative methods: past and present Jessie P.H. Poon Department of Geography, University at Buffalo-SUNY, Buffalo, New York 14262, USA
I
Introduction
In celebrating the 40th anniversary of Administrative Science Quarterly, Weick (1996) related the story of several firefighters who did not survive in a 1994 South Canyon fire because they had failed to drop the heavy tools they were carrying while trying to escape from the fire. In this second report, I will argue that the heavy tools assembled during the zenith of quantitative methods (QM) practice in the 1960s have largely been retooled. Such retooling reflects the field’s innovative adaptation and thereby modernization in human geography to the mutation and diversity in both methodological and substantive geographic problems. More specifically, the question of what constitutes the appropriate strategy for studying geographic phenomena has evolved from assigning quantitative methods an ontological position to analytical narratives that are more sensitive to writing geography from above and from below. Past practitioners of QM in human geography largely subscribed to the principle of methodological individualism, that is, the assumption that social or group outcomes may be explained by the behavior or action of individuals (Brodbeck, 1968). Investigation of social entities or group phenomena in space tended to be ontologically economical (e.g., the use of terms like ‘migrant’ or ‘market’) and these groups were often but not always analyzed through some form of statistical aggregates. The popular use of group entities is based on the difficulties of conducting geographic explanations based on individual sovereignty. However, the present construction of geographic knowledge by QM practitioners has become more sensitive to the constitutive parts, that is, individuals and their properties or attributes, than the social or geographic wholes themselves. Secondly, it is commonly perceived that the QM community in human geography is legislated by methodology and the model of reduction. Such a standpoint neglects the fact that the practice of human geographic research has increasingly required that the community be given the autonomy to pursue # Arnold 2004
10.1191/0309132504ph521pr
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‘reconstructed logic’ rather than scientific idealized logic, that is to say, the possibility of imperfect explanations in geographic ‘facts’.1 Models and analogies are not the only features that have scientific outcomes. Hence I will argue in this report that the evolution of the QM field is paralleled by a more critical attitude towards the scientific standard of completeness of theories and explanations under logical positivism and towards a more contextualized view where normative explanation is only one of potentially diverse explanatory outcomes. II Metaphorical thinking and scale Poststructuralist and postmodernist geographers adopt the stance that researchers should exercise more self-reflection and bring their own foundational assumptions to the surface. Scientific knowledge is charged to establish singular methodologies and meanings, yet social texts and geographic facts are contested terrain. Contemporary practitioners of QM find the notion that science is unified through methods rather disconcerting. While models and analogies continue to be important heuristic devices, metaphorical thinking and reasoning are also constitutively powerful and QM practitioners are as dependent on metaphors (e.g., scale, diffusion, cluster) as agnostics of QM. This dependence by both communities may reflect the fact that metaphors are part of the composition and expression of knowledge whether in the form of scientific or textual language. To illustrate this, the scale metaphor which is widely explored by both quantitative and ‘non-quantitative’ geographers will be discussed. More than half a dozen articles in this journal have been devoted to the scale metaphor among the non-QM community, including discourses on the production and politics of scale. While united by metaphorical thinking, the two communities, however, appear to be divided by incommensurable discourse-making. This is because the transfer of meaning of a metaphor frequently occurs through models and analogies in QM discourses. Metaphorical analysis in QM requires thinking about objects at a cognitive distance (Rosenthal, 1982). While critiques might protest that this distracts the observer from his or her own subjectivity, such thinking however has its merit because it reflects a ‘response to the doubtful’ (Dewey, 1960: 224 – 25). Objects that we adopt with the least objective attitude like water running out of a tap are also objects we respond to without analysis. Responding to the doubtful may have committed QM practice to a paradigm of the unconscious, but this oversimplifies the social practice of the field since models and metaphors are not a dispassionate exercise but involve a process that enables thinking about thinking (Rosenthal, 1982). Casetti’s (1997) expansion method, for example, encourages researchers to think about the theory of errors by providing a systematic way of asking questions about geographical contingency and contextual effects. Errors may arise from a misspecification of the model, but they may also reflect idiographic spatial effects. Through a series of expansions from an ‘initial’ to a ‘terminal’ equation, the model allows geographers to engage in heretical scientific research that involves outlier analysis (that is, idiographic space) since it adopts the philosophical stance that systems of equations capture a more nuanced reality when exogenous variables reflecting geographical contingency are taken into account.2 The utility of metaphorical thinking in QM may be seen in geographical discourses on scale. Thinking about scale among quantitative geographers involves translating the
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metaphor into statements of geographic facts that may be subjected to verification and empirical test, thereby allowing metaphorical reasoning (see Miller, 1979). The adequacy of the metaphor ‘scale’, however, is not always clearly decidable with singular stable meanings as critiques might charge. Indeed thinking about scale involves more than reduction; it also requires discovery about what is overlooked or neglected by metaphorical thinking. Traditionally, for example, geographical scale is associated with cartographic scale. Yet it has been known for some time that geographical patterns observed at one scale may be random at another. Reducing spatial forms and processes to classical or Euclidean geometry is associated with applications of ‘heavy’ tools that do not take into account the fact that variables may influence processes differently at different scales (Lam and Quattrochi, 1992). Metaphorical thinking has been extended into ‘scaling up’ and ‘scaling down’ problems that have in turn led to the development of multiscale models that enables formalization and analysis of processes which operate at different scales – for example, wavelet-based methods or more inferential procedures such as the framework provided by Kolaczyk and Huang (2001). Multiscale geographic phenomena are also increasingly analyzed using multilevel models (Orford, 2000; Huang and Clark, 2002). Data that have a hierarchical or clustered structure, or data that are collected from longitudinal designs, are potential candidates for multilevel modeling. The term ‘multilevel’ should be qualified since there is no single multilevel model but a variety of models that may be used to analyze scale, which in Orford’s case is achieved through a hedonic specification, and in Huang and Clark’s case a logistic regression. Alternatively known as the random coefficient model or hierarchical linear model, multilevel models may be viewed as a hierarchical system of regression equations. Put another way, a twolevel specification involves a micromodel at level 1 which is based upon individual data (e.g., at household or firm level), and, a macromodel at level 2 which is based on group or aggregated data (e.g., at city or state level). A level 1 equation adopts the form of Yij ¼ b0j þ bijXij þ eij where Y is the dependent variable and X the independent variable, i denotes the number of individuals and j is the group or aggregated unit. b0j and bij are outcome (dependent) variables on level 2 so that the coefficients are allowed to be varying and random, that is, b0j ¼ C0 þ U0j and b1j ¼ C1 þ U1j. The relevance of multilevel modeling is that it accounts for a more complex error structure and allows scale to be simultaneously conceptualized and modeled in terms of individual and group data that are hierarchically nested. Thinking about scale has further pushed the boundary of the scale metaphor. Scaling up and down problems have raised important questions on scale in the GIS and spatial analysis literature (see Quattrochi and Goodchild, 1997), and they are largely investigated in terms of a hierarchical ordering or ranking. Pereira (2002), however, has introduced an alternative way of thinking about scale by establishing a typology of scale relations that is more multidirectional and multidimensional. This typology involves the qualitative comparison of two components of scale, namely grain or the unit with which a phenomenon is measured and extent that measures the geographic area of a phenomenon. Retooling in scaling methods has also resulted in the modernization of old tools such as network analytic techniques that were popular in the 1960s and early 1970s. Such modernization allows a more complex scale representation such as social relations to be investigated in world city networks. Rather than focus on city geometry as the major expression of the world city network, Taylor (1999) redirects spatial fixity from the city to the level of the firm, more specifically
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service firms and their global interactions and relations. In so doing, four intercity relational matrices may be identified that permit deployment of traditional network analytical tools to measure world city relations. III Incompleteness and spatial diffusion models In the first report (Poon, 2003), I argued that human geographic research is characterized by imperfect knowledge, and further that, while spatial statistical knowledge is also conspicuously imperfect, it remains nonetheless an indispensable tool of the discipline because it helps compensate for the incompleteness of geographic explanations. It is apparent that, unlike the physical sciences, theories in social sciences suffer from what Brodbeck (1968) refers to as problems of incompleteness and openness. Closure and completeness occur when at any given time the values of any one variable may be computed using laws from the values of all other variables at any other time. Prediction is possible because influences within the system are known, explanation is complete and the system is closed. Completeness and closure are clearly much more difficult if not impossible to achieve in human geography. In the sections to follow, I will show that modeling among QM practitioners has evolved over time to address the absence of completeness with more effort being directed at ways to compensate for incompleteness. To illustrate this, I trace the past and present contours of space-time models. The durability of space-time models in contemporary human geography, pioneered by Hagerstrand (1970), represents a trend towards investigating the behavior of individuals composing the group across space and time. Interest on processes of behavioral change was paralleled if not preceded by dissidents in QM, who, dissatisfied with merely analyzing pattern and equilibrium state, began searching for better ways of studying dynamical systems. Catastrophe Theory illustrates early efforts at modeling temporal and geographical discontinuities rather than a series of linear equations (e.g., Wagstaff, 1977; Casetti, 1981). While Catastrophe Theory failed to gain widespread popularity among human geographers, it contributed nonetheless to an increased recognition that closure is not a realistic goal in quantitative modeling. Catastrophe Theory after all cannot really be used to predict because ‘it is a form of qualitative modeling, mathematically rigorous but not easily susceptible to quantification in many situations . . .’ (Wagstaff, 1977: 192). However, Catastrophe Theory highlighted the possibility of modeling local singularities within a global environment. Equally important, it shifted quantitative modeling to more process-orientated techniques because of its demand for longitudinal data. Focus on spatiotemporal transformations in Catastrophe Theory was superseded by an interest on space-time process dynamics and the latter is now investigated across an array of human geographic phenomena including human accessibility (Miller, 1999; Kwan, 1998; 1999), contagious diseases (Haggett, 1994), technology (Sugiura, 1993), migration (Waldorf and Franklin, 2002; Pellegrini and Fotheringham, 1999) and democracy (O’Loughlin et al., 1998). While space-time models have contributed to the embedding of geographical metaphors such as ‘diffusion’, ‘contagion’ and ‘hierarchy’, it has also seen two important recent developments in process modeling. First, previous commitment to a form of definable group or class as a relevant variable of social, political or
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economic process has moved to variables that take the occurrence of a particular kind of individual into account. This may be illustrated in accessibility studies using travel diary data (Kwan, 1998; 1999). Here, space-time accessibility is based on an individual’s reachable area. Accessibility is investigated in terms of the everyday lives of individuals that build upon person-to-person formulations rather than a more utilitarian-based minimizing model of travel effort towards a reference location, usually the destination. The possibility of research at individual rather than group level is aided by developments in geoprocessing capabilities since the level of detail and thereby computational complexity is more demanding of such data to which Kwan applies a GIS-algorithm. More importantly, it potentially allows geographers to be able to tell smaller stories that reflect the heterogeneous positions and experiences of people – in this case, the stories of White female accessibility in Ohio. This addresses at least part of the criticism leveled at quantitative geographers that they tell only large stories and engage in metanarratives. While Kwan relies on an empirical tradition in her use of travel diary data, Bian (2004) on the other hand employs a more agent-based formulation that models the processes and mechanisms by which contagious diseases spread. Like Kwan, she offers an individual as opposed to a group or ‘population-based’ spatial diffusion models to describe the spread of infectious diseases. Individual-based models are concerned with individual movements and the heterogeneous environments whereby individual interactions occur. This requires a shift in emphasis towards more discrete as opposed to continuous time periods or events. More importantly, it requires a more stochastic approach in modeling the spread of diseases. The incorporation of stochastic processes introduces the possibility of chance but is consistent with the theory of errors. On the other hand, the potential of dealing with individuals as opposed to macroscopic aggregates allows geographers to interrogate spatial experiences and interactions at the individual level indicating that not all quantitative work is about holism and organicism. An important development in the more open system research of contemporary quantitative geographers is therefore the increased use of microdata where analysis is at the level of the individual, household or firm so that spatial representation is more sensitive to gender or, in migration studies, family circumstances between husbands and wives (e.g., Clark and Withers, 2002; Cooke, 2003; Fan, 2002) and technological change at the plant level in economic geography (Rigby and Essletzbichler, 2000). In sum, not only have space-time geographers attempted to compensate for knowledge incompleteness with more individual-based models recently, but they have also exercised a greater sensitivity to the use of numerical data that combines with textual evidence to explain an event or process. An example of this may be found in Smallman-Raynor and Cliff’s (2001) study of the spread process of typhoid fever in the 1898 Spanish-American war. They supplemented space-time modeling with textual reports and the testimonies of relevant actors (e.g., physicians) to identify meaningful narratives that helped explain how and why a series of events established under spatial diffusion models came to be. IV Discussion and conclusion In describing the development of quantitative modeling in human geography, Casetti (1999) proposes that three modes of inquiry describe the field, namely,
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literary, academic and mathematic. A literary mode of inquiry includes descriptions and narratives, an academic mode of inquiry uses a systematic development of verbally and textually formulated explanations and theories, and a mathematical mode of inquiry establishes analytical structures that elicit relevant variables and the relationships between them. He further suggests that ‘The theories and research questions that emerge from the literary and the academic modes of enquiry constitute the two major sources of mathematical models’ (p. 334). Accordingly, this view interprets the main objective of QM to be the building of logical structures of geographic explanations that cull from a diversity of sources including textual sources but where the latter are ordered in favor of analytic outcomes. Such a view may be distinguished, for example, from the social practice of neoclassical economists, more specifically the new economic geographers, whom geographers have criticized for returning geographic inquiry back to equilibrium outcomes and static analysis (Martin, 1999). Indeed, as the review of the evolution of space-time models has revealed in this report, geographers are largely hesitant about the goal of closure, despite a trend to reintroduce this principle back to spatial analysis in the new economic geography.3 Space-time models have reaffirmed that dynamic processes are worth thinking about as the call for a more nonequilibrium way of explaining geographic structures and processes that began perhaps with Catastrophe Theory becomes more popular (Plummer and Sheppard, 2004). Geographic inquiry has also provided fertile ground for the cultivation of spatial metaphors. Metaphorical thinking and reasoning in QM reflects cognitive processes that, when developed into clear analysis, provides a basis for discovery. This is because metaphors need not be exact or proportional in the way that analogies in principle need to be (Miller, 1979), hence the significance of metaphors is in its breadth rather than the compressed meanings that quantitative discourses are said to create. I have illustrated this with scalar thinking in QM. The illumination that a spatial metaphor provides tends to operate through verification among quantitative geographers and the reasoning itself subjected to a critical rather than discursive process. In sum, the present direction in analytical formalization appears to be moving towards a more critical scrutiny of general linear reality with the effect that scientific ambitions have been downsized though not abandoned. The traditional chasm that separates literary and modeling modes of inquiry need not pose as barriers to intellectual trade. It is unlikely that human geography will return to a Braudelian revolution that has described more humanistic disciplines like history as some historians convert to a more analytical model of inquiry (Franzosi and Mohr, 1997); there is nonetheless the possibility of more middle ground between analytical models engaged in the production of stylized facts and system-level implications, and those that emphasize and overly build substance into models of individual behavior with the consequence that it is difficult to explain any outcome (see Baron and Hannan, 1994). Caricatures of the QM position have been to hold verification and empiricism as a sufficient condition for the reliability and generalization of geographic phenomena. Yet, as our colleagues in the physical sciences have reminded us about models of climate change, while the QM community may hold the view that testability through empiricism may be a necessary though not a sufficient condition for reliability, the reasoning leading to it is fraught with caveats (Schneider, 2001). Quantitative methodology too tells a story.
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Acknowledgements I would like to thank David Rigby and Peter Rogerson for their comments and discussion of an earlier draft of the report. Notes 1. On imperfect explanation, Haggett (1994: 18) writes that ‘. . . a little speculation and a willingness to make mistakes is a necessary though not sufficient ingredient for the predictive scientist . . . Perhaps historians of our discipline will look back on the last few decades and say that we erred not in making mistakes, but not in having the courage to make nearly enough.’ For a further elaboration of reconstructed logic and a critique of knowledge being evaluated in terms of scientific idealized logic, see Gunnell (1969). 2. The notion of geographical context under the expansion method can be, though not necessarily, modeled as an exogenous effect. Plummer and Sheppard (2004), however, have argued that space should be endogenized in geographical models. 3. Fujita and Krugman (2004: 141) write that ‘The goal of the new economic geography . . . is to devise a modeling approach, a story-telling machine that lets one discuss things like the economics of New York in the context of the whole economy’. The economist’s emphasis on general-equilibrium is consistent with big stories and metanarratives but it also reflects ‘how differently economists and traditional geographers treat/understand geographic space’ (p. 150). Furthermore, scientific practice has a different goal in that establishment of a microeconomic foundation is important in economic model formulation.
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