Quantitative methods I: Reckoning with uncertainty

In this report, I address the timeless topic of uncertainty. Quantitative methods are in flux and uncertainty provides a useful vantage point for assessing these changes. I draw particular attention to two forces at odds with each other: on the one hand, innovative methods and data that allow researchers to embrace uncertainty, and on the other, recent trends towards deliberate amplification of uncertainty, in the form of differential privacy methods which potentially limit our capacity for measuring and understanding the world around us. I close by highlighting some implications for users and producers of quantitative methods and data in geography.


I Introduction
If there is one certainty in quantitative methods, it is that uncertainty is always with us. Hidden behind the analytical curtains, hard-wired into data collection and interpretation, and fundamental to both methodological and conceptual development, uncertainty shapes every stage of the research process in quantitative human geography. Indeed, the existence of uncertaintyhere defined broadly as the gap that exists between real-world, 'true but unknown' values and relationships and what we are able to observe, given the methods and data availableis our raison d'être as researchers; were we able to observe true but unknown values directly, there would, after all, be no need for further investigation.
Of course, quantitative methods are not unique in this respect; from assorted perspectives and positions, all of geography wrestles in some fashion with uncertainty (Fusco et al., 2017). This universality is, in fact, one impetus for the focus on uncertainty in this report on Quantitative Methods. The ways in which quantitative methods approach uncertainty, and how it is implicated in contemporary advances and challenges, are meaningful not only to this particular corner of the discipline, but to all of geography.
If uncertainty is a constant in quantitative methods, why write about it now? The answer is that quantitative methods are in flux and one way of comprehending the changes that are occurring is through an uncertainty lens. For one thing, the range of methods employed in quantitative geography is undergoing rapid expansion, due to growth in data types and provenance but also related to subtle changes in the types of research done in the field. This expansion both maintains traditional relationships with uncertainty but also introduces new, important ways in which it matters.
Second, where arguably many traditional quantitative approaches subsumed uncertainty under a veneer of normative truthwhat Poon (2005) refers to as 'the monopoly of logical positivism as the central way of knowing'newer methods explicitly address and incorporate the uncertainty of reality. This is exciting research that bridges the conceptual and the methodological, often leveraging increased availability of high-resolution spatial data. Continued innovation is highly contingent on the availability of certain kinds of data.
However, data are also changing. This is, of course, well known where 'big data' are concerned (McAfee et al., 2012). This report focuses on other landmark shifts that are also occurring, as traditional data providers (such as governments) make existing uncertainty estimates more visible and, more recently, as they wrestle with new forms of strategic injections of uncertainty into data as a way of maintaining respondent confidentiality, whilst still aiming to provide high-quality data as inputs not only to research but also policymaking, election redistricting and localarea funding mechanisms. These efforts have particularly strong ramifications for geographers and others who depend on small-area data (such as census tracts or output areas). Uncertainty has never been more important.
In this first of three reports on quantitative research methods, all organized around the themes of flux and continuity in quantitative methods, 1 I outline the bedrock role of uncertainty in quantitative methods, emphasizing some principal elements. I then turn to recent research that specifically aims to reckon with uncertainty and reflect on the implications for the data we use, and potential challenges on the horizon.

II Uncertain foundations
To understand where we are, it helps to look at how we got here. Much has already been written about uncertainty in quantitative methodsit is an evergreen subject. This reflects its importance, but the range of perspectives adopted also underscores that it means different things to different people (Fusco et al., 2017). One important distinction is that made by Derbyshire (2020), who highlights the difference between epistemological uncertainty, or 'the accuracy of what we know presently', and ontological uncertainty, which, in my own paraphrasing, can be thought of as changes to the 'true but unknown' world. Most commentary on uncertainty in quantitative methods, including this report, is focused on the formerthe factors that create a gap between what we observe or model, and the actual phenomena we are studying. In GIScience, this includes concerns with error propagation (Heuvelink, 2002) and spatial information (Goodchild, 2018). Griffith (2018), one of many to list and classify potential sources of uncertainty, identifies the following: calculation, measurement, specification, sampling and stochastic. Wei and Murray (2012), in discussing uncertainty in spatial optimization methods, identify many of the same sources, but focus on the need to explicitly account for and measure uncertainty in existing models. From yet another standpoint, in this case, physical geography and environmental science, Brown (2010, p. 77) states that uncertainty can be defined as 'a state of confidence. Here, confidence is defined in the broadest sense of (degree of) trust or conviction in knowledge, which includes the narrower sense of "statistical confidence"'. This is a helpful definition, which stresses that uncertainty is about much more than statistics or models, and, in its essence, comes down to confidence in our results, as quantitative human geography researchers. The range of definitions of uncertainty is also important. It signifies that, although we may all arrive at the same destinationthe central role of uncertainty in quantitative methodsmany have travelled different routes to reach it.
Coming to terms with uncertainty, if we concur that it has many guises and impacts research in a variety of ways, can be difficult. O'Sullivan (2004), writing about complexity and human geography, suggests that acknowledging the (deterministic) uncertainty that arises from the unpredictability of systems highlights 'the futility of prediction' (p. 283) but also opens up new possibilities for models as 'thought experiments' or narrative tools. Stepping back and surveying the whole of quantitative methods, this insight suggests that uncertainty in our methods is not a deal-breaker, but rather a tool that can help produce unexpected insights. Or, in common parlance, uncertainty is a feature, not a bug.
Adopting Brown's (2010) definition of uncertainty a state of confidence in knowledgesome interesting components of uncertainty in quantitative methods emerge. Although they may not be as visible to those outside the field, these are topics that, among quantitative human geographers, animate discussion about reliability of research findings. Among them are: statistical uncertainty, sampling and representation uncertainty, construct uncertainty and the modifiable areal unit problem (which, like uncertainty, seems to always be with us).
The most obvious of these is statistical uncertainty. By their very nature, inferential statistics are about measuring confidence in knowledge; models attempt to approximate reality, but as abstractions they are never perfect. (And mainstream machinelearning methods are little better; classic estimates of error or uncertainty may be absent, but that does not mean the uncertainty is, too.) A still-common criticism of geographical models and modellers is that they tend to treat analytical results as truth. In reality, quantitative geography has long moved away from rigid adherence to logical positivist rules around the search for generalizable laws (see, e.g. Poon, 2004or Phillips, 2004. Instead, spirited conversations about model integrity are frequent and tend to revolve around the roles of model specification and fealty to underlying assumptions and interpretationall likely to undermine confidence and increase overall uncertainty. Beware the researcher who says they have proved something! Data matter, too. Smaller numbers of observations mean increased uncertainty about behaviours, preferences and interactions. Statistics can help quantify this uncertainty, but only up to a point, because the companion to sample size is representativenessthat the data observed capture the full realm of the phenomenon or population being studied. For quantitative human geographers, this has always entailed a double dose of uncertainty: social, but also spatial. Nationally representative samples, for example, may accurately (with some accepted level of uncertainty, of course) characterize a country as a whole, but not any one place. As quantitative geographers have expanded the types of data they employ to include forms of 'big data', an argument is sometimes made that large sample size obviates concern about uncertainty, however, the reality of lack of representativeness, selection bias and other forms of missingness mean that uncertainty is still thereonly perhaps more difficult to measure. Bigger data can entail bigger problems (Graham and Shelton, 2013).
Less talked about, where uncertainty in quantitative methods is concerned, is that which derives from the social construction of classifications and variables typically employed in quantitative analysis (and elsewhere in the discipline!). As (Robbin) puts it, in the opening of her Robbin, 1999 paper, 'The routine production of statistical information reinforces a sense that the measures are real, the properties of categories invariant, and their meaning unproblematic. The contrary is, however, the real-ity…' (p. 467). D'ignazio and Klein (2020) make a similar point where gender is concerned: 'And while the gender binary is one of the most widespread classification systems in the world today, it is no less constructed than the Facebook advertising platform or, say, the Golden Gate Bridge…all these structures were created by people: people living in a particular place, at a particular time, and who were influencedas we all areby the world around them'. Where race and ethnicity categories are concerned, this shortcoming is well known (Mateos et al., 2009); censuses, administrative data and surveys may allow for self-identification of race, for example, but the categories from which respondents must choose are not pre-ordained. This, in turn, implies that characteristics such as population composition, diversity, or segregation have a degree of uncertainty embedded in them, independent of choice of analytical method. The same holds for constructs like migration and, increasingly, for those we may have previously taken for granted, such as employment or occupation.
Often researchers have to grapple with applications that concern not only individuals, but also areal aggregates. This deepens the potential for uncertainty: not only from validity of inputs, but sample size, methodological approach, and the modifiable areal unit problem (MAUP), as well. MAUP is defined as a boundary and a zonation challenge (Openshaw and Taylor, 1981) but can be summarized in lay terms as, 'the choice of spatial units may affect results'. Ideally, researchers match spatial unit to the process under investigation; the distance between what is measured and what is actually occurring is minimized. In reality, researchers typically adopt the spatial units that are available and best match the hypothesized scale of the phenomenon being studied. The outcome is that a degree of uncertainty adheres to the findings.

III Uncertain methods
Recent research in quantitative human geography has capitalized on uncertainty and the ways in which it confounds our capacity to model the human experience uncertainty as a feature. Kwan's (2012Kwan's ( , 2018 contribution to our understanding of the 'uncertain geographic context problem' is an example of such research, but so is Brunsdon, Fotheringham, and Charlton's (1998) Geographically Weighted Regression (GWR), which acknowledges the uncertainty that underlies global regression estimates that presume spatial stationarity between determinants and outcomes. Indeed, over the past decade or so, uncertainty challenges related to spatial scale and context have been at the forefront of quantitative methods development. Representative examples in my own population subfield from a very large literature include: Reardon et al. (2008) and Fowler (2015) on multi-scalar segregation profiles, Clark et al. (2015) on segregation and diversity, the measurement of neighbourhood context (Andersson and Malmberg, 2014), and neighbourhood definition (Spielman and Logan, 2013).
A flurry of recent research expands on the quantification of uncertain exposure, context and neighbourhood, building off the consensus that administrative units such as census tracts, blocks, or output areas are insufficient proxies or containers for actual, spatial lived experience. Rather than amalgamations of units, buffers, or comparative analysis across multiple spatial scales, newer research pairs very high-resolution data with multi-scalar methods to more clearly home in on hypothesized neighbourhood or contextual effects. These methods do not necessarily depend on 'big data' in the sense that we have come to understand it, but rather 'sensitive data' in that location, movements and temporal variations are finely measured.
In fact, the lead role of the data is what distinguishes much of the recent innovative research working with uncertain contextsor, rather, data and methods have been cast as dual, highly co-dependent, leads. Victoriano, Paez, and Carrasco (2020), for example, use machine-learning methods to characterize mobility strategies. Their seemingly small set of participants (165) belies the complexity of the data: a 7-day travel diary for each results in 1128 days of data and over 16,000 trips and activities. Fowler et al. (2020) address contextual effects in segregationin many ways similar to research contributions noted above, except that they rely on secure access to individual-level census data for their analysis. Their emphasis is on what they term the 'contextual fallacy', the extent to which individuals in the same spatial unit (e.g. census tract) have different contextual experiences. In a similar vein, Petrović et al., (2018Petrović et al., ( , 2021 estimate multiscale contextual effects, but starting from individuals located at a resolution of 100 m by 100 m. Pearce (2018), on health and exposure, expands the conceptual uncertainty bounds to include both space and time, arguing that exposure accumulates over the life course, which is a function of both time (years lived), but also space (residential stability and mobility). As Pearce highlights, to more accurately account for potential exposure over the life course requires data on individual trajectories over very long timespans, but also environmental data over time and at sufficient spatial resolutions to be able to account for individual context. This is not primarily a methodological challenge, but a data challenge. Folch, Fowler and Mikaelian (2021) also consider how uncertainty in context and exposure can be measured, estimating air and water toxicity and child mobility, both over time and from day to day (home versus daycare location). Their analysis explicitly addresses questions around uncertainty: positional accuracy of children, relative to toxicity measures, neighbourhood context and child mobility. This is only one narrow slice of the data-method innovation nexus, but of course similar revolutions are occurring across the quantitative methods spectrum. Increased availability of location data, whether from devices or satellites, vastly expands our capacity for modelling the real world, which in turn demands novel methods for bridging theory and data. Uncertainty is a leitmotif: highly certain locational attributes, likely increased uncertainty on other dimensions such as representativeness, and the possibility of new methods that help render visible the uncertainties that surround so much of human behaviour, interaction and systems.
The tension between methodological advances, uncertainty and data requirements is nicely encapsulated in Petrović et al., (2020)

in this journal.
Speculating about what sorts of data would be necessary to test existing theories about the importance of neighbourhood effects for a range of outcomes, they land on the importance of quantitative methodsmulti-scalar measuresbut also microgeographic data. And herein lies the potential problem: contemporary quantitative methods permit more nuanced and sophisticated understanding of a range of geographic and social phenomena, thereby hopefully decreasing uncertainty, or increasing our confidence in knowledge. In parallel, however, the availability of high-resolution data required for these methods increasingly runs against the grain of heightened perception of possible loss of privacy and confidentiality. One solution to these concerns is to shift uncertainty onto the data via differential privacy tools, thereby protecting individuals. This deliberate insertion of error into existing data products has repercussions not only for quantitative methods, but all of geography and a range of civic stakeholders and policymakers, as well.

IV Uncertain data
There is a recent precedent for uncertain data to disrupt geographical research. In the United States, the bread and butter of geographical analysis has long been U.S. Census data products, whether microdata, reflecting individual characteristics, or summary files for a range of geographical units, from states to counties to census tracts, block groups and blocks. Until 2010, research relied on short form (full census counts but for limited variables) and long form (sample data on a range of questions, including migration, educational attainment and income).
Uncertainty was a factor for these estimates, even for the full count dataundercounts have always presented a challenge for particular sub-groups and places. Data tables for geographic areas presented only point estimates, however, and no margins of error (MOEs), even for data based on long form sample data. Thus, although researchers were in theory aware of data limitations, in practice the data were often treated as fact. This changed with the advent of the American Community Survey (ACS), which came fully online in 2005 and entirely replaced the long form for the 2010 decennial census. The ACS offers advantages over the long form: rather than providing important socio-economic updates once a decade, the ACS is a rolling survey that goes out to a sample of households every month. This provides timelier information and, for researchers, has helped ensure that analysis is not wildly out of date by the time results are published. 2 However, the ACS sample size is considerably smaller than that for the long form and this has had two important impacts.
First, for the first time, MOEs were published alongside estimates, disclosing to many researchers the very unstable (read: uncertain) ground upon which they were conducting their neighbourhood and local analyses. The visibility of uncertainty was problematicshould researchers and local stakeholders simply ignore the MOEs? As Jurjevich et al. (2018) have shown, many users of ACS data, including local planners and stakeholders, simply do not know how to interpret the uncertainty embedded in MOEs. Practically speaking, wider margins of error mean more uncertainty, so that it can be difficult to know what the 'true' neighbourhood characteristics of a place actually are. Spielman and Singleton (2015) propose composite or geodemographic measures for neighbourhoods that combine characteristics to provide more certainty for local areas, but this is a limited solution.
Second, and most importantly for quantitative spatial researchers, ACS uncertainty is not constant across spacesome areas have higher uncertainty in estimates than others. As Folch et al. (2016) document, uncertainty varies both locally and regionally and, crucially, appears to be related to the characteristics of places (more uncertainty in lower-income areas, for example). Whilst this has clear implications for those working with the full universe of areas, it can also affect basic understanding of individual places: 'For example, in census tract 190,602 in the Belmont Cragin neighbourhood of Chicago, Illinois, the number of children in poverty is somewhere between 9 and 965 (2006-2010 ACS estimates)' (Folch et al., 2016(Folch et al., : p. 1537. My point here is that methodological advances do not exist in a vacuum, but rather in concert with changes to data infrastructure. The example given here is from the United States but is not likely to be an isolated case, given evolving conversations around data and privacy in the age of big data and fast-growing private and public surveillance apparatuses. Whether data become newly uncertain or simply have a light shone on their pre-existing uncertainty (in the ACS it was sadly both), this impacts both the types of analytical advances that can be expected and the confidence we as researchers can have in our results. Moreover, where data and uncertainty are concerned, this is in many ways a bestcase scenario. Government providers have strict protocols for quality assessment; other data providers have no such responsibility to make uncertainty visible or prominent in their offerings.

V Uncertainty is dead; long live uncertainty
It is one thing to confront the imperfections of existing data sources, such as the American Community Survey. It is another entirely to face the prospect of data deliberately rendered more uncertain, and yet that is where we are. Quantitative methodological innovation works in tandem with increased sophistication of data, both 'big' and 'sensitive'. With these innovations comes increased risk of violation of individual privacy and confidentiality, as they are understood (and socially constructed) in today's society. This is a particular challenge for government data providers, many of whom have a legal obligation to guarantee confidentiality. Either our expectation of privacy must evolve or the data will have to.
The test case for this new reality is, again, the US Census Bureau. With the 2020 Census, the Bureau will introduce additional error to all statistics for areas below the state level, in a process termed differential privacy (Ruggles and Van Riper, 2021). This represents a major shift in data provision. 3 As Hawes (2020) puts it, 'Consumers of official statistics, particularly those who use data products that have been produced for a long time, are accustomed to the data looking a certain way, and to interpreting those data as the "ground truth." As such, they are unaccustomed to seeing population counts with fractional or negative values'.
Quantitative geographers may be among the most impacted researchers, given their focus on aggregate geographic data, but the Census Bureau Disclosure Avoidance System and differential privacy are likely to have wide-reaching effects for non-quantitative researchers and policymakers, as well. Preliminary research has suggested that uncertainty may be much higher for certain places and groups, for example, indicating decadal population loss when none has occurred, for small towns and indigenous areas (Wezerek and Van Riper, 2020) or mis-characterizing population counts and characteristics in political redistricting (Kenny et al., 2021). In estimating possible effects on county-to-county migration data, Winkler et al. (2021) find that uncertainty is potentially higher for Hispanic migrants, as well as the young and old. Smaller-population counties, rural areas and the Great Plains of the United States may also be differentially affected by more uncertain data. The effects of differential privacy on data provision may be widespread and pernicious. Evaluating the impacts of the COVID-19 pandemic on mortality rates, Hauer and Santos-Lozada (2021) emphasize the importance of reliable denominatorsthe source of which is often Census Bureau datain constructing age-specific mortality rates.
The Census Bureau differential privacy example is just one front on a wide-ranging debate about the high-resolution, fine-grained data that nourish the development of methods and theory in quantitative human geography, but it may be a harbinger of things to come. Although the preliminary assessments of the Bureau's algorithm suggest that the risk of disclosure is no higher than what would be expected at random (Ruggles and Van Riper, 2021), the direction of the prevailing wind is clear: the days of (relatively) easy access to reliable high-resolution data may be limited. And whilst this may be a boon to individual privacy, the downsides are also evident: not only increased uncertainty in data, which hampers development of quantitative methods and knowledge, but alsoquite likelyan entrenchment of alreadyunequal privileged access to high-quality data, and a reinforcement of societal inequities that mean some groups and places are better measured (and understood) than others.

VI Conclusions: With great data comes great responsibility
Research in quantitative human geography has blossomed over the past several years, as better data and computational ease have facilitated engagement with tricky questions that, theoretically anyway, have long entailed a high degree of uncertainty. This report has scarcely scratched the surface of dynamic and conceptually rich research currently published across the continuum of quantitative human geography. And yet, as this report has shown, new data-related dilemmas are emerging which, although they may not completely disrupt innovation, may very well introduce new forms of uncertainty. For example, big data require curation in order to be easily usable and the data engineering that underpins this curation is often opaque where it should be transparent (Arribas-Bel et al., 2021). In addition, where quantitative researchers have traditionally been consumers of data products, recent method developments, especially on uncertain context, indicate that we may soon be producers of bespoke geographies, such as neighbourhoods. How will we make clear the uncertainties embedded in these geographies?
Equally importantly, how can we contribute to conversations that help disentangle the needs of the few and the needs of the many, where data are concerned? High-quality data provision is not only about arcane model development and researcher privilege; it is also about research that feeds equitable policy development and visibility of under-represented groups. As D'ignazio and Klein (2020) emphasize: 'What gets counted counts'. Uncertainty is an intrinsic component of quantitative research, but new emphasis on differential privacy methodsalthough laudable from an individual privacy perspectiveposes very real risks not only for geographical research but also for disadvantaged groups and places who rely on accurate numbers and statistics as a form of representation.