Visualizing Statistical Significance of Disease Clusters Using Cartograms
Kronenfeld and Wong Int J Health Geogr (2017) 16:19 DOI 10.1186/s12942-017-0093-9 International Journal of Health Geographics
METHODOLOGY Open Access Visualizing statistical signifcance of disease clusters using cartograms Barry J. Kronenfeld1 and David W. S. Wong2*
Abstract Background: Health ofcials and epidemiological researchers often use maps of disease rates to identify potential disease clusters. Because these maps exaggerate the prominence of low-density districts and hide potential clus- ters in urban (high-density) areas, many researchers have used density-equalizing maps (cartograms) as a basis for epidemiological mapping. However, we do not have existing guidelines for visual assessment of statistical uncertainty. To address this shortcoming, we develop techniques for visual determination of statistical signifcance of clusters spanning one or more districts on a cartogram. We developed the techniques within a geovisual analytics framework that does not rely on automated signifcance testing, and can therefore facilitate visual analysis to detect clusters that automated techniques might miss. Results: On a cartogram of the at-risk population, the statistical signifcance of a disease cluster is determinate from the rate, area and shape of the cluster under standard hypothesis testing scenarios. We develop formulae to deter- mine, for a given rate, the area required for statistical signifcance of a priori and a posteriori designated regions under certain test assumptions. Uniquely, our approach enables dynamic inference of aggregate regions formed by combin- ing individual districts. The method is implemented in interactive tools that provide choropleth mapping, automated legend construction and dynamic search tools to facilitate cluster detection and assessment of the validity of tested assumptions. A case study of leukemia incidence analysis in California demonstrates the ability to visually distinguish between statistically signifcant and insignifcant regions. Conclusion: The proposed geovisual analytics approach enables intuitive visual assessment of statistical signifcance of arbitrarily defned regions on a cartogram. Our research prompts a broader discussion of the role of geovisual exploratory analyses in disease mapping and the appropriate framework for visually assessing the statistical signif- cance of spatial clusters. Keywords: Cartograms, Density equalizing maps, Disease mapping, Scan statistics, Geovisual analytics
Background confrmatory statistical techniques in which potential In epidemiological analysis, there is a long history of cluster regions must be designated a priori. Maps also mapping disease and related health and socio-demo- provide spatial context and present data in a visual form graphic data. Maps are used to explore spatial patterns that is accessible to the public. Te need to respond to and identify potential causal factors, which then may public concern has spurred development of automated warrant further investigation by cohort and case–control systems to construct maps of disease occurrence [2, 3]. studies [1]. Because human visual perception is adept at Tough maps can help identify disease clusters, there spatial pattern recognition, epidemiological maps may is much controversy over their use. Map readers likely facilitate detection of clusters that might be missed using see clusters that may just be the results of random vari- ation, or a mild degree of clustering of similar values. Yet *Correspondence: [email protected] further investigation to determine their statistical sig- 2 Geography and Geoinformation Science, George Mason University, nifcance requires substantial resources [4, 5]. For this 4400 University Drive, Fairfax, VA 22030, USA Full list of author information is available at the end of the article
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/ publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 2 of 13
reason, the communication of statistical uncertainty is an identify and assess clusters that are unknown in advance important problem in disease mapping [1, 2, 6]. and might not be detected by automated search tech- Although the problem of visualizing uncertainty is niques. Most standard tests of statistical signifcance broadly recognized by cartographers [7], it is particu- apply to a priori defned districts and are therefore prob- larly signifcant in disease mapping due to the rareness of lematic in an exploratory framework due to the issue of disease events and the non-uniform spatial distribution multiple hypothesis testing. To support estimation of the of the underlying population. Tese factors compound statistical signifcance of a posteriori defned districts, each other, as a standard map may place disproportion- we use scan statistics that provide a stricter test based on ate emphasis on geographically large but sparsely popu- the likelihood of a given cluster occurring anywhere on lated districts, and these same districts will tend to have the map under the null hypothesis. A measure of com- relatively unreliable observed disease rates as refected by pactness is further used to mitigate problems introduced their large standard errors or related statistics [1]. Map when irregularly-shaped clusters are considered. readers will therefore have difculty visually distinguish- ing between statistically signifcant clusters and areas Statistical signifcance in maps that have high disease rates but are not statistically sig- Te importance of communicating uncertainty and/or nifcant due to small sample size. statistical signifcance associated with disease rates is One method to avoid placing disproportionate visual widely appreciated in the epidemiological community. emphasis on geographically large but sparsely popu- While any location with a high rate represents a poten- lated districts is to use a density-equalizing map, or “car- tial area of concern, areas in which the observed rate is togram”. A cartogram is a map in which district shapes unlikely the result of random variation should be given are altered such that their size is proportional to a des- higher priority than those that fall within statistical con- ignated variable, typically population. Tis results in a fdence intervals [6, 13]. Despite this recognition, con- uniform density of the population variable, and so car- structing maps that efectively communicate observed tograms have also been referred to as density-equalizing rates and associated uncertainty simultaneously has maps [8]. When used as a basis for disease mapping, proven to be a difcult challenge [7]. Attempts to meet population cartograms reduce the size, and therefore this challenge have included both statistical and visual the visual prominence, of low-population districts in approaches, each of which has strengths and weaknesses. which high disease rates might occur due to chance alone Statistical approaches involve the construction of a sin- [9–12]. gle variable that communicates some degree of both the While visual emphasis can infuence human percep- estimated magnitude of risk (i.e. efect size) and associ- tion, a more concrete approach is needed to communi- ated uncertainty (i.e. signifcance). A simple approach is cate statistical uncertainty. In this paper, we extend the to map statistical signifcance values rather than crude cartogram visualization framework to develop a formal disease rates. Te drawback of this approach is that the visual assessment method for determining the statistical infuences of disease rate and sample size cannot be dis- signifcance of observed disease rates over user-defned tinguished [14]. For example, the same signifcance value regions. Our approach is based on the fact that statistical might indicate a very high disease rate in a small sample variance is normally an inverse function of population, or a slightly elevated disease rate in a large sample. An and therefore statistical signifcance is determinate from alternative is to use Bayesian techniques to adjust the the population and observed disease rate of a district. relative risk estimates on the basis of sample size and/ Given this, we propose using a cartogram to show popu- or neighborhood [14, 15]. In this approach, a prior risk lation combined with standard choropleth symbolization ratio estimate is constructed based on either the overall to show observed disease rates, thereby visually commu- rate or the rate observed in a predefned neighborhood nicating the necessary data to determine statistical signif- around each district. Tis prior value is then updated icance. Te resulting visual framework does not require based on the observed rate and sample size in each dis- use of coincident or overlapping symbols that have been trict, again sometimes including neighboring districts, shown to reduce map readers’ ability to efectively per- resulting in a posterior risk estimate that is moderated by form analysis tasks. sample size and neighborhood efects. Bayesian estima- An advantage of the proposed geovisual analytics tion techniques are now widely used in health mapping framework is that it supports on-the-fy determination [2], but map readers must be aware that the smoothing of statistical signifcance for user-defned regions con- efect of these techniques reduces efect size in favor of sisting of multiple districts or portions of districts. As stable estimates [1]. Specifcally, the rate estimate shown such, it supports truly exploratory analysis, exploiting the in each district will be a weighted average of the actual map reader’s visual perception and spatial cognition to rate in that district, rates in nearby districts and the Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 3 of 13
overall mean rate [14]. Te exact nature of this tradeof is many such districts, one or more district will inevitably dependent on the underlying model [15]. appear as a disease hot-spot even when the distribution In contrast to the statistical approach that combines of disease occurrence is random across the population [6]. information from the observed rate and sample size into Conversely, heavily populated districts may not be visually a single signifcance value, the visual approach attempts prominent if they have small areas, even if their disease to communicate two values (e.g. rates and signifcance rates are high and statistically signifcant. levels) simultaneously. Since using two maps to display Another limitation of current methods is that they only the two sets of values requires additional space or a scale encode uncertainty values at the level of the individual reduction of maps, a preferable method is to put both district. In practice, disease clusters may encompass mul- sets of values on a single map using symbol overlay. Sev- tiple districts. However, it is impossible to visually deter- eral health agencies produce online maps with texture mine the uncertainty or statistical signifcance associated overlays (e.g. stipple marks, diagonal or cross-hatch lines) with an aggregate region from symbols that encode val- on top of a choropleth (sequential color-coded) map. ues for each district individually. Tere is, however, signifcant variation in the type of symbol used and their meaning. For example, the United Methods Kingdom Small Area Health Statistics Unit’s Rapid Many researchers have proposed cartograms as a solu- Inquiry Facility (RIF) and the California Cancer Registry tion to the problem of variable population density in (CCR) both use prominent black dot stipple overlay, but health mapping [20, 21]. Tere have been two distinct in the RIF the overlay varies in density and indicates dis- approaches to cartogram usage in health research. In one tricts with rates that are more statistically signifcant [6], approach, disease-related events are used as the desig- whereas the CCR uses a single stipple density to indicate nated cartogram variable, creating an event cartogram. districts with less stable estimates [3]. On an event cartogram, districts with high numbers of Cartographic studies are mixed in their assessment events appear larger on the map. Recent studies have of the efectiveness of maps with overlapping symbols. used event cartograms to visualize global incidence of Some studies have found that users perform worse on drowning [22], mortality from ischaemic heart disease analysis tasks using a single map with uncertainty infor- [23], as well as global variations in research efort related mation overlaid onto each district than when uncertainty to health issues, including air pollution [24], trafc acci- is presented separately on an adjacent map [7]. In addi- dents [25], heat-related illness [26], yellow fever [27], tion, the approach is limited practically by the difculty infuenza [28] and silicosis [29]. Te visual efect of resiz- of placing texture symbols in small polygons/areas. How- ing countries on a world map based on the number of ever, other studies indicate that overlaid maps are more event occurrences is striking: countries with more events efective than adjacent maps to display both the mapped appear enlarged, while countries with fewer events are values and associated uncertainty statistics [16, 17]. An visually shrunken. alternative is to take advantage of the multi-dimensional Unfortunately, many problems emerge with event nature of color by using desaturation, transparency or cartograms when one realizes that map readers must blurring techniques to adjust an underlying sequential disentangle the efects of land area, population, event (e.g. light-to-dark) color scheme [7]. Some studies have frequencies and event rates. Any district with a high fre- found transparency to be efective at communicating quency of events relative to its land area will be enlarged uncertainty in a qualitative sense [18, 19], but the degree on an event cartogram. Tis will include not only dis- of precision with which users can decode transparency tricts with high event rates per population (i.e. event hot- into a numerical value are still poorly understood. spots), but also districts with high population per unit A related problem is the disproportionate visual empha- land area (i.e. population hot-spots). Since usually we are sis of districts based on their geographical area. Statisti- interested in detecting event hot-spots, only the former cal districts that are relatively large in area may be sparsely cases are of interest. A variation on this approach is to populated, and can dominate visual perception despite use event rates instead of event counts as the cartogram representing only a small percentage of the population. variable [22, 28, 29]. However, using rates discards popu- In the USA, for example, 75% of the population resides lation size information and can therefore over-emphasize in census tracts that occupy only 3.5% of the land area small population districts with high event rates that may (according to 2010 data from the U.S. Census Bureau). be the result of random variation. Te fact that small Tis highly uneven distribution exacerbates the problem countries do not appear dramatically enlarged on pub- of communicating uncertainty, since random efects will lished event rate cartograms is almost certainly due to be greater in the large but sparsely populated districts defciencies in cartogram algorithms that enlarge small that tend to dominate visual perception [1]. In a map with areas sufciently on the map [30]. Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 4 of 13
Te second approach difers from the frst in that the a layperson’s (i.e. informal) concept of signifcance. From underlying at-risk population, as opposed to the number a cartographic perspective, a given event rate is perceived of events, is designated as the cartogram variable, cre- as more indicative of a meaningful pattern if it occurs over ating an at-risk population cartogram. Event locations a larger map area. In formal hypothesis testing, the statis- or rates are then displayed on the top of the cartogram tical signifcance of an observed rate increases monotoni- using standard mapping techniques, such as dot-density cally with and is often (but not always) fully determinated mapping or sequential choropleth color-coding. In this by sample size. Tus, a possible approach to visualizing manner, the cartogram serves as a base map depicting uncertainty is to designate sample size as the population the domain within which health events occur, rather than variable of a cartogram, allowing the map reader to asso- as a measure of the health events themselves. For exam- ciate the area of a region with its statistical signifcance. ple, in one study sheep scrapie occurrence and sampling Given this general framework, the process of explora- rates were presented in dot maps and choropleth maps tory geovisualization begins with the designation of null on a cartogram of the sheep population by county in and alternative hypotheses. In what follows, we assume Great Britain [9]. Although the focus is on scrapie inci- a null hypothesis of equal probability of occurrence of a dence and detection, the cartogram allows map readers health event among all individuals in a population, and to quickly discern which counties have more sheep at a one-tailed alternative hypothesis of higher probability risk. At-risk population cartograms have also been used of occurrence in a given region. However, the method to help contextualize rates of obesity in Canada [10], HIV can be extended to other hypothesis testing scenarios incidence [11] and overall mortality [31] in Japan, and (e.g. testing for low probability of occurrence, diferences lung cancer cases in New York State [8]. in non-ratio values, etc.). Next, it is necessary to obtain Building on the same logic, at-risk population car- data representing the sample size of each district. In what tograms have also been used as a basis for statistical follows, we assume that all cases are reported and there- methods to detect disease clusters. In an early but illus- fore the efective sample size is equivalent to the at-risk trative example, reported cases of Wilm’s tumor were population. plotted on a population cartogram and the resulting Once the hypothesis framework and sample popula- pattern was compared with that produced by an equal tion variable have been established, geovisualization is number of randomly placed points [32]. Later research- supported by two complementary mathematical proce- ers applied formal tests of statistical signifcance to test dures that allow dynamic visual inference of the statis- the hypothesis that event locations on a cartogram of the tical signifcance of arbitrary regions on a cartogram of at-risk population are spatially random [12, 33, 34]. Te the at-risk population. Te frst procedure computes the logic of this approach is that the cartogram translates a minimum size of a region on the cartogram over which null hypothesis of random distribution within the popu- a user-designated rate would be statistically signifcant. lation into a spatially random distribution [35], thus con- Te second and converse procedure computes the mini- verting a Bernoulli process into an equivalent Poisson mum rate for which a region of a designated size would process. In this manner, at-risk population cartograms be statistically signifcant. Together, these procedures avoid the need for complex statistical methods to handle support a visual scanning process and related tools in spatial inhomogeneity [36]. which the map reader is able to estimate statistical sig- Although the above studies used cartograms as the nifcance of arbitrarily selected regions. basis for statistical hypothesis testing, these tests have Strictly speaking, both procedures entail the assump- been automated and are confrmatory in nature. We are tion that statistical signifcance is dependent only on not aware of any studies that have employed cartograms sample size and event rate. We begin with a simple sce- within a geovisual exploratory framework to visually nario in which this assumption holds, and then proceed communicate uncertainty or statistical signifcance. to a more realistic scenario in which it does not. Tis is an important omission, considering that epide- miological maps are probably more useful in exploratory Scenario one: A priori designated region settings than in formal hypothesis testing [1, 37]. With We frst consider a single region on a map designated this in mind, we seek methods to visually communicate a priori as a potential cluster. An example would be a uncertainty on a cartogram within a geovisual analytics citizen with no prior belief seeks to determine whether approach that takes advantage of human perception and or not a disease is prevalent in his or her county of resi- cognitive abilities to identify patterns and relationships in dence. Under the null hypothesis of random distribu- spatial data [38]. tion, the sampling distribution of the number of disease Our use of cartograms to communicate uncertainty is occurrences within the designated region can be deter- motivated by a natural association between map area and mined from its population. Let P and p denote the at-risk Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 5 of 13
populations overall and of the designated region, respec- than the observed scan statistic [39]. Determination of tively, and let R and r denote the observed event rates per this signifcance is a difcult problem, and most solutions person overall and in the designated region. Since the require computationally intensive Monte Carlo simula- map includes the designated region, we compare the des- tion as well as a priori designation of a large but fnite set ignated region with its complement rather than with the of scan window locations [39, 40]. To support real-time, entire map to avoid overlap between the regions under dynamic geovisual analysis, however, we seek an analytic comparison. Te hypothesis that the actual event rate in method for determining the signifcance of an a posteri- the designated region is higher than in the complement ori scan statistic without Monte Carlo simulation. to that region can be tested using a standard diference of Alm [41] proposes such a method that estimates the proportions test, with the following test statistic: signifcance of a free-moving rectangular scan window ( ) within a larger rectangular region. Specifcally, Alm’s for- r R . z − mulae estimate the probability under a spatially random = 1 1 (1) R(1 R) Poisson process that there will exist a rectangular window − p − P of fxed dimensions containing n or more events some- We solve for p to fnd the minimum population for which where within a larger rectangular region containing N a given rate is statistically signifcant: events. Our use of Alm’s formula is made possible due to 2 (1 ) the fact that the cartogram translates a Bernoulli process z PR R . p 2 − (2) into a Poisson process. To simplify, we assume a square = P(r R) z2R(1 R) − + − study region with area A and overall event density λ, and Since population density is uniform on the cartogram, a square moving scan window with area a. Alm [41] dem- the value of p can be used to determine the minimum onstrates that under the assumption of a random Poisson area on the cartogram for which an observed rate r would process, the probability φ that there exists at least one be statistically signifcant given a predefned signifcance scan window location with n or more events can be esti- threshold and corresponding z-score. Alternatively, mated by the function f(n), which is defned as follows: we can also solve for r to determine the minimum rate a µn 1 √a √A √a p a(n 1) (4a) for which a given population, and thus area on the car- = − n − − togram, would be statistically signifcant: 1 1 1 . a r R z R( R)( /p /P) (3) µn = + − − γn 1 √a √A √a (µn 1 µn)e− (4b) = − n − − − Although this framework has served as the basis for pre- vious health mapping applications [2, 3], the assumption (µn γn) φ f (n) 1 F a(n 1)e− + (4c) of an a priori designated region seems incompatible with ≈ = − − visual exploratory analysis, which by defnition entails where Fλa denotes the Poisson distribution function with searching for undiscovered event clusters that are mean λa. To support exploratory visual analysis, we seek unknown in advance. For this reason, we also develop an the minimum scan area a for which an observed rate n/a alternative framework based on a scan statistic, which would be statistically signifcant, and also the minimum does not assume an a priori designated region. rate n/a for which a scan window with area a would be statistically signifcant. We are unable to fnd an analytic Scenario two: A posteriori designated region solution to these problems. However, the probability φ Consider a scenario in which an investigator actively logically increases monotonically with both n and n/a. searches a map for clusters of events, with no specifc Terefore, an accurate solution can be obtained quickly region designated a priori. Te investigator will naturally using a binary search. A slight complication is that while seek to identify a region in which the event rate and/or Alm’s functional estimate f(n) appears valid for all input signifcance is maximal. Since the investigator is able to parameters in which φ ≪ 1, f(n) does not always pro- search all possible regions, an a posteriori test is needed. duce valid probability estimates when φ ≈ 1. Tus, it is A standard approach is to defne a scan statistic as the necessary to check the validity of input parameters to most extreme (i.e. highest) value of a designated cluster- avoid computational errors. Details on this process are ing metric observed for a scan window that is moved in a described in the Additional fle 1: Appendix. continuous fashion across the map, such that all possible placements of the scan window are examined. Statisti- Aggregate regions and compactness metric cal signifcance is defned as the probability, given a map Since sample size is additive, the above techniques may constructed under the null hypothesis of random spatial be applied to aggregate regions formed from more than distribution, of obtaining a scan statistic as high or higher one district. Tis is a distinct advantage in exploratory Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 6 of 13
analysis, but in the case of an a posteriori test the free- Basic (Microsoft Corp.), with mapping and statistical dom to form regions dynamically afects interpretation of functions from the DotSpatial and MathNet.Numerics signifcance values. Tis is because the distribution of a open source libraries. Te application reads population scan statistic is dependent on both the size and shape of count and rate data from a standard polygon shapefle. A the moving scan window. Tis renders the search space choropleth map is then constructed automatically using difcult to defne, and also the null distribution of the a diverging color scheme in which the central color rep- scan statistic difcult to compute. To mitigate this prob- resents the overall event rate. At present, the application lem, we support qualitative analysis by supplementing includes three functions for visualization and exploratory reported statistical signifcance under the assumption of analysis. All functions are responsive to user selections of a compact (square) scan window with a measure of com- hypothesis-testing framework (a priori vs. a posteriori) pactness to assess the squareness of a user-designated and signifcance threshold (p value). region. We use existing formulas [42] to calculate a shape Te frst function constructs a custom legend to sup- metric as a function of the moment of inertia Ig(P) of port visual scanning and detection of regions with sta- polygon P about its centroid g. If one considers a poly- tistically signifcant event rates. Te legend indicates the gon to be composed of infnitely many points, then Ig(P) minimum area over which an observed rate would be is the average squared distance from these points to g. considered statistically signifcant given the user-selected In a compact polygon, all locations in P will be relatively hypothesis-testing framework and signifcance threshold. close to g, resulting in a small value of Ig(P). A compact- Second, interactive spatial scan is supported using a ness index cmi for a polygon P as: square scan window that follows the movement of the Ig (C) mouse. Te size (i.e. population) of the scan window is cmi (5) adjustable. As the user scans the map, the event rate in = Ig (P) the scan window is computed as the weighted average where C denotes a circle equal in area to P [42]. Te value of the observed rates in districts that intersect the scan ranges from 0 (less compact) to 1 (more compact). Since window, weighted by the area (i.e. population) of the our formulation of statistical signifcance is based on a intersection. Statistical signifcance is then calculated square scan window, we modify this slightly by using a based on the scan window population and event rate and square (S) instead of a circle in the numerator: reported in real time to the user. Ig (S) Tird, exploration of potential clusters is supported cmi′ (6) by tools for defning custom regions through aggrega- = Ig (P) tion. Districts may be aggregated in any manner to form Te ratio of the moments of inertia of a square and circle contiguous or non-contiguous regions. Once a region is π π c cmi with equal area is a constant 3, so that mi′ = 3 . Te constructed, the population, average event rate and shape theoretical range of cmi′ is 0–1.047, with a value greater compactness of the region are reported to the user. than one indicating a region that is even more compact To illustrate the geovisual analysis framework, we than the square scan window. examine the distribution of age-adjusted leukemia inci- Te compactness metric is provided as a check on the dence rates for California counties from 2008 to 2013. validity of the test assumption. Te reported signifcance Data were extracted from the Surveillance, Epidemiol- estimate is considered more reliable if the value of cmi′ is ogy and End Results (SEER 2014-16) program [43]. A 1 closer to 1; to the degree that cmi′ < , the signifcance cartogram of California was constructed from SEER value should be treated with caution. For example, if a age-adjusted populations, reported in person-years, “region” is designated as the aggregate of non-adjacent using Cartogram Studio software for manual cartogram districts located at opposite corners of the study area, the construction [44]. Figure 1 shows the cartogram along- resulting value of cmi′ would be close to zero. Intuitively, side a conventional map, with the locations of major cit- this would indicate that the investigator apparently went ies labelled for reference. Apportionment error is 0.62%, to great lengths to form the region, and therefore the sta- meaning that 99.38% of the population is represented in tistical test has low validity. In this manner, cmi′ is used the correct county on the cartogram. as a surrogate variable to refect the extensiveness of the Choropleth representation of incidence rates on a search performed by the human investigator. conventional map is shown in Fig. 2. Seven classes were constructed by (a) assigning all counties with incidence Results rates below the overall state rate of 1.864 per 100,000 to A prototype software application was developed to sup- a single class, and (b) classifying the remaining counties port geovisual exploratory analysis within the above into six classes using the quantile method. Counties with framework. Te application was written in C# and Visual signifcantly high rates according to the a priori test are Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 7 of 13
Fig. 1 Ordinary map (a) and at-risk population cartogram (b) of California counties. County sizes on the cartogram are proportional to SEER data- base age-adjusted leukemia at-risk population values, reported in person-years. Selected cities are shown for reference outlined in blue. As a baseline for comparison, results of statistically signifcant areas for counties in each class, a Bernoulli spatial scan statistic with circular scan win- which can be compared to the actual counties to infer dow computed using SaTScan software [45] are also signifcance. For example, it can be inferred that the leu- shown in Fig. 2. kemia incidence rate in Los Angeles County, the larg- Tere are several limitations to the conventional map. est (most populated) county on the map, is moderately Te at-risk population of each county is not shown, so higher than the rest of the state but is statistically signif- map readers are likely to place cognitive emphasis on cant because the county is larger than the size of the cor- counties that are large in area instead of those with high responding legend symbol of the same color. Statistically populations. Also, only individual counties are con- signifcant counties are also highlighted in blue in Fig. 3. sidered as potential clusters, and it is impossible to tell While the geovisual analysis environment allows visual visually whether statistically signifcant regions could be inference of statistical signifcance for individual dis- formed from aggregating counties. Automatic cluster tricts, the true power of the framework comes from the detection using a spatial scan statistic is helpful but idio- ability to make inferences about aggregate regions. Note syncratic, as it is infuenced by the shape and size param- that if the component districts are in diferent categories, eters of the scan window and the spatial arrangement of there are multiple corresponding legend symbols and county centroids. For example, the only statistically sig- several cases must be considered. If the aggregate region nifcant cluster detected by SaTScan is centered on Inyo is larger than the symbol associated with the lowest rate County, whose rate is below average. Furthermore, two category of the component districts, then the aggregate adjacent counties with rates that are high and statistically rate is defnitely statistically signifcant. On the other signifcant (Los Angeles, Ventura) are excluded from the hand, if the aggregate region is smaller than the symbol cluster (Fig. 2). associated with the highest rate category of the compo- Figure 3 shows the same choropleth map on a car- nent districts then the aggregate rate is defnitely not sta- togram loaded into the geovisual analysis environment tistically signifcant. If neither of the above is true then with a legend built using the a priori signifcance test statistical signifcance cannot be defnitively determined at a threshold of α = 0.05. Te legend shows minimum from visual analysis alone. In Fig. 3, the combined area Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 8 of 13
scan window are reported dynamically as the mouse is moved, and the color of the scan window is modifed to refect the combined rate. Te statistics for scan window A show that the rate in this square region is not statisti- cally higher than the rest of the state (p = 0.102). Scan window B covers an area with a lower rate than scan win- dow A, but is larger and therefore statistically signifcant (p = 0.016). To aid in interpretation, the scan window is outlined in blue when the region it covers is statistically signifcant. It should be noted that the scan windows rep- resent square regions on the cartogram not the map. It would be desirable to transform the scan window into its actual shape in the real world, but this capability requires complete transformation formulae and is not yet imple- mented in our software. Two examples of custom defned aggregate regions are shown in Fig. 6, with signifcance values calculated using the a posteriori test. Under this stricter test, region A (formed by the six northern counties with above-average leukemia rates) is not statistically signifcant. Further- more, the compactness measure for this region is low, suggesting a large departure from the test assumption of a square scan window. However, a statistically signifcant Fig. 2 Standard choropleth map of leukemia incidence in California region can be constructed from several combinations counties, 2008–2013. Counties highlighted in blue outline have rates of counties in the southern portion of the state, includ- that are signifcantly higher than the remainder of the state according ing the combination comprising region B in Fig. 6. It is to the a priori test of signifcance at the 0.05 signifcance level notable that all such regions include Los Angeles county despite the fact that its leukemia rate is lower than many nearby counties. Tis suggests a broad cluster of mildly of (a) is clearly smaller than the legend symbol associated above average leukemia rates, as opposed to a small clus- with the higher rate (the largest symbol in the legend), ter of highly elevated rates. Region B in Fig. 6 also has a and so the aggregate rate of the region formed from these high compactness index, indicating that its apparent sig- two counties is not statistically signifcant. nifcance is not likely to be due to the relaxation of the Te choice of hypothesis framework has a large infu- assumption of a square scan window. ence on statistical signifcance. Figure 4 shows the same Discussion data using the a posteriori test based on a square scan window. It can be seen from the legend in Fig. 4 that Te cartogram geovisualization approach presented here the area required for each rate category to be statisti- difers from previous studies on uncertainty visualization cally signifcant increases substantially (as compared to in that the latter have focused on methods that apply to Fig. 3). Visually, it seems clear that none of the counties is individual districts in isolation, either by applying sym- individually larger than their corresponding legend sym- bol overlays or computing adjusted estimates of disease bol. Even aggregating the six northern counties with the rates in each district. Tese approaches do not provide highest rates is unlikely to result in a region larger than a means to determine uncertainty over larger aggregate the average size of the corresponding legend squares. In regions. If the power of geovisual analytics lies in the elic- the southern portion of the state where rates are lower, itation of human cognition and perceptual capabilities, matching map district colors to the legend suggests that a then our approach removes the narrow focus on individ- rather large area would be required to form a statistically ual districts and takes full advantage of human cognitive signifcant region. abilities to discern visual patterns in complex data. Two examples of the interactive scan process are Te ability to form regions from multiple districts or shown in Fig. 5 under the a priori test. In these examples, parts of districts in a highly fexible manner creates a scan windows of diferent sizes have been positioned truly exploratory environment that takes full advantage by the user over potential leukemia cluster locations. of human powers of pattern recognition. Clusters can Te population, combined rate and signifcance of each be identifed that would not be found by spatial scan Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 9 of 13
Fig. 3 Geovisual analysis environment with a priori test hypothesis and 0.05 signifcance threshold selected. Legend shows the minimum area for which the observed rate of an a priori designated map region would be signifcantly higher than the rest of the state
statistics. Alternatively, automatically detected clusters search space based on distance [39] or topological rela- can be modifed by removing districts with low rates tions [40], but reducing computation time remains a sub- or adding districts with high rates that were included/ stantial research problem and a barrier to adoption. excluded as an artefact of the circular or elliptical scan Furthermore, restrictions on the search space pre- windows in SaTScan. Tis is notable because the search clude many potential clusters, including those com- space for cluster detection problems is large and grows prised of non-contiguous or even distant districts, exponentially with the number of districts if all possible but which share a common property whose relevance aggregations are considered. If only contiguous regions might be detected by an expert. For example, a clus- are considered, the number of possible aggregations is ter of malnutrition formed from the counties of San reduced but is still quite large. For example, using the Francisco, Los Angeles and San Diego might suggest subgraph enumeration algorithm of Wernicke [46], urban causality, while elevated depression in selected 2,712,603 contiguous regions can be formed from ten or non-contiguous coastal counties might arouse suspi- fewer of California’s 58 counties. Tus, relying on algo- cion about the infuence of fog. In addition to expert rithmic search may be highly computational intensive knowledge on potential causal factors, investigators and, for larger datasets, practically infeasible. Existing may also carry knowledge of sub-district population scan techniques handle this problem by restricting the patterns that could be used to spot possible clusters Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 10 of 13
Fig. 4 Geovisual analysis environment with global scan hypothesis and 0.05 signifcance threshold selected. Legend shows the minimum area for which the presence of a square region with the given rate would indicate a statistically signifcant event cluster
that cross district boundaries. For example, in our illus- Another well-known drawback of cartograms is that trated examples a health researcher might have further they are visually unfamiliar and can be aesthetically expert knowledge on leukemia rates in diferent parts of unappealing. Tese aesthetics must be weighed against Los Angeles County that is not represented in the data, the simple logic of equating map area with statistical sig- which might infuence the interpretation of the cluster nifcance, which would not be possible without the use of at location B in Figs. 5 and 6. An exploratory framework a cartogram. Although we have not yet conducted formal allows such expert knowledge to be brought to bear. human subjects testing, our experience via presenting While the geovisual analytics framework presented here these maps to students and colleagues so far suggests that has many advantages, it has some drawbacks as well. most users are able to quickly learn the fundamental logic Although we use signifcance values to communicate of the proposed visualization framework. Tat is, users uncertainty on our epidemiological maps (as do many are able to visually identify regions and compare them to previous researchers), it should be kept in mind that legend entries to infer statistical signifcance. However, imputed statistical signifcance values in an explora- some details of the inferential processes are less likely to tory environment cannot be treated as confrmatory in be fully appreciated, including the efects of classifca- nature. Nevertheless, provision of signifcance values is tion and the appropriate assessment of regions formed an efective way to communicate uncertainty and allows from combining districts from diferent legend classes. prioritization of regions for further investigation. Tis is By automating the calculation of weighted averages, the in accordance with the exploratory role of map investi- interactive tools for scanning and region-building miti- gation in epidemiological research, as noted in previous gate these issues of interpretation, but further testing and studies. refnement of the visualization environment is needed. Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 11 of 13
Fig. 5 Illustration of two dynamically placed moving scan windows. Reported signifcance is calculated under the assumption of an a priori desig- nated region. Population sizes of the two regions (A and B) are in person-years
Conclusion population enable visual determination of statistical sig- Although cartograms have been employed previously nifcance. Furthermore, since sample size is additive, this in health mapping research, the inferential possibilities determination can be performed for any arbitrary region, aforded by equalizing the density of the at-risk popula- assuming a homogenous density within each region. tion have not been explored in a geovisualization setting. Te legend design and interactive scanning and region- We have demonstrated that cartograms of the at-risk building tools presented here work by translating Kronenfeld and Wong Int J Health Geogr (2017) 16:19 Page 12 of 13
Fig. 6 Illustration of two user-defned aggregate regions. Reported signifcance is calculated using the global scan test under the assumption of a square scan window
mathematical into visual relationships, allowing inference Availability of data and material Data used for the study are cited in [43] and the tools developed will be avail- of statistical signifcance from the size and observed rate able to the public through a website when the manuscript is accepted for of a region. publication (the url of the website will be provided upon acceptance).
Additional fle Funding This research was partly supported by the National Institutes of Health (NIH) under Award Number R01HD076020. Additional fle 1. Appendix: Checking the Validity of Alm’s Functional Estimate. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in pub- lished maps and institutional afliations. Authors’ contributions BK initiated the idea, developed the conceptual framework and tools, per- Received: 23 January 2017 Accepted: 2 May 2017 formed the test, and drafted the manuscript. DW helped refne the statistical- visual framework and was consulted and provided inputs at all stages. Both authors read and approved the fnal manuscript.
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