Maps, rates and small numbers: the geography trap

Year 3 / Module 33 Data visualisation in practice / Unit 7 / Maps, rates and small numbers: the geography trap

HCI3109 · Unit 7 of 10

A map encodes geography as well as data, so rates, areal units and the instability of small counts manufacture patterns that are not in the underlying process.

Maps, rates and small numbers: the geography trap

A map is unusual among common charts in that its layout is largely settled before the data arrives. Bars and scatterplots let the analyst choose both axes. A choropleth inherits its geometry from a boundary file drawn for administration. This seventh of ten units argues that the inheritance is the trap. A map encodes geography as well as data. Geography brings three decisions no encoding rule can make neutral: which areas the map is built from, which denominator divides each rate, and how few cases each area holds.

A better palette cannot repair a pattern manufactured at the boundary, the denominator or the count.

The map's geometry is decided before the data arrives, and the areal unit is a choice

Jenks and Caspall (1971) gave the problem its most general statement. They defined three uses a choropleth can serve: overview, tabular and boundary. They built techniques for measuring the error component of each, and proposed procedures for reducing it.

Beyer, Tiwari and Rushton (2012) set out five properties a disease map needs for public health use. It must control the population basis of spatial support for rates, and display rates continuously through space. It must keep maximum geographic detail, consider directly and indirectly age and sex adjusted rates, and place rates in a relevant context. The first three attack the administrative boundary directly, asking for a continuous rate field rather than a step at a line.

Anselin (1995) supplies the means of testing that second property. His local indicators of spatial association decompose a global measure such as Moran's I into the contribution of each observation. They are readable as local pockets of nonstationarity or as outliers, and Anselin evaluated them on conflict across African countries and in simulations. Once a pattern can be attributed to named places, the unit of aggregation starts looking like a hypothesis rather than a container.

A rate is a ratio, and the denominator is a decision

Counting cases and mapping rates are different acts, separated by a denominator the map rarely shows. Clayton and Kaldor (1987) rejected the standardised mortality ratio and its significance test as unsatisfactory for disease mapping. They proposed empirical Bayes estimators that compromise between a district's own ratio, the overall mean relative rate and a local mean of nearby areas. How far a district's estimate moves depends on how reliable its own count is.

Curtin and Klein (1995) set out the arithmetic that makes the denominator explicit. A directly standardised death rate weights the age-specific rates of the area by the age distribution of a single reference population. The note dates from 1995, when the United States centre used a 1940 standard million. The National Centre for Health Statistics replaced it with a projected year 2000 population for deaths from 1999 onward. Two areas with identical age-specific rates therefore get different adjusted rates whenever their age structures differ. The document is a methods note, so it establishes the procedure, not its effect on any reader.

Small numbers manufacture the extremes, and smoothing relocates rather than removes them

The centre of the unit is Gelman and Price (1999). They show that plotting observed rates fails when sample sizes vary. Very high and very low rates fall disproportionately in poorly sampled areas. Adjusting for small-sample noise introduces the opposite effect, so the highest adjusted rates fall disproportionately in well-sampled areas. In either case spatial variation in the parameter is confounded with spatial variation in sample size. Patterns appear where the underlying parameters have no spatial structure, and adjusted rates look too uniform where data are sparse. A top-ten list of districts by rate is largely a list of the smallest districts.

The standard remedy is shrinkage, and Clayton and Kaldor's (1987) estimator is a clear instance, pulling each district toward a local and a global mean. Gelman and Price's (1999) result is the necessary caution: because shrinkage depends on sample size, the fix relocates the artefact rather than removing it.

Cartograms trade one distortion for another

Gastner and Newman (2004) presented a diffusion-based method for density-equalising maps. It scales each region in proportion to a property such as population without overlapping regions, unreadable distortion or a strong dependence on the coordinate axes. They illustrated it on the 2000 United States presidential election, lung cancer cases in New York State and the geography of news stories. Nusrat and Kobourov's (2016) survey judges cartograms on statistical, geographical and topological accuracy. A cartogram encoding population by area pays for statistical accuracy with geographic and topological accuracy, so there is no free map, only a named trade.

Class breaks fail before the geometry is read

Brewer and Pickle (2002) compared seven choropleth classification methods with fifty-six subjects over nine series of United States mortality maps. Quantiles and minimum boundary error were best suited to general map-reading tasks. Natural breaks and a hybrid of equal intervals were less than seventy per cent as accurate. Matched legends across a series raised comparison accuracy by about twenty-eight per cent. The class break is therefore a design decision with a measurable effect on accuracy.

What a map of areas shows is not what people do

Robinson's (1950) analysis of the 1930 United States census remains the clearest demonstration that an aggregate relationship need not hold for the individuals inside the aggregate. The ecological correlation between foreign birth and illiteracy across states was about minus 0.53. The same relationship across individuals was about plus 0.12, so the sign reverses. Foreign-born residents tended to live in states whose native-born population was relatively literate. The paper reports a divisional basis value near minus 0.62 also, so the magnitude too depends on the aggregation. A district map shaded by a rate is an ecological object. A reader who takes it as a statement about residents commits the fallacy the map invites.

Interaction lets the reader change the map, not the geometry

Interactive cartography is mature, and its reviews are careful about when interaction should be offered. Roth (2013) reviewed the science of cartographic interaction around six questions: what it is, why, when, to whom, where and how it should be provided. He treats the static versus interactive choice and the productivity paradox as live design questions. Roth, Ross and MacEachren (2015) reported a user-centred design case study for GeoVISTA CrimeViz, a visual analytics application for criminal activity in space and time. They improved it through iterative user, utility and usability loops. A toggle between a count map and a rate map earns its place because it makes the denominator visible. A control that reclassifies a bad denominator hands the reader a dial, not an answer. The rate map also reaches the population it describes, so its reader may be in later life and on a metered connection.

The contested question: is smoothing a correction or a manufacture of pattern?

The first position holds that standardisation, smoothing and redrawn geometry are corrections. Clayton and Kaldor (1987) proposed empirical Bayes estimation because the raw standardised ratio is unreliable. Beyer, Tiwari and Rushton (2012) make age and sex adjustment, controlled spatial support and continuous rates part of what a disease map must do. Gastner and Newman (2004) fix the mismatch between area and population.

The second holds that every adjustment carries an artefact of its own. Gelman and Price (1999) show adjusted rates displaying patterns where the underlying parameters have none. Jenks and Caspall (1971) treat error as a property of the map's use, so no classing or smoothing step is free of it. Robinson (1950) shows aggregation can reverse the sign of a relationship. On this view a map is never the process, and an honest design states which transformation it applied and what it cost.

Both can be right, because they disagree about what a map is for: standardisation corrects a comparison and manufactures an estimate. What would settle it is not another map but an experiment, and half the design is published. Gelman and Price (1999) supply the simulation, two models with no spatial pattern. The missing half is the reader. A pre-registered study would simulate counts from a known spatial field. It would build raw, directly standardised and smoothed maps from the same counts, and state the denominator of every area as Beyer and colleagues (2012) require. It would then ask readers from the map's audience to recover the true rate, and to state the strongest case against their reading.

The two sides rest on different evidence. The correction side rests on statistical estimators and a methods argument (Clayton and Kaldor, 1987; Beyer and colleagues, 2012). The artefact side rests on a simulation of two models with no spatial pattern (Gelman and Price, 1999) and on a reanalysis of census data in which the sign reverses (Robinson, 1950). While the dispute is open, a designer should publish the denominator and the transformation applied, and pair every estimate with its count or an interval. The strongest objection is that no study here measures a reader, so the claim that a map manufactures a pattern rests on the mathematics of aggregation, not evidence about reading.

Where this is heading

Two directions are visible. The first moves away from the bounded choropleth toward continuous spatial support and reported denominators. That is what the first three properties of Beyer, Tiwari and Rushton (2012) require, and what Anselin's (1995) local indicators made operational. The second is cartographic interaction designed around a named user. That is Roth's (2013) six questions and Roth, Ross and MacEachren's (2015) loops, extended to cartograms by the guidelines with which Nusrat and Kobourov (2016) close their survey.

The sources this unit rests on

Each note states what the cited work did and found and what it adds to the argument. Identifiers and metadata were checked on 2 November 2026.

Anselin, L. (1995). Local Indicators of Spatial Association. Geographical Analysis, 27(2), 93-115. doi:10.1111/j.1538-4632.1995.tb00338.x Defined local statistics that decompose a global measure such as Moran's I into each observation's contribution, read as local pockets of nonstationarity or as outliers, and evaluated on conflict across African countries and in simulations. Makes a local pattern testable.
Beyer, K. M. M., Tiwari, C. and Rushton, G. (2012). Five Essential Properties of Disease Maps. Annals of the Association of American Geographers, 102(5), 1067-1075. doi:10.1080/00045608.2012.659940 Argued for five properties a disease map needs for public health use: control of the population basis for rates, rates continuous through space, maximum geographic detail and age and sex adjustment. The unit's constructive counterweight.
Brewer, C. A. and Pickle, L. W. (2002). Evaluation of Methods for Classifying Epidemiological Data on Choropleth Maps in Series. Annals of the Association of American Geographers, 92(4), 662-681. doi:10.1111/1467-8306.00310 Compared seven classification methods with fifty-six subjects over nine mortality map series; quantiles fitted general reading best and matched legends raised accuracy by about twenty-eight per cent. Shows the class break changes the reading.
Clayton, D. and Kaldor, J. (1987). Empirical Bayes Estimates of Age-Standardized Relative Risks for Use in Disease Mapping. Biometrics, 43(3), 671-681. doi:10.2307/2532003 Rejected standardised mortality ratios and their significance tests as unsatisfactory and proposed empirical Bayes estimators compromising between a district's own ratio, the overall mean rate and a local mean, weighted by each count's reliability. The canonical remedy, and why it depends on sample size.
Curtin, L. R. and Klein, R. J. (1995). Direct standardisation (age-adjusted death rates) (the published title uses the American spelling). Healthy People 2000 Statistical Notes, No 6 revised. National Centre for Health Statistics. doi:10.1037/e584012012-001 Sets out direct standardisation, in which age-specific rates are weighted by the age distribution of a standard population, with a 1940 standard million then used by the United States centre, and works through the computation for Healthy People 2000 objectives.
Gastner, M. T. and Newman, M. E. J. (2004). Diffusion-based method for producing density-equalizing maps. Proceedings of the National Academy of Sciences, 101(20), 7499-7504. doi:10.1073/pnas.0400280101 Presented a diffusion-based cartogram algorithm scaling regions to a property such as population without overlapping regions or unreadable distortion, illustrated on the 2000 United States election, lung cancer in New York State and news geography. The best case for redrawn geometry.
Gelman, A. and Price, P. N. (1999). All maps of parameter estimates are misleading. Statistics in Medicine, 18(23), 3221-3234. doi:10.1002/(sici)1097-0258(19991215)18:23<3221::aid-sim312>3.0.co;2-m Showed that observed rates confound parameter variation with sample size, and that adjusting for small-sample noise creates the opposite artefact, with the highest adjusted rates falling in well-sampled areas and patterns appearing where the parameters have none, using normal and Poisson models. Central evidence that smoothing relocates the artefact.
Jenks, G. F. and Caspall, F. C. (1971). Error on Choroplethic Maps: Definition, Measurement, Reduction. Annals of the Association of American Geographers, 61(2), 217-244. doi:10.1111/j.1467-8306.1971.tb00779.x Defined overview, tabular and boundary uses of choropleth maps, developed techniques for measuring the error component of each and set out procedures for reducing it. Establishes that map error is defined against a stated use.
Nusrat, S. and Kobourov, S. (2016). The State of the Art in Cartograms. Computer Graphics Forum, 35(3), 619-642. doi:10.1111/cgf.12932 Surveyed a century of cartogram research and organised it around three accuracy dimensions, statistical, geographical and topological, before setting design guidelines. Supplies the account of what a redrawn map trades away.
Robinson, W. S. (1950). Ecological Correlations and the Behaviour of Individuals (the published title is American). American Sociological Review, 15(3), 351-357. doi:10.2307/2087176 Compared correlations across 1930 United States census states with correlations across individuals, finding a state-level correlation between foreign birth and illiteracy of about minus 0.53 against an individual-level correlation of about plus 0.12, the sign reversing. A map of areas is not evidence about the people in them.
Roth, R. E. (2013). Interactive maps: What we know and what we need to know. Journal of Spatial Information Science, 6, 59-115. doi:10.5311/josis.2013.6.105 Reviewed cartographic interaction around six questions of what, why, when, to whom, where and how it should be provided, treating the static versus interactive choice as an open question. Frames interaction as a decision with costs.
Roth, R. E., Ross, K. S. and MacEachren, A. M. (2015). User-Centred Design for Interactive Maps: A Case Study in Crime Analysis (the published title is American). ISPRS International Journal of Geo-Information, 4(1), 262-301. doi:10.3390/ijgi4010262 Reported a user-centred design case study for GeoVISTA CrimeViz, a visual analytics application for criminal activity in space and time, improved through iterative user, utility and usability loops. Shows what interaction serving a named user requires.

Check your understanding & apply it · 4 questions, 1 essay

1A press office asks for the ten districts with the highest rate of a condition, expecting a list of the worst-affected places. The list is topped by the smallest districts. Which diagnosis does the unit's evidence support?

2A map series must use one classification method across all its maps, and a designer proposes natural breaks. Using Brewer and Pickle (2002), which prediction about general map-reading tasks is best supported?

3A councillor reads a district map shaded by the foreign-born rate and concludes that foreign-born residents are less literate, because the shaded states scored higher on illiteracy. Which diagnosis does the unit's evidence support?

4A designer proposes to smooth a small-area rate map and says the smoothing 'makes the map safer to read'. Which judgement does the unit's reading of the evidence support?

5Essay A public health authority wants a web page carrying a district map of the rate of a long-term condition across 320 districts. Case counts range from 4 in the smallest district to 900 in the largest, the populations differ by a factor of forty, and the press office has asked for a map that names the ten worst districts, believing these are isolated local problems. The page will be read on phones by journalists and councillors, most of whom will look for one second and leave. Using this unit's sources, say how you would build the map and its denominator, what you would refuse to show and why, how you would serve the readers who never interact, what you would predict before release that readers will most likely take wrongly, name the one check you would run, and then state the strongest counter-argument to your design and answer it.

I would start from the denominator, because with four cases in the smallest district any rate is unstable by construction, and Gelman and Price (1999) show that plotting observed rates places the extremes in the thinly populated areas while adjusting them moves the extremes to the well-sampled ones, so a bare rate map tells the reader about population size rather than about illness. I would therefore show no rate without its count, and I would not rank districts without that count beside them. I would use an age and sex adjusted rate, following Beyer, Tiwari and Rushton (2012), whose first properties ask for a controlled population basis, continuous display and maximum geographic detail, and I would follow Clayton and Kaldor (1987) in using empirical Bayes estimation so that small districts are pulled toward their neighbours, while making clear that this shrinkage depends on sample size and is a design decision rather than a correction I can assume. For classing I would use an equal-count method with a diverging, colour-vision-safe scheme rather than the data dependent optimum, because Brewer and Pickle (2002) found quantiles and minimum boundary error best for general map-reading tasks and the heavily optimised natural breaks less than seventy per cent as accurate for those tasks. I would refuse the ranking the press office asked for, and refuse any map that shades a raw rate without its count, because such a map invites the ecological fallacy Robinson (1950) demonstrated when the sign of a relationship reversed between the state and individual levels. For readers who never interact I would put a three-item static summary above the map, naming the denominator rule, the adjustment and the uncertainty, since the field evidence is that adding an on-ramp does not convert non-interacting readers. Before release I would predict two errors: that readers take the named districts as bad places rather than as small ones, which is the consequence of the denominator, and that they read the smoothed surface as measured fact when the shrinkage has borrowed strength from neighbours. The one check I would run is a short task-based session with readers resembling the audience, asking them to state the rate and its count for two districts, one large and one small, and then to give the strongest argument against the conclusion they drew, with the proportion who can supply it recorded. The strongest counter-argument is that omitting the ranking hides the authority's real finding and that smoothing may bury a genuine local cluster, since Anselin's (1995) local statistics exist precisely to detect local pockets of nonstationarity. That argument is right that local clusters matter, which is why the page should carry the local statistic and the count in the tooltip, and it is wrong only if it treats a small district's high rate as evidence of a cluster rather than as the expected behaviour of a small denominator. The counter-argument changes what the page discloses, not whether the denominator is shown.