Many census datasets such as the U.S. Census Bureau’s American Community Survey (ACS) data1 are based on surveys from small samples. This entails that the variables provided by the Census Bureau are only estimates with a level of uncertainty often provided as a margin of error (MoE) or a standard error (SE). Note that the Bureau’s MoE encompasses a 90% confidence interval2 (i.e. there is a 90% chance that the MoE range covers the true value being estimated). This poses a challenge to both the visual exploration of the data as well as any statistical analyses of that data.
One approach to mapping both estimates and SE’s is to display both as side-by-side maps.
While there is nothing inherently wrong in doing this, it can prove to be difficult to mentally process the two maps, particularly if the data consist of hundreds or thousands of small polygons.
Another approach is to overlay the measure of uncertainty (SE or MoE) as a textured layer on top of the income layer.
Or, one could map both the upper and lower ends of the MoE range side by side.
Attempting to convey uncertainty using the aforementioned maps fails to highlight the reason one chooses to map values in the first place: that is to compare values across a spatial domain. More specifically, we are interested in identifying spatial patterns of high or low values. What is implied in the above maps is that the estimates will always maintain their order across the polygons. In other words, if one polygon’s estimate is greater than all neighboring estimates, this order will always hold true if another sample was surveyed. But this assumption is incorrect. Each polygon (or county in the above example) can derive different estimates independently from its neighboring polygon. Let’s look at a bar plot of our estimates.
Note, for example, how Piscataquis county’s income estimate (grey point in the graphic) is lower than that of Oxford county. If another sample of the population was surveyed in each county, the new estimates could place Piscataquis above Oxford county in income rankings as shown in the following example:
Note how, in this sample, Oxford’s income drops in ranking below that of Piscataquis and Franklin counties. A similar change in ranking is observed for Sagadahoc county which drops down two counties: Hancock and Lincoln.
How does the estimated income map compare with the simulated income map?
A few more simulated samples (using the 90% confidence interval) are shown below:
There is no single solution to effectively convey both estimates and associated uncertainty in a map. Sun and Wong (Sun and Wong 2010) offer several suggestions dependent on the context of the problem. One approach adopts a class comparison method whereby a map displays both the estimate and a measure of whether the MoE surrounding that estimate extends beyond the assigned class. For example, if we adopt the classification breaks [0 , 20600 , 22800 , 25000 , 27000 , 34000 ], we will find that many of the estimates’ MoE extend beyond the classification breaks assigned to them.
Take Piscataquis county, for example. Its estimate is assigned the second classification break (20600 to 22800 ), yet its lower confidence interval stretches into the first classification break indicating that we cannot be 90% confident that the estimate is assigned the proper class (i.e. its true value could fall into the first class). Other counties such as Cumberland and Penobscot don’t have that problem since their 90% confidence intervals fall inside the classification breaks.
This information can be mapped as a hatch mark overlay. For example, income could be plotted using varying shades of green with hatch symbols indicating if the lower interval crosses into a lower class (135° hatch), if the upper interval crosses into an upper class (45° hatch), if both interval ends cross into a different class (90°-vertical-hatch) or if both interval ends remain inside the estimate’s class (no hatch).
Data uncertainty issues do not only affect choropleth map presentations but also affect bivariate or multivariate analyses where two or more variables are statistically compared. One popular method in comparing variables is the regression analysis where a line is best fit to a bivariate scatterplot. For example, one can regress “percent not schooled”” to “income”” as follows:
The \(R^2\) value associated with this regression analysis is 0.2 and the p-value is 0.081.
But another realization of the survey could produce the following output:
With this new (simulated) sample, the \(R^2\) value dropped to 0.07 and the p-value is now 0.322–a much less significant relationship then computed with the original estimate! In fact, if we were to survey 1000 different samples within each county we would get the following range of regression lines:
These overlapping lines define a type of confidence interval (aka confidence envelope). In other words, the true regression line between both variables lies somewhere within the dark region delineated by this interval.