1.1 A General Framework for Changes of Support

Destination unit here refers to the desired spatial unit given one’s theory, and source unit refers to the unit at which data are available. Consider two dimensions. The first, relative nesting, captures whether source units fall completely and neatly inside destination units. If perfectly nested, CoS problems become computationally simpler, and can sometimes be implemented without geospatial transformations (e.g., aggregating tables by common ID, like postal code). If units are not nested, CoS requires splitting polygons across multiple features, and reallocating or interpolating values. Nesting is a geometric concept, not a political one: even potentially nested units (e.g., counties within states) may appear non-nested when rendered as geospatial data features. Such discrepancies may be genuine (e.g., historical boundary changes), or driven by measurement error (e.g., simplified vs. detailed boundaries), or differences across sources.

The second dimension, relative scale, adds additional, useful information. It captures whether source units are generally smaller or larger than destination units. If smaller, transformed values will represent aggregation of measurements taken at source units. If larger, transformed values will entail disaggregation—a more difficult process, posing nontrivial EI challenges (Anselin and Tam Cho Reference Anselin and Tam Cho2002; King Reference King1997). Many practical applications represent hybrid scenarios, where units are relatively smaller, larger, or of similar size, depending on location. For instance, U.S. congressional districts in large cities are smaller than counties, but in rural areas they are larger than counties.

To illustrate, consider relative nesting and scale among three sets of polygons in Figure 1: (a) electoral precincts in the U.S. state of Georgia in 2014, (b) Georgia’s electoral constituencies (congressional districts) in 2014, and (c) regular hexagonal grid cells (half-degree in diameter) covering the state. Georgia is an interesting case study due to its diverse voting population and judicial history surrounding elections. Legal challenges and court rulings require that the state’s data on electoral boundaries be accurate and publicly available at granular resolution (Bullock III Reference Bullock2018).

Figure 1 Spatial data layers (U.S. state of Georgia).

Precincts report to constituencies during the administration of elections, so units in Figure 1a should be fully nested within and smaller than those in Figure 1b. The intersection in Figure 1d confirms that every precinct falls inside a larger constituency, and—except for small border misalignments on the Atlantic coast—the transformation does not split precincts into multiple parts. If constituency IDs are available for precincts, a change of support from (a) to (b) could be reduced to calculating group sums, a straightforward procedure.

The hexagonal grid in (c) presents more difficulty. The grid cells were drawn independently of the other two layers, and are not nested. A transformation from (b) to (c) requires splitting constituencies across multiple grid cells, and vice versa. The overlay in Figure 1e suggests that cells are smaller than most, but not all, constituencies (e.g., Atlanta metro area). Changes of support to and from (c) therefore require complex spatial operations, accounting for differences in scale and misalignment of boundaries.

While visual assessments of nesting and scale can be informative, they introduce subjectivity and are often infeasible. Small geometric differences are difficult to detect visually, and partial degrees of nesting are hard to characterize consistently. Visual inspections are also slow and not scalable for batch processing, which requires automated subroutines.

We thus propose two nonparametric measures of relative nesting and scale. Let $\mathcal {G}_S$ be a set of source polygons, indexed $i=1,\dots ,N_S$ , and let $\mathcal {G}_D$ be a set of destination polygons, indexed $j=1,\dots ,N_D$ . Let $\mathcal {G}_{S\cap D}$ be the intersection of these polygons, indexed ${i\cap j}=1,\dots ,N_{S\cap D} : N_{S\cap D}\geq \max (N_S,N_D)$ . Let $a_i$ be the area of source polygon i, let $a_j$ be the area of destination polygon j, and let $a_{i\cap j}$ be the area of ${i\cap j} : a_{i\cap j}\leq \min (a_i,a_j)$ .Footnote ³ Let $M_{S\cap D}$ be an $N_{S\cap D}\times 3$ matrix of indices mapping each intersection $i\cap j$ to its parent polygons i and j. $M_{i\cap D}$ is a subset of this matrix, indexing the $N_{i\cap D}$ intersections of polygon i (see Section A2 of the Supplementary Material for examples). Let $1(\cdot )$ be a Boolean operator, equal to 1 if “ $\cdot $ ” is true.

Our first measure—relative nesting ( $RN$ )—captures how closely source and destination boundaries align, and whether one set fits neatly into the other:

(1) $$ \begin{align} RN = \frac{1}{N_S}\sum_{i}^{N_S} \sum_{i\cap j}^{N_{i\cap D}}\left(\frac{a_{i\cap j}}{a_i} \right)^2, \end{align} $$

which reflects the share of source units that are not split across destination units. Values of 1 indicate full nesting (no source units are split across multiple destination units), and a theoretical lower limit of 0 indicates no nesting (every source unit is split across many destination units). $RN$ has similarities with the Herfindahl–Hirschman Index (Hirschman Reference Hirschman1945) and the Gibbs–Martin index of diversification (Gibbs and Martin Reference Gibbs and Martin1962), which the electoral redistricting literature has used to assess whether “communities of interest” remain intact under alternative district maps (Chen Reference Chen2010). A value of $RN=1$ , for example, corresponds to redistricting plans in which every source unit is assigned to exactly one destination unit (e.g., “Constraint 1” in Cho and Liu Reference Cho and Liu2016).

The second measure—relative scale ( $RS$ )—captures whether a CoS task is generally one of aggregation or disaggregation:

(2) $$ \begin{align} RS = \frac{1}{N_{S\cap D}}\sum_{{i\cap j}}^{N_{S\cap D}}1(a_i<a_j) \end{align} $$

which is the share of intersections in which source units are smaller than destination units. Its range is 0 to 1, where 1 indicates pure aggregation (all source units are smaller than intersecting destination units) and 0 indicates no aggregation (all source units are at least as large as destination units). Values between 0 and 1 indicate a hybrid (i.e., some source units are smaller, others are larger than destination units).

Table 1 reports pairwise $RN$ and $RS$ measures for the polygons in Figure 1, with source units in rows and destination units in columns. The table confirms that precincts are always smaller ( $RS=1$ ) and almost fully nested within constituencies ( $RN=0.98$ ), with the small difference likely due to measurement error. Precincts are also smaller than ( $RS=1$ ) and mostly nested within grid cells ( $RN=0.92$ ). At the opposite extreme, obtaining precinct-level estimates from constituencies or grid cells would entail disaggregation ( $RS=0$ ) into non-nested units ( $0.01\leq RN \leq 0.05$ ). Other pairings show intermediate values: hybrids of aggregation and disaggregation, where changes of support require splitting many polygons.

Table 1 Relative scale and nesting of polygons in Figure 1.

Before proceeding, let’s briefly consider how $RN$ and $RS$ relate to each other, with more details in the Supplementary Material. Note, first, the two measures are not symmetric (i.e., $RS_{S,D}\neq 1- RS_{D,S}$ , $RN_{S,D}\neq 1-RN_{D,S}$ ), except in special cases where the outer boundaries of $\mathcal {G}_S$ and $\mathcal {G}_D$ perfectly align, with no “underlapping” areas as in Figure 1e. Section A2 of the Supplementary Material considers extensions of these measures, including symmetrical versions of $RS$ and $RN$ , conditional metrics defined for subsets of units, measures of spatial overlap, and metrics that require no area calculations at all.

Second, as Table 1 suggests (and Section A2 of the Supplementary Material shows), the two measures are positively correlated. Relatively smaller units mostly nest within larger units. Yet because $RN$ is more sensitive to small differences in shape, area, and orientation, unlike $RS$ , it rarely reaches its limits of 1 or 0. Even pure aggregation (e.g., precinct-to-grid, $RS=1$ ) may involve integrating units that are technically not fully nested ( $RN=0.92$ ). One barrier to reaching $RN=1$ is that polygon intersections often result in small-area “slivers” due to minor border misalignments, but these can be easily excluded from calculation.Footnote ⁴

$RN$ and $RS$ can diverge, within a limited range, like in the precinct-to-grid example in Table 1. As Section A3 of the Supplementary Material shows using randomly generated maps, such divergence reflects the fact that the distributions of $RN$ and $RS$ have different shapes: $RS$ is bimodal, with peaks around $RS=0$ and $RS=1$ , while $RN$ is more normally distributed, with a mode around $RN=0.5$ . The relationship between the two measures resembles a logistic curve, where numerical differences are largest in the tails and smallest in the middle. Differences between the measures tend to be numerically small. We observe no cases, for instance, where $RS>0.5$ and $RN<0.5$ (or vice versa) for the same destination and source units.

2.1 Illustration: Changing the Geographic Support of Electoral Data

Our first illustration transforms electoral data across the polygons in Figure 1. We demonstrate generalizability through a parallel analysis of Swedish electoral data (Section A5 of the Supplementary Material).Footnote ⁶

The variable we transformed was Top-2 Competitiveness, scaled from 0 (least competitive) to 1 (most competitive):

(3) $$ \begin{align}\hspace{-40pt} \text{Top-2 Competitiveness}&=1-\text{winning party vote share margin} \qquad\qquad\qquad\qquad\qquad\end{align} $$

(4) $$ \begin{align}\hspace{30pt} &=\frac{\text{valid votes}-(\text{votes for winner}-\text{votes for runner-up})}{\text{valid votes}}. \end{align} $$

We obtained “true” values of competitiveness for precincts (Figure 1a) and constituencies (Figure 1b) from official election results, measuring party vote counts and votes received by all parties on the ballot (Kollman et al. Reference Kollman, Hicken, Caramani, Backer and Lublin2022). For grid cells (Figure 1c), we constructed aggregates of valid votes and their party breakdown from precinct-level results.

Our analysis did not seek to transform Top-2 Competitiveness directly from source to destination units. Rather, we transformed the three constitutive variables in Equation (4)—valid votes, and votes for the top-2 finishers—and reconstructed the variable after the CoS. In Section A6 of the Supplementary Material, we compare our results against those from direct transformation.

Because the purpose of much applied research is not univariate spatial transformation, but multivariate analysis (e.g., effect of $\mathbf {x}$ on $\mathbf {y}$ ), we created a synthetic variable, $y_i=\alpha + \beta x_i + \epsilon _i$ , where $x_i$ is Top-2 Competitiveness in unit i, $\alpha =1$ , $\beta =2.5$ , and $\epsilon \sim N(0,1)$ . We assess how changes of support affect the estimation of regression coefficients, in situations where the “true” value of that coefficient ( $\beta =2.5$ ) is known.

Figure 2 shows transformed values of Top-2 Competitiveness ( $\widehat {\mathbf {x}_{\mathcal {G}D}}^{(k)}$ ) alongside true values in destination units ( $\mathbf {x}_{\mathcal {G}D}$ ), where darker areas are more competitive. Figure 2a reports the results of precinct-to-constituency transformations (corresponding to “ $a \cap b$ ” in Figure 1d). Figure 2b reports constituency-to-grid transformations (“ $b \cap c$ ” in Figure 1e). Of these two, the first set of transformations (where $RN=0.98$ , $RS=1$ ) more closely resembles true values than the second set ( $RN=0.29$ , $RS=0.12$ ), with fewer missing values and implausibly smooth or uniform predictions.

Figure 2 Output from change-of-support operations (Georgia). : source features are polygons. $\bigodot $ : source features are polygon centroids.

Figure 3 reports fit diagnostics for the full set of CoS transformations across spatial units in Georgia (vertical axes), as a function of the transformations’ relative nesting and scale (horizontal axes). Each point corresponds the quality of fit for a separate CoS algorithm. The curves represent fitted values from a linear regression of each diagnostic on source-to-destination $RN$ and $RS$ coefficients. Gray regions are 95% confidence intervals.

Figure 3 Relative nesting, scale and transformations of election data (Georgia).

The results confirm that the accuracy of CoS transformations increase in $RN$ and $RS$ . RMSE is lower, correlation is higher, and OLS estimation bias is closer to 0 where source units are relatively smaller (Figure 3a) and more fully nested (Figure 3b). OLS bias is negative (i.e., attenuation) when $RN$ and $RS$ are small and shrinks toward 0 as they increase.

Our analysis also reveals substantial differences in relative performance of CoS algorithms. Overall, simpler methods like overlays and areal interpolation produce more reliable results. For example, median RMSE—across all levels of $RN$ and $RS$ —was 0.22 or lower for both types of simple overlays, compared to 0.43 for universal kriging. Median correlation for simple overlays was 0.85 or higher, compared to 0.12 for universal kriging. Simple overlays also returned the smallest OLS bias, with a median of $-0.15$ , compared to a quite severe underestimation of OLS coefficients for universal kriging ( $-2.4$ ). Similar patterns emerge in our analysis of Swedish electoral data (Section A5 of the Supplementary Material).

Because the Top-2 Competitiveness variable is a function of other variables, we considered how transformation quality changes when we transform this variable directly versus reconstructing it from transformed components. The comparative advantages of these approaches depend on the relative nesting and scale of source and destination units: indirect transformations perform better when $RN$ and $RS$ are closer to 1, direct transformations are preferable as $RN$ and $RS$ approach 0 (Section A6 of the Supplementary Material).

2.2 Illustration: Monte Carlo Study with Synthetic Polygons

To generalize, we performed Monte Carlo simulations with artificial boundaries and variables on a rectangular surface. This analysis compares fit diagnostics from the same CoS algorithms, over a broader set of transformations covering the full range of $RN$ and $RS$ .

We consider two use cases. First, we change the geographic support of an extensive variable, like population size or number of crimes. Extensive variables depend on the area and scale of spatial measurement: if areas are split or combined, their values must be split or combined accordingly, such that the sum of the values in destination units equals the total in source units (i.e., satisfying the pycnophylactic, or mass-preserving, property). Second, we change the support of an intensive variable, like temperature or elevation. Intensive variables do not depend on the size of spatial units; quantities of interest in destination units are typically weighted means. Two or more extensive variables can combine to create a new intensive variable, like population density or electoral competitiveness. In Section A6 of the Supplementary Material, we consider the merits of (re-)constructing these variables before versus after a CoS.

While real-world data rarely conform to a known distribution, we designed the simulated geographic patterns to mimic the types of clustering and heterogeneity that are common in data on political violence and elections (see examples in Section A7 of the Supplementary Material).

At each iteration, our simulations executed these steps:

1. 1. Draw random source ( $\mathcal {G}_S$ ) and destination ( $\mathcal {G}_D$ ) polygons. Within a rectangular bounding box $\mathcal {B}$ , we sampled a random set of $N_S$ points and created $N_S$ tessellated polygons, such that for any polygon $l_i \in \{1,\dots ,N_S\}$ corresponding to point $i \in \{1,\dots ,N_S\}$ , all points inside $l_i$ were closer to i than to any other point $-i$ . We did the same for $N_D$ destination polygons. The relative number of source-to-destination polygons ( $N_S$ : $N_D$ ) ranged from 10:200 to 200:10. Figure 4a shows one realization of $\mathcal {G}_S,\mathcal {G}_D$ , with $N_S=200,N_D=10$ .

   

   Figure 4 Examples of spatial data layers used in Monte Carlo study. Dotted lines are source units ( $\mathcal {G}_S$ ), solid lines are destination units ( $\mathcal {G}_D$ ).

2. 2. Assign “true” values of a random variable X to units in $\mathcal {G}_S$ and $\mathcal {G}_D$ . For extensive variables, we simulated values from an inhomogeneous Poisson point process (PPP). For intensive variables, we simulated values from a mean-zero Gaussian random field (GRF), implemented with the sequential simulation algorithm (Goovaerts Reference Goovaerts1997).Footnote ⁷ Figure 4b,c illustrates examples of each. In both cases, while we sought to imitate the spatially autocorrelated distribution of real-world social data, we also simulated spatially random values as benchmarks (see Section A7 of the Supplementary Material).Footnote ⁸ As before, we also created a synthetic variable $Y=\alpha + X\beta + \epsilon $ , with $\alpha =1$ , $\beta =2.5$ , $\epsilon \sim N(0,1)$ .

3. 3. Change the geographic support of X from $\mathcal {G}_S$ to $\mathcal {G}_D$ , using the CoS algorithms listed above. We then compared transformed values $\widehat {x_{\mathcal {G}D}}$ to the assigned “true” values $x_{\mathcal {G}D}$ , and calculated the same summary diagnostics as before, adding a normalized RMSE $\left ({\sqrt {\sum _j \frac {1}{N_{\mathcal {G}D}} (x_{j\mathcal {G}D} - \widehat {x_{j\mathcal {G}D}})^2}}/{\left (\max (x_{\mathcal {G}D})-\min (x_{\mathcal {G}D})\right )}\right )$ for extensive variables.

We ran this simulation for randomly generated polygons with $N_S \in [10,\dots ,200]$ and $N_D \in [10,\dots ,200]$ , covering cases from aggregation ( $N_S=200,N_D=10$ ) to disaggregation ( $N_S=10,N_D=200$ ). We repeated the process 10 times, with different random seeds.

To facilitate inferences about the relationship between nesting and the diagnostic measures, we estimated semi-parametric regressions of the form:

(5) $$ \begin{align} M_{km}&=f(RN_{km})+\text{Method}_k+\epsilon_{km}, \end{align} $$

where k indexes CoS algorithms, and m indexes simulations. $M_{km}$ is a diagnostic measure for CoS operation $km$ (i.e., [N]RMSE, correlation, OLS bias), $f(RN_{km})$ is a cubic spline of $RN$ (or $RS$ ), Method $_k$ is a fixed effect for each CoS algorithm, and $\epsilon _{km}$ is an i.i.d. error term. This specification restricts inferences to the effects of nesting and scale within groups of similar operations, adjusting for baseline differences in performance across algorithms.

Figure 5 reports predicted values of [N]RMSE, Spearman’s correlation, and OLS bias across all methods at different levels of $RN$ , for both (a) extensive and (b) intensive variables. Section A8 of the Supplementary Material reports results for the $RS$ coefficient, which generally align with these. As source units become more nested and relatively smaller, [N]RMSE and OLS bias trend toward 0, while correlation approaches 1. The primary difference between extensive and intensive variables is in the estimation of OLS coefficients. For extensive variables, we see attenuation bias, which becomes less severe as $RN$ approaches 1. For intensive variables, we see attenuation bias as $RN$ approaches 1, but inflation bias as $RN$ approaches 0.

Figure 5 Relative nesting and transformations of synthetic data.

Figure 6 shows the Monte Carlo results by CoS algorithm, for (a) extensive and (b) intensive variables. In each matrix, the first, left-most column reports average statistics for each algorithm, pooled over all values of $RN$ . The remaining columns report average statistics in the bottom decile of $RN$ (0%–10%), the middle decile (45%–55%), and the top decile (90%–100%). The bottom row presents median statistics across algorithms. Darker colors represent better-fitting transformations (i.e., closer to 0 for [N]RMSE and bias; closer to 1 for correlation).

Figure 6 Transformation quality at different percentiles of relative nesting.

First, the relative performance of all CoS algorithms depends, strongly, on $RN$ (and $RS$ , see Section A8 of the Supplementary Material). In most cases, the largest improvements in performance occur between the lowest and middling ranges. Where values of $RN$ are low (bottom 10%), most algorithms fare poorly. Performance improves as $RN$ increases, especially in the first half of the range. For example, median RMSE (intensive variables) is 1.02 for CoS operations with low $RN$ , 0.63 for intermediate values, and 0.35 for the top decile. Most algorithms will perform better even at middling levels of $RN$ —0.4 and up—than at lower levels.

Second, while no CoS algorithm clearly stands out, some perform consistently worse than others. For example, population weighting offers no discernible advantages in transforming intensive variables (but seemingly plenty of disadvantages) relative to simple area weighting. Ancillary data from covariates, these results suggest, do not always improve transformation quality. Centroid-based simple overlays also fare poorly throughout.

Third, some CoS algorithms are more sensitive to variation in $RN$ than others. For example, simple overlays produce credible results for extensive variables when $RN$ is high, while areal and population weighting results are more stable.

Are $RN$ and $RS$ redundant? After we condition on $RN$ , for example, does $RS$ add any explanatory value in characterizing the quality of CoS operations (i.e., measurement error and bias of transformed values)? As we show in Section A3 of the Supplementary Material, $RN$ is more strongly predictive of transformation quality than $RS$ when considering the two metrics separately. But $RS$ is not redundant; including information about both $RN$ and $RS$ accounts for more variation in transformation quality than does information about $RN$ alone.Footnote ⁹

In Section A3 of the Supplementary Material, we consider how divergence between $RN$ and $RS$ affects the relative performance of CoS methods. Not much, we conclude. The absolute proximity of $RS$ and $RN$ to 0 or 1 is far more predictive of transformation quality than their divergence.

Our simulations confirm that patterns from our analysis of election data—higher $RN$ and $RS$ are better—hold in more general sets of cases. These include changes of support between units of highly variable sizes and degrees of nesting, and transformations of variables with different properties and distributional assumptions.