MEASURING POLARIZATION/SEGREGATION

There is a very large literature on the measurement of spatial segregation—which is the equivalent of spatial polarization. Almost all of it (as in Glaeser and Ward’s Reference Glaeser and Ward2005 paper) uses single-number indices of polarization/segregation that are descriptive only; they lack a basis in formal inferential statistics, and so any judgement regarding their relative size is qualitative. ^{Footnote 2} If one wants to establish that polarization is greater at one date than at another, then a method is needed with which the statistical significance of any observed differences can be evaluated. Further, as clearly demonstrated by Carrington and Troske (Reference Carrington and Troske1997), most of those indices based on the variance in a distribution over a set of spatial units systematically inflate the identified level of segregation because they confound systematic and random variation, especially where the spatial units contain relatively small numbers of people.

Most studies of segregation/polarization using single number indices to assess its intensity are conducted at a single spatial scale only—such as counties in Bishop and Cushing’s (Reference Bishop and Cushing2008) book and in Lang and Pearson-Merkowitz’s (Reference Lang and Pearson-Merkowitz2015) parallel study to that reported here. But—as Glaeser and Ward (Reference Glaeser and Ward2005) indicate—there is a substantial literature exploring whether there is greater polarization in voting patterns at the state scale, and other work focuses on even broader scales, such as those of the nine census divisions. A full evaluation of any evolving spatial pattern thus requires exploration of trends at a variety of scales. However, as pointed out some decades ago (Duncan, Cuzzort, and Duncan Reference Duncan, Cuzzort and Duncan1961) but rarely taken into consideration since, any measure of segregation at one spatial scale necessarily incorporates its measure at any larger scales: if there is growing polarization at the state scale, for example, this is bound to be incorporated—to an unknown extent—at the county scale too and, as Jones et al. (Reference Jones, Johnston, Manley, Owen and Charlton2015) have argued, any measure of segregation at a micro-scale is likely to be over-stated if it does not “hold constant” that measure at a macro-scale within which the micro-scale units are nested (e.g. counties within states).

To take both of these major criticisms of the standard measures of segregation into account, in this article we apply a recently-developed method for the analysis of residential segregation in cities (Jones et al. Reference Jones, Johnston, Manley, Owen and Charlton2015), which subsequent work has demonstrated is clearly multi-scale in its organization (e.g. Manley et al. Reference Manley, Johnston, Jones and Owen2015). We have modelled the proportion voting Democrat with the denominator as the total Democrat plus Republican vote. Those rates are then modelled within a multi-level framework, to obtain estimates of the intensity of segregation at each scale, net of segregation at the next largest scale and taking account of binomial variation occasioned by the varying denominators that form the proportion of those who vote Democrat. (Full details of the modelling strategy are in Leckie et al. Reference Leckie, Pillinger, Jones and Goldstein2012.) Thus each of the polarization measures reported here is for a particular spatial scale, independent of any polarization at a larger scale within which the specified units are nested. Polarization at the county scale, for example, is measured net of any polarization at the state scale; we are evaluating whether there is polarization across the states and then, independently, whether there is further polarization across counties within states—testing whether any observed differences across counties simply reflect differences between the states within which they are located.

In work on multi-group segregation patterns—such of those of ethnic group residential segregation in cities—the derived segregation measure from this modeling approach is the Median Rate Ratio (MRR). Where just two categories are involved, such as voting either Republican or Democratic, we use the Median Odds Ratio (MOR), which can be interpreted in exactly the same way as odds ratios in logistic regressions. The MOR values are derived from the modeled (logit) variances in the rates at each scale, and they have associated Credible Intervals (CIs), which provide the degree of empirical support for the values of a parameter. We have used the 95% intervals to convey the uncertainty; as they are based on Bayesian estimation, they can be asymmetrical around the estimated MOR value.

MORs can be interpreted in the following way. Take a set of counties within a state for which we have the modeled rate. Take any pair of counties at random and calculate the ratio between the highest and lowest of the pair of modeled rates. Repeat this many times. The MOR is then the median value of the resultant distribution—the average difference between any pair of modelled rates; a value of 1 means that there is no polarization whatsoever. Further, because the measures are ratios, they can readily be compared: an MOR of 1.5 is 20% larger than one of 1.25, for example, and so in comparing two measures we can not only assess how much larger one is relative to the other but also, using the associated Bayesian credible intervals, whether the two are significantly different from each other.

Our chosen measure of the degree of spatial polarization is thus superior to the standard indices of spatial unevenness (such as the index of dissimilarity used by Glaeser and Ward Reference Glaeser and Ward2005) because it: is readily interpretable; has associated credible intervals that allow for rigorous and robust estimates of differences over time; and separately identifies—again using credible intervals allowing for robust testing—the intensity of any observed differences at a number of spatial scales (in this case, three) independent of patterns at each of the other scales. With it we can estimate with confidence whether polarization is greater at some dates rather than others, and at some scales rather than others—and the combination of those two. It allows for a clear test of the spatial polarization hypothesis.

SPATIAL POLARIZATION IN VOTING DEMOCRATIC AT THREE SCALES, 1992–2012

Any analysis of spatial polarization is necessarily arbitrary in its selection of the number and nature of scales to be included. In that reported here, we use three: ^{Footnote 3} the nine divisions used for reporting many statistical series by the US Bureau of the Census and which approximate to the political culture regions identified by Elazar (Reference Elazar1972); the 49 states (excluding Alaska, which is not divided into counties or equivalent smaller areas for the reporting of vote numbers); and the 3,077 counties—or county-equivalents—within those states. ^{Footnote 4} The variable whose spatial pattern is being modelled is the number of Democratic voters and the null hypothesis is that those voters are distributed across the counties (and thence the states and divisions) in proportion to the total number of Republican-plus-Democratic voters (i.e. we exclude votes for minor-parties). The data are a bespoke collection derived from the official returns published in each state after the relevant election. ^{Footnote 5}

The resulting MORs, with their associated CIs, are shown in figure 1. Those in the left diagram are for the nine census divisions; those in the central diagram are for the 49 states, net of any variations between divisions (i.e. the MORs are the average odds ratios in each year between pairs of states within each division—and so are net of any inter-division variation); and those in the right diagram are for counties within each state (i.e. the average ratio between pairs of counties selected at random within a state, and thus net of any inter-state, inter-division variation).

Figure 1

Polarization Trends at Three Spatial Scales

The polarization trends at the three spatial scales, showing the MOR values and their associated CIs. The MOR values for counties are net of any differences between states, and those for states are net of any differences between divisions.

All three diagrams provide clear evidence of growing polarization at the selected scales. There was greater polarization in Democratic voting over the period 1992–2012 across the nine divisions—that is, there was greater segregation of Democratic voters into some divisions relative to others. (The MOR value for 2012—1.55—was 29% larger than that for 1992—1.20.) There was greater polarization in Democratic voting over the period 1992–2012 between states within the nine divisions—that is, there was greater segregation of Democratic voters into some states within each division relative to others although that increased polarization was less than that which occurred at the divisional scale (a 12% increase to 1.36). And there was greater polarization in Democratic voting over the period 1992–2012 between the 3,077 counties within states—that is, there was greater segregation of Democratic voters into some counties relative to others within each state (the MOR increased by 14% to 1.69).

These clear upward trends at each scale vary in the statistical significance of the differences in the MOR values at any pair of dates. At the divisional scale, the CIs are large and—not surprisingly given the low number of observations (nine)—overlap: there is no convincing statistical evidence that polarization was significantly greater at any one later date compared to an earlier one—even the CIs for 2012 overlap slightly with those for 1992. However the estimates do not show trendless fluctuation as the pattern is one of consistent increasing polarization.

At the state scale, too, the number of observations (of states within divisions) is fairly small, and the CIs are relatively wide as a consequence – though much less so than at the divisional scale. The modelled MOR for 2012 is significantly greater than that for 1992, however, indicating that by the end of this relatively short period there was greater polarization in voting Democratic at that scale than there was at its beginning. The states were less divided–net of the divisional changes—in their support for Bill Clinton than they were for Barack Obama.

The modelled MOR values at the state scale are much smaller (and significantly so) than those shown in the diagram for the county scale. For counties—within-states, within-divisions—polarization was relatively high at the start of the period (an MOR of 1.48, 48% larger than a value of 1.0 which would indicate no polarization) and substantially larger still (at 1.69) 20 years later. Further, after the first two elections in the sequence, the MOR for each contest was statistically significantly larger than that for the previous election, giving very strong evidence of greater polarization over time at that finest of geographical scales analyzed.

It is also possible to decompose the total logit higher level variance (Browne et al. Reference Browne, Subramanian, Jones and Goldstein2005) between the three scales at each date, and the resulting percentages are shown in table 1. These indicate a substantial shift across the two decades, with the percentage of the variance between divisions more than doubling whereas that between-counties within states declined by a comparable amount; there was no change in the percentage associated with between-states within divisions. Across the six elections, therefore, whereas the differences at the macro-scale between divisions have become more accentuated, those at the micro-scale between counties within states have become less. In relative terms the fine-grained patterning at the start of the period—accounting for over two-thirds of the variance across the map—has become less important whereas the coarser-grained differences at the macro-scale of the nine divisions have become more pronounced. Thus while the absolute change as shown clearly in figure 1 has seen increasing polarization at all three levels, the greater proportion of it has been at the most macro level. But this does not mean that in absolute terms the between-county variation has decreased—quite the contrary. This difference between absolute and relative change poses a further set of questions to be addressed in exploring the reasons behind these changing and increasingly important electoral geographies.

Table 1

Decomposition of the Higher-level Variance into the Percentage of the Total at the Three Spatial Scales