Chapter 3 in King, Rosen, and Tanner, Ecological Inference: New Methodological Strategies

Using Ecological Inference for Contextual Research:

When Aggregation Bias Is the Solution as Well as the Problem

D. Stephen Voss

University of Kentucky

dsvoss@uky.edu

http://www.uky.edu/~dsvoss

Debate over Gary King's proposed solution to the ecological inference problem (King 1997) has begun to emerge in journals oriented toward statistics and political methodology. For the most part, this debate has operated at a theoretical level – evaluating properties of the estimators generated by King's method (as well as his supplementary software, EI and EzI). The tone of the debate has been rather critical of King's attempted advance in ecological inference (Herron and Shotts 2003; McCue 2001; Rivers 1998), and diagnostic trials introduced in this context have not produced favorable results (Anselin and Cho 2002; Cho 1998; Freedman et al. 1998). At the same time, substantive applications using King's method have begun to appear in social-science journals (e.g., Burden and Kimball 1998; Gay 2001; Gimpel and Schuknecht 2002; Liu 2001; Lublin and Voss 2002; Voss and Miller 2001). Not only have these manuscripts successfully cleared the peer-review process for numerous respectable outlets, the authors also report concrete diagnostic checks that indicate the relative success of EI at solving difficult estimation problems.

Disagreement between theoretical critiques and substantive applications is not unique to aggregate-data analysis. Critics generally compare a method against ideal statistical properties, so the incentive is to identify data that illustrate methodological problems. Analysts, by contrast, generally compare a method against convenient alternatives; the incentive is to defend whichever statistical tools they select. Unfortunately, the pressures that drive apart theory and practice worsen in the instance of ecological inference. Success depends heavily upon the amount of information lost in the aggregation process and the contextual knowledge that an analyst exploits when trying to recapture lost information. Critics face little difficulty identifying data that will lead methods of ecological inference astray, whereas knowledgeable analysts can tweak performance out of a statistical tool that it would not exhibit if applied naively. The result is a dissonance that can only confuse practitioners waiting for the methodological literature's final verdict on a statistical innovation.

My purpose in this chapter is to lay out, in relatively accessible terms, why King's solution to the ecological inference problem represents a path-breaking development that should fundamentally change scholarly standards for what is acceptable in aggregate-data analysis. However, I do not simply summarize, in less-technical vocabulary, the various statistical virtues that King already has documented on behalf of his approach (1997, 1999, 2000, 2002). Rather, I focus on EI as a substantive resource for conducting what social scientists customarily call "contextual research." King's approach helps the analyst probe theoretically important contextual effects that undermined previous methods of ecological inference because these methods had to assume them away. Unlike the conventional forms of aggregate-data analysis that preceded it, King's approach holds out some possibility for pulling apart compositional and contextual patterns – that is, for determining the extent to which social behavior varies from place to place because of changes in the balance of social groups composing each locale, and the extent to which social behavior varies because of how each social group reacts to the context in which it operates.1

Pursuing this argument requires four distinct steps. The body of the chapter begins by embedding ecological inference within the needs of contextual research to show why they are inextricably linked. The aggregation bias that spoils conventional ecological inference derives in large part from the very same contextual effects central to social theories of human behavior. The second section reviews why King's method is sensitive to aggregation bias, which in turn means that it provides leverage for capturing otherwise-elusive contextual effects. The third section explains why previous diagnostic tests do not give a good picture of EI's value for contextual research. The extant critiques analyze patently uninformative data and neglect to incorporate the sort of substantive insights that normally would be available to an analyst applying the method.2

The final section therefore exhibits King's approach in action to show how it is especially valuable for contextual research by analysts armed with informative data or with a significant background in the substance of their topics. In particular, it applies King's innovations to real data collected to address genuine research questions in racial voting behavior, and it illustrates the unprecedented success of his statistical tools. Techniques of ecological inference still have far to go, and numerous enhancements still await practical implementation (as illustrated by the many essays in this volume). But all of the sound and fury generated by this scholarly ferment – which will give numerous reminders why EI is imperfect – should not distract practitioners from the simple fact that King's method is far superior to the tools upon which both scholars and policy makers previously relied. The criticism of King should not serve as an excuse to continue using those older methods.

How Ecological Analysis Killed Ecological Inference

Contextual theories of political behavior generally characterize how individuals respond to their social environment. They recognize that individuals view political life through different lenses depending upon the communities they inhabit. People experience different pressures and face different incentives because of where they happen to reside. Different memories and different symbols shape their political lives, and the schema that structure this emotion-laden raw material may be shared narratives that have emerged from a geographically bounded stream of communication (Couto 1993). This complex, socially embedded perspective motivated many early empiricists in political science, including the "quintessential political scientist," V.O. Key, Jr. (Lucker 2001; Natchez 1985, 74, 94-99). It eventually faded in favor of highly individualistic approaches to political behavior grounded in social psychology, as epitomized by the Michigan model of voting behavior.3

Surveys encourage individualistic approaches to politics. They sample random people in isolation and ask for their subjective perceptions. Secondary analysis often explains opinion with opinion and preference with preference; the burden seldom falls on researchers to trace raw opinions back to any tangible source (Natchez 1985, 64-67). In particular, analysts rarely consider the strongest community attachments that individuals might have – family, friends, church, school, neighborhood, county, newspaper – because they are localized.4 Secondary analysts may not even possess the contextual information needed to allow such nuance. Political actors inherently appear as disembodied souls in work developed from such a data source. By contrast, aggregate statistics summarize the social environment shaping political behavior. They portray many of the external incentives and pressures that operate on political actors, especially those bounded by geography. With reliable methods of aggregate-data analysis, deeper insights about the social nature of politics would be possible, so the line between contextual analysis and aggregate-data analysis is hazy and to some extent their fates have always been tied together.

Sociologists recognized long ago the intimate connection between aggregation and attempts to contextualize human behavior. Indeed, the language we still use to discuss aggregate-data analysis reflects their awareness. They used biological metaphors to discuss how humans related to the pressures of a social environment, implicitly or explicitly comparing the process to a species interacting with its ecology. Ecological analysis, when applied to human behavior, meant studying contextual effects on how people acted and what they believed. Therefore, when aggregate-data analysis suffered a series of high- profile attacks that undermined confidence in available methods (e.g., Goodman 1953; Robinson 1950), the natural result was a decline in ecological or contextual studies.

Few early critics discussed the ironic tension between ecological inference and ecological analysis – the focus tended to be on fallacious conclusions rather than their substantive origin – yet it was precisely the importance of human ecology that made aggregate-data techniques for ecological inference so unreliable (Hauser 1970). Contextual behavior undermined the very data conventionally available to study political context. For this reason, the two most common approaches to aggregate-data analysis assume that contextual effects do not exist. They require analysts to believe that members of demographic groups act the same way everywhere; demographic variation in their communities bears no relation to how people behave. "Homogeneous unit analysis," for example, requires the researcher to assume that places where a group is numerically dominant will be representative of how the group acts in more diverse places (MacManus 1995, 45n). Linear "ecological regression," meanwhile, explicitly assumes that all systematic behavioral variation is compositional, none is contextual (see the Introduction [King, Rosen, and Tanner].5

A quick formal exploration illustrates why linear ecological regression is almost useless in the presence of contextual effects.6 Analysts regress an aggregate political outcome on the population density of each demographic group, and interpret the resulting coefficients as rates of behavior. Take the case of black (Xi) and white (1-Xi) voting for the left-wing candidate in an election (ViL). The assumed linear relationship, across multiple areal units "i", would take on the following form: 7

ViL	= bbXi + bw(1 – Xi) + ui

where ui is a complex residual to capture how voting differs from that predicted by the two coefficients.8 This form of the regression lacks an intercept, but transforming the model to a bivariate version with a constant term is simple:

ViL	= bbXi + bw – bwXi + ui

= bw + (bb – bw)Xi + ui

Both political outcomes (ViL) and racial densities (Xi) usually appear in aggregate data. One generally lacks the frequencies that would appear in a cross-tabulation, though: the proportion of blacks (bb) and whites (bw) who supported the left-wing candidate.

The lack of an index for either parameter illustrates the common assumption that ethnic rates of behavior are constant across areal units, which is what makes a least-squares method of estimation so convenient. When this assumption is false – when members of an ethnic group do not, on average, behave the same way everywhere – the data may contain aggregation bias. Estimates from this simple model can fall quite far from the truth because they attribute contextual variation in one group's behavior to the other group.

Consider a simple contextual effect, in which whites support the left-wing candidate at a higher rate when they live in heavily minority areas (Voss and Lublin 2001). The formula for left-wing support now contains a shifting parameter, biw, which might be a linear function of racial density:

ViL	= bb Xi + biw (1 –Xi) + ui

= biw + (bb – biw) Xi + ui	where biw = g0 + g1 Xi

Naive ecological regression, performed on data of this sort, will not estimate the desired quantities (i.e., bb and the weighted average of biw, presuming all other assumptions of the model were correct). Instead, it would turn up the following:

ViL	= bb Xi + biw (1 – Xi) + ui

= bb Xi + (g0 + g1Xi) (1 – Xi) + ui

= bb Xi + g0 (1 – Xi) + g1Xi (1 – Xi) + ui

= [bb + g1 (1 – Xi)] Xi + g0 (1 – Xi) + ui

The estimated white support for the left-wing candidate, therefore, will be g0 only – too low if the contextual effect g1 is positive. Similarly, the coefficient estimated for black density will be bb+g1(1–Xi), which would be too high. This is why analysts often estimate that Democratic support among blacks exceeds 100% (see King 1997, 16).

Of course, one can adjust the model for this simple case of parameter variation. Applying the distributive property and then regrouping provides:

= bb Xi + g1 Xi – g1 Xi2 + g0 – g0 Xi + ui

= g0 + (bb + g1 – g0) Xi – g1 Xi2 + ui

One can compute the three parameters of interest after running that regression. However, if white and black vote choices both vary with the racial density of the locale, either directly or indirectly, then linear ecological regression is impossible because the parameters are underidentified. We would need not bb but the weighted average of bib:

= bib Xi + g1 Xi – g1 Xi2 + g0 – g0 Xi + ui	where bib = r0 + r1Xi

= (r0 + r1Xi) Xi + g1 Xi – g1 Xi2 + g0 – g0 Xi + ui

= g0 + (r0 + g1 – g0) Xi + (r1 – g1) Xi2 + ui

It is impossible to compute the four parameters of interest that are now necessary.

The failure of conventional forms of ecological inference therefore amounts to more than just the danger of getting wrong answers, of committing an "ecological fallacy." For contextual analysis, they never provide any answers at all.

Enter the Method of Bounds

King's (1997) approach offers some hope of capturing systematic geographical variation through a method of statistical inference. This is not the place to summarize King's proposed solution to the ecological-inference problem; a review appears in the Introduction [King, Rosen, and Tanner]. However, the critical wrinkle for contextual analysis is King's use of the deterministic "method of bounds" (Duncan and Davis 1953). The basic insight is fairly simple. Consider a locale in which the Democratic candidate wins 50% of the vote and 25% of the voters are black. Even if every African American voted Democratic, whites must account for the remaining 25% of the Democratic candidate's votes. At least a third of whites mathematically must have voted that way (although more may have, if some blacks voted Republican). So the possible white vote in this hypothetical example is not between 0% and 100%; the minimum is 33%. Add up all the constraints on what is possible, maxima and minima, from areal unit to areal unit, and you may have quite a lot of information about larger electoral districts.

As the Introduction [King, Rosen, and Tanner] explains, EI identifies from the outset the complete set of possible rates of racial voting behavior – not for an electoral district as a whole, but for each areal unit it contains. Furthermore, EI goes beyond the simple method of bounds by considering the two rates of group behavior simultaneously. It takes into account the linear relationship between how one group could have behaved and how the other could have in order to produce the final vote totals (see Equation 5 in the Introduction [King, Rosen, and Tanner]). EI therefore constrains estimates of joint behavior, reducing the range of estimates for each areal unit to a series of exclusive pairs.

By exploiting the deterministic information in the data so thoroughly, EI has the potential to detect and adjust for contextual patterns. This is best understood using the "tomography plot," a graphical representation of the line segments for each areal unit in an analysis (see the first figure in the Introduction [King, Rosen, and Tanner], and its accompanying explanation). Figure 1 presents a plot containing line segments for a random selection of 25% of 1996 precincts in Georgia's second congressional district (see Voss and Lublin 2001).9 Horizontal lines correspond to precincts that are almost entirely white. They contain so few African Americans that we know quite precisely how whites behaved but almost nothing about the few blacks, which is why the slant of such a line only allows a small range of possible values on the Y-axis but any value on the X-axis. A segment becomes more vertical, however, as black voters increase proportionally. We are less sure how many whites supported the Democrat in these mixed counties, because the aggregate data also include a large black population.

(Figure 1 about here)

The narrow bounds that can appear for areal units often will push estimates from a basic EI analysis into a contextual pattern. This pattern alerts the analyst that the data contain aggregation bias. Figure 1 illustrates how the bounds inform ecological inference. The slanted lines often run up against the right edge of the unit square at a very high level of white support for black Democrat Sanford Bishop. The line segments cascade downward as they straighten out, though. All-white precincts are less likely to strike a high value on the right edge of the unit square. Their bounds indicate lower levels of white support for Bishop. White voters clearly supported the black Democrat at higher rates when they lived in a racially mixed context, and even the simplest versions of EI will capture this pattern by requiring each precinct estimate to be possible. Figure 2 shows how simple EI estimates run on these data follow a contextual pattern.

(Figure 2 about here)

Not all examples of aggregation bias will be as obvious as the one I have selected here – but EI can pick up patterns in a large sample that are too subtle for the naked eye to detect. This is especially true when researchers approach an analysis with theoretical reasons to anticipate the contextual effect. In instances when researchers can hypothesize why voting rates might correlate with racial density, an extended version of King's model allows the introduction of covariates to explain systematic variation in group behavior (see Equation 10 in the Introduction [King, Rosen, and Tanner]). The model picks up contextual variation in the bounds and projects that pattern onto the remaining areal units whose bounds were not as informative. Bounding the estimates for each areal unit, and building up estimates for a larger electoral district based on those bounds, therefore allows EI to function even if both racial groups exhibit some degree of contextual behavior. It is not underidentified the way linear regression would be.

Of course, being able to produce estimates does not guarantee they will be good ones. Their quality depends upon how much information the bounds provide. Critics point out that King's method may underestimate the strength of contextual patterns, which are suppressed by the aggregation process (Anselin and Cho 2002; Cho 1998; Rivers 1998). They worry about variation in estimates powered by residuals from a statistical analysis (Herron and Shotts 2003; McCue 2001). These are valid concerns. Researchers should understand the severity of the limits; methodologists should search for ways to alleviate the dangers. Nevertheless, the ability to detect even a healthy portion of that contextual variation takes King's method far beyond what practitioners were using for the five decades between consciousness of this problem (Robinson 1950) and King's attempted solution (1997).

"If It Is So Useful, Then Why All the Criticism?"10

King's critics have complained that he was not explicit enough about when EI will be reliable and when it will be unreliable (Freedman 1998, 1520). For the most part, these insights long precede King's book. An extensive literature has explored when data aggregation will throw away more or less information. King (1997, 88-89) also presented "scattercross graphs," as described by Leo Goodman (1959, 617-18), that provide a visual means of determining when the method of bounds will help make up for the limitations of ecological regression. Briefly, here are the "rules of thumb" that practitioners apparently need. Ecological inference will work better when:

1.	The group of interest either constitutes a large majority proportion of the overall population or is relatively segregated across areal units (which is more likely when areal units are very small).

2.	Outcomes of interest approach unanimity in particular areal units, especially the ones with highly mixed populations;

3.	Data contain numerous observations to insulate the estimation from errors due to chance;

4.	The grouping of interest identifies a politically distinct population with internally similar behavior that differs from the behavior of other groups – that is, across-group variation outweighs within- group variation);

5.	Within-group variation across areal units is effectively random, resulting from innumerable small sources rather than a few large but unmeasured influences;

6.	Any systematic variation in a group's behavior across areal units does not respond to the balance among groups in each place, nor does it respond to other variables that are correlated with group density;

7.	The researcher possesses sufficient knowledge of the subject matter to inform the estimation about possible sources of systematic within-group variation and to detect fallacious conclusions.

King's solution does not alter any of these generalizations. It still performs better with detailed data. It is still hindered by contextual patterns when bounds are uninformative. It still does not provide a panacea for analysts lacking substantive familiarity with their data.

Why King's Method Performs Better with Poor Data

The similarities between King's approach and ecological regression leads Douglas Rivers (1998, 442) to warn readers against thinking "that King has somehow relaxed the assumptions of ecological regression." He dismisses EI's "fanfare" and "notoriety" as undeserved because King's approach relies on assumptions that "include all those required for Goodman's estimator and then some." This critique is misleading at best. It implies that ecological regression is somehow a generalized form of King's approach, requiring fewer assumptions – which would make King's method more susceptible to faulty application. In fact, the reverse is the case. King's method is less vulnerable to faulty assumptions – not because the assumptions have become completely irrelevant, but because the method of bounds helps avoid analytical dependence on them.

Furthermore, some assumptions of ecological regression do not appear in King's method and therefore have been "relaxed" even in the sense that methodologists usually use the word (i.e., dropped). For example, EI does not assume that parameters are fixed. It does not assume that errors in prediction always have the same variance (i.e., the homoskedasticity assumption). By aggregating across areal units, EI computes direct estimates for the quantities of interest; it does not assume that the average aggregate behavioral pattern represents the average individual-level behavior pattern (Palmquist 2002). Nor, thanks to King's use of the truncated bivariate normal distribution and the method of bounds, does EI assume linearity across the full range of data.11

Finally, it is worth noting that practitioners using EI are less vulnerable to faulty assumptions in another way – which is that the software includes both graphical and statistical diagnostics that often will indicate when assumptions are wrong. Indeed, these diagnostics not only help experts check some of their basic ideas about the data-generation process, the graphical and statistical tools sometimes have an independent substantive value; presenting them can help clarify the results of an analysis.12 By contrast, aside from prior knowledge, about the only defense against faulty inference in ecological regression is observed nonlinearity in the relationship between racial density and outcome. Also, linear methods of ecological inference, from correlations to multiple regressions, all lack meaningful standard errors that researchers regularly consult and that indicate the uncertainty contained within estimates.13

A disadvantageous distribution of voters will inflate any bias contained in the aggregation process many times over (Palmquist and Voss 1996). Take one terrible case for ecological inference: when an analyst wishes to probe gender differences in behavior. Gender differences are especially resistant to aggregate-data analysis, unless one can incorporate outside information (see Chapter 6 [Corder and Wolbrecht]). Men and women usually live together – aside from places containing prisons, military bases or other institutions that might skew the normal gender balance – so their populations are almost evenly divided. Furthermore, the fluctuations in gender that do occur across electoral units are almost completely irrelevant to politics; they are small, mostly random noise. Yet practitioners may not notice the uselessness of naive ecological inferences based on data such as gender aggregates. Ecological correlations give a false sense of security, whereas King's software provides no such cover; standard errors advertise when an analyst tries to get milk from a stone.14

Why King's Method Performed Poorly with Horrible Data

Gary King's book (1997) already tests his method in real data. These examples illustrate the method's effectiveness, or at least its ability to warn analysts when something has gone wrong, although they are limited in quantity (Freedman et al. 1998, 1520). By contrast, critics have subjected King's approach to other diagnostic tests that point toward severe limitations (Anselin and Cho 2002; Cho 1998; Freedman et al. 1998, 1999). It may seem superfluous, at this point, to continue providing examples of the method in action – since success in a few data sets "does not prove anything" (Rivers 1998, 443) and his model "works on some datasets but not others" (Freedman et al. 1998, 1520).

Nevertheless, the available slate of trials contain certain drawbacks that seem to call for a reconsideration. The initial trials were not representative of how King's method would work in most situations – let alone how it would perform in particularly useful situations. Leaving aside simulated data, King's method has been tested against:

•	Aggregate data produced from a survey conducted during the 1984 California general election, in which the analyst attempted to estimate racial differences in education across 30 precincts (Cho 1998, 153-60).

•	Aggregate data produced from a survey conducted during a San Francisco city council race, in which the analyst attempted to estimate differences between Chinese and non-Chinese voters in their support for candidate Thomas Hsieh across 37 precincts (Cho 1998, 160-61).

•	Aggregate data produced from an exit poll conducted in Stockton during the 1988 presidential primary, in which the analyst attempted to estimate Jesse Jackson's support among Hispanics and non-Hispanics across 39 sample precincts (Freedman et al. 1998).

•	Census-tract data from Los Angeles that contained 1980 demographics, in which the analyst attempted to estimate education, income, and home ownership for Hispanics and non-Hispanics across 1,409 tracts (Freedman et al. 1998).

•	Census-tract data from Los Angeles that contained 1988 election statistics, in which the analyst attempted to estimate party registration for Hispanic and non-Hispanic registered voters (Freedman et al. 1998).

•	County data from Texas that contained incidents of strokes, in which the analysts attempted to estimate stroke rates by gender (Anselin and Cho 2002, 287-88).

How do these trials stack up against the conditions under which an analyst should expect optimal performance from an aggregate-data analysis technique? The data are notable, as a group, for the extent to which they deviate from conditions under which one ought to try ecological inference – and generally the few diagnostics they report from EI indicate as much.

Consider the following comparison of these data against the enumeration of optimal circumstances listed earlier. As far as a reader can judge, these trials apply King's method to data with very few homogeneous units. Blacks never exceed a third of the precinct population in Cho's first example (see the giant standard errors in her initial analysis). The Chinese only exceed majority status in a handful of her precincts in the second example (see the slope of lines in the tomography plot as well as the large standard errors). The stroke analysis uses gender as its demographic category, probably the toughest case in ecological inference because of sharply limited compositional variation. Most of Freedman's examples use census tracts, long known for how seldom they approach homogeneity (Myers 1954). With his exit-poll data, the standard error on Hispanics is more than four times as large as the one for non-Hispanics, suggesting that the data provided limited information about the minority group.15

Three of the examples include only a few dozen observations each. In some of these cases, it is unlikely that a researcher would have had to stop with so little data. For example, even if one were interested only in Jesse Jackson's Hispanic support in Stockton, the analysis would be stronger if it could borrow strength from related cases of Hispanic voting behavior nearby. Cho's education example not only includes few cases, it obviously requires covariates, which will tax the data more than usual.

Most of the trials attempt to predict socioeconomic data, which are known to vary widely within groups. Black education rates in Cho's first example jump from 1% to 90% across precincts! More importantly, socioeconomic status strongly shapes people's localized residential choices (see, e.g., Biggar and Martin 1976), so minorities who reside in mixed-race or mixed-ethnicity settings are likely to have much higher status than those in segregated ghettoes. The stroke data, meanwhile, were selected precisely because they exhibited significant spatial autocorrelation.16 These examples were certain to be characterized by systematic variation within groups and by community-based similarities across them.

All of the political trials look at urban areas and yet make no reference to African Americans – who stand apart for their heavy Democratic party support. Blacks simply fall into the non-Hispanic or non-Chinese categories, along with white Anglos. Anyone knowledgeable about racial politics should realize that combining whites and blacks into one group violates King's distributional assumptions, and probably introduces severe within-group variance while suppressing across-group variance.

None of the examples uses King's extended model to capture the contextual effect of group density directly. Only one uses the extended model at all: Cho introduces two covariates, income and age, while predicting racial education rates. Income helps, age hurts – which leads her to conclude that an analyst has no idea when a covariate might improve the situation. Yet EI's diagnostics clearly show that the income covariate was the right thing to do. In both the basic EI model and the version with an age covariate, standard errors from EI were so large that they swamped the estimates themselves. After Cho adds the income covariate, though, the model suddenly behaved: it revised the estimate of black education rates upward by 50% and returned much smaller standard errors. One suspects that likelihood scores from the maximum-likelihood estimates also indicated the desirability of an income covariate.

People who study subjects empirically should not reinvent the wheel. A scholarly literature usually provides enough descriptive information that one can assess in advance the likely success of any particular inference, as I have done retrospectively with the extant trials. I severely doubt that scholars familiar both with ecological inference and with race/ethnicity would have been surprised by the EI's poor performance in these hostile tests. Furthermore, a knowledgeable scholar would have balked at most of the biggest estimation errors even without the truth to guide choices: 91% of Hispanics as Democratic registrants, which will happen rarely across the country; standard errors so huge that the estimates are not substantively interesting; an estimate that whites with a college degree outnumbered blacks 4-to-1 when in fact blacks were more likely to have graduated from college. Even if it is true that theory and previous research provide "no clear way of distinguishing good models from bad models," they provide a lot more guidance than these particular trials brought to bear.

Applying EI with Informative Data or Contextual Knowledge

To illustrate how King's approach outperforms conventional methods of ecological inference by incorporating and then summarizing contextual information, I will apply his parametric model to the study of racial voting behavior in the U.S. South. In each case the goal is to fill in cross-tabulations between aggregate race data and aggregate political outcomes, thereby producing accurate estimates of voting behavior by race – exactly the sort of estimates ecological inference commonly would be called upon to produce in real voting or policy research. Before presenting the application, though, I should explain why racial voting behavior is a critical case for ecological inference, as well as why southern data are particularly helpful for assessing a method's contribution.

Why is race a good choice? First, racial categories are advantageous because they are highly segregated at low levels of aggregation but not terribly segregated at higher ones. They allow a test of King's method in a variety of situations. Second, racial behavior tends to be contextual, and in theoretically important ways. The nature of these contextual effects links specific substantive concerns of political scientists, sociologists, social psychologists, quantitative historians, constitutional-law scholars, and anthropologists, among others (e.g., Kousser 1974; Pettigrew 1985; Taylor 1998; Voss 2001). Third, voting-rights specialists depend upon ecological estimation. The U.S. Supreme Court declared racial bloc voting to be one component of a justiciable "vote dilution" claim in Thornbugh v. Gingles (478 U.S. 30-
108), and litigants simply do not have state or local survey data available to establish or ward off constitutional claims. Analysts require some workable method of ecological inference to recapture data lost due to the secret ballot.

Of course, one could argue that selecting race stacks the deck in King's favor. Racial groups are more polarized in their partisan behavior than just about any other demographic category. Nevertheless, racial polarization is a fourth reason why race is a critical case for ecological inference. The survey- research revolution has not liberated those who study racial politics in the same way it has freed up work in other subfields. Politically sensitive topics resist objective measurement (Kuklinski, Cobb and Gilens 1997; Traugott and Price 1992), so aggregate data naturally play a significant role in the study of race. Anyway, party choice is not the only form of racial voting behavior; registration and turnout do not polarize the races nearly as much. Race is a good choice for assessing the success of King's method.

Why southern data? First, the South makes sense because, among places with a substantial black population, that region has less residential segregation than the rest of the country (Massey and Denton 1993). Estimation will be harder than usual. Second, my own research tends to focus on southern politics (e.g., Lublin and Voss 2000a, 2000b, 2003; Voss 1996), so I need not apply King's method blindly. Third, not only did southern states fuel the contextual study of racial voting (Heard 1952; Key 1949; Matthews and Prothro 1966), they are the site of most redistricting litigation and thus typify the conditions under which King's method would shape public policy. Finally, and most importantly, a string of voting-rights cases have convinced the South's public officials that they need to invest in precise data collection. Sometimes the data reach such high quality that they include answers to questions that, in other locales, would require some form of statistical inference to probe.

County-Level Racial Voting Behavior

George Wallace's 1968 presidential campaign may have dealt the death blow to aggregate-data studies of voting behavior. White support for Wallace was so contextual, with support high in exactly the places where blacks were most numerous, that it completely skewed the techniques of ecological inference available at the time. Furthermore, the Wallace vote was only available to researchers at relatively high levels of aggregation, so aggregate data were (and are) quite poor.

An analysis by Schoenberger and Segal (1971, 585), for example, implied that Wallace received heavy black support in his 1968 presidential bid. The authors did not embrace this absurd conclusion, correctly attributing the findings to Wallace's heavy white support in the Black Belt, but lacked any statistical basis for their interpretation or any real estimates of white behavior. Not even additional methodological refinements, intended to counteract the "ecological fallacy," could remove their reliance on prior knowledge, or provide reliable estimates of how each race actually voted (Wasserman and Segal 1973, 179). The complete inadequacy of aggregate-data analysis for contributing to this important topic of interest helped illustrate why the discipline needed to move toward surveys in the short term. The Wallace vote has even stumped researchers using more modern methods, including King's solution (Palmquist 2002). It is therefore about as tough a real-life test as King's method could face.

Louisiana reports parish-level (i.e., county-level) voter registration broken down by race. If we consider the vote as a three-stage process–the decision to register, the decision to vote, the vote choice–the first stage of that process does not require estimation in this rare instance. Louisiana data are particularly useful for two other reasons. First, the state's registration data contained lots of idiosyncratic variation in 1968, all of which violates EI's assumptions. Some parishes purged their voter rolls of dead weight more frequently than others.17 Some hosted federal election examiners eager to expand the voting rolls.18 Second, all evidence indicates that very few blacks voted for Wallace. This strong prior on what the black estimate ought to be means that I can check King's method at multiple levels of racial behavior. It is precisely the combination of difficult data with known answers that makes the Louisiana case worthwhile as a test of King's method.

For my initial run, I did not take advantage of EI's more advanced features, such as modeling parameters of the truncated bivariate normal to vary with other relevant quantities. I want to compare the estimates from this simple EI analysis both to the real numbers and to estimates produced by other methods of aggregate-data analysis. The first row of Table 1 therefore reports the real racial registration rates: 53.6% of blacks and 77.3% of whites. I then report what an analyst lacking the truth would know to be true: the bounds for the state computed from state-level marginals, and the bounds for the state computed from parish-level marginals. The state bounds alone do not constrain black registration estimates at all, and only pinch possible white registration rates to a span of roughly 35 percentage points. The bounds based upon parish-level limitations squeeze the range of valid estimates for whites even more, and rule out the more extreme levels of black registration as well. The truth in this case clearly falls near to the midpoints between the bounds: 75.9% for whites and 57.5% for blacks.

(Table 1 about here)

Naive linear ecological regression produces quite impossible estimates: 86% registration for whites (more than two large standard deviations from the truth), and 57% registration for blacks. For the entire state, this implies a faulty 78.5% statewide registration rate. Faced by obviously incorrect implications, an analyst would know that ecological regression had failed. One solution would be to use a weighted analysis (Palmquist and Voss 1997, 13). Therefore I repeated the ecological regression using weighted least squares. The white registration estimate then comes quite close–within a percentage point of the truth–but the black registration rate rises to more than six percentage points away. The overall estimates are still impossible, since they represent a 73.2% statewide registration rate.

Ecological regression does not produce parish-level estimates – 77.9% of whites and 60% of blacks presumably registered in each parish – but we can evaluate these figures as surrogate parish estimates by comparing them to the parish bounds that an analyst would know even without Louisiana's racial data. Together the estimates are impossible in all but, at best, two or three parishes (as indicated by how few of the tomography lines would cross that point). Even looking one race at a time, the white registration rate is impossible for 33 parishes, too low for 27 and too high for 6. The black registration rate is impossible for 14 parishes. That the bounds are so active signifies more than just the failure of linear regression; it also suggests that contextual effects may be biasing the estimates.

Another common method of analysis is to look at all-black units to determine black behavior and all-white units to determine white behavior. Unfortunately, only four of the 59 parishes used in my EI estimation contain more than a 90% white voting-age population, and none are more than 90% black.19 The limits of this homogeneous unit approach are therefore obvious. The registration rate in the four white parishes, computed as a weighted average, comes to 85.4% percent–far higher than what could be true statewide. All-white parishes were not representative. The neighborhood model, which assumes that all races behave the same way in particular electoral units but differ across them, is similarly unimpressive. It estimates almost equal registration rates for blacks and whites when the true results are much more polarized.

Even a simple EI estimation clearly outperforms every method except weighted ecological regression. It estimates that 76.7% of whites registered, within a percentage point of the truth, and that 55.2% of blacks registered, within 2 percentage points of the truth. Given the wide variation from one parish to the next, these estimates are fantastic. The one drawback is that standard errors are very large. This is a reassuring drawback, though, since it means EI's standard errors reflect the high degree of uncertainty much better than those from naive ecological regression.

This simple EI run also predicts the true white registration rates in each parish amazingly well. Figure 3 presents a scatterplot of the EI estimates against the true white registration rates (a circle's radius indicates size of the parish's white population). The solid slanted line, flanked by an 80% confidence interval, represents where cases would fall when estimates were exactly correct. Circles to the left of the line mean estimates were too low, those to the right that estimates were too high. As the graph shows, the bulk of parishes fall right on or around the solid line, indicating a wonderful fit with the real answers. The exception is a handful of very small parishes with white voting estimates that are too low. In these rural parishes, black registration was below 25%, a known legacy of Jim Crow that no simple EI analysis captures because it violates the model's assumptions.

(Figure 3 about here)

EI partially adapts to the presence of aggregation bias, even when the researcher makes no overt attempt to model that bias, because requiring estimates to be possible will force them to follow contextual behavior in informative data. These data are far from informative. Nevertheless, consistent with previous research (Matthews and Prothro 1966), the black registration estimates on average decrease 2.4 percentage points with a 10 percentage-point shift in black density. The basic version of EI therefore picked up some, but not all, of the true contextual pattern of aggregation bias. As the black density increased 10 percentage points, the real black registration rates decreased 4.1 percentage points, and the white rate climbed 1.4 points.

I tried several refinements to the estimation – drawn from my knowledge of the southern politics literature, not from my knowledge of the true numbers. Two corrections to the basic model did appear useful– judging, for example, from how much they improved the log-likelihood score. I allowed parameters to react to the black density of the population, in keeping with previous literature. I also included a dummy variable for two areas historically known for their racial sensitivity: the Mississippi Delta parishes in the state's northeast corner, and segregationist Judge Leander Perez' Plaquemines Parish. These refinements improved estimates, although the uncertainty remained high. Estimates for both races were within half a percentage point of the truth, and better reflected the aggregation bias in the true data. The black registration estimates decrease 3 percentage points, on average, with a 10 percentage-point increase in black density, and the white estimates inch upward (whereas in the simple analysis they remained static).

In sum, my EI estimation of Louisiana's 1968 parish registration rates is imperfect. Standard errors are quite conservative, given the amazing precision of the estimates, and a few intransigent parishes that violate the model assumptions do stump the program. Yet the main lesson of this exploration is that EI is resilient even in the most troublesome cases one is likely to face in real racial voting data – especially when one is armed with substantive knowledge. Nor, when faced by a tough case, does EI necessarily underestimate the uncertainty contained in its best estimates.

Normally the next stage would be to estimate turnout, using the racial registration estimates as a base (King 1997, chap. 15). However, since I possess the right answers for the initial estimation stage, I will use those instead. Estimating turnout, therefore, follows an identical procedure to that used estimating registration in the previous test case. I know how many registered whites and blacks appear in each parish, and how many people voted, but I don't know how many of each voted. For this analysis, I allowed white turnout to increase as black density increased, since Wallace's anti-segregation message played very well among Black Belt whites. I estimate that 79.5% of registered whites voted in the presidential election, whereas 59.2% of registered blacks did so. The findings resemble the 50-60% turnout rates estimated for Louisiana's registered blacks in 1966 (Campbell and Feagin 1975, 136).

Of course, at this point most researchers might be willing to assume, say, a 1.5% rate of support for Wallace among black voters; turnout was the real source of uncertainty. But I'll carry it through for purposes of testing King's method, since it is exactly this stage of analysis that foiled old inference techniques soon after Wallace's campaign; they all returned estimates of black support for Wallace much too high to be realistic. The procedure is almost identical to the previous stage. We know the distribution of votes, and we possess estimates of turnout for each race (from the last stage), but we don't know the voting preferences.20

A simple EI analysis, with no embellishments, produces rather disheartening results: approximately one quarter of African-American voters allegedly supported George Wallace (analysis not shown). Even this analysis is better than other ecological studies of the Wallace vote, which often implied greater support from blacks than for whites. Furthermore, this is a case in which prior knowledge can inform estimation without assuming the answers outright. EI can estimate aggregation bias informed by the contextual hypothesis that whites in parishes threatened by desegregation would be more supportive of Wallace.21 This simple refinement removes the absurd numbers, as no previous statistical analysis of the aggregate Wallace vote has ever been able to do. Now 1.8% of blacks apparently backed Wallace, either by intent or not (which is surprisingly realistic given the large standard error of 6.2). And we have by far the best available estimate of Wallace's white support from each Louisiana parish, which comes to 57.1% at the state level.22

Precinct-Level Racial Voting Behavior

Precinct-level race and election data are widely available, especially from Southern states that collect such information to document their compliance with the Voting Rights Act. For most states, the racial data come from aggregating Census voting-age population figures to the level of electoral units, as Lublin and Voss (2003) did in the Federal Election Project for 2000. Other states, such as North Carolina or Alabama, actually collect racial registration data, asking registrants to declare their racial category and aggregating those figures. Two states, Louisiana and South Carolina, even keep track of turnout by race.

The availability of precinct-level data is the key to why King's method will work so consistently with modern racial voting studies. Residential segregation is still the norm in American society. A third of African Americans live around few whites (Massey and Denton 1993, 75-77), and many whites live in neighborhoods with only token black presence. The aggregation process discards relatively little information about these segregated voters, as far as producing racial estimates is concerned, limiting the scope of ecological inference necessary to produce accurate estimates. This sort of segregation is less severe in the South, where interracial contact has always been more common than in hyper-segregated northern cities, yet still appears when data are measured at low levels.

Take Louisiana's racial registration figures for 1992. Out of 3,998 usable precincts, 11.6% are entirely uniform: 440 without a single African American registered to vote, 25 with nothing but African Americans. So we know the exact racial characteristics of 150,000 voters; no estimation is necessary and no uncertainty present.23 A third of the state's precincts contained almost no blacks (i.e., fewer than 5% of registered voters), such that more than half of the white population resides in "homogeneous precincts." Another 260 precincts were almost exclusively black, containing more than a quarter of the African-American population. Using a lower standard, in which only 90% of the population must be uniform, means that 58% of Louisiana voters appeared in segregated locales where we have little doubt about what one racial category is doing.

The result of all this segregation is that, in many precincts, we know roughly how the dominant race behaved–a certainty that greatly informs statewide estimates as well. Leaving aside the uniform precincts, where we have no doubt at all, the range of possible white turnout is less than 10 percentage points in 1,829 precincts. The range of possible black turnout is equally narrow in 343 more. Thanks to the bounds imposed by racial registration rates in these informative precincts, white turnout for the state must have been between 75% and 88.9%, and black turnout between 51.5% and 87.9%. But results near these extremes would entail rather implausible behavior, the sort of thing a researcher with substantive expertise would know about, so it would be a safe bet estimates were somewhere within an even narrower range (Flanigan and Zingale 1985).24 The bounds work similarly for voting choices. Thanks to bounds imposed by racial turnout rates in each precinct, the white vote for Clinton must have been between 29.5% and 44%, the black vote between 54.3% and 97.7%, with the plausible results falling within even narrower ranges. The neighborhood model cannot be correct.

Segregation is one condition common to racial data that assists King's method. Another condition that adds to the certainty is extreme behavior.25 When the phenomenon being studied approaches consensual levels, such as 0% or 100% of residents engaging in a particular activity, then we know with fair certainty how people of all races behaved there. Racial voting behavior frequently runs up against this sort of boundary, since variance in voter choice can be quite high, with 9 in 10 black voters backing the Democratic party and rural whites heavily endorsing Republicans. In the Louisiana case, we see such extreme behavior with turnout as well, because most registered voters go to the polls.

The implications should be clear. Embedding the ecological inferences within known bounds on voting behavior does much of the work required to produce decent racial voting estimates. King's method therefore milks both the quality of contemporary data and the segregated nature of American society for the information they make available. His approach to ecological inference does add to the precision, especially when coupled with whatever substantive knowledge the researcher brings to bear, and the method's assumptions do not drive the results as much as King's critics suggest.

Nevertheless, it is worth observing how well King's method performs with such highly informative data. In particular, I will show how successful EI is at predicting racial turnout in Louisiana, the sort of thing a researcher normally must estimate (e.g., it is the first stage of "double regression"). Naturally Louisiana is a unique place, so one may be hesitant to extrapolate from there to other states. However, for purposes of gauging the success of King's method, there's no reason to think it poses a particularly easy trial. Segregation is not exceptionally high in Louisiana's cities, and turnout rates are not extraordinary. It also may seem unreliable to assume that, if EI works well predicting turnout, it will perform equally well predicting vote choices for which the truth cannot be known. Here too, however, my trial is more representative than one might assume. The main reason EI thrives is racial segregation, and that operates equivalently on both turnout and vote data.26 Furthermore, voting behavior is more extreme than turnout in the contemporary period, so EI should get more purchase with votes.

I will use turnout from the 1991 Louisiana gubernatorial open primary, although results are quite similar with other elections from the period.27 The actual statewide turnout rates were 65.2% for blacks and 74.7% for everyone else, underscoring the faultiness of using a one-stage analysis (i.e., population straight to vote choice without estimating turnout as an intermediate step). The state-level bounds are not particularly informative; we have no idea what the black turnout rate would be, and the white rate could be anything greater than 61.6% (all figures appear in Table 2). If the absolute bounds are imposed individually on each precinct and added up to statewide totals, however, white turnout must be between 67.6% and 82.4%, while black turnout must fall between 44.7% and 83.9%. The midpoints of these bounds would indicate white turnout of roughly 75%, and black turnout of 64.3%. The former is off the truth by only 0.3%, while the latter misses the mark by just under a percentage point, so once again the bounds are pointing us in the right direction. There's no guarantee for success shooting from the hip this way, but it does reinforce Flanigan and Zingale's point (1985, 82-83) that the plausible bounds are usually narrower than the absolute bounds.28

(Table 2 about here)

The attractive precinct-level data permit most methods to approximate the truth. The neighborhood model, for example, estimates that white turnout was 73.8%, black turnout 67.4%. Naive ecological regression places those figures at 75.9% and 63.7% respectively. The white turnout estimate is 1.2 percentage points (or almost 19,000 voters) from the truth, one of the worst provided by any method despite the highly informative data on whites. The estimates also imply a statewide turnout rate that was impossibly high. The worst of the simple methods is homogeneous precinct analysis. I tried it two ways this time: once a simple precinct average, the other weighting that average by the number of registered voters in each precinct. Both are faulty, because white turnout is notably high in all-white precincts, and black turnout notable low in all-black precincts.

Weighted ecological regression does a nice job. It estimates that 75.3% of whites turned out, off by less than a percentage point. The black estimate is more disappointing, since it falls almost 2 percentage points from the truth, but given the less informative data for blacks such an estimate still seems fairly strong. Furthermore, this time the joint estimates do not imply impossible statewide turnout rates. A simple EI run produces roughly the same estimates as weighted ecological regression: 75.4% of whites and 63.3% of blacks. The basic results do not indicate any aggregation bias: estimates of white and black turnout do not correlate with racial density. This is a case in which the EI diagnostics by themselves would leave a researcher satisfied.

However, I have a theoretical reason to expect that white turnout should decline in heavily black areas. Whites in integrated settings tend to have lower socioeconomic status than their segregated counterparts, and socioeconomic resources are an important determinant of political activity (Verba, Schlozman, and Brady 1995, 513, 527). I therefore ended with a more complex EI analysis, allowing white turnout to vary as black density changed, without specifying the likely direction of the relationship.29 Once again, EI did not detect all of the aggregation bias. As black population density increases 10 percentage points, white turnout declines 1.5 percentage points, on average. The estimated contextual effect is only a third as large, and therefore too low. Nevertheless, what bias EI did detect helped improve the estimates relative to the alternatives. The resulting estimates were 74.9% turnout for whites, 64.5% for blacks. In both cases, these estimates are closer to the truth than those from any other method used. The white estimate was only off by a fifth of a percentage point, and the black estimate by less than a percentage point as well.30

The hypothesis that white mobilization rates change with racial density emerges from a real-life research agenda. One theory suggests that whites mobilize when faced with a large black population in their locale (see Voss 2000 for a literature review). By contrast, some scholars have noted a decline in white Anglo turnout in minority-dominated electoral districts (Gay 2001; Barreto, Segura, and Woods 2002). They interpret this decline as some sort of white demoralization caused by their minority status in elections. This is a clear case of competing contextual theories. The evidence from Louisiana suggests that ecological inferences, such as those reported by Gay (2001), should cooperate with testing relatively strong contextual hypotheses. EI picked up enough of the true pattern that a test for ecological effects would have been strong and statistically significant in the correct direction. On the other hand, the results from Louisiana – along with my analysis of Georgia and Florida (Voss and Lublin 2001) – reveal that those studying the effect of minority-dominated electoral districts need to rethink their interpretations. White turnout declines in heavily minority precincts regardless of whether the larger electoral environment is dominated by whites or blacks. EI's sensitivity to contextual effects therefore has contributed to an important substantive debate, debunking the white backlash idea yet calling into question the conventional wisdom on why white turnout declines in majority-minority districts.

The preceding analysis shows that precinct-level data are extremely useful for determining statewide behavior. Even fairly simple estimation methods come within a percentage point of true white behavior, and within two percentage points of true black behavior. The Louisiana data allow another test of EI's performance, however, which is to check the quality of precinct-level estimates it produces. EI might use the precinct data to generate excellent statewide estimates, but still be way off for individual areal units.31 As it turns out, however, EI's estimates are amazingly accurate even at the precinct level. The estimates correlate heavily with the true white turnout rates: a coefficient of .93 when weighted by the number of registered whites, and .98 when weighted by the inverse of EI's reported precinct standard error (see Burden and Kimball 1998, 539). The figures for blacks are naturally slightly lower, given the more limited information on black behavior, but still impressive: .87 when weighted by the number of registered blacks, and .94 when weighted by EI's reported level of confidence. It is worth noting that the cost of any survey with this level of local precision across an entire state would be astronomical, even if respondents could be trusted to report their turnout properly.

The EI estimation I ran took no notice of which precincts fell in which parishes. Although EI certainly would allow the researcher to adjust estimates based upon county traits, none of my estimations took advantage of this option. Indeed, the estimation routine did not in any way attempt to optimize fit with parish behavior. It is worth investigating, therefore, how well the EI precinct estimates aggregate up to parish values. I created parish-level white (black) turnout estimates by averaging the figures for all precincts in a parish, weighted by the number of whites (blacks) contained in each.32 The parish-level estimates end up almost as accurate as the statewide estimate, correlating with the truth at .99 for whites, .97 for blacks. The largest error for any parish's white turnout estimate is 2.3 percentage points, the largest error among blacks 3.6 percentage points. The average parish error is, of course, much smaller–under a half of a percentage point for both races (see Figure 4). Again, this level of success is astounding when considered in light of what surveys of similar precision would require.

(Figure 4 about here)

One major concern with King's EI is that, like other methods of ecological inference, it may not pick up enough of the information missing in aggregate data to represent contextual effects accurately. Cho's critique (1998), for example, creates a hypothetical aggregate data sets from survey data, and shows that King's method does not pick up the aggregation bias contained in her particular sample. Is his method equally limited for racial voting studies? This application indicates that, because of the virtues of precinct-level voting data, King's method performs quite well. For example, I regressed EI's parish-level estimates on the parish black density, controlling for the true estimates. If EI falls prey to aggregation bias, errors in predicting the white voting rate should change with the black density and vice versa. However, the results do not indicate a statistically significant connection between the parish error and the racial demographics (analysis not shown); there is no evidence aggregation bias has made much difference to the parish-level figures.

Moving Beyond Aggregate Data

Some of King's critics reject ecological inference altogether, preferring the use of surveys. Of course, surveys have serious problems, going beyond sampling error to include refusals, deception and questionnaire confusion. They are plagued with numerous sorts of bias (see, e.g., Burden 2000, 390; Palmer and Duch 2001), including an inability to measure racial attitudes reliably without taking special precautions (Kuklinski, Cobb and Gilens 1997; Traugott and Price 1992). Almost all of the variables measured by a survey are contemporaneous, and separating out the causal order between two variables is difficult at best (see Hetherington and Globetti 2002 for one recent attempt). However, it is worth considering how EI might stand up against survey analysis for the contextual study of voting behavior.

Most surveys either ask how people will vote in advance or ask how they voted afterward. Respondents asked this question after the fact misreport their voting at exceptionally high levels, and those asked before the fact misreport whether they will really vote at all. Presumably the best survey of how people voted is an exit poll, since it only captures those who have voted and it catches them at a time when they have made the final decision but have not been contaminated by knowledge of the election results. Certainly it is common for journal reviewers, when faced with an aggregate-data analysis, to resist the conclusions unless they come accompanied by exit-poll analysis. Political scientists frequently use exit polls as a data source in studies of voting behavior, sometimes even using the data to assess contextual patterns (Carsey 1995; Highton 2002). Are exit polls a serious rival to ecological inference?

Below the presidential level, this is doubtful. Exit polls are rare except in particularly high-profile campaigns. More importantly, exit polls resemble other surveys in that they provide little leverage at testing behavior below the state level because they collect few observations in any one subdivision. They typically sample precincts to optimize overall estimates; nothing in their method should produce data meaningful at a lower level of aggregation. Even areal units as large as congressional districts may contain few respondents. For example, only 34 people from one precinct appeared in the 1996 Voter News Service data for Georgia's 2nd congressional district. The 11th district received slightly more coverage: four precincts with a combined 151 people. These numbers are not sufficient to test contextual theories. Nor is the sampling frame used in an exit poll likely to produce reliable estimates of behavior for demographic groups. Indeed, data weighted to maximize state-level accuracy may produce group-level estimates worse than what unweighted data would offer.

Compare the two methods as applied to 1996 congressional voting in Georgia. Because precinct- level data are available, aggregate-data analysis using King's method has a great deal of information with which to work. Georgia does not report racial registration, so estimates must build from the entire population of adults, for whom turnout rates are not so high. This limitation widens the bounds compared to those found in Louisiana, but segregation is nevertheless the norm. Out of 2,641 usable precincts, not a single black adult appeared in 109 of them. Roughly 43% of Georgia's white adults and 15.5% of black adults live in precincts where their race predominates (i.e., more than 95% of population). Lower the threshold to 90%, and 61.4% of whites and 23.3% of blacks inhabit homogeneous precincts. The bounds are therefore helpful. Statewide racial turnout for the 1992 presidential contest must have fallen between 53.8% and 71.5% among whites adults, 24.1% and 79.3% among black adults. With 1992 congressional voting, similarly, the bounds on white turnout were tiny in four of Georgia's 11 districts (i.e., the range of possible white turnout was less than 10 percentage points). And the range only exceeded 20 percentage points in the three heavily black districts.

The Georgia VNS exit poll, by contrast, is completely unreliable below the state level. It does a wonderful job predicting the statewide presidential vote once data are weighted: 45.7% Clinton, 46.9% Dole, 6.3% Perot. When estimates are broken up by congressional district and by race, though, they are quite poor. It becomes clear that some combination of the sampling frame used by VNS and response bias among those polled results in invalid estimates of racial voting behavior by congressional district. Table 3 allows a district-by-district comparison between the VNS exit-poll results and the lower bound on white support for Clinton.33 Of the 11 district-level estimates of white behavior, six are impossibly low. All but one of those errors exceeds five percentage points.

(Table 3 about here)

These errors are consistent, not random, and therefore extremely unlikely to result from sampling error. Clinton simply could not have won as many votes as he did in these districts if so few non-blacks supported him, and the errors tend to be worst in the districts where we have the most aggregate information–that is, in the whiter districts. In Gingrich's Sixth, for example, the exit poll sample of 143 whites (when weighted) reported giving 23.3% of the vote to Clinton. However, looking at the precinct distribution of whites and votes, at least 29.1% of white voters must have done so. In the lily-white Ninth, the exit poll estimate drawn from 99 whites is 23.2% for Clinton, when the actual white vote clearly was within half a percentage point of 34%. Exit polls are far off from what was possible, let alone from the truth. Polls containing this kind of error below the state level cannot assist with contextual analysis. Ecological estimates, while imperfect, sure seem better than any alternative for analyzing turnout or party choice at the ballot box. EI offers a critical corrective to political science's dependence on surveys.

Conclusion

Gary King's proposed solution to the ecological inference problem possesses all the virtues of methods previously used to analyze aggregate data. It takes advantage of the tangible information found in segregated locales, as homogeneous unit analysis does, since it precludes impossible results and "borrows strength" from the homogeneous places to estimate behavior in the mixed ones (King 1997, 106- 112). Similarly, it allows two-stage analysis much as ecological regression will, with the additional virtue of producing measures of uncertainty proportional to the real errors in estimation. Like the neighborhood model, the extended version does not require an analyst to ignore contextual patterns found in aggregate data. But it combines all of these traits into a unified model that constrains estimates according to the method of bounds and therefore is sensitive to both compositional and contextual patterns in the data. It produces low-level estimates with far more precision than surveys typically make available. EI is thus an important advance in the ability of researchers to conduct contextual analysis.

King's critics have ignored the vast improvement EI offers to scholars interested in contextualizing social behavior. Their applications did not give a fair representation of how EI usually performs. In particular, the applications presented in this chapter indicate that EI has taken us far enough to study voting behavior on exactly the sorts of questions – turnout and racial behavior – for which survey questions are unreliable and survey data limited in their contextual information. I tested King's EI using two Louisiana data sets in which racial behavior was known. One was a 1968 parish-level data set where I knew racial rates of voter registration among black and white adults and also held a strong prior about racial voting preferences. The other was a 1991 precinct-level data set where I knew the racial turnout rates among those registered. These cases indicated that King's EI performs admirably, whether the comparison is (1) with other methods of aggregate-data analysis, (2) with the truth in data sets for which estimates can be evaluated, or (3) with the likely assistance that surveys might provide. King's EI run on precinct-level data produced estimates that were extremely close to the truth at the statewide, parish and precinct levels. It also picked up some, but not all, of the contextual pattern contained in these data. The parish-level analysis, while clearly based on poor data, not only resulted in fairly precise estimates of voter registration, it also avoided the error committed by just about every aggregate-data analysis of the Wallace vote: the tendency to estimate high black support for the segregationist. Therefore, despite the criticism King's approach has faced and the limits it shows when applied to concocted data sets, it is unparalleled when applied to the actual sort of data needed for analyzing important social issues such as racial voting patterns.

As I have illustrated, the quality of EI estimates responds to an analyst's ability to anticipate contextual patterns and inform the software of these expectations – for example, by including covariates that capture the contextual effect directly (see Herron and Shotts 2003 and the exchange that has followed). Of course, estimates shaped by a practitioner's instincts, field research, or theoretical arguments may face a greater burden of proof than estimates generated by a plug-and-chug equation that externalizes judgment. But developments within Bayesian statistics have created, even among quantitative researchers, a growing sense that analysts should not approach empirical inference under the pretense that they know nothing about the subjects they study. King's solution to the ecological inference problem fits comfortably within this trend of developing methods that succeed when analysts can incorporate what they have learned, a Bayesian approach that has received even greater development since the publication of his book (see Chapter 1 [Wakefield]). My applications, guided by the relevant substantive literature, suggest that King's approach outperforms every commonly used alternative for ecological inference, incorporating the advantages of all of them while escaping the pitfalls of each. There is no reason, aside from ease of use, that would justify rejecting King's ecological-inference method in favor of the conventional competition.

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