Estimating the likelihood of fallacious ecological inference: linear ecological regression in the presence of context effects

In situations where the only reliable data source is electoral data at the aggregate level for a geographic unit such as voting precincts, social scientists have sought to use ecological regression techniques to recreate the voting behavior of particular groups without committing the ecological fallacy of the sort warned of by Robinson (1950). Until quite recently, the most common use of ecological regression techniques was for the analysis of historical data, e.g., determining the social groups that supported the Nazis. In the US, however, in a development little known to political geographers, within the past decade, social science expert witnesses have testified in hundreds of cases involving minority voting rights—cases where a central issue has been to ascertain the extent of white/Anglo support for black or Hispanic (or more recently, Asian-American) candidates in elections involving candidates of more than one race/ ethnic background. One issue in several recent voting rights cases has been whether linear regression methods can validly be used to model racial voting patterns, with experts for defendant jurisdictions claiming that the presence of contextual effects may invalidate the assumptions underlying linear models and give rise to an ecological fallacy. For data on voting patterns in small geographic units such as voting precincts, we show that the presence of a plausible type of race-related contextual effect will, in general, lead to a quadratic relationship between minority share of the electorate and the share of the vote received by the minority candidate. However, when either a substantial proportion of the electorate is located in racially homogeneous or near homogeneous precincts (as often occurs given the patterns of residential concentration in US cities), or when the contextual effects are small, we show that the estimates of the key parameters of interest in voting rights litigation, namely the proportions of minority and non-minority voters who voted for the minority candidate(s), will be very well approximated by a bivariate linear model. In such circumstances, fallacies of ecological inference can be avoided with near certainty, despite the fact that the linear model that has been used to fit the data omits potentially important variables and regression diagnostics for it would show heteroskedasticity, while a quadratic regression would provide coefficients that are essentially uninterpretable in terms of the parameter of interest. This result has potentially broader implications for the reliability of ecological inference because it demonstrates that, under plausible assumptions, a simple ecological model can yield accurate results despite theoretical errors in model specification—because those errors prove to be of little or no practical significance.