Isolating Spatial Autocorrelation, Aggregation Bias, and Distributional Violations in Ecological Inference: Comment on Anselin and Cho

Few better ways of checking and improving statistical methods exist than having other researchers go over your results, and so I especially appreciate the efforts in Anselin and Cho (2002), hereinafter AC. In this note, I make two main points. First, AC’s numerical findings from its empirical example and simulations contradict no prior research. The article’s one empirical example violates EI’s spatial independence and no aggregation bias assumptions, according to AC, and thus offers no evidence of the independent effects of either. In AC’s Table 1, Goodman’s regression gives one answer 11.2 times smaller than the truth and logically impossible (the percentage of males having strokes is 6.5% fewer than there are males) and another 8.5 times larger than the truth. Yet, EI gives answers that are 1.41 and 0.74 times the truth. This pattern is common and occurs for a reason: Although aggregation bias can cause Goodman’s regression to be biased to any degree, EI’s potential bias, although not guaranteed to be zero, is strictly limited, and hence more robust.1 Similarly, in simulations with autocorrelation levels set considerably higher than any published ecological inference application, AC’s numerical results still confirm that spatial autocorrelation has modest effects. Results in the article’s Tables 2 and 3 are similar to and often smaller than those in King (1997, Ch. 9; 2000). As with heteroskedasticity or autocorrelation in linear regression, AC find that EI is unbiased in the presence of spatial autocorrelation, and it has a proportionately larger variance than data without autocorrelation. Because finding autocorrelation is another way of saying the data contain less information, this is precisely as it should be. The only issue is the extent to which EI’s uncertainty estimates miss this loss of information, but AC does not address this issue. My second main point is that AC ignores the role of the bounds in EI and is mistaken when it claims that “setting aside consideration of the role of the bounds does not have a material consequence on [sic] our discussion.” The advantage of EI comes precisely from combining the only two approaches that had been used in practice prior to EI—Goodman’s regression