Ecological Inference for 2 × 2 Tables (with Discussion)

Abstract

Summary.A fundamental problem in many disciplines, including political science, sociology and epidemiology, is the examination of the association between two binary variables across a series of 2 2 tables, when only the margins are observed, and one of the margins is fixed. Two unobserved fractions are of interest, with only a single response per table, and it is this non-identifiability that is the inherent difficulty lying at the heart of ecological inference. Many methods have been suggested for ecological inference, often without a probabilistic model; we clarify the form of the sampling distribution and critique previous approaches within a formal statistical framework, thus allowing clarification and examination of the assumptions that are required under all approaches. A particularly difficult problem is choosing between models with and without contextual effects. Various Bayesian hierarchical modelling approaches are proposed to allow the formal inclusion of supplementary data, and/or prior information, without which ecological inference is unreliable. Careful choice of the prior within such models is required, however, since there may be considerable sensitivity to this choice, even when the model assumed is correct and there are no contextual effects. This sensitivity is shown to be a function of the number of areas and the distribution of the proportions in the fixed margin across areas. By explicitly providing a likelihood for each table, the combination of individual level survey data and aggregate level data is straightforward and we illustrate that survey data can be highly informative, particularly if these data are from a survey of the minority population within each area. This strategy is related to designs that are used in survey sampling and in epidemiology. An approximation to the suggested likelihood is discussed, and various computational approaches are described. Some extensions are outlined including the consideration of multiway tables, spatial dependence and area-specific (contextual) variables. Voter registration-race data from 64 counties in the US state of Louisiana are used to illustrate the methods.

eco:RPackage for Ecological Inference in 2×2 Tables

“…The model, which is similar to the ones proposed by King (1997) and Wakefield (2004), assumes the independence between W i and X i and thus no contextual effect. The maximum likelihood (ML) estimates of µ and Σ can be computed via the EM algorithm.…”

“…For example, one might conduct a survey in randomly selected counties to obtain such information. Sometimes, a small scale survey can be conducted to get rough estimates of W i for some counties, and incorporating such auxiliary information can also be helpful (Wakefield 2004). In the eco package, it is straightforward to incorporate such information into the estimation of both parametric and nonparametric models (see Section 3.5).…”

Improving ecological inference using individual‐level data

“…The obvious choice of a at prior for with a large variance, combined with (1), leads to a marginal prior distribution for p ij (for ÿxed covariate values) that is heavily biased towards 0 and 1 [15]. Instead we choose a logistic prior for with location 0 and scale 1, and 1= 2 ∼ Gamma(1; 0:01), which lead to an approximately uniform marginal prior for p ij .…”

“…For a binary exposure, as discussed by Wakeÿeld [15], the accuracy of ecological inference depends on the between-group variability of the proportions of individuals exposed in each group. For example, if only 10 per cent of every group are smokers, then data consisting of only the total number of smokers in the group and the corresponding number of outcomes contain a lot of information about the risk of outcome in the non-smoking population, and very little information about the minority smoking group.…”

“…If the individual-level exposures were assumed to be ÿxed and known, for example socioeconomic indicators from a census of the full population, then an extension of the convolution likelihood suggested by Wakeÿeld [15] for 2 × 2 tables would be more appropriate. This is constructed by conditioning on individual-level exposures.…”