Bayesian and Frequentist Inference for Ecological Inference: The R×C Case

Abstract

In this paper we propose Bayesian and frequentist approaches to ecological inference, based on R 3 C contingency tables, including a covariate. The proposed Bayesian model extends the binomial-beta hierarchical model developed by KING, ROSEN and TANNER (1999) from the 2 3 2 case to the R 3 C case. As in the 2 3 2 case, the inferential procedure employs Markov chain Monte Carlo (MCMC) methods. As such, the resulting MCMC analysis is rich but computationally intensive. The frequentist approach, based on ®rst moments rather than on the entire likelihood, provides quick inference via nonlinear least-squares, while retaining good frequentist properties. The two approaches are illustrated with simulated data, as well as with real data on voting patterns in Weimar Germany. In the ®nal section of the paper we provide an overview of a range of alternative inferential approaches which trade-off computational intensity for statistical ef®ciency.

The 2008 Malaysian Elections: An End to Ethnic Politics?

“…This is the strategy that I adopt here. A generalization of EI to handle more than two ethnic groups (see Rosen et al 2001) yields both an estimate of support for BN parties by ethnic group and a measure of the uncertainty associated with that estimate. Table 10 presents a list of estimates broken down by party contesting and allows differential turnout rates in each district to affect the estimates.…”

Cross-race misaggregation: Its detection, a mathematical decomposition, and Simpson’s Paradox.

“…Stratified analyses may provide more valid inferences (Hruschka & Hackman, 2014). If lower-level data (e.g., stratified by race) are unavailable, these can sometimes be estimated from aggregated data (i.e., ecological inference) using methods developed in political science (King, 1997;Rosen, Jiang, King, & Tanner, 2001). …”

Ignoramus, Ignorabimus? On Uncertainty in Ecological Inference

“…King proposes that these conditional probabilities have a truncated normal distribution. Brown and Payne (1986) and Rosen et al (2001) use the more natural assumption that the conditional probabilities have a Dirichlet distribution with mean…”

“…In EI problems, however, they are not. This problem, which may be called inferential indeterminacy, is much more serious than the fact that distributional assumptions of, for example, ecological regression models of King (King 1997;King, Rosen, and Tanner 1999;Rosen et al 2001) or Brown and Payne (1986) cannot be checked or that under certain circumstances an ecological regression model like King's may be susceptible to ''aggregation bias'' (Openshaw andTaylor 1979, 1981;Cho 1998;Steel, Beh, and Chambers 2004). For example, one may want to make predictions about the number of African-Americans who will turn out to vote in a specific voting district.…”

“…Their approach consists of an extension of the ecological regression model (12), which allows the conditional probabilities p jjik to vary in different units k according to a Dirichlet distribution with mean parameters E(p jjik ) 5 p jji and precision parameter h 0 . According to Brown and Payne (1986), both the mean and the dispersion parameters can be estimated from the marginal tables n Ájk and n iÁk , the mean parameters based on generalized least squares and the precision parameter based on the variance of the residuals of the regression of n Ájk on n iÁk (Rosen et al [2001] consider estimation of this model via MCMC and a simpler model without a Dirichlet mixing distribution estimated via nonlinear least squares). Since the precision parameter h 0 is estimated from the residuals of this regression, this model may account for model departures with respect to those aspects in which model (12) is more restrictive than the Johnston-Pattie model (8).…”