Deprivation, ill-health and the ecological fallacy

The use of ecological studies in explaining the relationship between deprivation and ill-health is widespread in many health applications. However, inferences drawn from these studies about individuals are susceptible to serious bias known as the ecological fallacy. Our paper demonstrates the ecological fallacy effect in this context but also shows how it can be considerably reduced by taking into account different population structures at the aggregate level. Two regression analyses of limiting long-term illness are performed, one at the individual level and one at the electoral ward level, using the 1991 UK census sample of anonymized records and the small area statistics. The analyses compare several measures of deprivation including the standard Carstairs index, with the separate variables which make up the indices, to determine their effectiveness in explaining rates of illness. Two of the deprivation scores are constructed using latent variable modelling techniques which enable a score to be generated at the individual level as well as at the ward level. It is shown that, given the right choice of socioeconomic variables and taking into account the age structure of the population, it should be possible to construct a single aggregate deprivation index that will explain most of the variation in rates of illness across the study region.     

Bayes methods in the ecological fallacy cntext:estimation of individual correlation from aggregate data

The ecological fallacy is related to Simpson's paradox (1951) where relationships among group means may be counterintuitive and substantially different from relationships within groups, where the groups are usually geographic entities such as census tracts. We consider the problem of estimating the correlation between two jointly normal random variables where only ecological data (group means) are available. Two empirical Bayes estimators and one fully Bayesian estimator are derived and compared with the usual ecological estimator, which is simply the Pearson correlation coefficient of the group sample means. We simulate the bias and mean squared error performance of these estimators, and also give an example employing a dataset where the individual level data are available for model checking. The results indicate superiority of the empirical Bayes estimators in a variety of practical situations where, though we lack individual level data, other relevant prior information is available.     

Improving The Efficiency of The Case-Crossover Design

The case-crossover design has been used by many researchers to study the transient effect of an exposure on the risk of a rare outcome. In a case-crossover design, only cases are sampled and each case will act as his/her own control. The time of failure acts as the case and non failure times act as the controls. Case-crossover designs have frequently been used to study the effect of environmental exposures on rare diseases or mortality. Time trends and seasonal confounding may be present in environmental studies and thus need to be controlled for by the sampling design. Several sampling methods are available for this purpose. In time-stratified sampling, disjoint strata of equal size are formed and the control times within the case stratum are used for comparison. The random semi-symmetric sampling design randomly selects a control time for comparison from two possible control times. The fixed semi-symmetric sampling design is a modified version of the random semi-symmetric sampling design that removes the random selection. Simulations show that the fixed semi-symmetric sampling design improves the variance of the random semi-symmetric sampling estimator by at least 35% for the exposures we studied. We derive expressions for the asymptotic variance of risk estimators for these designs, and show, that while the designs are not theoretically equivalent, in many realistic situations, the random semi-symmetric sampling design has similar efficiency to a time-stratified sampling design of size two and the fixed semi-symmetric sampling design has similar efficiency to a time-stratified sampling design of size three.