Alleviating Linear Ecological Bias and Optimal Design with Subsample Data

Abstract

Summary. In this paper, we illustrate that combining ecological data with subsample data in situations in which a linear model is appropriate provides three main benefits. First, by including the individual level subsample data, the biases associated with linear ecological inference can be eliminated. Second, by supplementing the subsample data with ecological data, the information about parameters will be increased. Third, we can use readily available ecological data to design optimal subsampling schemes, so as to further increase the information about parameters. We present an application of this methodology to the classic problem of estimating the effect of a college degree on wages. We show that combining ecological data with subsample data provides precise estimates of this value, and that optimal subsampling schemes (conditional on the ecological data) can provide good precision with only a fraction of the observations.

Interpreting meta‐regression: application to recent controversies in antidepressants’ efficacy

“…$${\gamma }_1={\overline{\beta }}_1+\frac{{cov}_{\theta }false({\mu }_x,{\beta }_{0}false)+E_{\theta }false[{false({\mu }_x-{\overline{\mu }}_{x}false)}^{2}false({\beta }_1-{\overline{\beta }}_{1}false)false]+{\overline{\mu }}_x{\text{cov}}_{\theta }false({\mu }_x,{\beta }_{1}false)}{{var}_{\theta }false({\mu }_{x}false)},$$ which extends the finite mixture model result of [20, page 182] to the general mixture setting. Based on (12), there is no clear relation between the meta-regression slope γ 1 and the average within-study slope β̄ 1 .…”

Ecologic Studies Revisited

“…The obvious approach is to collect a random sample of individuals within areas. For a continuous outcome, Raghunathan et al (54) show that moment and maximum likelihood estimates of a common within-group correlation coefficient will improve when aggregate data are combined with individual data within groups, and Glynn et al (24) derive optimal design strategies for the collection of individual-level data when the model is linear. With a binary nonrare outcome, the benefits have also been illustrated (68,74).…”

Spatial parasite ecology and epidemiology: a review of methods and applications

“…The only way to truly overcome the problem of ecological bias therefore is to supplement aggregate data with samples of data at the individual level, which on their own may be too sparse to accurately capture geographic variation but can provide an indication of intra-unit variation. Several theoretical methods have been proposed to do this, which can be used to address bias and separate individual and contextual effects when either the outcome or the exposure measure is available at an ecological level (Prentice and Sheppard, 1995; Steel and Holt, 1996; Lasserre et al 2000; Best et al 2001; Wakefield and Salway, 2001; Glynn et al 2008). The so-called aggregate data method, for example, estimates individual-level exposure effects by regressing population-based disease rates on covariate data from survey samples in each population group (Prentice and Sheppard, 1995).…”