4.2 Ecological bias

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  Suppose that we wish to describe the relationship between an outcome variable $Y$ and an exposure $x$;

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  imagine that we collect data $(x_{ij},Y_{ij})$, where $i$ denotes groups (areas), and $j$ denotes individuals within groups.

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  Figure LABEL:fig:ecological1 shows the relationship between $x$ and $Y$ using synthetic data from individuals in five groups:

Unnumbered Figure: Link

Unnumbered Figure: Link

Unnumbered Figure: Link

Caption: Showing the potential relationships between $x$ and $Y$, that might arise depending on the choice of model. In each plot, the red dots are the data points, $(x_{ij},Y_{ij})$. Broadly speaking, these show a strong negative trend, but within this negative trend, there are 5 groups of individuals and within each group, the trend between $x$ and $Y$ is slightly positive. In the top plot, only the individual level data are shown. In the bottom left plot, in addition to the individual level data, the large blue dots are the group-level averages $({\overline{x}}_i,{\overline{Y}}_i)$, where ${\overline{x}}_i$ is the mean of the $x$s in the $i$th group and ${\overline{Y}}_i$ is the mean of the $Y$s in the $i$th group. In the bottom right plot, the group level averages are now shown as hollow blue circles and two regression lines have been added. The dashed blue line corresponds to a regression on the group level averages (model 1 in the text), whereas the five solid red lines are from a regression model that respects both the group and individual-level structures in the data (model 3 in the text). Model 1 would lead one to conclude that the relationship between $Y$ and $x$ is strongly negative, whereas Model 3 captures the positive trend between $Y$ and $X$ within each group.

Consider three possible models for the above synthetic data:

1. 1.

   Common individual-level regressions

   	$Y_{ij}$	$=\alpha +\beta x_{ij}+Z_{ij}$	$Z_{ij}\sim \mathrm{iid}\mathit{\hspace{1em}}\mathrm{N}(0,{\sigma }^2)$
   	${\overline{Y}}_i$	$=\alpha +\beta {\overline{x}}_i+{\overline{Z}}_i$	${\overline{Z}}_i\sim \mathrm{iid}\mathit{\hspace{1em}}\mathrm{N}(0,{\sigma }^2/n_i)$

   This model is not correct.

2. 2.

   Separate individual-level regressions

   	$Y_{ij}$	$={\alpha }_i+\beta x_{ij}+Z_{ij}$	$Z_{ij}\sim \mathrm{iid}\mathit{\hspace{1em}}\mathrm{N}(0,{\sigma }^2)$
   	${\overline{Y}}_i$	$={\alpha }_i+\beta {\overline{x}}_i+{\overline{Z}}_i$	${\overline{Z}}_i\sim \mathrm{iid}\mathit{\hspace{1em}}\mathrm{N}(0,{\sigma }^2/n_i)$

   This model is correct, but not identifiable from group-level data (we need individual-level data for each group to estimate the group’s slope).

3. 3.

   An additional group-level covariate

   	$Y_{ij}$	$={\alpha }_i+\beta x_{ij}+Z_{ij}$	$Z_{ij}\sim \mathrm{iid}\mathit{\hspace{1em}}\mathrm{N}(0,{\sigma }^2)$
   	${\alpha }_i$	$=\alpha +\gamma u_i+Z_i^{*}$	$Z_i^{*}\sim \mathrm{iid}\mathit{\hspace{1em}}\mathrm{N}(0,{\tau }^2)$

   1. (a)

      if $u_i\ne {\overline{x}}_i$, then

      $${\overline{Y}}_i=\alpha +\beta {\overline{x}}_i+\gamma u_i+(Z_i^{*}+{\overline{Z}}_i)$$

      This model is correct, and is identifiable if $u_i$ is known, and

   2. (b)

      if $u_i={\overline{x}}_i$, then

      $${\overline{Y}}_i=\alpha +(\beta +\gamma ){\overline{x}}_i+(Z_i^{*}+{\overline{Z}}_i)$$

      This model is implicitly assumed in an ecological regression of ${\overline{Y}}_i$ on ${\overline{x}}_i$ hence the estimand in ecological regression is $\beta +\gamma$, not $\beta$.

Ecological bias is closely related to the phenomenon known as Simpson’s paradox, or Yule’s paradox. Wakefield (2004) provides a full discussion of ecological inference and related topics.

Let $A_i$ be small spatial regions $Y_i$ be counts in $A_i$. Let $\{z(x):x\in {\bigcup }_i{A}_i\}=$ be a spatially varying risk factor.

1. 1.

   Individual-level model

   Cases form an inhomogeneous Poisson process,

   $$\lambda (x)={\lambda }_0(x)\exp \{\beta z(x)\}$$

2. 2.

   Area-level model

   	$Y_i$	$\sim$	$\mathrm{Poisson}({\mu }_i)$
   	${\mu }_i$	$=$	${\displaystyle {\int }_{x\in A_i}}{\lambda }_0(x)\exp \{\beta z(x)\}dx$

   It is common practice to assume a model of the form

   $${\mu }_i={\overline{\lambda }}_{0i}\exp (\beta {\overline{z}}_i)$$

   where ${\overline{\lambda }}_{0i}$ and ${\overline{z}}_i$ are the averages over the small area $A_i$ of ${\lambda }_0(x)$ and $z(x)$, respectively. Note that this is strictly incorrect, except under rather special conditions.

For problems of this kind, the suggested strategy is to:

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  specify the model at the individual level

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  derive the resulting joint probability distribution for area-level data

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  check that parameters of interest are identifiable from area-level data

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  make the required inferences

See, for example, Prentice and Sheppard (1995) and Sheppard and Prentice (1995). Li et al. (2012) and Taylor et al. (2013) discuss a data-augmentation scheme (under modelling assumptions) to enable an individual-level model to be fitted to aggregated data under the slightly more complex scenario that $\lambda (x)$ is stochastic.