Sensitivity analysis on the ecological bias for Seoul tuberculosis data

Abstract

In ecological studies, researchers often try to convey the analysis results to individual level based on aggregate data. In order to do this correctly, the possibility of ecological bias should be studied and addressed. One of the key ideas used to address the ecological bias issue is to derive the ecological model from the individual model and to check whether the parameter of interest in the individual model is identifiable in the ecological model. However, the procedure depends on unverifiable assumptions, and we recommend checking how sensitive the results are to these unverifiable assumptions. We analyzed the tuberculosis data that was collected in Seoul in 2005 using a spatial ecological regression model for the aggregate count data with spatial correlation, and found that the deprivation index is likely to have a small positive effect on the occurrence risk of tuberculosis in individual level in Seoul. We considered this finding in various aspects by performing in depth sensitivity analyses. In particular, our findings are shown to be robust to the distribution assumptions for the individual exposure and missing binary covariate across various scenarios.

Appendix

In Sect. 7, we assume that $x_{ij}|z_{ij} \sim N(x_{i}+\psi z_{ij},s^2_{i})$. Note that

$$\begin{aligned} E(x_{ij})=E(E(x_{ij}|z_{ij}))=x_i + \psi E(z_{ij})=x_{i} +\psi m_{i} \end{aligned}$$

where $m_{i}$ denotes the smoking rate in ith dong. The difference of mean deprivation indices between two different dongs (i and $i'$) becomes

$$\begin{aligned} E(x_{ij})-E(x_{i'j})=(x_{i}-x_{i'})+\psi (m_{i}-m_{i'}). \end{aligned}$$

Therefore, we have

$$\begin{aligned} \psi =\frac{E(x_{ij})-E(x_{i'j})-(x_{i}-x_{i'})}{(m_{i}-m_{i'})}. \end{aligned}$$

Take two dongs where $m_{i}>m_{i'}$. If we assume that $\psi >0$ and $x_{i}$ monotonically increases with $m_{i}$, then

$$\begin{aligned} \psi \le \frac{E(x_{ij})-E(x_{i'j})}{(m_{i}-m_{i'})}. \end{aligned}$$