Imputing Unmeasured Explanatory Variables in Environmental Epidemiology With Application to Health Impact Analysis of Air Pollution

This paper presents the results of a reconsideration of earlier work that finds an association between daily hospital admissions for respiratory distress and daily concentrations of sulphate (lag 1) as well as daily maximum concentrations of ozone (lags 1 and 3). These associations are found even after clustering the data by hospital of admission and accounting for the effects of temperature. We use an adaptation of their generalized estimating equation technique for clustered data, that daily data being for southern Ontario summers from 1983 to 1988. Like them, we adjust for daily maximum temperatures. However, unlike the earlier work returned to ours includes daily average humidity as a potential explanatory variable in our model. Our analysis also differs from theirs in that we cluster the data by census subdivision to reduce the risk of confounding pollutant levels with population size within regions. Moreover, we log-transform the explanatory variables and then high-pass filter the resulting data. We also deviate from the earlier analysis by taking account of measurement error incurred in using surrogate measures of the explanatory variables. To do so we use new methodology designed for our study but of potential value in other applications. That methodology requires a spatial predictive distribution for the unmeasured explanatory variables. Each day about 700 missing measurements for each of these variables can then be imputed over the geographical domain of the study. With these imputations we get a measure of imputation error through the covariance of the predictive distribution. Along with the predictive distribution we require an impact model to link-up with the predictive distribution. We describe that model and show how it uses the imputed measurements of the missing values of the explanatory variables. We also show how through that model, uncertainty about these values is reflected in our analysis and in commensurate uncertainties in the inferences made. Apart from its substantive objectives, our analysis serves to test the new methods with the earlier results serving as a foil. The reassuring qualitative agreement between our findings and the earlier results seems encouraging.