  Causal inference for air pollution mixtures is an increasingly important
issue with appreciable challenges. When the exposure is a multivariate mixture,
there are many exposure contrasts that may be of nominal interest for causal
effect estimation, but the complex joint mixture distribution often renders
observed data extremely limited in their ability to inform estimates of many
commonly-defined causal effects. We use potential outcomes to 1) define causal
effects of air pollution mixtures, 2) formalize the key assumption of mixture
positivity required for estimation and 3) offer diagnostic metrics for
positivity violations in the mixture setting that allow researchers to assess
the extent to which data can actually support estimation of mixture effects of
interest. For settings where there is limited empirical support, we redefine
causal estimands that apportion causal effects according to whether they can be
directly informed by observed data versus rely entirely on model extrapolation,
isolating key sources of information on the causal effect of an air pollution
mixture. The ideas are deployed to assess the ability of a national United
States data set on the chemical components of ambient particulate matter air
pollution to support estimation of a variety of causal mixture effects.
