  The deconfounder was proposed as a method for estimating causal parameters in
a context with multiple causes and unobserved confounding. It is based on
recovery of a latent variable from the observed causes. We disentangle the
causal interpretation from the statistical estimation problem and show that the
deconfounder in general estimates adjusted regression target parameters. It
does so by outcome regression adjusted for the recovered latent variable termed
the substitute. We refer to the general algorithm, stripped of causal
assumptions, as substitute adjustment. We give theoretical results to support
that substitute adjustment estimates adjusted regression parameters when the
regressors are conditionally independent given the latent variable. We also
introduce a variant of our substitute adjustment algorithm that estimates an
assumption-lean target parameter with minimal model assumptions. We then give
finite sample bounds and asymptotic results supporting substitute adjustment
estimation in the case where the latent variable takes values in a finite set.
A simulation study illustrates finite sample properties of substitute
adjustment. Our results support that when the latent variable model of the
regressors hold, substitute adjustment is a viable method for adjusted
regression.
