  In many scenarios, the observational data needed for causal inferences are
spread over two data files. In particular, we consider scenarios where one file
includes covariates and the treatment measured on one set of individuals, and a
second file includes responses measured on another, partially overlapping set
of individuals. In the absence of error free direct identifiers like social
security numbers, straightforward merging of separate files is not feasible, so
that records must be linked using error-prone variables such as names, birth
dates, and demographic characteristics. Typical practice in such situations
generally follows a two-stage procedure: first link the two files using a
probabilistic linkage technique, then make causal inferences with the linked
dataset. This does not propagate uncertainty due to imperfect linkages to the
causal inference, nor does it leverage relationships among the study variables
to improve the quality of the linkages. We propose a hierarchical model for
simultaneous Bayesian inference on probabilistic linkage and causal effects
that addresses these deficiencies. Using simulation studies and theoretical
arguments, we show the hierarchical model can improve the accuracy of estimated
treatment effects, as well as the record linkages, compared to the two-stage
modeling option. We illustrate the hierarchical model using a causal study of
the effects of debit card possession on household spending.
