  The regression discontinuity design (RDD) is a quasi-experimental design that
can be used to identify and estimate the causal effect of a treatment using
observational data. In an RDD, a pre-specified rule is used for treatment
assignment, whereby a subject is assigned to the treatment (control) group
whenever their observed value of a specific continuous variable is greater than
or equal to (is less than) a fixed threshold. Sharp RDDs occur when guidelines
are strictly adhered to and fuzzy RDDs occur when the guidelines or not
strictly adhered to. In this paper, we take a rigorous decision theoretic
approach to formally study causal effect identification and estimation in both
sharp and fuzzy RDDs. We use the language and calculus of conditional
independence to express and explore in a clear and precise manner the
conditions implied by each RDD and investigate additional assumptions under
which the identification of the average causal effect at the threshold can be
achieved. We apply the methodology in an example concerning the relationship
between statins (a class of cholesterol-lowering drugs) and low density
lipoprotein (LDL) cholesterol using a real set of primary care data.
