Instrumental Variables

An instrumental variable is an odd name for a thing, particularly such an important thing.  The term refers to an observable characteristic of individuals in the data that has the following two properties:
1. It determines (at least to some extent) the treatment the individual receives.
2. It is unrelated to the observed outcome of interest (except possibly through the treatment selection).

The randomization procedure in a randomized controlled trial is an instrumental variable.  The randomization procedure directly determines which treatment arm the patient is allocated into and is, by definition, unrelated to the outcome from trial.  In a study into the effectiveness of mammograms the invitation to receive a mammogram is an instrumental variable.  Women were allocated to either receive an invitation or not receive an invitation at random.  The invitation was at random and unrelated to the treatment outcomes.  For the invitation to be an instrument it must be that women who received the invitation were more likely to have mammography than women who did not receive the treatment.

Consider the graph to the right.  We are interested in measuring the casual relationship between chemotherapy (X) and survival (Y) (in the graph it is the blue line from X to Y).  The observed relationship between X and Y may confounded by the unobserved characteristic (U).  When we observe the statistical relationship between X and Y we may be actually estimating the relationship between U and Y.

The instrumental variable is represented in the graph by Z.  The graph shows that Z causally affect X (the blue line from Z to X) but does not affect Y through some other relationship.  There is no arrow between Z and U or between Z and Y.


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