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13.2 Threats to Validity of Experiments 527

experimenter knows whether the treatment was actually received (for example,
whether the trainee attended class), and the treatment actually received is
recorded as Xi. With partial compliance, there is an element of choice in whether
the subject receives the treatment, so Xi will be correlated with ui even if initially
there is random assignment. Thus, failure to follow the treatment protocol leads
to bias in the OLS estimator.

If there are data on both treatment actually received (Xi) and on the initial
random assignment, then the treatment effect can be estimated by instrumental
variables regression. Instrumental variables estimation of the treatment effect
entails the estimation of Equation (13.1)—or Equation (13.2) if there are control
variables—using the initial random assignment (Zi) as an instrument for the treat-
ment actually received (Xi). Recall that a variable must satisfy the two conditions
of instrument relevance and instrument exogeneity (Key Concept 12.3) to be a
valid instrumental variable. As long as the protocol is partially followed, then the
actual treatment level is partially determined by the assigned treatment level, so
the instrumental variable Zi is relevant. If initial assignment is random, then Zi is
distributed independently of ui (conditional on Wi, if randomization is conditional
on covariates), so the instrument is exogenous. Thus, in an experiment with ran-
domly assigned treatment, partial compliance, and data on actual treatment, the
original random assignment is a valid instrumental variable.

This instrumental variables strategy requires having data on both assigned
and received treatment. In some cases, data might not be available on the treat-
ment actually received. For example, if a subject in a medical experiment is pro-
vided with the drug but, unbeknownst to the researchers, simply does not take it,
then the recorded treatment (“received drug”) is incorrect. Incorrect measure-
ment of the treatment actually received leads to bias in the differences estimator.

Attrition. Attrition refers to subjects dropping out of the study after being ran-
domly assigned to the treatment or control group. Sometimes attrition occurs for
reasons unrelated to the treatment program; for example, a participant in a job
training study might need to leave town to care for a sick relative. But if the rea-
son for attrition is related to the treatment itself, then the attrition results in bias
in the OLS estimator of the causal effect. For example, suppose that the most able
trainees drop out of the job training program experiment because they get out-of-
town jobs acquired using the job training skills, so at the end of the experiment
only the least able members of the treatment group remain. Then the distribution
of unmeasured characteristics (ability) will differ between the control and treat-
ment groups (the treatment enabled the ablest trainees to leave town). In other
words, the treatment Xi will be correlated with ui (which includes ability) for those