  Health care decisions are increasingly informed by clinical decision support
algorithms, but these algorithms may perpetuate or increase racial and ethnic
disparities in access to and quality of health care. Further complicating the
problem, clinical data often have missing or poor quality racial and ethnic
information, which can lead to misleading assessments of algorithmic bias. We
present novel statistical methods that allow for the use of probabilities of
racial/ethnic group membership in assessments of algorithm performance and
quantify the statistical bias that results from error in these imputed group
probabilities. We propose a sensitivity analysis approach to estimating the
statistical bias that allows practitioners to assess disparities in algorithm
performance under a range of assumed levels of group probability error. We also
prove theoretical bounds on the statistical bias for a set of commonly used
fairness metrics and describe real-world scenarios where our theoretical
results are likely to apply. We present a case study using imputed race and
ethnicity from the Bayesian Improved Surname Geocoding (BISG) algorithm for
estimation of disparities in a clinical decision support algorithm used to
inform osteoporosis treatment. Our novel methods allow policy makers to
understand the range of potential disparities under a given algorithm even when
race and ethnicity information is missing and to make informed decisions
regarding the implementation of machine learning for clinical decision support.
