  Background: With the ever-increasing amount of medical imaging data, the
demand for algorithms to assist clinicians has amplified. Unsupervised anomaly
detection (UAD) models promise to aid in the crucial first step of disease
detection. While previous studies have thoroughly explored fairness in
supervised models in healthcare, for UAD, this has so far been unexplored.
  Methods: In this study, we evaluated how dataset composition regarding
subgroups manifests in disparate performance of UAD models along multiple
protected variables on three large-scale publicly available chest X-ray
datasets. Our experiments were validated using two state-of-the-art UAD models
for medical images. Finally, we introduced a novel subgroup-AUROC (sAUROC)
metric, which aids in quantifying fairness in machine learning.
  Findings: Our experiments revealed empirical "fairness laws" (similar to
"scaling laws" for Transformers) for training-dataset composition: Linear
relationships between anomaly detection performance within a subpopulation and
its representation in the training data. Our study further revealed performance
disparities, even in the case of balanced training data, and compound effects
that exacerbate the drop in performance for subjects associated with multiple
adversely affected groups.
  Interpretation: Our study quantified the disparate performance of UAD models
against certain demographic subgroups. Importantly, we showed that this
unfairness cannot be mitigated by balanced representation alone. Instead, the
representation of some subgroups seems harder to learn by UAD models than that
of others. The empirical fairness laws discovered in our study make disparate
performance in UAD models easier to estimate and aid in determining the most
desirable dataset composition.
