  The exploration of biomarkers, which are clinically useful biomolecules, and
the development of prediction models using them are important problems in
biomedical research. Biomarkers are widely used for disease screening, and some
are related not only to the presence or absence of a disease but also to its
severity. These biomarkers can be useful for prioritization of treatment and
clinical decision-making. Considering a model helpful for both disease
screening and severity prediction, this paper focuses on regression modeling
for an ordinal response equipped with a hierarchical structure.
  If the response variable is a combination of the presence of disease and
severity such as \{{\it healthy, mild, intermediate, severe}\}, for example,
the simplest method would be to apply the conventional ordinal regression
model. However, the conventional model has flexibility issues and may not be
suitable for the problems addressed in this paper, where the levels of the
response variable might be heterogeneous. Therefore, this paper proposes a
model assuming screening and severity prediction as different tasks, and an
estimation method based on structural sparse regularization that leverages any
common structure between the tasks when such commonality exists. In numerical
experiments, the proposed method demonstrated stable performance across many
scenarios compared to existing ordinal regression methods.
