  Regression analysis under the assumption of monotonicity is a well-studied
statistical problem and has been used in a wide range of applications. However,
there remains a lack of a broadly applicable methodology that permits
information borrowing, for efficiency gains, when jointly estimating multiple
monotonic regression functions. We introduce such a methodology by extending
the isotonic regression problem presented in the article "The isotonic
regression problem and its dual" (Barlow and Brunk, 1972). The presented
approach can be applied to both fixed and random designs and any number of
explanatory variables (regressors). Our framework penalizes pairwise
differences in the values (levels) of the monotonic function estimates, with
the weight of penalty being determined based on a statistical test, which
results in information being shared across data sets if similarities in the
regression functions exist. Function estimates are subsequently derived using
an iterative optimization routine that uses existing solution algorithms for
the isotonic regression problem. Simulation studies for normally and binomially
distributed response data illustrate that function estimates are consistently
improved if similarities between functions exist, and are not oversmoothed
otherwise. We further apply our methodology to analyse two public health data
sets: neonatal mortality data for Porto Alegre, Brazil, and stroke patient data
for North West England.
