  When there are resource constraints, it is important to rank or estimate
treatment benefits according to patient characteristics. This facilitates
prioritization of assigning different treatments. Most existing literature on
individualized treatment rules targets absolute conditional treatment effect
differences as the metric for benefits. However, there can be settings where
relative differences may better represent such benefits. In this paper, we
consider modeling such relative differences that form scale-invariant contrasts
between conditional treatment effects. We show that all scale-invariant
contrasts are monotonic transformations of each other. Therefore we posit a
single index model for a particular relative contrast. Identifiability of the
model is enforced via an intuitive $l_2$ norm constraint on index parameters.
We then derive estimating equations and efficient scores via semiparametric
efficiency theory. Based on the efficient score and its variant, we propose a
two-step approach that consists of minimizing a doubly robust loss function and
a subsequent one-step efficiency augmentation procedure to achieve efficiency
bound. Careful theoretical and numerical studies are provided to show the
superiority of the proposed approach.
