TABLE 1.

A compendium of the main notation in our methodology by section.

Notation	Description
Framework (Section 3.1)
Y	The random variable (r.v.) for the outcomes for the subjects
$\mathit{X}$, $\mathcal{X}$	The r. v. for the observed measurements for the subjects, its support
A	The r. v. for the treatment
$T_1,T_2$	The two treatments, shorthand for their codes, zero and one
d or $d\left[f\right]$	The decision function; this function maps observed measurements to treatment
V or $V\left[d\right]$	The value of the decision function, the average outcome over all patients if this decision is used to allocate treatment
$d^{\text{*}}$	The unknown optimal decision function i.e. the one with highest V
$d_0$	A naive, baseline, business-as-usual or null decision function
${\mu }_{I_0}$	The unknown improvement of d over $d_0$; it is the difference of values
The RCT data (Section 3.2.1)
n	The number of subjects in the randomized comparative trial (RCT)
p	The number of measurements assessed on each subject
$x_i$	The vector of p measurements for the ith subject
$x_{i,j}$	The jth measurement for the ith subject
$\mathit{X}$	The $n\times p$ matrix of all measurements for all subjects
$\mathit{A}$	The vector of treatments for all the n subjects
$\mathit{y}$	The vector of outcomes for all the n subjects
The response model (Section 3.2.2)
f	The function that relates the p measurements and A to the response
$U_i$	The r. v. for the unknown covariates for subject i
$\xi \left(x_i,A_i,U_i\right)$	The function that computes misspecification in the response
${\mathrm{ℰ}}_i$	The r. v. for irreducible noise for the ith subject
${\beta }_j$	The linear coefficient for the jth measurement when $A=0$
${\gamma }_j$	The additional linear coefficient for the jth measurement when $A=1$
Out of sample estimation and validation (Section 3.4)
${\mathit{X}}_{\text{train}},{\mathit{y}}_{\text{train}}$	The subset of the data used to create the fit of f
${\mathit{X}}_{\text{test}},{\mathit{y}}_{\text{test}}$	The subset of the data used to validate the fit of f
$\widehat{f}$, $\widehat{d}$, $\widehat{V}$, ${\widehat{I}}_0$	The finite-sample estimates of f, d, V, ${\mu }_{I_0}$
${\overline{y}}_{set}$	The arithmetic average of $\left\{y_i:i\in set\right\}$
${\widehat{\beta }}_j$, ${\widehat{\gamma }}_j$	The finite-sample estimates of ${\beta }_j$, ${\gamma }_j$
Inference (Section 3.5)
B	The number of bootstrap samples
$\tilde{\mathit{X}},\tilde{\mathit{y}}$	A sample of the rows of $\mathit{X},\mathit{y}$ with replacement
${\tilde{I}}_{0,b}$	The bth estimate of ${\mu }_{I_0}$ in the bootstrap
α	The size of the hypothesis test
Personalization of future subjects’ treatments (Section 3.6)
${\mathit{x}}_{\mathit{*}}$	A future subject (not part of the RCT)