Medidas internacionales ante la aplicación ineficaz de las disposiciones de la

We now consider the precision medicine setting. As outlined in Section 4.2, in this setting we observe the data{Xi, Ai, Ri}, i = 1, . . . , n, whereXi ∈ X ⊂ Rp is a vector of covariates independent of treatment,Ai ∈ {−1,1}is the assigned treatment, andRi ∈ Y ⊂ Ris a clinical

1.25 1.50 1.75 2.00 0 5 10 15 Gamma value Delta v alue a 0 25 50 75 100 0 5 10 15 Gamma value Rank of Delta v alue b

Figure 4.14: Values of (a)∆i`and (b) rank(∆i`)by value ofγ`forn = 100observations and L= 10values ofγ. Note that the average magnitude of∆i`rises asγ` rises, visible in (a) but the amount of large relative change in∆i`drops. The dropoff in large crossing lines is more clearly visible in (b), where the scale is held constant.

reward. Following conventions in the field, we assume that higher values of the clinical reward indicate more favorable rewards; as in Zhou et al. (2017), we will allow rewards to be real-valued rather than strictly non-negative. The modelP estimates an ITRπ : X → {−1,1}, a function that takes a value in the covariate space and recommends a value of treatment. We will index πbyP for the remainder of this section to stress the dependence of the estimated ITR on the modelP. In particular,P estimates an ITR that is optimal in the sense of maximizing the value V attainable within a class of policies, whereV(π) = _Eπ[Y]is the expected reward obtained by following the policyπ. The natural summary of model performance for this setting is given byΨ(P) = ˆV(πP),whereVˆ is a consistent estimator ofV; in particular, we useVˆ(πP) =

n X i=1 RiI{Ai =πP(Xi)}/ n X i=1

I{Ai =πP(Xi)}. AsΨ(P)∈R,a natural choice of distance metric δis squared univariate distance. We assume that potential mismeasurement occurs symmetrically at the level of the clinical reward up to a prespecified amountγ—that is,Γ(Ri) = [Ri−γ, Ri+ γ], i= 1, . . . , n.

In our numerical experiment, we setn = 100andp = 8. We generatedX ∼ U(−1,1)and A ∼ Bernoulli(1₂)independent ofX. We mimicked data setup 1 from the simulations of Zhou et al. (2017), settingµ(X) = 1 +X1 +X2 + 2X3 + 0.5X4 andν(X) = 1.8(0.3−X1 −X2)

and then generatingR ∼ N(µ(X) +ν(X),1). As discussed in that paper, this creates a moderate treatment effect size with a true linear decision boundary that depends onX1andX2alone. We

use Residual Weighted Learning (RWL) as our modelP, and we setγ = 2.

We examine the influence of measurement error in this setting through a series of three graphs. Figure 4.15, which displays the ITR assignment byR,X1, andX2, demonstrates that

our RWL model is estimating the true linear decision boundary well. Figure 4.16, which shows the values of∆ifor each of then= 100observations byR,X1, andX2, demonstrates that RWL

does not strictly seek either points near the decision boundary or points near the extremes of X, instead finding what appears to be a mixture of such points. And Figure 4.17, which demon- strates which points switch ITR assignment from the observed ITR to the minimally and maxi- mally different ITR, in terms of estimated valueVˆ, demonstrates why the point with the largest

Figure 4.15: RWL ITR assignments from the precision medicine simulation described in Section 4.4.2, by values ofX1,X2, andR. Note the decision boundary’s approximate linearity inX1 and

X2, which matches the true data-generating mechanism despite the presence of noise covariates.

∆i value has high measurement influence: when it is mismeasured, observations with large re- ward magnitudes switch ITR assignment.

Note that, in this experiment, the ITR with maximal perturbation has higher estimated value than the observed (Vˆ(πP) = 2.26, Vˆ(π(imax)) = 2.49). This is not at all required, though—the ITR with maximal perturbation under measurement may achieve worse value instead. If gains and losses in value are not equally weighted—if achieving a worse value than the observed is more costly than failing to achieve a higher value than the observed, for instance—users of this method may wish to calibrate∆to consider these cost weights.

Figure 4.16: Values of∆computed via RWL from the precision medicine simulation described in Section 4.4.2, by values ofX1,X2, andR. Deeper shades of blue correspond to lower∆

values, while brighter shades of red correspond to higher∆values. Note that high∆values do appear near the decision boundary, and near the extremes ofX, but are not confined to these locations deterministically.

Figure 4.17: Depiction of ITR assignment switching between the observed, minimal impact, and maximal impact measurement errors. Black points have the same ITR assignment in all three ITRs. Red points switch assignment between the observed and maximal impact ITRs, blue points switch assignment between the observed and minimal impact ITRs, and green points switch assignment beween the observed and both the minimal and maximal impact ITRs. In this case, the minimal impact ITR departs only slightly from the observed because the blue points essentially balance each other out in terms of reward, leaving only the impact of the green points shared by the maximal impact ITR; meanwhile, the maximal impact ITR gains an additional high-reward point. Befitting this scenario, we observeVˆ(πP) = 2.26,Vˆ(πP(i_min)) = 2.32, and

ˆ V(πP_(i