Que es un manejo de objeciones

To construct our estimator, πopt, we make three standard causal inference as-

sumptions (Robins et al. 2000, Zhang et al. 2012b): (C1) consistency,

(Y, Z) = {Y∗(A), Z∗(A)}; (C2) positivity, for each a there exists > 0 such that

P(A =a∣X,W) ≥; (C3) ignorability, [{Y∗(a), Z∗(a)} ∶ a∈ {−1,1}] ⊥ A∣X,W. The

consistency assumption implies that the individual’s observed outcome is the potential outcome associated with the observed exposure; the positivity assumption implies there is a positive probability of receiving all possible treatments; the ignorability assump- tion implies that patients receiving all possible treatments have equal distributions of experiencing the potential outcomes. We further assume (C4) (A, Y, Z) ⊥ E∣X,W,

which states that the treatment assignment and responses are independent of a pa- tient’s preference. This important assumption is reasonable in the single stage scenario because the treatments are assigned randomly and if outcomes were affected by the preferences, treatments would be obsolete. Under these stated assumptions, it can be shown (Schulte et al. 2014) that the optimal individualized treatment can then be defined in terms of the predefined utility function:

πopt(x,w) =arg max

a∈{−1,1}E [U(Y, Z;E)∣X =x,W =w, A=a] =arg max a∈{−1,1}E [h(G)Y + {1−h(G)}Z∣X =x,W =w, A=a] =arg max a∈{−1,1} [_E{h(G)∣X =x,W =w}_E(Y∣X =x,W =w, A=a) + [1−_E{h(G)∣X=x,W =w}]_E(Z∣X=x,W =w, A=a) ] =arg max a∈{−1,1} [µ(x,w)Q_Y(x,w,a) + [1−µ(x,w)]Q_Z(x,w, a)], where QY(x,w, a) = _E(Y∣X = x,W = w, A = a), QZ(x,w, a) = _E(Z∣X = x,W =

w, A=a)andµ(x,w) =_E{h(G) ∣X =x,W =w}. Now letQ̂_Y,n(x,w, a),Q̂_Z,n(x,w, a)

and̂µ_n(x,w)be respective estimators ofQ_Y(x,w,a),Q_Z(x,w, a)andµ(x,w). Then,

̂

πn(x,w) =arg max

a∈{−1,1}[̂µn

(x,w) ̂Q_Y,n(x,w, a) + {1− ̂µ_n(x,w)} ̂Q_Z,n(x,w, a)]

This implies that estimationπopt_(x,w)can be broken down into three estimators: ̂

QY,n(x,w, a),Q̂_Z,n(x,w, a) and ̂µ_n(x,w)

To define _Q̂

means that the itemized response information contained inW as it relates to the out- comes is sufficiently captured in the covariates X and need not be included in our estimator. Then, we construct linear working models as QY(x,w, a;γ_Y) = x⊺_Y,0γ_Y,0 +

ax⊺

Y,1γY,1 and QZ(x,w, a;γZ) =x ⊺

Z,0γZ,0+ax ⊺

Z,1γZ,1, where x`,j for`=Y, Z and j =0,1

are known feature vectors constructed from x and γY, γZ are unknown parameter vectors. Note that we assume a linear working model, but this form is note re- quired and was just assumed for simplicity. Let ̂γ_Y,n and ̂γZ, n be the maximum

likelihood estimates such that ̂γ_Y,n = arg min_{γY E}{Y −Q_Y(X,W, A;γ_Y)}2 and ̂γ_Z,n = arg minγZE{Z−QZ(X,W, A;γZ)}

2_{. Then, utilizing these estimates, we find that} ̂

QY,n(x,w, a;̂γ_Y)andQ̂_Z,n(x,w, a;̂γ_Z)are the estimators ofQ_Y(x,w, a)andQ_Z(x,w, a).

Constructing and estimator ̂µ_n(x,w) is sufficiently more difficult because E is

latent and therefore unobservable. The first step in defining this estimator is finding an estimate for Φ(E). To develop a latent preference model, we assume that E ⊥X∣W.

As we will see, this assumption simplifies the construction of the estimator and is reasonable ifX does not contain any additional information about patient preferences beyond that contained in W. Since Wj is a binary response variable, we can assume that, conditional on the unobservede,Wj are independent Bernoulli random variables. We define the generating model for W as P(W_j∣E = e) = ₁₊exp_exp{fj(e)}

{fj(e)}, where fj is

constrained to be a monotone increasing function for j = 1, . . . , p. To solve this we

employ an EM algorithm. In the estimation step we create starting values forE using the standard Rasch model from traditional item response theory (Rasch 1961; 1980), which we discussed at length in Section 3.2.2. We note that in the EM algorithm these starting values will be replaced with estimates of ê_n as the algorithm progresses

iteratively.

With these starting values, e0, we start the maximization step by finding _f̂

j(e0)

{t_1,e, . . . , t_kE,e} are quantiles of e0. For computational and asymptotic simplicity, we

impose piecewise linear splines and let πj(e0) = {πj,₁(e0), πj,₂(e0), . . . , πj,K

E(e

0_)}T be the set of splines of order 2. We then estimate fj by f̂j,n(e0) = π_j(e0)T. Define the

linear component asπj,k =a_j,k+b∗_j,ke0 for k=1, . . . , K_E. The linear parametersa_j,k, b_j,k

for a fixedj and allk=1, . . . , K_E are simultaneously estimated by solving the following

equation using nonlinear programming (Ghalanos and Theussl 2015, Ye 1987):

πj,k(e0) =a_j,k+b∗_j,ke0 ∀ k∈ {1, . . . , K_G} where logit{P(W_j =1∣E=e)} =πj(e(0))

s.t.

aj,k>0, b_j,k >0 ∀ k∈ {1, . . . , K_E}

aj,k+b_j,kt_k,E=a_j,k+1+b_j,k+1t_k,E ∀ k∈ {1, . . . , K_E}.

Because the Wj are independent for j ∈1, . . . , p, this process is repeated for allj.

Let βE = {a_1,1, b_1,1, . . . , a_1,KE, b_1,KE, . . . , a_p,1, b_p,1, . . . , a_p,KE, b_p,KE}. Once we have

estimates _β̂

E for βE, we can create an estimator of e, ̂en. Given β̂E and a marginal distribution for the latent patient preferences, the conditional distribution of E given W =w is proportional to p(w∣h)p_h(h). This can be approximated using a Metropolis

Hastings algorithm. However, because we assume thatµ(x,w)does not depend on x,

it is less computationally intensive to apply a method of moments estimator. We let

̂e_n denote the solution to ∑p_j=1̂bj,lexpit(̂aj,l+ ̂b_j,le) = ∑p_j=1̂bj,lwj ∀ l =1, . . . , K_E, where expit(u) = exp(u)/ {1+exp(u)}. This estimator for e, ̂en, provides similar estimates

burdensome.

Finally, recall that we only collect the contentment information, B, to develop our rule. We need to incorporate that information into the estimate of h( ̂G), which

serves as linear trade off weight. To do this, we assume the distribution of B is de- pendent on the composite outcome, which is a function combination of the outcomes and the preference information as U(Y, Z;̂e_n) = h( ̂G)Y + {1−h( ̂G)}Z. Hence, we

are interested in is an estimator for h( ̂G) that optimizes that patients contentment.

We assume that logit{P(B = 1)} =r{U(Y, Z;̂e_n)}, where r is a monotone increasing

function. Let _Ĝ _{= Φ(̂e}

n), then, similar to what we have previously done, we fit this

model using monotone spline smoothing except this time we simultaneously fit h( ̂G)

andr{U(Y, Z;̂e_n)}as piecewise linear splines. LetK_Gbe the number of knots such that

the knots {t_1,G, . . . , tK

G,G} are quantiles of

̂

G. Under these specifications, we impose splines πh( ̂G) and π_r{U(Y, Z;̂e_n)} where π_h( ̂G) = {π_h,1( ̂G), π_h,2( ̂G), . . . , π_h,KG( ̂G)}

and πr{U(Y, Z;̂e_n)} = [π_r,1{U(Y, Z;̂e_n)}, π_r,2{U(Y, Z;̂e_n)}, . . . , π_r,KG{U(Y, Z;̂e_n)}] for

πr,k{U(Y, Z;̂e_n)} =δ_1,k+δ_2,kU(Y, Z;̂e_n) ∀ k∈ {1, . . . , K_G} where logit{P (B=1)} =πr{U(Y, Z;̂en)} and U(Y, Z;̂e_n) =h( ̂G)Y + {1−h( ̂G)}Z, πh,k( ̂G) =α_1,k+α_2,kĜ ∀ k∈ {1, . . . , K_G} where h( ̂G) =πh( ̂G) s.t. δ2,k >0, α_2,k>0 ∀ k∈ {1, . . . , K_G} α1,1=0 α1,KG+α_2,KG =1 α1,k+α_2,kt_k,G =α_1,k+1+α_2,k+1t_k,G ∀ k∈ {1, . . . , K_G} δ1,k+δ2,ktk,G =δ1,k+1+δ2,k+1tk,G ∀ k∈ {1, . . . , KG}

Then, the estimator of µ(x,w) is ̂µn(x,w) = ̂h( ̂G).

5.3 Simulation Study

The results of the simulation study are included for 3 knots, which has been shown to be sufficient by He and Shi (1998). For completeness, similar results for 5 knots are included in the Appendix B.1. We chose 3 knots because additional knots do not provide enough additional information to justify a more complex model with a burdensome computation.