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On Testing Conditional Qualitative Treatment Effects Chengchun Shi - - PowerPoint PPT Presentation

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On Testing Conditional Qualitative Treatment Effects Chengchun Shi Department of Statistics North Carolina State University Joint work with Wenbin Lu and Rui Song July 30, 2017 . . . . . . . . . . . . . . . . . . . . . .

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On Testing Conditional Qualitative Treatment Effects

Chengchun Shi Department of Statistics North Carolina State University Joint work with Wenbin Lu and Rui Song July 30, 2017

Chengchun Shi (NCSU) CQTE July 30, 2017 1 / 20

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A few words on causal inference

Data A: Treatment (0 or 1) X: Covariates Y : Observed outcome (usually the larger the better)

Chengchun Shi (NCSU) CQTE July 30, 2017 2 / 20

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A few words on causal inference

Data A: Treatment (0 or 1) X: Covariates Y : Observed outcome (usually the larger the better) Y ∗(a): Potential outcome a = 0, 1

Chengchun Shi (NCSU) CQTE July 30, 2017 2 / 20

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A few words on causal inference

Data A: Treatment (0 or 1) X: Covariates Y : Observed outcome (usually the larger the better) Y ∗(a): Potential outcome a = 0, 1 Objective Identify the optimal regime dopt to reach the best clinical outcome

Chengchun Shi (NCSU) CQTE July 30, 2017 2 / 20

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A few words on causal inference

Data A: Treatment (0 or 1) X: Covariates Y : Observed outcome (usually the larger the better) Y ∗(a): Potential outcome a = 0, 1 Objective Identify the optimal regime dopt to reach the best clinical outcome Maximize EY ∗(d) = E[d(X)Y ∗(1) + {1 − d(X)}Y ∗(0)] d : X → {0, 1}.

Chengchun Shi (NCSU) CQTE July 30, 2017 2 / 20

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Q, Contrast and Value function Q(x, a) = E[Y |X = x, A = a],

Chengchun Shi (NCSU) CQTE July 30, 2017 3 / 20

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Q, Contrast and Value function Q(x, a) = E[Y |X = x, A = a], τ0(x) = Q(x, 1) − Q(x, 0),

Chengchun Shi (NCSU) CQTE July 30, 2017 3 / 20

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Q, Contrast and Value function Q(x, a) = E[Y |X = x, A = a], τ0(x) = Q(x, 1) − Q(x, 0), V (d) = EY ∗(d) = E[d(X)Y ∗(1) + {1 − d(X)}Y ∗(0)].

Chengchun Shi (NCSU) CQTE July 30, 2017 3 / 20

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Q, Contrast and Value function Q(x, a) = E[Y |X = x, A = a], τ0(x) = Q(x, 1) − Q(x, 0), V (d) = EY ∗(d) = E[d(X)Y ∗(1) + {1 − d(X)}Y ∗(0)]. Optimal treatment regime SUTVA, no unmeasured confounders

Chengchun Shi (NCSU) CQTE July 30, 2017 3 / 20

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Q, Contrast and Value function Q(x, a) = E[Y |X = x, A = a], τ0(x) = Q(x, 1) − Q(x, 0), V (d) = EY ∗(d) = E[d(X)Y ∗(1) + {1 − d(X)}Y ∗(0)]. Optimal treatment regime SUTVA, no unmeasured confounders

  • ptimal treatment regime

dopt(x) = I(τ0(x) ≥ 0).

Chengchun Shi (NCSU) CQTE July 30, 2017 3 / 20

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There are two types of clinically “important” variables.

Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

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There are two types of clinically “important” variables.

Predictive variables are those involved in τ0(x).

Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

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There are two types of clinically “important” variables.

Predictive variables are those involved in τ0(x). Prescriptive variables are those involved in dopt(x).

Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

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There are two types of clinically “important” variables.

Predictive variables are those involved in τ0(x). Prescriptive variables are those involved in dopt(x).

Predictive variables have interactions with the treatment.

Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

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There are two types of clinically “important” variables.

Predictive variables are those involved in τ0(x). Prescriptive variables are those involved in dopt(x).

Predictive variables have interactions with the treatment. Prescriptive variables have qualitative interactions with the treatment.

Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

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There are two types of clinically “important” variables.

Predictive variables are those involved in τ0(x). Prescriptive variables are those involved in dopt(x).

Predictive variables have interactions with the treatment. Prescriptive variables have qualitative interactions with the treatment. Prescriptive variables ⊆ predictive variables.

Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

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There are two types of clinically “important” variables.

Predictive variables are those involved in τ0(x). Prescriptive variables are those involved in dopt(x).

Predictive variables have interactions with the treatment. Prescriptive variables have qualitative interactions with the treatment. Prescriptive variables ⊆ predictive variables. Predictive variables ̸⊆ prescriptive variables.

Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

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A tiny example: τ0(x) = exp(−x1)x2, for {x1, x2, . . . , xp} ∈ [−1, 1]p.

Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

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A tiny example: τ0(x) = exp(−x1)x2, for {x1, x2, . . . , xp} ∈ [−1, 1]p. dopt(x) = I{exp(−x1)x2 ≥ 0} = I(x2 ≥ 0).

Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

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A tiny example: τ0(x) = exp(−x1)x2, for {x1, x2, . . . , xp} ∈ [−1, 1]p. dopt(x) = I{exp(−x1)x2 ≥ 0} = I(x2 ≥ 0). x1 and x2 are the predictive variables.

Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

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A tiny example: τ0(x) = exp(−x1)x2, for {x1, x2, . . . , xp} ∈ [−1, 1]p. dopt(x) = I{exp(−x1)x2 ≥ 0} = I(x2 ≥ 0). x1 and x2 are the predictive variables. x2 is the prescriptive variable.

Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

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A tiny example: τ0(x) = exp(−x1)x2, for {x1, x2, . . . , xp} ∈ [−1, 1]p. dopt(x) = I{exp(−x1)x2 ≥ 0} = I(x2 ≥ 0). x1 and x2 are the predictive variables. x2 is the prescriptive variable. x1 and x2 have interactions with the treatment.

Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

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A tiny example: τ0(x) = exp(−x1)x2, for {x1, x2, . . . , xp} ∈ [−1, 1]p. dopt(x) = I{exp(−x1)x2 ≥ 0} = I(x2 ≥ 0). x1 and x2 are the predictive variables. x2 is the prescriptive variable. x1 and x2 have interactions with the treatment. x2 has qualitative interaction with the treatment.

Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

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Gunter et al. (2011) proposed an S-score method for quantifying the magnitude of the qualitative treatment effects.

Chengchun Shi (NCSU) CQTE July 30, 2017 6 / 20

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Gunter et al. (2011) proposed an S-score method for quantifying the magnitude of the qualitative treatment effects. Chang et al. (2015) and Hsu (2017) proposed nonparametric tests for testing the qualitative treatment effects.

Chengchun Shi (NCSU) CQTE July 30, 2017 6 / 20

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Gunter et al. (2011) proposed an S-score method for quantifying the magnitude of the qualitative treatment effects. Chang et al. (2015) and Hsu (2017) proposed nonparametric tests for testing the qualitative treatment effects. Here, we focus on the conditional qualitative treatment effects (CQTE).

Chengchun Shi (NCSU) CQTE July 30, 2017 6 / 20

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Gunter et al. (2011) proposed an S-score method for quantifying the magnitude of the qualitative treatment effects. Chang et al. (2015) and Hsu (2017) proposed nonparametric tests for testing the qualitative treatment effects. Here, we focus on the conditional qualitative treatment effects (CQTE).

Formalize the notion of CQTE and present equivalent representations

  • f no CQTE.

Chengchun Shi (NCSU) CQTE July 30, 2017 6 / 20

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Gunter et al. (2011) proposed an S-score method for quantifying the magnitude of the qualitative treatment effects. Chang et al. (2015) and Hsu (2017) proposed nonparametric tests for testing the qualitative treatment effects. Here, we focus on the conditional qualitative treatment effects (CQTE).

Formalize the notion of CQTE and present equivalent representations

  • f no CQTE.

Propose a testing procedure for testing the existence of CQTE.

Chengchun Shi (NCSU) CQTE July 30, 2017 6 / 20

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Gunter et al. (2011) proposed an S-score method for quantifying the magnitude of the qualitative treatment effects. Chang et al. (2015) and Hsu (2017) proposed nonparametric tests for testing the qualitative treatment effects. Here, we focus on the conditional qualitative treatment effects (CQTE).

Formalize the notion of CQTE and present equivalent representations

  • f no CQTE.

Propose a testing procedure for testing the existence of CQTE. Develop a variable selection procedure for selecting prescriptive variables in a sequential order.

Chengchun Shi (NCSU) CQTE July 30, 2017 6 / 20

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Conditional qualitative treatment effects (CQTE)

B and C are two disjoint subsets of [1, . . . , p].

Chengchun Shi (NCSU) CQTE July 30, 2017 7 / 20

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Conditional qualitative treatment effects (CQTE)

B and C are two disjoint subsets of [1, . . . , p]. X B and X C are the subsets of X.

Chengchun Shi (NCSU) CQTE July 30, 2017 7 / 20

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Conditional qualitative treatment effects (CQTE)

B and C are two disjoint subsets of [1, . . . , p]. X B and X C are the subsets of X. X C have CQTE given X B, if there exists some nonempty subsets C1, C2 and B such that (i) Pr { (X B, X C) ∈ B × C1 } > 0 and Pr { (X B, X C) ∈ B × C2 } > 0. (ii) For any xC

1 ∈ C1, xC 2 ∈ C2 and xB ∈ B, we have

arg max

a

E { Y ∗(a)|X B = xB, X C = xC

1

} ̸= arg max

a

E { Y ∗(a)|X B = xB, X C = xC

2

} .

Chengchun Shi (NCSU) CQTE July 30, 2017 7 / 20

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Unconditional qualitative treatment effects B = ∅.

Chengchun Shi (NCSU) CQTE July 30, 2017 8 / 20

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Unconditional qualitative treatment effects B = ∅. CQTE = QTE.

Chengchun Shi (NCSU) CQTE July 30, 2017 8 / 20

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Unconditional qualitative treatment effects B = ∅. CQTE = QTE. Let τ C

0 (xC) = E{τ0(X)|X C = xc}.

No CQTE means τ C

0 (X C) ≥ 0, a.s or τ C 0 (X C) ≤ 0, a.s.

Chengchun Shi (NCSU) CQTE July 30, 2017 8 / 20

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Unconditional qualitative treatment effects B = ∅. CQTE = QTE. Let τ C

0 (xC) = E{τ0(X)|X C = xc}.

No CQTE means τ C

0 (X C) ≥ 0, a.s or τ C 0 (X C) ≤ 0, a.s.

Conditional qualitative treatment effects B is not empty.

Chengchun Shi (NCSU) CQTE July 30, 2017 8 / 20

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Unconditional qualitative treatment effects B = ∅. CQTE = QTE. Let τ C

0 (xC) = E{τ0(X)|X C = xc}.

No CQTE means τ C

0 (X C) ≥ 0, a.s or τ C 0 (X C) ≤ 0, a.s.

Conditional qualitative treatment effects B is not empty. Assume X B have QTE.

Chengchun Shi (NCSU) CQTE July 30, 2017 8 / 20

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Unconditional qualitative treatment effects B = ∅. CQTE = QTE. Let τ C

0 (xC) = E{τ0(X)|X C = xc}.

No CQTE means τ C

0 (X C) ≥ 0, a.s or τ C 0 (X C) ≤ 0, a.s.

Conditional qualitative treatment effects B is not empty. Assume X B have QTE. CQTE measures whether X C are “important” in decision making given X B.

Chengchun Shi (NCSU) CQTE July 30, 2017 8 / 20

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For any D ⊆ [1, . . . , p], define τ D

0 (xD) = E{τ0(X)|X D = xD},

dD

  • pt(x) = I(τ D

0 (xD) > 0).

Chengchun Shi (NCSU) CQTE July 30, 2017 9 / 20

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For any D ⊆ [1, . . . , p], define τ D

0 (xD) = E{τ0(X)|X D = xD},

dD

  • pt(x) = I(τ D

0 (xD) > 0).

Let W = B ∪ C. Define ERW ,B =    0, if τ W

0 (X) = 0, a.s.

E[|dW

  • pt(X) − dB
  • pt(X)|I{τ W

0 (X W ) ̸= 0}]

Pr{τ W

0 (X W ) ̸= 0}

,

  • therwise,

VDW ,B = V (dW

  • pt) − V (dB
  • pt)

= E ( τ W

0 (X W )[I{τ W 0 (X W ) ≥ 0} − I{τ B 0 (X B) ≥ 0}]

) .

Chengchun Shi (NCSU) CQTE July 30, 2017 9 / 20

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Theorem (Charaterization of No CQTE) Under certain conditions, the followings are equivalent: (i) X c doesn’t have CQTE given X B.

Chengchun Shi (NCSU) CQTE July 30, 2017 10 / 20

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Theorem (Charaterization of No CQTE) Under certain conditions, the followings are equivalent: (i) X c doesn’t have CQTE given X B. (ii) VDW ,B = 0.

Chengchun Shi (NCSU) CQTE July 30, 2017 10 / 20

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Theorem (Charaterization of No CQTE) Under certain conditions, the followings are equivalent: (i) X c doesn’t have CQTE given X B. (ii) VDW ,B = 0. (iii) ERW ,B = 0.

Chengchun Shi (NCSU) CQTE July 30, 2017 10 / 20

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Theorem (Charaterization of No CQTE) Under certain conditions, the followings are equivalent: (i) X c doesn’t have CQTE given X B. (ii) VDW ,B = 0. (iii) ERW ,B = 0. (iv) For any xW such that τ W

0 (xW ) ̸= 0, dW

  • pt(x) = dB
  • pt(x).

Chengchun Shi (NCSU) CQTE July 30, 2017 10 / 20

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Theorem (Charaterization of No CQTE) Under certain conditions, the followings are equivalent: (i) X c doesn’t have CQTE given X B. (ii) VDW ,B = 0. (iii) ERW ,B = 0. (iv) For any xW such that τ W

0 (xW ) ̸= 0, dW

  • pt(x) = dB
  • pt(x).

(v) For any fixed xB, τ W

0 (xB, xC) ≥ 0 for any xC or τ W 0 (xB, xC) ≤ 0 for

any xC.

Chengchun Shi (NCSU) CQTE July 30, 2017 10 / 20

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X1, X2 are independently generated from Unif[−2, 2]. Does X2 has CQTE given X1?

Chengchun Shi (NCSU) CQTE July 30, 2017 11 / 20

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X1, X2 are independently generated from Unif[−2, 2]. Does X2 has CQTE given X1? τ0(x1, x2) = x1x2

2.

Chengchun Shi (NCSU) CQTE July 30, 2017 11 / 20

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X1, X2 are independently generated from Unif[−2, 2]. Does X2 has CQTE given X1? τ0(x1, x2) = x1x2

2.

No CQTE.

Chengchun Shi (NCSU) CQTE July 30, 2017 11 / 20

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X1, X2 are independently generated from Unif[−2, 2]. Does X2 has CQTE given X1? τ0(x1, x2) = x1x2

2.

No CQTE. τ0(x1, x2) = x1 max(x2, 0).

Chengchun Shi (NCSU) CQTE July 30, 2017 11 / 20

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X1, X2 are independently generated from Unif[−2, 2]. Does X2 has CQTE given X1? τ0(x1, x2) = x1x2

2.

No CQTE. τ0(x1, x2) = x1 max(x2, 0). No CQTE.

Chengchun Shi (NCSU) CQTE July 30, 2017 11 / 20

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X1, X2 are independently generated from Unif[−2, 2]. Does X2 has CQTE given X1? τ0(x1, x2) = x1x2

2.

No CQTE. τ0(x1, x2) = x1 max(x2, 0). No CQTE. τ0(x1, x2) = x1(x2 − 2).

Chengchun Shi (NCSU) CQTE July 30, 2017 11 / 20

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X1, X2 are independently generated from Unif[−2, 2]. Does X2 has CQTE given X1? τ0(x1, x2) = x1x2

2.

No CQTE. τ0(x1, x2) = x1 max(x2, 0). No CQTE. τ0(x1, x2) = x1(x2 − 2). No CQTE.

Chengchun Shi (NCSU) CQTE July 30, 2017 11 / 20

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Hypothesis testing: H0 : X C doens’t have CQTE given X B, versus H1 : X C has CQTE given X B.

Chengchun Shi (NCSU) CQTE July 30, 2017 12 / 20

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Hypothesis testing: H0 : X C doens’t have CQTE given X B, versus H1 : X C has CQTE given X B. Such hypothesis assesses the incremental value of X C in optimal treatment decision making conditional on X B.

Chengchun Shi (NCSU) CQTE July 30, 2017 12 / 20

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Hypothesis testing: H0 : X C doens’t have CQTE given X B, versus H1 : X C has CQTE given X B. Such hypothesis assesses the incremental value of X C in optimal treatment decision making conditional on X B. If H0 holds, estimate dB

  • pt(x).

Chengchun Shi (NCSU) CQTE July 30, 2017 12 / 20

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SLIDE 56

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hypothesis testing: H0 : X C doens’t have CQTE given X B, versus H1 : X C has CQTE given X B. Such hypothesis assesses the incremental value of X C in optimal treatment decision making conditional on X B. If H0 holds, estimate dB

  • pt(x).

Otherwise, estimate dW

  • pt(x).

Chengchun Shi (NCSU) CQTE July 30, 2017 12 / 20

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SLIDE 57

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Let f W be the probability density function of X W .

Chengchun Shi (NCSU) CQTE July 30, 2017 13 / 20

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SLIDE 58

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Let f W be the probability density function of X W . Estimate τ W

0 (xW )f W (xW ) by

τ W

n (xW ) = 1

n

n

∑

i=1

(Ai πi − 1 − Ai 1 − πi ) YiK W

hW (xW − X W i

), for some multivariate kernel function kW

hW with the bandwidth hW .

Chengchun Shi (NCSU) CQTE July 30, 2017 13 / 20

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SLIDE 59

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Testing procedure

SW ,B

n

= ∫

xW ∈ΩW τ W n (xW ){dW n (xW ) − dB n (xB)}I(xW /

∈ ˆ E)dν(xW ), where ˆ E = { xW :

  • τ W

n (xW )

ˆ f W (xW )

  • ≤ ηn,
  • τ B

n (xB)

ˆ f B(xB)

  • ≤ ηn

} , for some sequence ηn → 0. Here, ˆ f W and ˆ f B are the kernel density estimators of f W and f B, respectively.

Chengchun Shi (NCSU) CQTE July 30, 2017 14 / 20

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SLIDE 60

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Testing procedure (Cont’d)

Let ˆ F = {xW : |τ W

n (xW )/ˆ

f W (xW )| ≤ ηn, |τ B

n (xB)/ˆ

f B(xB)| > ηn}. The test statistic is defined by T W ,B

n

= { {√nSW ,B

n

− ˆ an( ˆ F)}/ˆ σn( ˆ F), if ν( ˆ F) ̸= 0, {√nSW ,B

n

− ˆ an(ΩW )}/ˆ σn(ΩW ),

  • therwise.

We reject the null when T W ,B

n

> zα.

Chengchun Shi (NCSU) CQTE July 30, 2017 15 / 20

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SLIDE 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem (Consistency) Under certain conditions, when H0 is true, we have lim Pr(T W ,B

n

> zα) ≤ α, for 0 < α ≤ 0.5.

Chengchun Shi (NCSU) CQTE July 30, 2017 16 / 20

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SLIDE 62

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem (Consistency) Under certain conditions, when H0 is true, we have lim Pr(T W ,B

n

> zα) ≤ α, for 0 < α ≤ 0.5. If Pr{τ W

0 (X W ) = 0} = 0, we have

lim Pr(T W ,B

n

> zα) = 0.

Chengchun Shi (NCSU) CQTE July 30, 2017 16 / 20

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SLIDE 63

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem (Consistency) Under certain conditions, when H0 is true, we have lim Pr(T W ,B

n

> zα) ≤ α, for 0 < α ≤ 0.5. If Pr{τ W

0 (X W ) = 0} = 0, we have

lim Pr(T W ,B

n

> zα) = 0. When H1 is true, we have lim Pr(T W ,B

n

> zα) → 1.

Chengchun Shi (NCSU) CQTE July 30, 2017 16 / 20

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SLIDE 64

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem (Consistency) Under certain conditions, when H0 is true, we have lim Pr(T W ,B

n

> zα) ≤ α, for 0 < α ≤ 0.5. If Pr{τ W

0 (X W ) = 0} = 0, we have

lim Pr(T W ,B

n

> zα) = 0. When H1 is true, we have lim Pr(T W ,B

n

> zα) → 1. Theorem (Informal statement) Under certain conditions, T W ,B

n

has non-negligible powers against some nonstandard n−1/2 local alternatives.

Chengchun Shi (NCSU) CQTE July 30, 2017 16 / 20

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SLIDE 65

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Simulation models: Y = 1 − X1 − X2 2 + Aφ1(X1)φ2(X2) + e,

Table: Simulation results.

VD = 0 VD = 4% VD = 8% VD = 12% α level α level α level α level n 0.05 0.1 0.05 0.1 0.05 0.1 0.05 0.1 Scenario 1 300 4.3% 6.0% 24.0% 34.0% 58.7% 68.1% 82.2% 87.5% 600 1.5% 3.3% 36.7% 45.5% 75.8% 83.3% 95.7% 97.3% Scenario 2 300 7.0% 11.1% 23.8% 32.7% 60.5% 69.3% 88.2% 92.3% 600 5.5% 10.0% 31.0% 41.8% 83.0% 90.5% 98.3% 99.5% Scenario 3 300 3.8% 6.5% 37.3% 48.5% 76.3% 79.7% 92.7% 94.7% 600 2.7% 6.7% 52.5% 61.8% 99.2% 100% 99.8% 99.8% Scenario 4 300 6.2% 9.8% 39.8% 47.7% 79.2% 87.3% 94.8% 96.7% 600 5.2% 8.8% 59.3% 68.2% 96.8% 98.3% 99.5% 99.5% Scenario 5 300 5.2% 9.7% 29.3% 40.5% 68.0% 76.3% 94.0% 96.8% 600 5.3% 9.5% 36.7% 45.5% 75.8% 83.3% 95.7% 97.3%

Chengchun Shi (NCSU) CQTE July 30, 2017 17 / 20

slide-66
SLIDE 66

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Set B = ∅. In Step 1, for each variable i, define the set Wi = {i} and

calculate the p-value pi for each test statistic T Wi,B. Stop if mini pi > α. Include the variable that gives the smallest p-value in the set B, i.e, B ← {arg min

i

pi}.

Chengchun Shi (NCSU) CQTE July 30, 2017 18 / 20

slide-67
SLIDE 67

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Set B = ∅. In Step 1, for each variable i, define the set Wi = {i} and

calculate the p-value pi for each test statistic T Wi,B. Stop if mini pi > α. Include the variable that gives the smallest p-value in the set B, i.e, B ← {arg min

i

pi}.

2 In Step 2, for each variable i /

∈ B, define Wi = B ∪ {i} and calculate the p-value pi for each test statistic T Wi,B. Stop if mini pi > α. Include the variable that gives the smallest p-value, B ← B ∪ {arg min

i

pi}.

Chengchun Shi (NCSU) CQTE July 30, 2017 18 / 20

slide-68
SLIDE 68

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Set B = ∅. In Step 1, for each variable i, define the set Wi = {i} and

calculate the p-value pi for each test statistic T Wi,B. Stop if mini pi > α. Include the variable that gives the smallest p-value in the set B, i.e, B ← {arg min

i

pi}.

2 In Step 2, for each variable i /

∈ B, define Wi = B ∪ {i} and calculate the p-value pi for each test statistic T Wi,B. Stop if mini pi > α. Include the variable that gives the smallest p-value, B ← B ∪ {arg min

i

pi}.

3 Continue the second step until it stops. Output B. Chengchun Shi (NCSU) CQTE July 30, 2017 18 / 20

slide-69
SLIDE 69

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduce the notion of CQTE and present several equivalent representations of No CQTE.

Chengchun Shi (NCSU) CQTE July 30, 2017 19 / 20

slide-70
SLIDE 70

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduce the notion of CQTE and present several equivalent representations of No CQTE. Propose a testing procedure for testing No CQTE.

Chengchun Shi (NCSU) CQTE July 30, 2017 19 / 20

slide-71
SLIDE 71

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduce the notion of CQTE and present several equivalent representations of No CQTE. Propose a testing procedure for testing No CQTE. Develop a procedure for selecting prescriptive variables in sequential

  • rder.

Chengchun Shi (NCSU) CQTE July 30, 2017 19 / 20

slide-72
SLIDE 72

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduce the notion of CQTE and present several equivalent representations of No CQTE. Propose a testing procedure for testing No CQTE. Develop a procedure for selecting prescriptive variables in sequential

  • rder.

Extend the testing procedure to a high-dimensional setting.

Chengchun Shi (NCSU) CQTE July 30, 2017 19 / 20

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SLIDE 73

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Thank you!

Chengchun Shi (NCSU) CQTE July 30, 2017 20 / 20