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

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

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

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

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 d opt to reach the best clinical outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 2 / 20

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 d opt to reach the best clinical outcome Maximize E Y ∗ ( d ) = E[ d ( X ) Y ∗ (1) + { 1 − d ( X ) } Y ∗ (0)] d : X → { 0 , 1 } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 2 / 20

Q, Contrast and Value function Q ( x , a ) = E[ Y | X = x , A = a ], . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 3 / 20

Q, Contrast and Value function Q ( x , a ) = E[ Y | X = x , A = a ], τ 0 ( x ) = Q ( x , 1) − Q ( x , 0), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 3 / 20

Q, Contrast and Value function Q ( x , a ) = E[ Y | X = x , A = a ], τ 0 ( x ) = Q ( x , 1) − Q ( x , 0), V ( d ) = E Y ∗ ( d ) = E[ d ( X ) Y ∗ (1) + { 1 − d ( X ) } Y ∗ (0)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 3 / 20

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

Q, Contrast and Value function Q ( x , a ) = E[ Y | X = x , A = a ], τ 0 ( x ) = Q ( x , 1) − Q ( x , 0), V ( d ) = E Y ∗ ( d ) = E[ d ( X ) Y ∗ (1) + { 1 − d ( X ) } Y ∗ (0)]. Optimal treatment regime SUTVA, no unmeasured confounders optimal treatment regime d opt ( x ) = I ( τ 0 ( x ) ≥ 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 3 / 20

There are two types of clinically “important” variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

There are two types of clinically “important” variables. Predictive variables are those involved in τ 0 ( x ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

There are two types of clinically “important” variables. Predictive variables are those involved in τ 0 ( x ). Prescriptive variables are those involved in d opt ( x ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

There are two types of clinically “important” variables. Predictive variables are those involved in τ 0 ( x ). Prescriptive variables are those involved in d opt ( x ). Predictive variables have interactions with the treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 4 / 20

There are two types of clinically “important” variables. Predictive variables are those involved in τ 0 ( x ). Prescriptive variables are those involved in d opt ( 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

There are two types of clinically “important” variables. Predictive variables are those involved in τ 0 ( x ). Prescriptive variables are those involved in d opt ( 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

There are two types of clinically “important” variables. Predictive variables are those involved in τ 0 ( x ). Prescriptive variables are those involved in d opt ( 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

A tiny example: τ 0 ( x ) = exp( − x 1 ) x 2 , for { x 1 , x 2 , . . . , x p } ∈ [ − 1 , 1] p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

A tiny example: τ 0 ( x ) = exp( − x 1 ) x 2 , for { x 1 , x 2 , . . . , x p } ∈ [ − 1 , 1] p . d opt ( x ) = I { exp( − x 1 ) x 2 ≥ 0 } = I ( x 2 ≥ 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

A tiny example: τ 0 ( x ) = exp( − x 1 ) x 2 , for { x 1 , x 2 , . . . , x p } ∈ [ − 1 , 1] p . d opt ( x ) = I { exp( − x 1 ) x 2 ≥ 0 } = I ( x 2 ≥ 0). x 1 and x 2 are the predictive variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

A tiny example: τ 0 ( x ) = exp( − x 1 ) x 2 , for { x 1 , x 2 , . . . , x p } ∈ [ − 1 , 1] p . d opt ( x ) = I { exp( − x 1 ) x 2 ≥ 0 } = I ( x 2 ≥ 0). x 1 and x 2 are the predictive variables. x 2 is the prescriptive variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

A tiny example: τ 0 ( x ) = exp( − x 1 ) x 2 , for { x 1 , x 2 , . . . , x p } ∈ [ − 1 , 1] p . d opt ( x ) = I { exp( − x 1 ) x 2 ≥ 0 } = I ( x 2 ≥ 0). x 1 and x 2 are the predictive variables. x 2 is the prescriptive variable. x 1 and x 2 have interactions with the treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

A tiny example: τ 0 ( x ) = exp( − x 1 ) x 2 , for { x 1 , x 2 , . . . , x p } ∈ [ − 1 , 1] p . d opt ( x ) = I { exp( − x 1 ) x 2 ≥ 0 } = I ( x 2 ≥ 0). x 1 and x 2 are the predictive variables. x 2 is the prescriptive variable. x 1 and x 2 have interactions with the treatment. x 2 has qualitative interaction with the treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengchun Shi (NCSU) CQTE July 30, 2017 5 / 20

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