Estimating time-varying causal effect Introduction moderation in - PowerPoint PPT Presentation

Estimate causal excursion effects T.Qian Estimating time-varying causal effect Introduction moderation in mobile health with binary Conditional on H t outcomes Estimator Simulation BariFit Tianchen Qian Extension Joint work with Daniel Almirall, Predrag Klasnja, Hyesun Yoo, Summary Susan Murphy References Department of Statistics Harvard University February 18, 2019 1 / 38

Estimate causal BariFit MRT excursion effects T.Qian Introduction • A micro-randomized trial (MRT) to promote weight Conditional maintenance among people who received bariatric surgery. on H t • Data collected from: Estimator Simulation • Fitbit tracker (step count) BariFit • user self-report (weight, calories intake) Extension • mHealth intervention components: Summary • daily step goals References • actionable activity suggestions reminders to track food intake • • ... • This talk: assess the effect of 2 / 38

Estimate causal Data in an MRT excursion effects T.Qian Introduction Conditional on H t • On each individual: O 1 , A 1 , Y 2 , . . . , O T , A T , Y T +1 . Estimator • t : decision point. Simulation BariFit • A t : treatment indicator at decision point t . Extension • O t : observation accrued between decision point t − 1 and Summary decision point t . References • History H t = ( O 1 , A 1 , Y 2 , . . . , O t ): information accrued prior to decision point t . 3 / 38

Estimate causal Decision Points t excursion effects T.Qian Introduction Conditional on H t Estimator • Times at which a treatment might be provided Simulation BariFit • Times that the treatment is likely to be beneficial Extension • BariFit: food track reminder may be sent every morning. Summary t = 1 , 2 , . . . , 112 (112 days) References 4 / 38

Estimate causal Treatment indicator A t excursion effects T.Qian Introduction Conditional on H t • Whether a treatment is provided at decision point t Estimator Simulation • (What type of treatment) BariFit • Here we assume binary ( A t ∈ { 0 , 1 } ) Extension • Randomization probability p t ( H t ) := P ( A t = 1 | H t ) Summary References • BariFit: whether a text message of food track reminder is sent. p t ( H t ) = 0 . 5. 5 / 38

Estimate causal Proximal outcome Y t +1 excursion effects T.Qian Introduction Conditional on H t • Outcome measured after decision point t (assumed to be Estimator Simulation binary here) BariFit • Something that the treatment is directly targeting Extension • BariFit: whether the individual completes food log on that Summary day References • Note the subscript! 6 / 38

Estimate causal Observation O t excursion effects T.Qian Introduction Conditional on H t Estimator • Observation accrued between decision point t − 1 and Simulation decision point t . BariFit • O 1 includes baseline variables. Extension • BariFit: Fitbit tracker (step count) Summary user self-report (e.g., weekly weight) References baseline variables (e.g., age, gender) 7 / 38

Estimate causal Availability I t excursion effects T.Qian Introduction • Treatment A t can only be delivered at a decision point if Conditional an individual is available. on H t Estimator • Available: I t = 1; unavailable: I t = 0. I t ∈ O t . Simulation • Safety and ethical consideration: e.g., an individual is BariFit unavailable for a physical activity suggestion message if Extension she is driving. Summary • Treatment effect is defined conditional on availability. References (later) • BariFit: for food track reminder, individuals are always available. • Availability is different from adherence! 8 / 38

Estimate causal Conceptual models excursion effects T.Qian • Data: O 1 , A 1 , Y 2 , . . . , O T , A T , Y T +1 Introduction • H t = ( O 1 , A 1 , Y 2 , . . . , O t ) Conditional on H t • Usually data analysts fit a series of models Estimator Simulation Y t +1 ‘ ∼ ’ g ( H t ) T α + β 0 A t , BariFit Extension Y t +1 ‘ ∼ ’ g ( H t ) T α + β 0 A t + β 1 A t S t , Summary . . . References • g ( H t ): summaries from H t ; “control variables” • S t : potential moderators (e.g., day in the study) • β 0 , β 1 : quantities of interest • ‘ ∼ ’: e.g., logit or log for binary Y 9 / 38

Estimate causal Goal excursion effects T.Qian Introduction Conditional on H t • Develop statistical methods to model and estimate the Estimator treatment effect Simulation BariFit • Be consistent with the scientific understanding of the β Extension coefficients Summary References • Use control variables g ( H t ) for noise reduction in a robust way 10 / 38

Estimate causal Potential outcomes excursion effects T.Qian Introduction Conditional on H t • To mathematize the problem, we use potential outcomes Estimator Simulation notation (e.g., Rubin (1974)) BariFit • Define ¯ a t = ( a 1 , . . . , a t ) where a 1 , . . . , a t ∈ { 0 , 1 } Extension • O t (¯ a t − 1 ): O t that would have been observed if individual Summary received treatment history ¯ a t − 1 . References • Similarly, Y t +1 (¯ a t ), H t (¯ a t − 1 ) 11 / 38

Estimate causal Causal excursion effect excursion effects T.Qian Introduction Y t +1 ( ¯ A t − 1 , 1) Conditional on H t Estimator Simulation BariFit Extension Summary References 12 / 38

Estimate causal Causal excursion effect excursion effects T.Qian Introduction Y t +1 ( ¯ A t − 1 , 1) Conditional on H t Y t +1 ( ¯ A t − 1 , 0) Estimator Simulation BariFit Extension Summary References 12 / 38

Estimate causal Causal excursion effect excursion effects T.Qian Introduction E { Y t +1 ( ¯ A t − 1 , 1) } Conditional on H t E { Y t +1 ( ¯ A t − 1 , 0) } Estimator Simulation BariFit Extension • Contrasting two excursions: following ¯ A t − 1 , then receive Summary treatment ( A t = 1) vs. no treatment ( A t = 0) at time t . References 12 / 38

Estimate causal Causal excursion effect excursion effects T.Qian Introduction E { Y t +1 ( ¯ A t − 1 , 1) | S t ( ¯ A t − 1 ) } Conditional on H t E { Y t +1 ( ¯ A t − 1 , 0) | S t ( ¯ A t − 1 ) } Estimator Simulation BariFit Extension • Contrasting two excursions: following ¯ A t − 1 , then receive Summary treatment ( A t = 1) vs. no treatment ( A t = 0) at time t . References • S t ( ¯ A t − 1 ) ⊂ H t ( ¯ A t − 1 ): a vector of summary variables chosen from H t ( ¯ A t − 1 ). • Effect is marginal over all variables in H t ( ¯ A t − 1 ) that are not in S t ( ¯ A t − 1 ) 12 / 38

Estimate causal Causal excursion effect excursion effects T.Qian Introduction E { Y t +1 ( ¯ A t − 1 , 1) | S t ( ¯ A t − 1 ) , I t ( ¯ A t − 1 ) = 1 } Conditional on H t E { Y t +1 ( ¯ A t − 1 , 0) | S t ( ¯ A t − 1 ) , I t ( ¯ A t − 1 ) = 1 } Estimator Simulation BariFit Extension • Contrasting two excursions: following ¯ A t − 1 , then receive Summary treatment ( A t = 1) vs. no treatment ( A t = 0) at time t . References • S t ( ¯ A t − 1 ) ⊂ H t ( ¯ A t − 1 ): a vector of summary variables chosen from H t ( ¯ A t − 1 ). • Effect is marginal over all variables in H t ( ¯ A t − 1 ) that are not in S t ( ¯ A t − 1 ) • Conditional on being available: I t ( ¯ A t − 1 ) = 1. 12 / 38

Estimate causal Causal excursion effect excursion effects T.Qian Introduction log E { Y t +1 ( ¯ A t − 1 , 1) | S t ( ¯ A t − 1 ) , I t ( ¯ A t − 1 ) = 1 } Conditional on H t E { Y t +1 ( ¯ A t − 1 , 0) | S t ( ¯ A t − 1 ) , I t ( ¯ A t − 1 ) = 1 } Estimator Simulation BariFit Extension • Contrasting two excursions: following ¯ A t − 1 , then receive Summary treatment ( A t = 1) vs. no treatment ( A t = 0) at time t . References • S t ( ¯ A t − 1 ) ⊂ H t ( ¯ A t − 1 ): a vector of summary variables chosen from H t ( ¯ A t − 1 ). • Effect is marginal over all variables in H t ( ¯ A t − 1 ) that are not in S t ( ¯ A t − 1 ) • Conditional on being available: I t ( ¯ A t − 1 ) = 1. 12 / 38

Estimate causal Examples excursion effects T.Qian Introduction • S t ( ¯ Conditional A t − 1 ) = 1: average treatment effect on H t Estimator log E { Y t +1 ( ¯ A t − 1 , 1) | I t ( ¯ A t − 1 ) = 1 } Simulation E { Y t +1 ( ¯ A t − 1 , 0) | I t ( ¯ A t − 1 ) = 1 } BariFit Extension Summary • S t ( ¯ A t − 1 ) = (1 , day in study) References log E { Y t +1 ( ¯ A t − 1 , 1) | day t , I t ( ¯ A t − 1 ) = 1 } E { Y t +1 ( ¯ A t − 1 , 0) | day t , I t ( ¯ A t − 1 ) = 1 } 13 / 38

Estimate causal Identifiability assumptions excursion effects T.Qian Assumption (consistency) Introduction Conditional The observed data equals the potential outcome under on H t observed treatment assignment: O t = O t ( ¯ A t − 1 ) for every t . Estimator Simulation BariFit Extension Summary References 14 / 38

Estimate causal Identifiability assumptions excursion effects T.Qian Assumption (consistency) Introduction Conditional The observed data equals the potential outcome under on H t observed treatment assignment: O t = O t ( ¯ A t − 1 ) for every t . Estimator Simulation Assumption (positivity) BariFit Extension For every t , for every possible history H t with I t = 1, Summary P ( A t = a | H t , I t = 1) > 0 for a ∈ { 0 , 1 } . References 14 / 38