Assessing Proximal and Lagged Moderated Effects in Mobile Health - PowerPoint PPT Presentation

Assessing Proximal and Lagged Moderated Effects in Mobile Health University of Washington Biostatistics Department Seattle, WA November 2, 2016 Audrey Boruvka, 1 Daniel Almirall, 1 Katie Witkiewitz, 2 and Susan A. Murphy 1 1 University of Michigan and 2 University of New Mexico

Outline 1. Three examples: BASICS Mobile, HeartSteps and Sense2Stop 2. What does data from a micro-randomized trial look like? 3. Proximal and lagged moderated effects 4. Estimating the proximal and lagged moderated effects 5. Simulation experiments 6. A data example using BASICS Mobile 1 / 34

BASICS-Mobile Example (College Drinking) PI: Katie Witkiewitz Smartphone-based intervention to curb heavy drinking and smoking in college students Data Collected EMA up to 3x/day (morning, aftern., eve) Intervention Frequency Up to 2x/day (afternoon, evening) Intervention Content Mindfulness-based message vs general health information (binary treatment) Intervention Availability Based on answering an EMA Typical Question Is the effect of providing a mindfulness-based intervention (vs GHI) on subsequent smoking rate moderated by increase in need to self-regulate? 2 / 34

HeartSteps Example (Physical Activity) PI: Pedja Klasjna Wearable activity-tracker + smartphone-based intervention to encourage physical activity Data Collected "Continuously" + EMA each evening Intervention Frequency Up to 5x/day (before work, lunch, 2pm, after work, eve) Intervention Content Delivers vs does not deliver (binary treatment) contextually relevant activity suggestion via the smartphone Intervention Availability Not in vehicle, not exercising, not "snooze" the app, phone on Typical Question Does time-of-day or the busyness influence the effect of suggesting an activity on step count? 3 / 34

Sense2Stop Example (Smoking Cessation) PI: Bonnie Spring Wearable chest-strap + wrist-band + smartphone-based intervention to sense stress and reduce smoking Data Collected "Continuously" + EMA Intervention Frequency 3x/day on average; with 50% chance of happening when stressed and 50% chance of happening when not stressed Intervention Content Deliver or not deliver prompt (binary treatment) via smartphone to use one of 3 stress-management apps Intervention Availability Not in vehicle, ≥ 60min since intervention, ≥ 10min since EMA, cannot have uncertain stress classification, phone on Typical Question Will delivering the message be more effective than not delivering the message in times of stress? In times of no stress? Or equally effective in either? 4 / 34

Data from a Micro-randomized Trial t treatment occasion X t individual and contextual characteristics at t A t binary treatment at t Y t + 1 continuous response following t and before t + 1 H t history through t : ( ¯ X t , ¯ Y t , ¯ A t − 1 ) Data in temporal order looks like this X 1 , A 1 , Y 2 , . . . , X t , A t , Y t + 1 , . . . , X T , A T , Y T + 1 ←←←←←← H t , A t , Y t + 1 , . . . ρ t ( 1 | H t ) is known randomization probability P ( A t = 1 | H t ) that generates A t 5 / 34

Example Data Structure BASICS Mobile X t − 1 Y t + 1 X t A t − 1 A t . . . . . . t − 1 t Morning Afternoon Evening Morning 6 / 34

Proximal moderated effect A t on Y t + 1 Y t + 1 (¯ a t ) response, had the treatments ¯ a t been provided S 1 t (¯ a t − 1 ) vector of candidate moderators from the history through t , H t , had the treatments ¯ a t − 1 been provided The proximal treatment effect is � � Y t + 1 ( ¯ A t − 1 , 1 ) − Y t + 1 ( ¯ A t − 1 , 0 ) | S 1 t ( ¯ E A t − 1 ) . A contrast of the average Y t + 1 under treatment 1 vs the average Y under treatment 0, conditional on S 1 t . Think of S 1 t (¯ a t − 1 ) as a "State" of particular interest 7 / 34

Proximal moderated effect A t on Y t + 1 A proximal treatment effect is � � Y t + 1 ( ¯ A t − 1 , 1 ) − Y t + 1 ( ¯ A t − 1 , 0 ) | S 1 t ( ¯ E A t − 1 ) . S 1 t ( ¯ a t − 1 ) is low-dimensional, pre-selected by scientist. It can be the "empty set". It can include time trends. Proximal effect is averaged over any variables in H t not represented in S 1 t . The definition depends on distribution of (past) treatments in the data. 8 / 34

What does a Proximal Effect look like? 9 / 34

Lagged moderated effect A t on Y t + 2 A lagged treatment effect is � � Y t + 2 ( ¯ A t − 1 , 1 , A a t = 1 t + 1 ) − Y t + 2 ( ¯ A t − 1 , 0 , A a t = 0 t + 1 ) | S 2 t ( ¯ E A t − 1 ) . t + 1 = A t + 1 ( ¯ A a t = a A t − 1 , a ) S 2 t (¯ a t − 1 ) is again a low-dimensional, pre-selected by scientist Delayed effect is averaged over any variables in H t not represented in S kt Delayed effect is averaged over future treatment A a t t + 1 . Here, lag = 2. 10 / 34

General case: Lag k treatment effects � � Y t + k ( ¯ A t − 1 , 1 , A a t = 1 t + 1 , . . . , A a t = 1 E t + k − 1 ) � � Y t + k ( ¯ t + k − 1 ) | S kt ( ¯ A t − 1 , 0 , A a t = 0 t + 1 , . . . , A a t = 0 − E A t − 1 ) . where t + 1 denotes A t + 1 ( ¯ A a t = a A t − 1 , a ) , t + 2 denotes A t + 2 ( ¯ A t − 1 , a , A t + 1 ( ¯ A a t = a A t − 1 , a )) , and so on. k pre-selected by scientist; interest in multiple k ’s is OK. S kt (¯ a t − 1 ) is again a low-dimensional, pre-selected by scientist for examining the lag k effect (could differ by k ) 11 / 34

Identification (Effects in terms of observed data) Under sequential randomization, consistency and positivity assumptions The proximal treatment effect is � � Y t + 1 ( ¯ A t − 1 , 1 ) − Y t + 1 ( ¯ A t − 1 , 0 ) | S 1 t ( ¯ E A t − 1 ) = E [ E [ Y t + 1 | A t = 1 , H t ] − E [ Y t + 1 | A t = 0 , H t ] | S 1 t ] � I ( A t = 1 ) Y t + 1 � � − I ( A t = 0 ) Y t + 1 � = E � S 1 t , � ρ t ( 1 | H t ) 1 − ρ t ( 1 | H t ) where ρ t ( 1 | H t ) = Pr ( A t = 1 | H t ) is the probabilities used to randomize sequentially. Lagged treatment effects can be identified similarly. 12 / 34

Extension of the Structural Nested Mean Model Treatment “blip" of a t versus stochastic A t (¯ a t − 1 ) on Y t + k is µ t , t + k ( H t ( ¯ A t − 1 ) , ¯ A t − 1 , a t ) � � Y t + k ( ¯ t + k − 1 ) | H t ( ¯ A t − 1 , a t , A a t = a t + 1 , . . . , A a t = a = E A t − 1 ) � � Y t + k ( ¯ A t − 1 , A t , A t + 1 , A t + k − 1 ) | H t ( ¯ − E A t − 1 ) � � Y t + k ( ¯ A t ) | A t = a t , H t ( ¯ = E A t − 1 ) by Seq Ign. & Cons. � � Y t + k ( ¯ A t ) | H t ( ¯ − E A t − 1 ) 13 / 34

Extension of the Structural Nested Mean Model Treatment “blip" of a t versus stochastic A t (¯ a t − 1 ) on Y t + k is µ t , t + k ( H t ( ¯ A t − 1 ) , ¯ A t − 1 , a t ) � � Y t + k ( ¯ t + k − 1 ) | H t ( ¯ A t − 1 , a t , A a t = a t + 1 , . . . , A a t = a = E A t − 1 ) � � Y t + k ( ¯ A t − 1 , A t , A t + 1 , A t + k − 1 ) | H t ( ¯ − E A t − 1 ) � � Y t + k ( ¯ A t ) | A t = a t , H t ( ¯ = E A t − 1 ) by Seq Ign. & Cons. � � Y t + k ( ¯ A t ) | H t ( ¯ − E A t − 1 ) Our lagged effects are expected contrasts of the “blips": � � µ t , t + k ( H t ( ¯ A t − 1 ) , ¯ A t − 1 , 1 ) | S kt ( ¯ E A t − 1 ) � � µ t , t + k ( H t ( ¯ A t − 1 ) , ¯ A t − 1 , 0 ) | S kt ( ¯ − E A t − 1 ) � � � Y t + k ( ¯ � S kt ( ¯ A t − 1 , 1 , A a t = 1 t + 1 , . . . , A a t = 1 = E t + k − 1 ) A t − 1 ) � � � Y t + k ( ¯ � S kt ( ¯ A t − 1 , 0 , A a t = 0 t + 1 , . . . , A a t = 0 − E t + k − 1 ) A t − 1 ) , 13 / 34

The Notion of Availability Not all individuals are available for treatment at all time points (e.g., Wang et al. 2012; Robins 2004). For simplicity, we define this in terms of the observed data. E [ E [ Y t + k | A t = 1 , I t = 1 , H t ] | I t = 1 , S kt ] − E [ E [ Y t + k | A t = 0 , I t = 1 , H t ] | I t = 1 , S kt ] � 1 ( A t = 1 ) Y t + 1 � − 1 ( A t = 0 ) Y t + 1 � � = E � I t = 1 , S kt , ρ t ( 1 | H t ) 1 − ρ t ( 1 | H t ) Note that I t = 1 is not a static subpopulation; and we expect prior treatment to effect it. 14 / 34

Modeling assumptions We consider linear models for each lag k effect E [ E [ Y t + k | A t = 1 , I t = 1 , H t ] | I t = 1 , S kt ] − E [ E [ Y t + k | A t = 0 , I t = 1 , H t ] | I t = 1 , S kt ] = f k ( t , S kt ) ⊺ β k . Recall k = 1 is the proximal effect. These models do not constrain each other across k (Robins, Rotnitzky and Scharfstein 2000, Theorem 8.6). We assume these treatment effect models are correct. 15 / 34

Modeling assumptions: Example 1 Suppose S kt is the null set. Here, the analyst is interested in a marginal effect that could vary over time; for example, E [ E [ Y t + 1 | A t = 1 , I t = 1 , H t ] | I t = 1 ] − E [ E [ Y t + 1 | A t = 0 , I t = 1 , H t ] | I t = 1 ] = β k 1 + β k 2 t + β k 3 t 2 . 16 / 34

What does a Proximal Effect look like? 17 / 34

Modeling assumptions: Example 2 Suppose S kt = Stress t is binary. Here, an example model is E [ E [ Y t + 1 | A t = 1 , I t = 1 , H t ] | I t = 1 , Stress t ] − E [ E [ Y t + 1 | A t = 0 , I t = 1 , H t ] | I t = 1 , Stress t ] = β k 1 + β k 2 Stress t + β k 3 t + β k 4 t Stress t . Here, the moderators of interest to the scientist are time and Stress. 18 / 34