Micro-Randomized Trials & mhealth. S.A. Murphy NRC 8/2014

Transcription

1 Micro-Randomized Trials & mhealth S.A. Murphy NRC 8/2014

2 mhealth Goal: Design a Continually Learning Mobile Health Intervention: HeartSteps + Designing a Micro-Randomized Trial 2

3 Data from wearable devices that sense and provide treatments State at j th decision time (high dimensional) Action at j th decision time (treatment) Y j : Response (time-varying response) 3

4 Examples 1) Decision Times (Times at which a treatment can be provided.) 1) Regular intervals in time (e.g. every 10 minutes) 2) At user demand HeartSteps includes two sets of decision times 1) Momentary: Approximately every hours 2) Daily: Each evening at user specified time. 4

5 Examples 2) State S j 1) Passively collected (location, weather, busyness of calendar, social context, activity on device) 2) Actively collected (answers to questions) HeartSteps includes activity recognition (walking, driving, standing/sitting), weather, location, calendar, adherence, step count, self-report (usefulness, burden, self-efficacy, etc.), whether momentary intervention is on. 5

6 Examples 3) Actions A j 1) Treatments that can be provided at decision time 2) Whether to provide a treatment HeartSteps includes two types of treatments 1) Momentary Lock Screen Recommendation 2) Daily Activity Planning 6

7 Examples 3) Actions A j 1) Treatments that can be provided at decision time 2) Whether to provide a treatment HeartSteps includes two types of treatments 1) Momentary Lock Screen Recommendation, 2) Daily Activity Planning 7

8 Daily Activity Planning 8

9 Momentary Lock Screen Recommendation 9

10 Examples 4) Proximal Response Y j HeartSteps: Activity (step count) between decision times or daily activity. 10

11 Scientific Goals 1) Assess if there are proximal causal effects of the actions on the response. 2) Assess if there are delayed causal effects; assess if the proximal/delayed causal effects vary by particular state variables. 3) Construct a treatment policy, aka Just-in-Time Adaptive Intervention that inputs state and outputs actions. 4) Construct an online training algorithm that will result in a Continually Updating Just-in-Time Adaptive Intervention 11

12 Today s Focus 1) Assess if there are proximal causal effects of the actions on the reward. 2) Assess if there are delayed causal effects; assess if the proximal/delayed causal effects vary by particular state variables. 3) Construct a treatment policy, aka Just-in-Time Adaptive Intervention that inputs state and outputs actions. 4) Construct an online training algorithm that will result in a Continually Updating Just-in-Time Adaptive Intervention 12

13 Proposed Experimental Design: Micro-Randomized Trial Randomize between actions at decision times Each person may be randomized 100 s of times. These are sequential, full factorial, designs. 13

14 Why Micro-Randomization? Factorial designs are the gold standard when collecting data to build a treatment involving many components Actions are often intended to have a proximal effect. randomization is the gold standard in providing data to assess a causal effect Sequential randomization will enhance quality of many interesting subsequent data analyses. 14

15 Justifying the Sample Size for a Micro-Randomized Trial Focus on whether to provide a Momentary Lock Screen Recommendation, e.g. A j {0,1} Randomization in HeartSteps P[A j =1]=.4 Size to Detect a Proximal Causal Effect 15

16 Proximal Causal Effect Recall that Y j is the proximal response recorded after action A j A j is only delivered if the momentary intervention is on at time j. Set R j =1 if the momentary intervention is on at time j, otherwise R j =0 16

17 Proximal Causal Effect Define Ā j ={A 1,A 2,...,A j },ā j ={a 1,a 2,...,a j } Define Y j (ā j ) to be the observed response, Y j if Ā j =ā j, e.g., Y j =Y j (Ā j ) Define R j (ā j 1 ) to be the observed intervention on indicator if Ā j 1 =ā j 1 17

18 Proximal Causal Effect Define the Proximal Causal Effect at time j as E [ Y j (Āj 1,1) Y j (Āj 1,0) R j (Āj 1)=1 ] What does this estimand mean? 18

19 Proximal Causal Effect The randomization implies that E [ Y j (Āj 1,1) Y j (Āj 1,0) R j (Āj 1)=1 ] = E [ Y j R j =1,A j =1 ] E [ Y j R j =1,A j =0 ] Put β j =E [ Y j R j =1,A j =1 ] E [ Y j R j =1,A j =0 ] 19

20 Change Notation t th time on j th day: β tj =E [ Y tj R tj =1,A tj =1 ] E [ Y tj R tj =1,A tj =0 ] 20

21 Test for Sample Size Calculation We construct a test statistic for H 0 :β tj =0, t,j A simple approach is parameterize β tj =β 0 +β 1 (j 1)+β 2 (j 1) 2 and test H 0 :β i =0,i=0,1,2 21

22 Alternative for Sample Size Calculation One calculates a sample size to detect a given alternative with a given power. Alternative: H 1 :β i =d i σ,i=0,1,2 σ 2 where is the residual variance. 22

23 Test Statistic for Sample Size Calculation Residual variance is σ 2 =VAR ( Y tj E[Y tj R tj =1] ) 23

24 Specify Alternative for Sample Size Calculation Scientist specifies standardized d i s initial proximal treatment effect: d 0, average proximal effect over trial duration: 1 5J J j=1 5 t=1( d0 +d 1 (j 1)+d 2 (j 1) 2), and day of maximal proximal effect: We solve for d i s. d 1 2d 2 24

25 Test Statistic for Sample Size Calculation The model E[Y tj R tj =1]= α tj +γ tj R tj +β tj R tj (A tj p tj ) where p tj is the randomization probability p tj is.4 25

26 Test Statistic for Sample Size Calculation Use GEE to fit model E[Y tj R tj =1]= α tj +γ tj R tj +β tj R tj (A tj p tj ) where β tj =β 0 +β 1 (j 1)+β 2 (j 1) 2 You select parameterization of α tj, γ tj 26

27 Test Statistic for Sample Size Put Calculation Y =(Y 11,...,Y 51,Y 12,...,Y 5J ) T p is the total number of parameters (p >3); X is the associated design matrix (5J by p) N is sample size Last 3 columns of X are R tj (A tj p tj ),R tj (A tj p tj )(j 1), R tj (A tj p tj )(j 1) 2 27

28 Test Statistic for Sample Size Calculation You choose your favorite correlation matrix associated with Var(Y X). ˆΣ= ( N ˆσ 2 N X T i X i ) 1 { N X T i }( Corr(Y N i X i )X i X T i X i ) 1 i=1 i=1 i=1 K is 3 by p matrix picking out β coefficients 28

29 Test Statistic for Sample Size Calculation GEE test statistic is Nˆβ T (KˆΣK T ) 1ˆβ The asymptotic distribution is a Chi- Squared on 3 degrees of freedom with non-centrality parameter: d T (Σ β ) 1 d 29

30 Test Statistic for Sample Size Calculation The asymptotic distribution is a Chi- Squared on 3 degrees of freedom with non-centrality parameter: d T (Σ β ) 1 d Σ β only depends on polynomials in (j-1), R tj the distribution of and on the variance of the sequential randomizations. 30

31 Test Statistic for Sample Size Calculation The asymptotic distribution of the test statistic does not depend on the form of α tj, γ tj aaaaaaa or on the correlation matrix. The asymptotic distribution does depend on the distribution of R tj 31

32 Standardized d i s initial proximal effect: d 0 =0 average proximal effect: 1 5J J j=1 5 t=1 Example ( d0 +d 1 (j 1)+d 2 (j 1) 2) day of maximal proximal effect: d 1 2d 2 =28 P[R tj =1 A jt,j 1 =0]=.9, P[R tj =1 A t,j 1 =1]=.8 32

33 Average Proximal Effect P[R tj =1 A jt,j 1 =0]=.9, P[R tj =1 A t,j 1 =1]=.8 33

34 Average Proximal Effect P[R tj =1 A jt,j 1 =0]=.7, P[R tj =1 A t,j 1 =1]=.5 34

35 Scientific Goals 1) Assess if there are proximal causal effects of the actions on the response. 2) Assess if there are delayed causal effects; assess if the proximal/delayed causal effects vary by particular state variables. 3) Construct a treatment policy, aka Just-in-Time Adaptive Intervention that inputs state and outputs actions. 4) Construct an online training algorithm that will result in a Continually Updating Just-in-Time Adaptive Intervention 35

36 Collaborators: P. Liao, P. Klasnja, A. Tewari if you have questions! 36