Evaluating time-varying treatment effects in hybrid SMART-MRT designs

Submitted to the Annals of Applied Statistics EV ALU A TING TIME-V AR YING TREA TMENT EFFECTS IN HYBRID SMAR T -MR T DESIGNS B Y M E N G B I N G L I 1 , a , I N BA L B I L L I E N A H U M - S H A N I 2 , c A N D W A L T E R D E M P S E Y 1 , b 1 Department of Biostatistics, University of Mic higan, a mengbing@umich.edu ; b wdem@umich.edu 2 Institute for Social Resear ch Univer sity of Michigan, c inbal@umich.edu Recently a ne w experimental approach – the hybrid e xperimental design (HED) – was introduced to enable in vestigators to answer scientiﬁc ques- tions about building behavioral interventions in which human-deli vered and digital components are integrated and adapted on multiple timescales—slow (e.g., e very fe w weeks) and fast (e.g., ev ery few hours), respectiv ely . An in- creasingly common HED inv olves the integration of the sequential, multi- ple assignment, randomized trial (SMAR T) with the micro-randomized trial (MR T), allowing in vestigators to answer scientiﬁc questions about potential synergistic effects of digital and human-delivered interventions. Approaches to formalize these questions in terms of causal estimands and associated data analytic methods are limited. In this paper , we formally deﬁne and assess these syner gistic ef fects in hybrid SMAR T -MR Ts on both proximal and dis- tal outcomes. Practical utility is shown through the analysis of M-Bridge, a hybrid SMAR T -MR T aimed at reducing binge drinking among ﬁrst-year col- lege students. 1. Introduction. An adapti ve interv ention is an interv ention approach that guides ho w dynamic information about the indi vidual should be used in practice to make intervention decisions about the type, intensity , and modality of intervention deliv ery ( Collins, Murphy and Bierman, 2004 ; Nahum-Shani et al., 2012 ). The goal is to address the unique and chang- ing needs of individuals in a resource efﬁcient manner ( Nahum-Shani and Almirall, 2019 ). Adv ances in digital technologies hav e enabled the rapid – e.g., ev ery minute ( Battalio et al., 2021 ) – adaptation of interventions in real time to meet the immediate needs of indi vid- uals in daily life. In behavioral health, mobile apps and wearable de vices ha ve presented ne w opportunities for adapting interventions to the individual’ s rapidly changing state (e.g., emotions) and context (e.g., location) to improve positive behaviors (e.g., physical acti vity , mental health) or reduce ne gati ve ones (e.g., alcohol use, smoking) ( Klasnja et al., 2018 ; Gustafson et al., 2014 ; Ben-Zee v et al., 2013 ; Riley , Obermayer and Jean-Mary, 2008 ). These just-in-time adaptive interventions (JIT AIs; Nahum-Shani and Murphy (2025 )) are deli vered via automated, digital services (e.g., mobile devices), offering sev eral advantages ov er human-deliv ered alternativ es, including access, affordability , capacity to deliv er com- plex intervention protocols with high ﬁdelity , and the ability to address fast-changing con- ditions in ev eryday life ( Nahum-Shani and Naar, 2023 ; Mohr , Zhang and Schueller, 2017 ; Nahum-Shani et al., 2018 ; Lattie, Stiles-Shields and Graham, 2022 ; V olko w and Blanco, 2023 ). Howe ver , suboptimal engagement represents a major barrier to the effecti veness of digital services ( Mohr , Cuijpers and Lehman, 2011 ; Schueller, T omasino and Mohr, 2017 ; Y ardley et al., 2016 ). Human-delivered services (e.g., by clinical staff) tend to be more en- gaging and produce larger effects ( Mohr , Cuijpers and Lehman, 2011 ; Schueller , T omasino K eywor ds and phrases: Dynamic treatment regimes, Hybrid Experimental Designs, Sequential, Multiple As- signment Randomized T rials, Micro-randomized trials, Causal inference. 1 2 and Mohr, 2017 ; Ritterband et al., 2009 ). Ho wev er , these services are adapted on a relati vely slo w timescale (e.g., every fe w weeks or months), are prone to inconsistent implementation, and are often more expensi ve and b urdensome. Thus, integrating digital technologies with human-deli vered support has enormous potential to increase the reach and impact of services for pre vention and treatment in chronic illness populations. Existing experimental designs and related data-analytic methods can be used to answer questions either about ho w to best employ components that are sequenced and adapted at relati vely slow timescales (e.g., monthly) or about how to best employ components that are sequenced and adapted at much f aster timescales (e.g., daily). Ho wev er , these methodologies do not accommodate sequencing and adaptation of components at multiple timescales. Re- cently , the hybrid experimental design (HED) was introduced to close this gap. HEDs provide a ﬂexible framew ork to address this need by accommodating sequential randomization at both fast and slo w timescales. Data from HEDs can be then be used by researchers to answer sci- entiﬁc questions about ho w to optimally blend digital and human-deliv ered intervention com- ponents ( Nahum-Shani and Naar, 2023 ; Nahum-Shani et al., 2022 ). In this paper , we focus on a particular type of hybrid design, the SMAR T -MR T hybrid design ( Nahum-Shani et al., 2024 ). The sequential multiple assignment randomized trial (SMAR T) implements sequential randomizations at slo wer timescales ( Kidwell and Almirall, 2023 ). Data from a SMAR T can be used to ev aluate dynamic treatment regimens (DTRs; Liu, Zeng and W ang (2014 )), also called standard adaptiv e interventions in the behavioral literature ( Nahum-Shani and Murphy , 2025 ), in which intervention components are adapted on a relativ ely slow timescale. These DTRs deﬁne decision rules at each decision point, tailored to indi viduals’ time-v arying char - acteristics and intermediate outcomes. The micro-randomized trial (MR T) in volv es frequent randomizations at f ast timecales resulting in hundreds or thousands of decision points ( Klas- nja et al., 2015 ; Dempsey et al., 2017 ). Data from an MR T is used to assess effect moder- ation of digital interventions to inform JIT AIs in which components are adapted in a fast timescale ( Nahum-Shani et al., 2022 ). The SMAR T -MR T hybrid design allows researchers to answer questions about how best to integrate digital intervention components that adapt rapidly with human-deliv ered components that adapt on a lower timescale. Current data an- alytic methods for SMAR T -MR T hybrid designs, howe ver , focus only on separate analysis of the two components, treating the other component as a potential moderator of the others ef fectiv eness but ignoring potential synergistic effects. Effect moderation analysis ( Boruvka et al., 2018 ) is insufﬁcient as it conditions on post-treatment variables (i.e., variables mea- sured after baseline SMAR T randomization) and cannot be used to assess synergistic ef fects. T o fully realize the potential of inte grating digital technologies and human-deli vered support, formal deﬁnitions of syner gistic effects and associated data analytic methods are critically needed. 1.1 The M-Bridge Study and Existing Analyses. The M-Bridge study employs a SMAR T -MR T hybrid design to reduce hea vy drinking and related risks among ﬁrst-year col- lege students ( P atrick et al., 2020 ). The SMAR T in volv ed tw o stages of randomization. First, students (N=591) were randomly assigned (with a 1:1 ratio) to one of two times for deliv ering an initial web-based intervention combining personalized normati ve feedback with bi-weekly self-monitoring of alcohol use: early (before the start of the f all semester), or later (during the ﬁrst month of the fall semester). Second, participants who self-identiﬁed as heavy drinkers based on the bi-weekly self-monitoring (n=158; 26.7%) were classiﬁed as non-responders and were re-randomized (1:1 ratio) to one of two strategies designed to bridge them to more intense treatment: either an email with a v ailable alcohol use intervention resources, or an in vitation to interact with an online health coach. Self-monitoring ceased once a participant was identiﬁed as a non-responder . Those not identiﬁed as heavy drinkers (i.e., responders) HYBRID SMAR T -MR T DESIGN 3 continued with self-monitoring alone. The MR T in volv ed bi-weekly randomization of those in the self-monitoring conditions to two types of prompts (1:1 ratio) encouraging partici- pants to self-monitor their alcohol use: either a prompt emphasizing beneﬁts to oneself (i.e., self-interest prompt) or a prompt emphasizing beneﬁts to other (pro-social prompt). The study design of M-Bridge allo ws researchers to answer scientiﬁc questions about ho w to best blend three intervention components, two of which are deli vered on a relativ ely slow timescale (i.e., the initial web-based intervention and subsequent bridging strate gies) and one deli vered on a faster timescale (i.e., bi-weekly self-monitoring prompts). Existing analyses of the M-Bridge study , ho wev er , are limited to analyzing either SMAR T ( P atrick et al., 2020 ) or MR T ( Carpenter et al., 2023 ) data in isolation. W e refer to these causal ef fects as mar ginal effects to emphasize that they marginalize over the other intervention component. W e refer to causal effects that jointly consider the two components as syner gistic effects to emphasize that they may look at contrasts in one component while keeping the other component to a ﬁxed le vel. In the statistical literature, various methods ha ve been dev eloped to analyze the causal ef- fects of interventions in SMAR Ts and MR Ts in separate contexts. For the analysis of data collected from a SMAR T , marginal mean models for estimating optimal dynamic treatment regimes (DTRs) hav e been established ( Murphy et al., 2001 ; Murphy , 2005 ; Orellana, Rot- nitzky and Robins, 2010 ; Chakraborty and Murphy , 2014 ). Building on this line of research, Nahum-Shani et al. (2012 ) introduced the weighted and replicated (WR) approach for an- alyzing data from SMAR Ts where only a subset of individuals are re-randomized in the second stage of a SMAR T . Sample size calculation and power analysis for different types of SMAR Ts using the WR method hav e been introduced ( See wald et al., 2020 ). For the analysis of data collected from an MR T , e xisting methods focus on estimating the time-varying causal excursion effect of binary treatments ( Dempsey et al., 2015 ; Liao et al., 2016 ). The weighted and centered least squares (WCLS; Boruvka et al. (2018 )) is regarded as the benchmark method used for estimating moderated causal excursion ef fects for a continuous outcome, with an extension to a binary outcome proposed in Qian et al. (2021 ). Shi, W u and Dempse y (2023 ) and Shi and Dempse y (2025 ) impro ve asymptotic ef ﬁciency of WCLS by incorporat- ing auxiliary v ariables and machine learning prediction algorithms respectiv ely . The foundation for a data analytic method speciﬁc to the hybrid SMAR T -MR T was laid out by Nahum-Shani and Naar (2023 ). The current literature on data analytic methods for hybrid SMAR T -MR Ts, howe ver , has three improtant gaps: (1) there is no formal statement of marginal and syner gistic causal ef fects in a SMAR T -MR T within a causal frame work; (2) there is no robust statistical method that has both statistical consistency guarantees and ensures powerful test statistics for synergistic and marginal effects; and (3) there is no com- prehensi ve approach that simultaneously estimates the syner gistic and marginal ef fects. 1.2 Our Contrib utions. In this paper , we propose a nov el data-analytic method for ana- lyzing data from hybrid SMAR T -MR T studies that addresses these three gaps. Our four main contributions are summarized as follows. First, we formally deﬁne a set of causal estimands of scientiﬁc interest when analyzing data from a SMAR T -MR T hybrid design. These esti- mands include the interaction effects of human-deliv ered and digital components as well as main effects of one component av eraging o ver the other . Second, a set of estimating equations is proposed that allow for simultaneous estimation of all causal estimands. Our method b uilds upon the WCLS method ( Boruvka et al., 2018 ) for MR Ts and the WR method ( Nahum-Shani et al., 2012 ) for SMAR Ts. W e lev erage similar ideas as in Shi and Dempsey (2025 ) to incor - porate time-varying cov ariates to improve efﬁcienc y relativ e while av oiding potential causal bias when incorporating post treatment variables. This leads to efﬁciency gains ov er WR methods for analyzing main DTR effects. Third, we build a nov el frame work to incorporate 4 eligibility when analyzing causal effects. Prior methods, such as WCLS, condition on indi- viduals who are “eligible” to receive MR T treatment at a gi ven decision point. Instead, we propose to av erage ov er eligibility status in estimating the treatment ef fects. F ourth, we apply our method to the M-Bridge study and draw scientiﬁc conclusions about the effects of rela- ti vely slow timescale (web-based interventions and subsequent bridging strategies) and fast timescale (self-monitoring prompts) intervention components in reducing binge-drinking. Our analysis reveals potential synergistic ef fects and serv es as a data analytic framew ork for future analysis of SMAR T -MR T hybrid studies. The rest of the paper is org anized as follows. Section 2 describes a typical hybrid SMAR T - MR T design and introduces notation. Section 3 lays out the modeling assumption and infer- ence framework. Section 5 compares estimation performances of the proposed and alternativ e approaches via simulation studies. Section 6 applies the proposed method to the M-Bridge study . Section 7 concludes with a brief discussion on study limitations and future directions. 2. Preliminaries. Moti vated by the M-Bridge study , we start by introducing our gen- eral notation for a two-stage hybrid SMAR T -MR T . Adaptation to similar hybrid SMAR T - MR T designs is possible and will be discussed below . Speciﬁcally , we will discuss additional considerations for application of our proposed approach to the M-Bridge study in Section 6 . T able 5 in the Supplementary Materials summarizes all notation for deﬁning the data, estimands, and estimators related to a SMAR T -MR T hybrid design. 2.1 Study Design and Notation. Let X 0 denote a vector of baseline cov ariates. At the beginning of Stage 1 of the study , an individual is randomly assigned to the ﬁrst-stage in- tervention Z 1 ∈ Z 1 := {− 1 , 1 } . During Stage 1, the individual is subsequently randomized to an interv ention A t ∈ { 0 , 1 } at each time point t = 1 , . . . , t ∗ − 1 . T ransition to Stage 2 occurs at time t = t ∗ , at which time individuals are randomized to a second-stage interven- tion Z 2 ∈ Z 2 . Here Z 2 can be {− 1 , 1 } or {− 1 , 0 , 1 } depending on whether we consider an unrestricted or a restricted SMAR T design, i.e., whether ev ery indi vidual is re-randomized or not (see Figure 2 for three common SMAR T designs). During Stage 2, the individual is randomized to intervention A t ∈ { 0 , 1 } at each time point t = t ∗ + 1 , . . . , T . Individual and contextual information at the t -th time point is represented by X t and is measured before recei ving intervention A t . The proximal response, denoted Y t +1 , is observed after receiving intervention A t . Additionally , let Y (with no subscript) be the distal outcome at the end of the study . The proximal response measures near-term impact of intervention components, while the distal outcome is designed to measure longer-term impact of the sequence of interven- tions. The collection of observed data is X 0 |{z} Baseline , Z 1 , X 1 , A 1 , Y 2 , . . . , X t ∗ − 1 , A t ∗ − 1 , Y t ∗ − 1 | {z } Stage 1 , Z 2 , X t ∗ +1 , A t ∗ +1 , Y t ∗ +1 , . . . , X T , A T , Y T +1 | {z } Stage 2 . Figure 1 presents a restricted two-stage hybrid SMAR T -MR T design. T able 5 provides a summary of the above notation. F or bre vity , we refer to ( Z 1 , Z 2 ) as the short-time scale (STS) intervention component and ( A 1 , . . . , A T ) as the fast-time scale (FTS) interv ention component. An ov erbar denotes a sequence of random variables (uppercase letters) or realized values (lo wercase letters) through a speciﬁc interv ention occasion. F or example, s A t = ( A 1 , . . . , A t ) , s X t = ( X 0 , X 1 , . . . , X t ) , and s Y t +1 = ( Y 2 , . . . , Y t +1 ) . For notational con venience, s Z t de- notes the sequence of FTS interventions in stages prior to time t , i.e., s Z t = Z 1 if t < t ∗ and s Z t = ( Z 1 , Z 2 ) if t ≥ t ∗ . The complete history of observ able information up to t is H t =  s X t , s Z t , s A t , s Y t  . HYBRID SMAR T -MR T DESIGN 5 Fig 1: A two-stage SMAR T -MR T hybrid design over T time points. Baseline information is collected time 0. The circled Rs represents randomization e vents. Upper panel: the restricted SMAR T component of the hybrid design where only non-responders are re-randomized in Stage 2. Lo wer panel: the MR T component of the hybrid design that randomizes a binary intervention at each time point. Next, we introduce the randomization scheme used in a SMAR T -MR T . In the embedded SMAR T , the Stage 1 randomization probability is P ( Z 1 = z 1 | X 0 ) for z 1 ∈ Z 1 , i.e., the ran- domization depends only on baseline information. In Stage 2, the randomization probability is P ( Z 2 = z 2 | R ) for z 2 ∈ Z 2 , i.e., only depends on binary response status R which is a de- terministic function of the observ ed history up to time t ∗ − 1 , H t ∗ − 1 . For the embedded MR T component, the FTS intervention randomization probability is P ( A t = a | H t ) = p t ( a | H t ) for a ∈ { 0 , 1 } , i.e., depends on the observed history which includes prior STS interv entions. In the M-Bridge study , recall that ﬁrst-stage STS interventions are early ( Z 1 = 1) and later ( Z 1 = − 1) with a 1:1 ratio, i.e., P ( Z 1 = 1 | X 0 ) = 1 / 2 . FTS interventions are digital SI prompts ( A t = 1 ) and PS prompts ( A t = 0 ) deliv ered prior to biweekly self-monitoring sur- ve ys with probability p t (1 | H t ) = 0 . 5 . The binary response indicator R is whether a student is classiﬁed as a non-hea vy drinker based on self-monitoring surveys: hea vy drinkers ( R = 0 ) are considered non-r esponders to Z 1 . For heavy drinkers, second-stage STS interv entions are a resource email ( Z 2 = 1 ) or an online health coach ( Z 2 = − 1 ) also with equal probability . In other words, P ( Z 2 = z 2 | R = 0) = 1 / 2 for z 2 = − 1 , 1 and P ( Z 2 = 0 | R = 1) = 1 . The proximal outcome Y t +1 is the maximum number of alcoholic drinks consumed within 24 hours during the past two weeks of the t -th SM survey . The distal outcome Y is the cumula- ti ve number of alcoholic drinks consumed throughout the study . R E M A R K 2.1. The embedded SMAR T in the M-Bridge study is a restricted SMAR T design – type (II) in Figure 2 below . Other commonly used SMAR T designs are illustrated in Figure 2 ; see Patrick et al. (2020 ); Nahum-Shani et al. (2023 ) for more examples. The method proposed in this paper applies to all these SMAR T designs in a hybrid SMAR T -MR T study , although we focus on type (II) based on the moti vating M-Bridge study . 2.2 P otential outcomes and dynamic tr eatment r egimes. T o deﬁne causal estimands, we adopt the potential outcome framew ork ( Rubin, 1974 ; Robins, Rotnitzky and Scharfstein, 2000 ). Denote Y t +1 ( s z t , s a t ) as the potential outcome for the proximal response under a spe- ciﬁc STS intervention sequence s z t and FTS intervention sequence s a t up to time t . Similarly , 6 Fig 2: Three commonly used two-stage SMAR T designs in See wald et al. (2020 ). Non-circled Rs and NRs are short for responders and non-responders. Circled Rs represent randomization. let A t ( s z t , s a t − 1 ) , X t ( s z t , s a t − 1 ) , and H t ( s z t , s a t − 1 ) be the potential outcomes for the FTS inter - vention, co variates, and history , respectiv ely . c A dynamic treatment regime (DTR) is a sequence of decision rules s d = ( d 1 , d 2 ) em- bedded in the SMAR T . The decision rule d 1 is a mapping from X 0 to the ﬁrst-stage STS intervention space Z 1 , and d 2 is a mapping from H t ∗ − 1 to Z 2 . Let D = { ¯ d = ( d 1 , d 2 ) : d k ∈ Z k , k = 1 , 2 } be the collection of all possible DTRs. The potential proximal outcome under a regime s d and FTS intervention sequence s a t is deﬁned as Y t +1  s d, s a t  = X z 1 ∈Z 1 1 { z 1 = d 1 ( X 0 ) } X z 2 ∈Z 2 1 { z 2 = d 2 ( H t ∗ − 1 ) } Y t +1 ( s z , s a t ) . (1) While our proposed methodology can be applied broadly to estimands under any DTR, in line with secondary analyses of SMAR Ts ( See wald et al., 2020 ) we will focus on DTRs where d 1 does not depend on X 0 , and d 2 only depends on response status. In the M-Bridge study , the restricted SMAR T design contains four DTRs that D = {− 1 , 1 } × 2 and an individual has equal probability P ( s d = ( l 1 , l 2 )) = 1 / 2 × 1 / 2 = 1 / 4 for ∀ ( l 1 , l 2 ) ∈ D to be consistent with any of the DTRs. 2.3 Causal Estimands. In this section, we deﬁne causal estimands and build associated estimators focusing on the sequence of proximal outcomes. Similar estimands and estimators can be deﬁned and built for distal outcomes; ho wev er , these require nuanced considerations around delayed effects as was discussed in Qian (2025 ). Focusing on proximal outcomes helps to clarify ho w we deﬁne and estimate synergistic effects which can then be translated in future work to similar syner gisitic effects for distal outcomes. W e motiv ate our causal estimands from four scientiﬁc questions about the effects of the slo w-time scale (STS) and fast-time scale (FTS) intervention components on proximal out- comes (bi-weekly reported maximum drinks) in the M-Bridge study: (1) Fixing the user to a prompt emphasizing beneﬁts to oneself (self-interest prompt), is it better to initiate the web- based intervention early and use a resource email to bridge non-responders, or initiate the web-based intervention later and use an online coach? (2) Fixing the user to receiv e an early intervention initiation and an email-based bridging strategy , is it more effecti ve to deliver a self-interest prompt or a pro-social prompt? (3) A veraging o ver prompt type (self-interest and pro-social), is it more ef fectiv e to initiate the intervention early and use a resource email to bridge non-responders, or initiate the intervention later and use a resource email? (4) A v er- aging across all STS intervention sequences, is a self-interest prompt more effecti ve than a HYBRID SMAR T -MR T DESIGN 7 pro-social prompt in a giv en week? Questions (1) and (2) correspond to the synergistic ef- fects between the STS and FTS intervention components, which capture how the impact of one intervention component (e.g., timing of the web-based intervention) interacts with the other intervention component (e.g., the type of prompt deli vered). Questions (3) and (4) cor- respond to the main effects of one interv ention, which is the difference in the mean outcome between different levels of an intervention, averaged across all other intervention components ( Collins, Dziak and Li, 2009 ). Additional scientiﬁc questions about the proximal interaction ef fects in this hybrid design are listed in T able 6 of Appendix F . Synergistic effects (Questions 1 and 2) are distinct from moderation effects ( Boruvka et al., 2018 ; Dempsey et al., 2017 ) as moderation analyses conditions on pre vious interventions while our questions consider ﬁxed re gimes. Question 2, for example, is the ef fect of self-interest prompt versus pro-social prompt under a ﬁxed dynamic treatment regime (DTR), which is distinct from an effect that is conditional on the STS interventions deli vered to an individual. W e deﬁne the causal estimands motiv ated by these questions as follo ws: (I.D) ( I nteraction effect for D TRs) The proximal ef fect comparing two DTRs s d versus s d ′ when assigning a ﬁxed FTS interv ention A t = a at time point t : (2) E  Y t +1  s d, ( s A t − 1 , a )  − Y t +1  s d ′ , ( s A t − 1 , a )  | X 0  . (I.A) ( I nteraction effect for A t ) The marginal proximal effect of FTS interventions at time point t , i.e., A t = 1 versus A t = 0 , when assigning a ﬁxed DTR s d : (3) E  Y t +1  s d, ( s A t − 1 , 1)  − Y t +1  s d, ( s A t − 1 , 0)  | X 0  . (A.D) ( A veraged effect for D TRs) The marginal proximal ef fect of two DTRs s d versus s d ′ at time point t , averaging o ver FTS interv entions: (4) E  Y t +1  s d, s A t  − Y t +1  s d ′ , s A t    X 0  . (A.A) ( A veraged effect for A t ) The marginal proximal effect of FTS interventions at time point t , i.e., A t = 1 versus A t = 0 , a veraging ov er all DTRs: (5) X ( l 1 ,l 2 ) ∈D P ( s d = ( l 1 , l 2 )) E  Y t +1  ( l 1 , l 2 ) , ( s A t − 1 , 1)  − Y t +1  ( l 1 , l 2 ) , ( s A t − 1 , 0)    X 0  . The expectations are taken with respect to the distribution of the potential history H t ( s d, s A t − 1 ) giv en X 0 , under the DTRs and MR T randomization probability . The causal es- timands av erage ov er past FTS interventions s A t − 1 and time-varying variables in the history except for a subset of baseline v ariables X 0 . This a veraging mitig ates the large space of FTS intervention sequences s a t ∈ { 0 , 1 } T due to large T relativ e to the number of observations. This approach is taken from the existing MR T literature ( Boruvka et al., 2018 ; Shi, W u and Dempsey, 2023 ; Qian et al., 2021 ) which also averages ov er prior interventions when deﬁning “causal excursion effects”. While it is possible to deﬁne causal effects for speciﬁc interven- tion sequences of short length, we focus on average effects to maintain interpretability and av oid the comple xity that comes with modeling and estimating ef fects ov er such intervention sequences. Extensions to handle such complexity are possible; see Shi and Dempsey (2025 ) for ho w to estimate effects of interv ention sequences on proximal outcomes. W e ne xt express the proximal ef fects in terms of the observed data, by assuming positi vity , consistency , and sequential ignorability ( Robins, 1994 , 1997 ): A S S U M P T I O N 2.2 . (Causal Identiﬁcation) • Positi vity: P ( Z 1 = z 1 | X 0 ) > 0 , P ( Z 2 = z 2 | R, Z 1 ) > 0 , and P ( A t = a | H t ) > 0 almost e verywhere for all z 1 ∈ Z 1 , z 2 ∈ Z 2 , a ∈ A . 8 • Consistency: for each t ≤ T ,  X t ( s Z t , s A t − 1 ) , A t ( s Z t , s A t − 1 ) , Y t +1 ( s Z t , s A t )  = { X t , A t , Y t +1 } . • Sequential ignorability (SI): (a) { X t +1 ( s z t , s a t ) , A t +1 ( s z t , s a t ) , Y t +1 ( s z t , s a t ) , . . . , Y T +1 ( s z T , s a T ) } ⊥ ⊥ s Z t | H t \ { s Z t } (b) { Y t +1 ( s z t , s a t ) , X t +1 ( s z t , s a t ) , A t +1 ( s z t , s a t ) , . . . , Y T +1 ( s z t , s a T ) } ⊥ ⊥ A t | H t . The positi vity assumption implies that an individual has a positi ve probability to follow any DTR and receiv e any FTS interv ention gi ven the history . In a hybrid SMAR T -MR T study , the SMAR T and MR T randomization probabilities are known at all decision points t = 1 , . . . , T . The consistency assumption subsumes Rubin’ s Stable Unit T r eatment V alue Assumption (SUTV A) that no interference exists between indi viduals ( Rubin, 1980 ). SI(a) and (b) are commonly seen in standard SMAR T and MR T studies. By study design, SI is automatically satisﬁed. Under Assumption 2.2 , we hav e E  Y t +1  s d t , ( s A t − 1 , a )    X 0  = E [ E [ E [ E [ Y | H t , A t = a ] | H t ∗ − 1 , Z 2 = d ( H t ∗ − 1 )] | X 0 , Z 1 = d 1 ( X 0 )] | X 0 ] (6) = E " X z 1 ∈Z 1 X z 2 ∈Z 2 1 { z 1 = d 1 ( X 0 ) } 1 { z 2 = d 2 ( H t ∗ − 1 ) } P ( Z 1 = z 1 , Z 2 = z 2 | H t ∗ − 1 ) 1 { A t = a } p t ( a | H t , s Z = s z ) Y      X 0 # . (7) The proof of ( 6 ) and ( 7 ) can be found in Appendix B . 3. Estimation and Inference. 3.1 Modeling Assumptions. In the following, we propose a method to jointly estimate marginal interaction and main effects of the DTRs and FTS interventions. Based on ( 6 ), we assume that the expectation of the proximal outcome giv en STS and FTS intervention assignments takes the form E  Y t +1  s d, ( s A t − 1 , a )  = ( a − ρ ) f t ( s d ) ⊤ β + m t ( s d ) ⊤ η , (8) where f t ( s d ) and m t ( s d ) are a p - and q -dimensional vector functions of DTRs s d , respectiv ely . Here, ρ ∈ (0 , 1) is a ﬁxed pseudo-centering probability for FTS interventions. As will be discussed in Section 3 , ρ should be chosen to stabilize the estimation of ( β , η ) . In addition, by deﬁnition of s d , f t and m t should depend on only d 1 if t < t ∗ , but may depend on d 1 and d 2 if t ≥ t ∗ . Moreo ver , while the regression coef ﬁcients β and η are constant ov er time, time- v arying effects may be included through time-dependent components (e.g., a linear term in time d 1 t ) into f t and m t . If we are interested in the conditional expectation given baseline v ariables X 0 , i.e., E  Y t +1  s d, ( s A t − 1 , a )    X 0  , we may incorporate X 0 into f t and m t . The functions f t and m t may differ depending on whether the marginal effect of DTRs, av eraged ov er s A t , is expected to remain constant across stages. As will be illustrated in Ex- ample 3.1 , we set f t = m t when both MR T and SMAR T randomization probabilities are constant (see Equation ( 13 )). In contrast, Simulation Scenario II in Secti on 5 considers MR T randomization probabilities that depend on prior STS interventions, leading to stage-speciﬁc marginal effects of d 1 av eraged over s A t . T o accommodate this, m t includes stage-speciﬁc intercepts and d 1 coef ﬁcients, while f t shares these terms across stages. Although stage- speciﬁc terms could also be included in f t as well in Example 3.1 , doing so would reduce ef ﬁciency of estimating the proximal intervention effects, since stage-speciﬁc terms would be estimated using only data from the corresponding stage. W e thus maintain separate spec- iﬁcations for f t and m t to reﬂect stage-wise dif ferences of the contributions of DTRs to the proximal ef fects while preserving estimation efﬁcienc y . HYBRID SMAR T -MR T DESIGN 9 Using ( 8 ), the interaction ef fects and the av erage ef fects of FTS interventions discussed in Section 2.3 can be expressed using the coef ﬁcients ( β , η ) as the follo wings: (I.D)  (1 − ρ ) f t ( s d ) − ρf t ( s d ′ )  ⊤ β +  m t ( s d ) − m t ( s d ′ )  ⊤ η (9) (I.A) f t ( s d ) ⊤ β (10) (A.A) X ( l 1 ,l 2 ) ∈D P ( s d = ( l 1 , l 2 )) f t (( l 1 , l 2 )) ⊤ β . (11) On the other hand, the a verage effects of DTRs marginalized o ver FTS interv entions, in (A.D) , may not be directly attainable from ( 8 ) if the MR T randomization probability p t ( a | H t ) depends on time-varying moderators in the history . In fact, η is interpreted as the proximal effects comparing DTRs av eraging ov er FTS interventions, as if all FTS inter- ventions were randomized with probability p t (1 | H t ) = ρ (see Example 3.1 ). Therefore, we obtain this ef fect directly by projecting onto the space of m t ( s d ) as E  Y t +1  s d, ( s A t − 1 , A t )  = m t ( s d ) ⊤ γ , (12) where γ ∈ R q . Here the coefﬁcient γ is interpreted as the proximal ef fect comparing DTRs av eraging ov er FTS interventions, under the actual MR T randomization probability by study design. See Section C of the Appendix for more detailed discussion. W e now giv e a simple e xample to illustrate the model and interpretation of the coefﬁcients. E X A M P L E . Consider the two-stage SMAR T -MR T hybrid design shown in Figure 1 . All indi viduals are randomized with equal probabilities to one of the two Stage-1 STS inter- ventions, and only non-responders are randomized with equal probabilities to Stage-2 STS interventions. As a result, each individual is assigned to one of the four DTRs with equal probability . T o illustrate k ey estimands of interest in hybrid designs, we consider a simpliﬁed working model for the mar ginal expectation of the proximal outcome: E  Y t +1  s d = ( l 1 , l 2 ) , ( s A t − 1 , a t )  = ( a t − 1 / 2) ( β 0 + β 1 l 1 + β 2 δ t l 2 + β 3 δ t l 1 l 2 ) + η 0 + η 1 l 1 + η 2 δ t l 2 + η 3 δ t l 1 l 2 E [ Y t +1 (( d 1 , d 2 ) = ( l 1 , l 2 ) , ¯ A t ))] = γ 0 + γ 1 l 1 + γ 2 δ t l 2 + γ 3 δ t l 1 l 2 . (13) where δ t = I { t > t ∗ } is a Stage 2 indicator . These models are simpliﬁed and time-in variant for illustrativ e purposes. In practice, time-varying components can be accommodated (e.g., replace β 1 d 1 with β 1 d 1 + β ′ 1 d 1 t ). Alternati vely , e ven if ( 13 ) is misspeciﬁed for true time- v arying effects, we may still vie w ( 13 ) as a working model targeting summaries of the time- v arying ef fects. For (I.D) , the effect of comparing ¯ d = (1 , 1) versus ¯ d ′ = ( − 1 , 1) when A t = 1 at t > t ∗ is gi ven by β 1 + β 3 + 2 η 1 + 2 η 3 . F or (I.A) , the effect of A t = 1 versus A t = 0 when ¯ d = (1 , 1) at t ≥ t ∗ is gi ven by β 1 + β 3 . F or (A.A) , the effect of A t = 1 versus A t = 0 av eraging ov er DTRs at t > t ∗ is gi ven by X l 1 ,l 2 ∈{− 1 , 1 } 2 P ( s d = ( l 1 , l 2 ))  1 − 1 2  ( β 0 + β 1 l 1 + β 2 l 2 + β 3 l 1 l 2 ) + η 0 + η 1 l 1 + η 2 l 2 + η 3 l 1 l 2 −  0 − 1 2  ( β 0 + β 1 l 1 + β 2 l 2 + β 3 l 1 l 2 ) − ( η 0 + η 1 l 1 + η 2 l 2 + η 3 l 1 l 2 )  = β 0 . The effect of DTRs ¯ d = (1 , 1) versus ¯ d ′ = ( − 1 , 1) av eraging ov er all past FTS interventions is then 2 γ 1 + 2 γ 3 . T able 6 in Appendix F expresses additional marginal ef fects of interest. 10 3.2 Estimation and Infer ence. W e now describe our two-step approach for estimation of the parameters θ := ( β , η , γ ) . The estimation procedure is designed to le verage the hy- brid SMAR T -MR T structure by incorporating both time-varying and stage-speciﬁc interven- tion assignments. Broadly , Step 1 estimates the proximal intervention effects ( β , η ) using a weighted estimating equation inspired by WCLS, while we improv e efﬁcienc y by incorpo- rating auxiliary variables that moderate intervention effects. Step 2 estimates the effects of DTRs embedded in γ , by regressing predicted outcomes from Step 1 on the regime indica- tors, using a method inspired by the WR approach. Overall, we appropriately propagate the uncertainty in the two steps by deri ving the asymptotic distribution for θ . Step 1: Estimating ( β , η ) . W e begin by modeling the marginal expectation of the proximal outcome Y t +1 as in ( 8 ). Let g t ( H t ) denote an r -dimensional vector function of the history H t that will be used as control variables. Let S t denote a set of auxiliary variables (dimension l ≤ r ) that are believ ed to be effect moderators. W e assume S t ⊆ g t ( H t ) and S t ∩ f t ( ¯ d ) = ∅ to ensure the auxiliary v ariables serve as valid augmentation terms. T o appropriately handle the hybrid design, we assign each observation two types of weights. The ﬁrst is a SMAR T weight associated with a DTR ¯ d as W S s d = 1 { Z 1 = d 1 ( X 1 ) } 1 { Z 2 = d 2 ( H t ∗ − 1 ) } P ( Z 1 | X 0 ) P ( Z 2 | R ) , (14) which accounts for the indi vidual’ s consistency with regime ¯ d = ( d 1 , d 2 ) under the SMAR T component of the design. The second is an MR T weight associated with A t deﬁned as W M t = e p t ( A t ) p t ( A t | H t ) , (15) where the numerator e p t ( A t ) = ρ A t (1 − ρ ) 1 − A t and ρ ∈ (0 , 1) is the constant in ( 8 ). This MR T weight adjusts for the randomization probability in the MR T component at time t . W e then deﬁne the following weighted and centered estimating equation U 1 ( α 0 , α 1 , β , η ) = X ¯ d ∈D T X t =1 W S s d W M t h Y t +1 −  g t ( H t ) − µ t, s d ( H t )  ⊤ α 0 − ( A t − ρ )  f t ( s d ) ⊤ β + ( S t − ψ t ( s d )) ⊤ α 1  − m t ( s d, s ) ⊤ η i     g t ( H t ) − µ t, s d ( H t ) ( A t − ρ ) f t  s d  ( A t − ρ )( S t − ψ t ( s d )) m t  s d      . (16) Here, µ t, ¯ d ( H t ) and ψ t ( ¯ d ) are centering functions designed to mak e estimating equations for the intervention ef fect parameters ( β , η ) orthogonal to the nuisance parameters ( α 0 , α 1 ) . Speciﬁcally , the two centering functions satisfy: E  W S ¯ d W M t ( g t ( H t ) − µ t, ¯ d ( H t )) | ¯ Z  = 0 , E  W S ¯ d W M t ( A t − ρ ) 2 ( S t − ψ t ( ¯ d )) | ¯ Z  = 0 . While v arious choices for the centering functions are a vailable, we assume a con venient choice in the rest of this paper as (17) µ t, s d ( H t ) = P T t =1 W S s d g t ( H t ) P T t =1 W S s d , ψ t ( s d, s ) ≡ ψ = P ¯ d ∈D P T t =1 W S s d e p t (1)(1 − e p t (1)) S t P ¯ d ∈D P T t =1 W S s d e p t (1)(1 − e p t (1)) . The estimates ( b α 0 , b α 1 , b β , b η ) are then obtained by solving P n U 1 ( α 0 , α 1 , β , η ) = 0 . HYBRID SMAR T -MR T DESIGN 11 R E M A R K 3.1 (Centering). The control v ariables in ( 16 ) are chosen such that g t ( H t ) is a working model for E  W S s d W M t Y t +1 | H t  . Unlike the WCLS method, which uses g t ( H t ) ⊤ α without centering, our approach centers g t ( H t ) around its conditional mean gi ven the DTRs. This centering is essential for unbiased estimation of the interaction ef fects comparing DTRs at a ﬁxed A t = a (see (I.D) ), ev en if the nuisance model for E  W S s d W M t Y t +1 | H t  is misspec- iﬁed, as sho wn in Section 5 . Similarly , we impose an orthogonality condition on the auxiliary v ariables to ensure unbiased b ut more ef ﬁcient estimation of ( β , γ ) under a time-varying moderator S t . The idea of incorporating auxiliary variables has been discussed by Shi, W u and Dempsey (2023 ) to account for time-varying intervention effects in MR T studies. Or- thogonality ensures consistent causal estimation while allowing us to incorporate control v ariables g t ( H t ) and auxiliary v ariables S t that can improv e statistical efﬁcienc y . R E M A R K 3.2 (W eights for Hybrid Designs). The SMAR T weight W S s d in ( 16 ) ac- counts for the SMAR T component, similar to the WR method ( Seewald et al., 2020 ). In a two-stage SMAR T as Example 3.1 where R denotes responder status, the weight is W S s d = I { Z 1 = d 1 } ( R + I { Z 2 = d 2 } (1 − R )) P ( Z 1 | X 0 ) P ( Z 2 | R ) . For non-responders, the numerator is 1 since their data align with only one DTR, whereas responders contribute to multiple consistent DTRs (e.g., ( Z 1 , Z 2 ) = (1 , 0) aligns with s d = (1 , 1) and (1 , − 1) ). The MR T weight W M t = e p t ( A t ) p t ( A t | H t ) resembles the WCLS approach ( Boruvka et al., 2018 ). W e set the numerator e p t ( A t ) to a constant to tar get marginal intervention effects, in contrast to WCLS’ s use of moderator- dependent weights for conditional ef fects. Step 2: Estimating γ . Having obtained predictions of the expected marginal proximal out- come from Step 1 deﬁned by b Y t +1 := ( a t − ρ ) f t ( s d ) ⊤ b β + m t ( s d ) ⊤ b η , (18) we next project these predictions onto the space of m t ( s d ) to directly estimate the marginal intervention effects of DTRs av eraging ov er FTS interventions. Instead of regressing on the observed proximal outcome Y t +1 , we use the predicted outcomes from Step 1 to ensure com- patibility with the estimating equation ( 16 ). The second weighted and centered estimating function is U 2 ( γ ) = X ¯ d ∈D T X t =1 W S s d h b Y t +1 − m t ( s d ) ⊤ γ i m t  s d  , (19) and the estimator b γ solves P n U 2 ( γ ) = 0 . Denote the true parameters as θ ∗ = ( β ∗ , η ∗ , γ ∗ ) . The next proposition states that we can obtain consistent estimators using our joint estimation method. P RO P O S I T I O N 3.3. Suppose that the causal assumption 2.2 and modeling assumptions ( 8 ) hold. Then (1) The estimator ( b β , b η ) is consistent and asymptotically normal. Speciﬁcally , (20) √ n  b β b η  −  β ∗ η ∗  d − → N  0 , [ E B 1 ] − 1 [ E M 1 ] [ E B 1 ] − 1  , where B 1 =  P T t =1 ( A t − ρ ) f t ( ¯ d ) m t ( ¯ d ) ⊤ P T t =1 m t ( ¯ d ) m t ( ¯ d ) ⊤  ⊗ 2 , and M 1 = U 1 ( α ∗ , β ∗ , η ∗ ) ⊗ 2 . A consistent estimator of the asymptotic v ariance is giv en by [ P n B 1 ] − 1 [ P n M 1 ] [ P n B 1 ] − 1 . 12 (2) The estimator b γ is consistent and asymptotically normal. Speciﬁcally , (21) √ n ( b γ − γ ∗ ) d − → N  0 , [ E B 2 ] − 1 [ E M 2 ] [ E B 2 ] − 1  , where B 2 = P T t =1 m t ( ¯ d ) ⊗ 2 , M 2 =  U 2 ( γ ∗ ) + Q Ω − 1 U 1 ( α ∗ , β ∗ , η ∗ )  ⊗ 2 , with Q =  0 r × p P T t =1 ( A t − ρ ) f t ( ¯ d ) m t ( ¯ d ) ⊤ P T t =1 m t ( ¯ d ) m t ( ¯ d ) ⊤  , Ω =   P T t =1 g t ( H t ) − µ t, s Z t ( H t ) P T t =1 ( A t − ρ ) f t ( ¯ d ) P T t =1 m t ( ¯ d )   ⊗ 2 . A consistent estimator of the asymptotic v ariance is giv en by [ P n B 2 ] − 1 [ P n M 2 ] [ P n B 2 ] − 1 . 4. Eligibility . In a hybrid SMAR T -MR T study , both slow-time scale (STS) and fast- time scale (FTS) intervention components restrict randomization to a subset of interv en- tion options for scientiﬁc, ethical, or practical considerations. For the embedded SMAR T , re-randomization is restricted to different intervention options based on response status. In the M-Bridge study , for example, non-response was determined through biweekly self- monitoring. Participants who self-identiﬁed as a heavy drinker were classiﬁed as a heavy drinker . Non-responders had randomization restricted to two bridging strategies. F or the em- bedded MR T , indi viduals are often determined to be ineligible to recei ve an FTS intervention at a particular decision point because interv ention deliv ery is inappropriate, unethical, or un- safe ( Klasnja et al., 2015 ). In the M-Bridge study , whenever a student is ﬂagged as a heavy drinker based on the response to second, third, or fourth self-monitoring survey , the student will transition to Stage 2 at which time they will be considered “ineligible” to receiv e MR T prompts. This restriction implies that the FTS interv ention assignment depends on the pre vi- ously observed proximal outcomes. Existing analysis of MR T data condition on eligibility to deﬁne ef fects among those av ail- able at a particular decision point ( Boruvka et al., 2018 ; Qian et al., 2021 ). Since we are primarily interested in synergistic ef fects, we cannot condition on eligibility as this will lead to causal bias due to conditioning on a post-treatment variable. W e propose to focus on a causal excursion ef fect that marginalizes over , instead of conditions on, eligibility status. This is ke y for the analysis of the M-Bridge study and requires redeﬁning the tw o FTS inter- vention options and associated causal estimands as deﬁned in Section 2.3 . Assume that the measurements X t prior to the t -th time point contain the individual’ s eligibility status, which is denoted by I t = 1 if the individual is eligible and I t = 0 if ineligible. The potential outcome of eligibility depends on the STS and FTS interventions and can be written as I t ( s Z t , s A t − 1 ) . In contrast to Section 2.1 , here we will incorporate decision rule notation to make explicit that MR T treatment A t is not deli vered under ineligibility . Deﬁne an FTS intervention func- tion as D ( a, i ) = ai for treatment a ∈ { 0 , 1 } , eligibility status i ∈ { 0 , 1 } . Note that D ( a, i ) equals 1 when the individual is eligible and treatment is deliv ered, and equals 0 otherwise. The potential proximal response under a particular DTR s d and MR T treatment sequence s a t is then Y t +1  s d,  s a t − 1 , D ( a t , I t ( s d, s a t − 1 ))  . W e now incorporate a v ailability into the deﬁnition of the marginal proximal effect. The marginal proximal ef fects in (I.D) and (I.A) become: Q(I.Z.EL) ( I nteraction ef fect for Z t marginalized over el igibility) The marginal proximal ef fect of two DTRs s d versus s d ′ when a ﬁxed MR T treatment A t = a is assigned: (22) E  Y t +1  s d t ,  s a t − 1 , D ( a, I t ( s d, s a t − 1 ))  − Y t +1  s d ′ t ,  s a t − 1 , D ( a, I t ( s d ′ , s a t − 1 ))  . HYBRID SMAR T -MR T DESIGN 13 Q(I.A.EL) ( I nteraction effect for s A t marginalized ov er el igibility) The marginal proximal ef fect of MR T treatments A t = 1 versus A t = 0 when a ﬁxed DTR s d is assigned: (23) E  Y t +1  s d,  s a t − 1 , D (1 , I t ( s d, s a t − 1 ))  − Y t +1  s d,  s a t − 1 , D (0 , I t ( s d, s a t − 1 ))  . Assuming consistency , positivity , and sequential ignorability , the marginal proximal out- come can be expressed using observed data similarly to ( 6 ) and ( 7 ). Estimation requires minor modiﬁcations to the proposed estimation strategy in Section 3 . First, we control for eligibil- ity by setting the auxiliary v ariable S t := I t as in ( 16 ). At ineligible decision points, we code A t = ρ in ( 16 ) to eliminate the MR T effect term. 5. Simulation. W e next ev aluate our proposed method through extensi ve simulations moti vated from the M-Bridge study . 5.1 Simulation Setup and Baseline Methods. W e consider a two-stage hybrid SMAR T - MR T study spanning ov er T = 50 days with the second stage starting on day t ∗ = 14 . The embedded SMAR T follows the restricted design as described in Example 3.1 . The data gen- eration model is based on Boruvka et al. (2018 ) and Seewald et al. (2020 ) with minor ad- justments made to demonstrate the necessity of our proposed method in a hybrid design. W e assume that we observe a state v ariable X t ∈ {− 2 , 2 } whose transition dynamics is given by p ( X t = 2 | A t − 1 , H t − 1 ) = expit ( − A t − 1 + 0 . 1 + 0 . 2(1 − R ) δ t Z 2 ) , where δ t = I { t ≥ t ∗ } . T wo scenarios with dif ferent MR T randomization probability and responder probability are considered. In scenario I , we let p t (1 | H t ) = 0 . 5 be a constant, and P ( R = 1 | H t ) = 0 . 6 I { Z 1 = 1 } + 0 . 45 I { Z 1 = − 1 } dependent only on Z 1 . In scenario II , we let p t (1 | H t ) be DTR-dependent and P ( R = 1 | H t ) dependent on the time-varying state X t . See Appendix G.1 for the detailed setup. In both scenarios, we assume that individuals are av ailable at all time points. The proximal outcome is generated as Y t +1 = 0 . 5 e X t + 0 . 1( A t − 1 − p t − 1 (1 | H t − 1 ))+ ( A t − p t (1 | H t ))( β ∗ 0 + β ∗ 1 Z 1 + β ∗ 2 Z 2 + β ∗ 3 δ t Z 1 Z 2 + β ∗ 4 e X t + β ∗ 5 e X t Z 1 )+ γ ∗ 0 + γ ∗ 1 Z 1 + γ ∗ 2 δ t Z 2 + γ ∗ 3 δ t Z 1 Z 2 + γ ∗ 4 e X t Z 1 + γ ∗ 5 δ t ( R − p ( R = 1 | H t )) + ϵ t (24) where δ t = I { t > t ∗ } , and e X t = X t − E [ X t | A t − 1 , H t − 1 ] is the centered state. The resid- ual error ϵ t follo ws an AR(1) Gaussian process with ϵ t ∼ N (0 , 0 . 5) and Corr( ϵ t , ϵ u ) = 0 . 5 | u − t | / 2 . The coef ﬁcients are set as β ∗ = (0 . 4 , − 0 . 3 , 0 . 2 , − 0 . 1 , 0 . 4 , 0 . 2) and γ ∗ = (0 , 0 . 2 , − 0 . 1 , − 0 . 1 , 0 . 2 , 0 . 2) , such that the proximal response depends on the assigned DTR, FTS interv entions at the current and pre vious time points, current state, and their interaction, as well as the responder status. W e generate data with sample sizes N = 100 , 400 , and repeat for 500 replications. An analytic expression of the true mar ginal ef fects is provided in Appendix G.3 . In terms of estimation, the numerator of the MR T weight is e p t (1) = 1 2 , and the control v ariables include X t and X t Z 1 . The marginal model used in scenario I takes the same form as ( 13 ). In scenario II, since the MR T randomization probabilities are different in Stages 1 and 2, we set the marginal model as E [ Y t +1 (( d 1 , d 2 ) , ¯ A t − 1 , a ))] = ( a − 1 / 2) ( β 0 + β 1 d 1 + β 2 δ t d 2 + β 3 δ t d 1 d 2 ) + ( η 0 + η 1 d 1 )(1 − δ t ) + ( η 3 + η 4 d 1 + η 5 d 2 + η 6 d 1 d 2 ) δ t . (25) For the interaction ef fects in (I.D) and (I.A) , we only e valuate the performance of the proposed method due to the lack of existing alternati ves. F or the a verage ef fects, we compare 14 against WCLS for (A.D) and WR for (A.A) . The detailed setup for the WCLS and WR analyses are described in Appendix G.2 . In all comparisons, we focus on the av erage biases, 95% co verage probabilities (CP), and the relati ve ef ﬁciency (ratio of the asymptotic v ariance of baseline ov er the proposed method). 5.2 Simulation Results. T ables 1 and 2 report the results under sample size N = 100 of scenarios I and II, respectiv ely; additional results under N = 400 are displayed in T ables 7 and 8 of the Appendix. As seen in T able 1c , the hybrid method obtains unbiased interaction ef fects (I.D) between two DTRs under a ﬁxed MR T treatment A t = a , in both Stages 1 and 2. Centering control v ariables X t and X t Z 1 , by their conditional e xpectations giv en the STS interventions as in ( 17 ), is the k ey to ensuring unbiased estimates. The cov erage probabilities achie ve the nominal 95% lev el. As for the interaction ef fects (I.A) between FTS interv entions under a ﬁx ed DTR, ro ws (1) - (6) of T able 1a suggest that the hybrid method obtains unbiased estimates and achieves the nominal 95% co verage probabilities. Howe ver , WCLS is subject to biased point estimates because WCLS ignores the restricted embedded SMAR T in a hybrid design, not accounting for the fact that only non-responders are re-randomized in Stage 2. This reason also e xplains biased estimates and low co verage probabilities obtained by WCLS for the ef fects (A.A) between FTS interventions av eraging ov er DTRs, as sho wn in ro ws (7)- (8) of T able 1a . Finally , the estimated effects between DTRs averaging ov er FTS interventions produced by the hybrid method and WR ha ve nearly zero biases, as demonstrated by T able 1b . The good performance of WR in point estimates is anticipated because the restricted SMAR T design is taken into consideration by the SMAR T weights (see ( 37 ) of the Appendix). On the other hand, the hybrid method has higher mean relati ve efﬁcienc y (mRE; the mean standard error of relativ e efﬁciency (sdRE) is also displayed). In particular , the hybrid method achieves 4% ∼ 26% efﬁcienc y gain compared to WR in scenario I and at most 8% efﬁciency gain in scenario II. The higher efﬁciency results from orthogolizing time-v arying moderators X t and X t Z 1 to reduce v ariance of the estimates. 6. A pplication to M-Bridge Study . 6.1 Study Design and Questions. In this section, we focus on a subset of N = 428 stu- dents from the M-Bridge study . Details of dataset construction as part of the lar ger M-Bridge study are provided in Appendix H . Recall that the M-Bridge study employs a SMAR T -MR T hybrid design where the SMAR T in volv ed two stages of randomization – ﬁrst-stage random assignment (1:1 ratio) early or late timing of an initial web-based intervention using person- alized normati ve feedback (PNF), with non-responders being re-randomized (1:1 ratio) to an email with ineligible alcohol use intervention resources (Email), or an in vitation to interact with an online health coach (Coach). Non-response was determined based on identifying as a heavy drinker via bi-weekly self-monitoring and therefore can occur at weeks 2, 4, 6, or 8. Those not identiﬁed as heavy drinkers (i.e., responders) continued with self-monitoring alone. The MR T in volv ed bi-weekly randomization of those in the self-monitoring condition to two types of prompts (1:1 ratio) encouraging participants to self-monitor their alcohol use: either a self-interest (SI) prompt or a pro-social (PS) prompt. Since not all students were randomized to a prompt at ev ery time point, we adopt the eligibility frame work in Section 4 . A student was ineligible ( I t = 0 , t = 2 , 3 , 4 ) to recei ve an MR T interv ention A t if the student was identiﬁed as a heavy drinker at any time point before t ; otherwise I t = 1 . W e point out that dif ferent from Example 3.1 where the MR T spans both stages, the M-Bridge only in volv es MR T in Stage 1. W e are interested in (a) the main and interaction between MR T and SMAR T intervention ef fects in Stage 1, and (b) the main SMAR T interv ention effects in Stage 2 (since MR T does not span Stage 2), HYBRID SMAR T -MR T DESIGN 15 T A B L E 1 Mar ginal effect estimation comparisons among thr ee methods in simulation scenario I, where the MRT randomization pr obability . Sample size N = 100 . (a) Comparison of effects between A t = 1 and 0, for a ﬁx ed DTR or averaging o ver DTRs. Hybrid WCLS Stage Condition T rue Bias SE CP Bias SE CP 1) 1 Fix d 1 = 1 0.1 0 0.05 0.96 -0.03 0.06 0.99 2) 1 Fix d 1 = − 1 0.7 0 0.05 0.97 0.06 0.05 0.75 3) 2 Fix s d = (1 , 1) 0.14 0 0.06 0.97 -0.01 0.07 0.97 4) 2 Fix s d = (1 , − 1) 0.06 0 0.06 0.97 -0.04 0.11 0.98 5) 2 Fix s d = ( − 1 , 1) 0.86 0 0.06 0.98 0.08 0.09 0.83 6) 2 Fix s d = ( − 1 , − 1) 0.54 0 0.06 0.96 0.04 0.06 0.9 7) 1 A veraging DTR 0.4 0 0.03 0.97 0.02 0.04 0.92 8) 2 A veraging DTR 0.4 0 0.03 0.97 0.02 0.04 0.92 (b) Comparison of effects on the proximal outcome between DTRs a veraging over FTS interv entions. Hybrid WR Stage Contrast T rue Bias SE CP Bias SE CP mRE sdRE 1) 1 d 1 = 1 vs -1 0.4 0 0.06 0.96 0 0.06 0.98 1.21 0.12 2) 2 s d = (1 , 1) vs (1, -1) -0.16 0 0.07 0.95 0 0.07 0.95 1.04 0.23 3) 2 s d = (1 , 1) vs (-1, 1) 0.32 0 0.08 0.98 0 0.08 0.98 1.06 0.13 4) 2 s d = (1 , 1) vs (-1, -1) 0.32 0 0.08 0.97 0 0.08 0.97 1.10 0.14 5) 2 s d = (1 , − 1) vs (-1, 1) 0.48 -0.01 0.07 0.96 -0.01 0.08 0.97 1.20 0.17 6) 2 s d = (1 , − 1) vs (-1, -1) 0.48 0 0.07 0.95 0 0.08 0.97 1.26 0.18 7) 2 s d = ( − 1 , 1) vs (-1, -1) 0 0.01 0.06 0.96 0.01 0.07 0.97 1.06 0.12 (c) Comparison of effects on the proximal outcome between DTRs for a ﬁx ed ﬁxed MR T treatment. Hybrid Stage Contrast T rue Bias SE CP 1) 1 d 1 = 1 vs -1 0.7 0 0.06 0.98 2) 2 s d = (1 , 1) vs (1, -1) -0.2 0 0.09 0.95 3) 2 s d = (1 , 1) vs (-1, 1) 0.68 0 0.09 0.98 4) 2 s d = (1 , 1) vs (-1, -1) 0.52 0.01 0.09 0.97 5) 2 s d = (1 , − 1) vs (-1, 1) 0.88 -0.01 0.08 0.97 6) 2 s d = (1 , − 1) vs (-1, -1) 0.72 0 0.08 0.97 7) 2 s d = ( − 1 , 1) vs (-1, -1) -0.16 0.01 0.08 0.95 8) 1 d 1 = 1 vs -1 0.1 0 0.07 0.96 9) 2 s d = (1 , 1) vs (1, -1) -0.12 0 0.08 0.93 10) 2 s d = (1 , 1) vs (-1, 1) -0.04 0 0.09 0.96 11) 2 s d = (1 , 1) vs (-1, -1) 0.12 0 0.09 0.97 12) 2 s d = (1 , − 1) vs (-1, 1) 0.08 -0.01 0.08 0.94 13) 2 s d = (1 , − 1) vs (-1, -1) 0.24 0 0.08 0.95 14) 2 s d = ( − 1 , 1) vs (-1, -1) 0.16 0 0.07 0.96 av eraging ov er the study population. The associated estimands are outlined in T able 3 . W e let are f t ( s d ) = (1 , d 1 , e t, e td 1 ) and m t ( s d ) = (1 , d 1 , δ t d 2 , δ t d 1 d 2 ) , where e t denotes the week of SM surve ys centered by mean and scaled by standard de viation, and δ t = I { t ≥ 4 } is an indicator 16 of whether a student is in Stage 2. In estimation, we set e p t = 0 . 5 . The control v ariables include baseline sex (female/male) and whether the student pledges Greek life. 6.2 Results. The estimated treatment effects are displayed in Figure 3 , and coefﬁcient estimates are reported in T able 4 . Our ﬁrst question was “What is the effect of Early versus Late PNF timing on maximum number of drinks when ﬁxing the MR T intervention to either PS or SI prompt. At week 2, Figure 3 (a) shows that the effect at week 2 is positi ve when ﬁxing to SI prompt and negati ve when ﬁxing to PS prompt. This suggests the beneﬁt of Early PNF timing relati ve to Late PNF timing may be synergistic with PS prompts at least initially . On the other hand, the effect is anti-synergistic with SI prompts in Stage 1, i.e., weeks 2 and 4 effects are positi ve. Both of these effects diminish to zero by week 8. Our second question was “What is the effect on maximum number of drinks for an SI prompt compared to a PS prompt in Stage 1, giv en a ﬁxed initial timing of PNF?” Figure 3 (b) shows that the effects when PNF timing is ﬁxed to Early is negligible and does not vary over weeks. When PNF timing is ﬁxed to Late, the SI prompts lead to reduction in the maximum number of drinks reported relativ e to PS prompts during week 2, but that the effect changes sign by week 8. Our ﬁnal question was “What is the ef fect on the maximum number of alcoholic drinks of the four embedded DTRs, av eraging ov er the MR T intervention component?” Figure 3 (c) presents the estimated marginal mean outcome for each of the four DTRs. The DTR that starts with Late PNF and provides Coaching for non-responders led to the highest marginal mean outcome of 4.5 (95% CI: 3.30 - 5.83), while the DTR that starts with Late PNF and provides a resource email for non-responders was associated with the lowest marginal mean outcome o 3.4 (95% CI: 2.25–4.62). −1.0 −0.5 0.0 0.5 1.0 2 4 6 8 Week Maximum number of drinks MRT prompt Prosocial Self−interest (a) Ef fect of Early vs Late timing of PNF gi ven ﬁxed bridging strate gy . −1 0 1 2 3 2 4 6 8 W eek Maximum number of drinks Intervention timing Early Late (b) Effect of SI vs. PS prompt giv en ﬁxed initial timing of PNF . 3 4 5 6 Early ,Coach Early ,Email Late,Coach Late,Email DTR Maximum number of drinks (c) Marginal mean outcome for a ﬁxed uni ver- sal intervention timing combined with bridg- ing strategy . Fig 3: Estimated trajectories in the M-Bridge study , where the proximal outcome is the max- imum number of alcoholic drinks a student consumed within a 24-hour period over the past two weeks. HYBRID SMAR T -MR T DESIGN 17 7. Discussion. In this paper , we proposed a novel statistical framework for analyzing data from hybrid SMAR T -MR T studies, enabling the joint estimation of the synergistic and marginal causal ef fects of interventions operating at multiple timescales. Our method in- tegrates information from both SMAR T and MR T components in a hybrid SMAR T -MR T , allo wing researchers to assess how long-term adapti ve intervention strategies interact with just-in-time support. In the the M-Bridge study , our method captures the combined ef fects of preventi ve strategies and JIT AIs on reducing binge drinking among ﬁrst-year college stu- dents. This approach provides a more comprehensiv e understanding of intervention ef fec- ti veness than existing methods, which typically analyze SMAR T and MR T data in isolation. Our work has se veral limitations. First, our analysis focuses on mar ginal effects, providing population-le vel estimates of intervention impact rather than conditional effects that account for individual heterogeneity . Future research is needed to de velop methods that accommo- date personalized treatment ef fects and examine mediation mechanisms. Second, we did not explicitly handle missing data, e xcept for cases related to participant av ailability in the MR T component. In practice, missingness may occur due to loss to follow-up, incomplete survey responses, or disengagement from intervention components. Future work should explore ro- bust imputation strate gies and in verse probability weighting methods to address missingness, particularly in longitudinal hybrid trial settings where data sparsity can be an issue. Data A vailability and Code. The data that support the ﬁndings of this study are pro- tected under a Data and Materials Distrib ution Agreement (DMD A). Access to the appli- cation data is av ailable upon request. Code to reproduce simulations is av ailable online at http://github .com/limengbinggz/ddtlcm . A. NO T A TION SUMMAR Y T able 5 summarizes all relev ant notation used in deﬁning the data, estimands, and estima- tors for SMAR T -MR T hybrid designs. B. IDENTIFICA TION RESUL TS For simplicity , we omit the baseline variables X 0 from the conditional expectations. W e ﬁrst prove the ﬁrst equation ( 6 ). For simplicity , we denote the history prior to the ﬁrst SMAR T treatment assignment as H 0 = ( X 0 , X 1 ) . By sequential ignorability SI(a) in Assuption 2.2 , we hav e E  Y t +1  s d t , ( s A t − 1 , a )    X 0  = E  Y t +1  s d t , ( s A t − 1 , a )    H 0  = E  E  Y t +1  s d t , ( s A t − 1 , a )    H 0 , Z 1 = d 1 ( H 0 )  . Recall that by consistency , H t ∗ ( Z 1 , s A t ∗ − 1 ) = H t ∗ and H t ( s Z , s A t − 1 ) = H t . E  E  Y t +1  s d, ( s A t − 1 , a )    H 0 , Z 1 = d 1 ( H 0 )  = E  E  E  Y t +1  s d, ( s A t − 1 , a )    H t ∗ − 1 ( Z 1 , s A t ∗ − 1 )    H 0 , Z 1 = d 1 ( H 0 )  = E  E  E  Y t +1  s d, ( s A t − 1 , a )    H t ∗ − 1    H 0 , Z 1 = d 1 ( H 0 )  = E  E  E  Y t +1  s d t , ( s A t − 1 , a )    H t ∗ − 1 , Z 2 = d 2 ( H t ∗ − 1 )    H 0 , Z 1 = d 1 ( H 0 )  = E  E  E  E  Y t +1  s d, ( s A t − 1 , a )    H t ( s Z , s A t − 1 )    H t ∗ − 1 , Z 2 = d 2 ( H t ∗ − 1 )    H 0 , Z 1 = d 1 ( H 0 )  = E  E  E  E  Y t +1  s d, ( s A t − 1 , a )    H t    H t ∗ , Z 2 = d 2 ( H t ∗ − 1 )    H 0 , Z 1 = d 1 ( H 0 )  , 18 where the ﬁrst and third equations follow from sequential ignorability SI(b), and the second and fourth equations follo w from consistency . F ollowing the same reasoning, the last equation becomes E  E  E  E  Y t +1  s d, ( s A t − 1 , a )    H t , A t    H t ∗ , Z 2 = d 2 ( H t ∗ − 1 )    H 0 , Z 1 = d 1 ( H 0 )  = E  E  E  E  Y t +1  s d, ( s A t − 1 , A t )    H t , A t = a    H t ∗ , Z 2 = d 2 ( H t ∗ − 1 )    H 0 , Z 1 = d 1 ( H 0 )  = E { E { E [ E [ Y t +1 | H t , A t = a ] | H t ∗ , Z 2 = d 2 ( H t ∗ − 1 )] | H 0 , Z 1 = d 1 ( H 0 ) }} , follo wing SI(b) and consistency . Next, we show ( 7 ). For simplicity , we will omit the base- line v ariable X 0 from E  Y t +1  s d, ( s A t − 1 , a )    X 0  , only focusing on E  Y t +1  s d, ( s A t − 1 , a )  . By sequential ignorability , we kno w that E  Y t +1  s z , ( s A t − 1 , a )  1 { z 1 = d 1 ( H 0 ) } 1 { z 2 = d 2 ( H t ∗ − 1 ) } 1 { A t = a }   H t  equals E  Y t +1  s z , ( s A t − 1 , a )    H t  E [1 { z 1 = d 1 ( H 0 ) } 1 { z 2 = d 2 ( H t ∗ − 1 ) } 1 { A t = a }| H t ] . W e then hav e E  Y t +1  s d, ( s A t − 1 , a )  = X z 1 ∈Z 1 X z 2 ∈Z 2 E  Y t +1  s z , ( s A t − 1 , a )  1 { z 1 = d 1 ( X 0 ) } 1 { z 2 = d 2 ( H t ∗ − 1 ) }  = X z 1 ∈Z 1 X z 2 ∈Z 2 E  E  Y t +1  s z , ( s A t − 1 , a )  1 { z 1 = d 1 ( X 0 ) } 1 { z 2 = d 2 ( H t ∗ − 1 ) }   H t  = X z 1 ∈Z 1 X z 2 ∈Z 2 E  E  Y t +1  s z , ( s A t − 1 , a )    H t  · E [1 { z 1 = d 1 ( H 0 ) } 1 { z 2 = d 2 ( H t ∗ − 1 ) }| H t ] 1 { A t = a } P ( s Z = s z | H t ) p t ( a | H t , s Z = s z )  = E " E " X z 1 ∈Z 1 X z 2 ∈Z 2 1 { z 1 = d 1 ( X 1 ) } 1 { z 2 = d 2 ( H t ∗ − 1 ) } P ( Z 1 = z 1 | X 1 ) P ( Z 2 = z 2 | Z 1 = z 1 , H t ∗ − 1 ) 1 { A t = a } p t ( a | H t , s Z = s z ) Y      H t ## = E " X z 1 ∈Z 1 X z 2 ∈Z 2 1 { z 1 = d 1 ( X 1 ) } 1 { z 2 = d 2 ( H t ∗ − 1 ) } P ( Z 1 = z 1 | X 1 ) P ( Z 2 = z 2 | Z 1 = z 1 , H t ∗ − 1 ) 1 { A t = a } p t ( a | H t , s Z = s z ) Y # . C. RA TION ALE FOR 3.3 From Example 3.1 , we can see that β in ( 8 ) can be interpreted as the MR T effect giv en a ﬁxed DTR. On the other hand, η has a more complicated interpretation that it embeds the comparison of proximal effects between DTRs av eraging over MR T treatments, as if the MR T randomization probability were ρ for A t = 1 . T o see this, note that E A t ∼ ρ  Y t +1  s d t , ( s A t − 1 , A t )    X 0  = E A t ∼ ρ n ( A t − ρ ) f t ( s d t , X 0 ) ⊤ β + m t ( s d t , X 0 ) ⊤ η o = ρ h (1 − ρ ) f t ( s d t , X 0 ) ⊤ β + m t ( s d t , X 0 ) ⊤ η i + (1 − ρ ) h (0 − ρ ) f t ( s d t , X 0 ) ⊤ β + m t ( s d t , X 0 ) ⊤ η i = m t ( s d t , X 0 ) ⊤ η . Hence, E A t ∼ ρ  Y t +1  s d t , ( s A t − 1 , A t )  − Y t +1  s d ′ t , ( s A t − 1 , A t )    X 0  =  m t ( s d t , X 0 ) − m t ( s d ′ t , X 0 )  ⊤ η , which is a different quantity than what the scientiﬁc HYBRID SMAR T -MR T DESIGN 19 question (A.D) is concerned with. Ho wev er , we point out that we only care about the mar ginal ef fects comparing MR T treatments or DTRs, which are linear combi- nations of the coefﬁcient ( β , η ) , rather than the coefﬁcient itself. In addition, Let H 0 t ( s d t , s A t − 1 ) ⊆ H t ( s d t , s A t − 1 ) denote a subset of the potential history which MR T ran- domization probability depends on. Note that by assumption 2.2 , the MR T randomization probability P ( A t = a | H 0 t ( s d t , s A t − 1 )) = P ( A t = a | H 0 t ) =: p t ( a | H 0 t ) . Then ( 4 ) corre- sponds to E  Y t +1  s d t , ( s A t − 1 , A t )    X 0  = Z H 0 t 1 X a =0 E  Y t +1  s d t , ( s A t − 1 , a )    X 0  P ( a | H 0 t ( s d t , s A t − 1 )) dP ( H 0 t ( s d t , s A t − 1 )) (26) = Z H 0 t 1 X a =0 h ( a − ρ ) f t ( s d t , X 0 ) ⊤ β + m t ( s d t , X 0 ) ⊤ η i p t ( a | H 0 t ) dP ( H 0 t ) . (27) Although p t ( a | H 0 t ) is kno wn in a h ybrid design study , the inte gral requires knowledge about the distribution of the potential history . This creates difﬁculty in ev aluating ( 4 ) directly . On the other hand, notice that ( 27 ) can be further written as E  Y t +1  s d t , ( s A t − 1 , A t )    X 0  = R H 0 t P 1 a =0 ( a − ρ ) f t ( s d t , X 0 ) p t ( a | H 0 t ) dP ( H 0 t ) R H 0 t P 1 a =0 m t ( s d t , X 0 ) p t ( a | H 0 t ) dP ( H 0 t ) ! ⊤  β η  (28) =: m t ( s d t , X 0 ) ⊤ γ , (29) where m t ( s d t , X 0 ) is an r -dimensional function of s d t and X 0 that combines the shared terms in the design matrices in ( 28 ), and γ is a v ector of coef ﬁcients. This allows us to directly model the e xpected potential proximal outcome of a DTR a veraging over MR T treatment sequences. The coef ﬁcient γ has the interpretation of the comparison of proximal ef fects be- tween DTRs a veraging ov er MR T treatments, under the actual MR T randomization probabil- ity by study design. This is because E  Y t +1  s d t , ( s A t − 1 , A t )  − Y t +1  s d ′ t , ( s A t − 1 , A t )    X 0  =  m t ( s d t , X 0 ) − m t ( s d ′ t , X 0 )  ⊤ γ . D. PR OOF OF PROPOSITION 3.3 Recall that we obtain the estimator ( b β , b γ ) by solving P n U ( β , γ ) = 0 , where U ( β , γ ) = X s d ∈D T X t =1    W S s d W M t  Y t +1 − ( g t ( H t ) − µ t, ¯ z ( H t )) ⊤ α − ( A t − e p t (1 | H t )) f t  s d, x 0  ⊤ β − m t  s d, x 0  ⊤ γ    ( g t ( H t ) − µ t, ¯ Z ,R ( H t )) ( A t − e p t ( A t | H t )) f t  s d, x 0  m t ( ¯ Z t )      , where µ P n ,t ( H t ) = P n P s d ∈D W S s d g t ( H t ) P n P s d ∈D W S s d . W e make the following re gularity and moments conditions. A S S U M P T I O N D.1 . (1) All entries in { Y t +1 , g t ( H t ) } T t =1 hav e ﬁnite fourth moments. 20 (2) Deﬁne µ t ( H t ) = E " P s d ∈D W S s d g t ( H t ) # E " P s d ∈D W S s d # = 1 D E " P s d ∈D W S s d g t ( H t ) # , where D = |D | is the num- ber of DTRs embedded in he SMAR T design. Also denote U 2 ( µ t ) = P s d ∈D W S s d ( µ t ( H t ) − g t ( H t )) . The centering control variable satisﬁes µ P n ,t ( H t ) = µ t ( H t ) + o p (1) , and √ n ( µ P n ,t ( H t ) − µ t ( H t )) = D − 1 P n U 2 ( µ t ) + o p (1) . Therefore, √ n ( µ P n ,t ( H t ) − µ t ( H t )) con verges in distribution to a normal distribution with mean 0 and variance D − 2 E  U ⊗ 2 2  , which has ﬁnite entries. Under standard regularity and moments conditions, we have the solution ( b α, b β , b γ ) ⊤ − ( α, β , γ ) ⊤ con verge asymptotically in distribution to a Gaussian distrib ution with mean 0 as n → ∞ , where ( α, β , γ ) is the solution to equation E U ( β , γ ) = 0 . The v ariance of the asymptotic Gaussian distribution is gi ven by h E ˙ U ( β , γ ) i − 1  E U ( β , γ ) ⊗ 2  h E ˙ U ( β , γ ) i − 1 , where E h ˙ U ( β , γ ) i = E           P s d ∈D T P t =1 W S s d e p t (1 | H t )(1 − e p t (1 | H t )) f t  s d, x 0  ⊗ 2 0 0 P s d ∈D T P t =1 W S s d m t  s d, x 0  ⊗ 2           , ζ t =  Y t +1 − ( g t ( H t ) − E [ g t ( H t ) | ¯ Z ]) ⊤ α − ( A t − e p t (1 | H t )) f t  s d, x 0  ⊤ β − m t  s d, x 0  ⊤ γ  , E  U ( β , γ ) ⊗ 2  = E   X s d ∈D T X t =1 W S s d W M t ζ t  ( A t − e p t ( A t | H t )) f t  s d, x 0  m t ( ¯ Z t )    ⊗ 2 . A consistent estimator of the asymptotic variance is giv en by h P n ˙ U ( b β , b γ ) i − 1 h P n U ( b β , b γ ) i ⊗ 2 h P n ˙ U ( b β , b γ ) i − 1 . Next, we show that the limit ( β , γ ) equals the true causal coef ﬁcient ( β ∗ , γ ∗ ) . F or simplic- ity , we abbreviate the centered control variable g t ( H t ) − E [ g t ( H t ) | ¯ Z ] as g c t ( H t ) . W e ﬁrst solve for α that satisﬁes E      X s d ∈D T X t =1 W S s d W M t  Y t +1 − g c t ( H t ) ⊤ α − ( A t − e p t (1 | H t )) f t  s d, x 0  ⊤ β − m t  s d, x 0  ⊤ γ     g c t ( H t )   = 0 . Note that E   X s d ∈D T X t =1 W S s d W M t ( A t − e p t (1 | H t )) f t  s d, x 0  ⊤ β g c t ( H t )   = 0 , HYBRID SMAR T -MR T DESIGN 21 E   X s d ∈D T X t =1 W S s d W M t m t  s d, x 0  ⊤ γ g c t ( H t )   = 0 . Hence α = E   X s d ∈D T X t =1 W S s d g c t ( H t ) ⊗ 2   − 1 E   X s d ∈D T X t =1 W S s d g c t ( H t ) E  W M t Y t +1 | H t    . W e next solve for β that satisﬁes E      X s d ∈D T X t =1 W S s d W M t  Y t +1 − g c t ( H t ) ⊤ α − ( A t − e p t (1 | H t )) f t  s d, x 0  ⊤ β − m t  s d, x 0  ⊤ γ     ( A t − e p t (1 | H t )) f t  s d, x 0    = 0 . Note that E   X s d ∈D T X t =1 W S s d W M t g c t ( H t ) ⊤ α ( A t − e p t (1 | H t )) f t  s d, x 0    = 0 , E   X s d ∈D T X t =1 W S s d W M t m t  s d, x 0  ⊤ γ ( A t − e p t (1 | H t )) f t  s d, x 0    = 0 , E  W S s d W M t Y t +1 ( A t − e p t (1 | H t )) f t  s d, x 0  = W S s d e p t (1 | H t )(1 − e p t (1 | H t ))  E  Y t +1 | ¯ Z = ¯ z , A t = 1  − E  Y t +1 | ¯ Z = ¯ z , A t = 0  | {z } δ ¯ z A , (30) E h W S s d W M t ( A t − e p t (1 | H t )) 2 f t  s d, x 0  ⊤ β f t  s d, x 0  i = W S s d e p t (1 | H t )(1 − e p t (1 | H t )) f t  s d, x 0  ⊤ β f t  s d, x 0  . Therefore, we hav e β = E   X s d ∈D T X t =1 W S s d e p t (1 | H t )(1 − e p t (1 | H t )) f t  s d, x 0  ⊗ 2   − 1 E   X s d ∈D T X t =1 W S s d e p t (1 | H t )(1 − e p t (1 | H t )) δ ¯ z A f t  s d, x 0    . (31) Under Assumption 2.2 , we ha ve E  Y t +1  ¯ z , ( ¯ A t − 1 , 1)  − Y t +1  ¯ z , ( ¯ A t − 1 , 0)  = E  Y t +1 | ¯ Z = ¯ z , A t = 1  − E  Y t +1 | ¯ Z = ¯ z , A t = 0  = δ ¯ z A . Therefore, If the model ( 8 ) is correct, we hav e β = E   X s d ∈D T X t =1 W S s d e p t (1 | H t )(1 − e p t (1 | H t )) f t  s d, x 0  ⊗ 2   − 1 22 E   X s d ∈D T X t =1 W S s d e p t (1 | H t )(1 − e p t (1 | H t )) f t  s d, x 0  f t  s d, x 0  ⊤ β ∗   = β ∗ . Similarly we can solve for γ that satisﬁes E      X s d ∈D T X t =1 W S s d W M t  Y t +1 − g c t ( H t ) ⊤ α − ( A t − e p t (1 | H t )) f t  s d, x 0  ⊤ β − m t  s d, x 0  ⊤ γ     m t ( ¯ Z )   = 0 , which results in γ = E   X s d ∈D T X t =1 W S s d m t ( ¯ Z ) ⊗ 2   − 1 E   X s d ∈D T X t =1 W S s d m t ( ¯ Z ) E  W M t Y t +1 | H t    . E. COMP ARISON BETWEEN CA USAL ESTIMANDS A CCOUNTING FOR A V AILABILITY Let S t := I t and the centered a vailability be I c t = I t − ψ t , where ψ t = P ¯ d ∈D P T t =1 W S s d e p t (1)(1 − e p t (1)) I t P ¯ d ∈D P T t =1 W S s d e p t (1)(1 − e p t (1)) . W e consider the form of α 1 . Recall that 0 = E   X ¯ d ∈D T X t =1 W S s d W M t h Y t +1 − g c t ( H t ) ⊤ α 0 − ( A t − ρ )  f t ( s d, x 0 ) ⊤ β + I c t α 1  − m t ( s d, s ) ⊤ η i ( A t − ρ ) I c t i . (32) Then ( 32 ) equals 0 = E    X ¯ d ∈D T X t =1 W S s d W M t h Y t +1 − g c t ( H t ) ⊤ α 0 − ( A t − ρ )  f t ( s d, x 0 ) ⊤ β + I c t α 1  − m t ( s d, s ) ⊤ η i ( A t − ρ ) I c t    = E    X ¯ d ∈D T X t =1 W S s d W M t h E [ Y t +1 | H t , A t , I t ] − g c t ( H t ) ⊤ α 0 − ( A t − ρ )  f t ( s d, x 0 ) ⊤ β + I c t α 1  − m t ( s d, s ) ⊤ η i ( A t − ρ ) I c t    = E    X ¯ d ∈D T X t =1 W S s d W M t h E [ Y t +1 | H t , A t , I t ] − ( A t − ρ )  f t ( s d, x 0 ) ⊤ β + I c t α 1 i ( A t − ρ ) I c t    HYBRID SMAR T -MR T DESIGN 23 = E    X ¯ d ∈D T X t =1 W S s d 1 X i =0 1 X a =0 h W M t E [ Y t +1 | H t , A t = a, I t = i ] − ( a − ρ )  f t ( s d, x 0 ) ⊤ β + ( i − ψ t ) α 1 i ( a − ρ )( i − ψ t ) P ( A t = a | H t , I t = i ) P ( I t = i | H t )    , where the last equality averages over A t and I t . When I t = 0 , the MR T treatment A t is not randomized and we let A t = e p t (1) = ρ in this case. Therefore P ( A t = ρ | H t , I t = 0) = 1 and P ( A t = a | H t , I t = 0) = 0 for a ∈ { 0 , 1 } . The last equality then simpliﬁes to 0 = E    X ¯ d ∈D T X t =1 W S s d h ( E [ Y t +1 | H t , A t = 1 , I t = 1] − E [ Y t +1 | H t , A t = 0 , I t = 1]) −  f t ( s d, x 0 ) ⊤ β + (1 − ψ t ) α 1 i e p t (1)(1 − e p t (1))(1 − ψ t ) P ( I t = 1 | H t )    = E    X ¯ d ∈D T X t =1 W S s d h ( E [ E [ Y t +1 | H t , A t = 1 , I t = 1] − E [ Y t +1 | H t , A t = 0 , I t = 1] | s Z , I t = 1]) −  f t ( s d, x 0 ) ⊤ β + (1 − ψ t ) α 1 i e p t (1)(1 − e p t (1))(1 − ψ t ) P ( I t = 1 | H t )    Thus α 1 = E   X s d ∈D T X t =1 W S s d e p t (1)(1 − e p t (1))(1 − ψ t ) 2 P ( I t = 1 | H t )   − 1 E   X s d ∈D T X t =1 W S s d  δ ¯ Z ,I =1 A − δ ¯ Z A  (1 − ψ t ) P ( I t = 1 | H t )   , (33) where δ s Z ,I =1 A = E [ E [ Y t +1 | H t , A t = 1 , I t = 1] − E [ Y t +1 | H t , A t = 0 , I t = 1] | s Z , I t = 1] , δ s Z A = E  E [ Y t +1 | H t , A t = 1] − E [ Y t +1 | H t , A t = 0]   s Z  . The contrast δ s Z ,I =1 A − δ s Z A represents the causal effect of the MR T treatment for a ﬁxed DTR attributed to accounting for av ailability status. If av ailability I t is conditionally mean inde- pendent of the MR T treatment effect given s Z , then δ s Z ,I =1 A = δ s Z A and α 1 = 0 . In the M-Bridge study , howe ver , av ailability is predictive of the SMAR T treatment in the second stage. This is because if I t = 0 for any t ∈ { 2 , 3 , 4 } when a student is classiﬁed as a hea vy drink er , then Z 2 must be a resource email (1) or an online health coach ( − 1) , while I t = 1 for all t ∈ { 1 , 2 , 3 , 4 } implies Z 2 = 0 . Therefore, based on Proposition 4.4 of Shi, W u and Dempsey (2023 ), accounting for av ailability status improv es efﬁciency when estimating the causal ef- fect. On the other hand, ( 33 ) suggests that the proposed estimator for the hybrid design accounts for av ailability in a different manner than WCLS ( Boruvka et al., 2018 ). Recall that WCLS 24 estimates the causal excursion effect of MR T treatments by conditioning on av ailable time points and assumes the model E [ E [ Y t +1 | H t , A t = 1 , I t = 1] − E [ Y t +1 | H t , A t = 0 , I t = 1] | I t = 1] = f ( X 0 ) ⊤ β W C LS . Using the WCLS method, β W C LS = E " T X t =1 e p t (1)(1 − e p t (1)) f t ( x 0 ) ⊗ 2 I t # − 1 E " T X t =1 W S s d e p t (1)(1 − e p t (1)) E [ E [ Y t +1 | H t , A t = 1 , I t = 1] − E [ Y t +1 | H t , A t = 0 , I t = 1] | I t = 1] f t ( x 0 ) I t   . The corresponding parameter in the hybrid method takes the form as in ( 31 ): β = E   X s d ∈D T X t =1 W S s d e p t (1)(1 − e p t (1)) f t  s d, x 0  ⊗ 2   − 1 E   X s d ∈D T X t =1 W S s d e p t (1)(1 − e p t (1)) δ s Z A f t  s d, x 0    . Therefore, β W C LS only uses observations at time points when A t is deli vered and yields a conditional ef fect giv en av ailability . In contrast, β uses observations at all time points and yields a marginal ef fect a veraging out av ailability which is accounted for as an auxiliary v ariable via ( 33 ). HYBRID SMAR T -MR T DESIGN 25 T A B L E 2 Mar ginal effect estimation comparisons among thr ee methods in simulation scenario II, where the MRT randomization pr obability depends on the DTR assignment. Sample size N = 100 . (a) Comparison of effects on the proximal outcome between A t = 1 and 0, for a ﬁxed DTR or averaging over DTRs. Hybrid WCLS Stage Condition T rue Bias SE CP Bias SE CP 1) 1 Fix d 1 = 1 0.1 0 0.05 0.94 -0.02 0.07 0.92 2) 1 Fix d 1 = − 1 0.7 0 0.05 0.94 0.06 0.05 0.81 3) 2 Fix s d = (1 , 1) 0.15 0 0.07 0.95 -0.01 0.07 0.93 4) 2 Fix s d = (1 , − 1) 0.05 0 0.08 0.93 -0.03 0.13 0.9 5) 2 Fix s d = ( − 1 , 1) 0.89 0 0.08 0.96 0.07 0.1 0.86 6) 2 Fix s d = ( − 1 , − 1) 0.51 0 0.07 0.93 0.04 0.07 0.91 7) 1 A veraging DTR 0.4 0 0.04 0.95 0.02 0.04 0.92 8) 2 A veraging DTR 0.4 0 0.04 0.95 0.02 0.04 0.92 (b) Comparison of effects on the proximal outcome between DTRs a veraging over FTS interv entions. Hybrid WR Stage Contrast T rue Bias SE CP Bias SE CP mRE sdRE 1) 1 d 1 = 1 vs -1 0.4 0 0.06 0.95 0 0.06 0.95 1 0 2) 2 s d = (1 , 1) vs (1, -1) -0.2 0 0.08 0.95 0 0.08 0.95 1 0 3) 2 s d = (1 , 1) vs (-1, 1) 0.3 0 0.08 0.95 0 0.08 0.95 1 0 4) 2 s d = (1 , 1) vs (-1, -1) 0.3 0 0.08 0.96 0 0.08 0.96 1 0 5) 2 s d = (1 , − 1) vs (-1, 1) 0.5 0 0.08 0.92 0 0.08 0.92 1 0 6) 2 s d = (1 , − 1) vs (-1, -1) 0.5 0 0.08 0.94 0 0.08 0.94 1 0 7) 2 s d = ( − 1 , 1) vs (-1, -1) 0 0 0.07 0.93 0 0.07 0.93 1 0 (c) Comparison of effects on the proximal outcome between DTRs for a ﬁx ed MR T treatment. Hybrid Stage Contrast T rue Bias SE CP 1) 1 d 1 = 1 vs -1 0.62 0 0.11 0.91 2) 2 s d = (1 , 1) vs (1, -1) -0.24 0 0.11 0.96 3) 2 s d = (1 , 1) vs (-1, 1) 0.46 0 0.09 0.94 4) 2 s d = (1 , 1) vs (-1, -1) 0.48 0 0.1 0.95 5) 2 s d = (1 , − 1) vs (-1, 1) 0.7 -0.01 0.1 0.93 6) 2 s d = (1 , − 1) vs (-1, -1) 0.73 0 0.11 0.94 7) 2 s d = ( − 1 , 1) vs (-1, -1) 0.03 0 0.08 0.94 8) 1 d 1 = 1 vs -1 0.02 0 0.1 0.89 9) 2 s d = (1 , 1) vs (1, -1) -0.14 0 0.09 0.93 10) 2 s d = (1 , 1) vs (-1, 1) -0.28 0 0.11 0.92 11) 2 s d = (1 , 1) vs (-1, -1) 0.12 0 0.1 0.93 12) 2 s d = (1 , − 1) vs (-1, 1) -0.14 0 0.1 0.91 13) 2 s d = (1 , − 1) vs (-1, -1) 0.27 0 0.09 0.92 14) 2 s d = ( − 1 , 1) vs (-1, -1) 0.4 0 0.1 0.96 26 T A B L E 3 Causal estimands for the M-Bridge study Scientiﬁc Question Causal Estimand (1) What is the effect on maximum number of drinks for an SI prompt compared to a PS prompt in Stage 1, giv en a ﬁxed initial timing of PNF d 1 ∈ {− 1 , 1 } ? E  Y t +1  d 1 ,  s a t − 1 , D (1 , I t ( d 1 , s a t − 1 ))  − Y t +1  d 1 ,  s a t − 1 , D (0 , I t ( d 1 , s a t − 1 ))    X 0  , t = 2 , 3 , 4 . (2) What is the effect on maximum number of drinks for early compared to late timing of PNF in Stage 1, giv en a ﬁxed prompt type A t = a ? E  Y t +1  1 ,  s a t − 1 , D ( a, I t (1 , s a t − 1 ))  − Y t +1  − 1 ,  s a t − 1 , D ( a, I t ( − 1 , s a t − 1 ))    X 0  , t = 2 , 3 , 4 . (3) What is the effect on maximum number of drinks for an SI prompt compared to a PS prompt in Stage 1, av eraging over initial timings of PNF? X l 1 ∈{− 1 , 1 } P ( d 1 = l 1 ) E  Y t +1  l 1 ,  s a t − 1 , D (1 , I t ( l 1 , s a t − 1 ))  − Y t +1  l 1 ,  s a t − 1 , D (0 , I t ( l 1 , s a t − 1 ))    X 0  , t = 2 , 3 , 4 . (4) What is the effect on maximum number of drinks for early compared to late timing of PNF , a veraging ov er prompt types? E  Y t +1  1 ,  s A t − 1 , D ( A t , I t (1 , s A t − 1 ))  − Y t +1  − 1 ,  s A t − 1 , D ( A t , I t ( − 1 , s A t − 1 ))    X 0  . T A B L E 4 Estimated coefﬁcients and 95% conﬁdence intervals in M-Bridg e data analysis Parameter V ariable Estimate 95% CI β 0 ( a t − 0 . 5) 0.47 (0.36, 0.57) β 1 ( a t − 0 . 5) d 1 0.02 (-0.09, 0.12) β 2 ( a t − 0 . 5) t 0.46 (0.3, 0.61) β 3 ( a t − 0 . 5) d 1 t -0.44 (-0.6, -0.29) γ 0 Intercept 3.99 (3.96, 4.01) γ 1 d 1 -0.01 (-0.04, 0.02) γ 2 δ t d 2 0.20 (-0.17, 0.57) γ 3 d 1 δ t d 2 -0.36 (-0.73, 0.01) T A B L E 5 Notation Summary for Hybrid SMART -MRT Design Notation Description t ∗ T ime of entering Stage 2. X 0 Pre-treatment baseline cov ariates. X t Contextual and indi vidual information at time t . Z 1 , Z 2 SMAR T treatments in Stages 1 and 2, respectively . s Z t Sequence of SMAR T treatments up to time t ( s Z t = Z 1 for t < t ∗ , s Z t = ( Z 1 , Z 2 ) for t ≥ t ∗ ). A t MR T treatment at time t . Y t +1 Proximal response subsequent to treatment A t . s A t , s X t , s Y t Sequence of MR T treatments, conte xtual information, and proximal responses up to time t , respectiv ely . H t =  s X t , s Z t , s A t , s Y t  Complete history of observed data up to time t . Y t +1 ( s z t , s a t ) Potential outcome for proximal response under speciﬁc SMAR T and MR T treatment sequences. R T ailoring v ariable for Stage 2 SMAR T treatment (e.g., responder status). s d = ( d 1 , d 2 ) Dynamic treatment regime (DTR), where d 1 and d 2 are decision rules for SMAR T treatments in Stages 1 and 2. f t ( s d ) p -dimensional vector functions of DTRs s d and baseline v ariables X 0 m t ( s d ) q -dimensional vector functions of DTRs s d and baseline v ariables X 0 S t r -dimensional time-v arying effect moderator ψ t ( s d ) r -dimensional centering function of S t HYBRID SMAR T -MR T DESIGN 27 F . SCIENTIFIC QUESTIONS FOR A HYBRID DESIGN W e list additional scientiﬁc questions of interest to a hybrid SMAR T -MR T study in T able 6 . 28 T A B L E 6 Scientiﬁc questions about pr oximal effects and model par ameters for a hybrid SMART -MRT design Scientiﬁc question T ype Estimand Model parameters in Example 3.1 (a) Does the effects of two DTRs, e.g., s d = (1 , 1) vs (1 , − 1) , on the proximal outcome at time t ≥ t ∗ differ for a ﬁxed MR T treat- ment A t = 1 ? Interaction E  Y t +1  s d = (1 , 1) , ( s A t − 1 , 1)  − Y t +1  s d = (1 , − 1) , ( s A t − 1 , 1)  ( β 0 + β 1 + 2 β 2 + 2 β 3 ) / 2 + 2 η 2 + 2 η 3 (b) Does the effect of MR T treatment on the proximal outcome at time t < t ∗ vary by ﬁrst-stage regimes, d 1 = 1 vs d 1 = − 1 ? Interaction E  Y t +1  d 1 = 1 , ( s A t − 1 , 1)  − Y t +1  d 1 = 1 , ( s A t − 1 , 0)  − E  Y t +1  d 1 = − 1 , ( s A t − 1 , 1)  − Y t +1  d 1 = − 1 , ( s A t − 1 , 0)  2 β 1 (c) Does the effect of MR T treatment on the proximal outcome at time t > t ∗ vary by DTRs, e.g., s d = (1 , 1) vs (1 , − 1) ? Interaction E  Y t +1  s d = (1 , 1) , ( s A t − 1 , 1)  − Y t +1  s d = (1 , 1) , ( s A t − 1 , 0)  − E  Y t +1  s d = (1 , − 1) , ( s A t − 1 , 1)  − Y t +1  s d = (1 , − 1) , ( s A t − 1 , 0)  2 β 2 + 2 β 3 (d) Do the ef fects of MR T treatments dif fer on the proximal outcome at time t > t ∗ av er- aging ov er DTRs? Main X ¯ d ∈D P ( ¯ d ) { E  Y t +1  s d, ( s A t − 1 , 1)  − Y t +1  s d, ( s A t − 1 , 0)  β 0 (e) Do the ef fects of tw o DTR, e.g., s d = (1 , 1) and (1 , 1) , differ on the proximal outcome at time t > t ∗ av eraging over MR T treat- ments? Main E  Y t +1  s d = (1 , 1) , s A t  − Y t +1  s d = (1 , − 1) , s A t  2 γ 2 + 2 γ 3 HYBRID SMAR T -MR T DESIGN 29 G. ADDITION AL DET AILS OF SIMULA TIONS G.1 Detailed Simulation Scenarios. In scenario II, we let p t (1 | H t ) = p t (1 | Z 1 , R, Z 2 ) = 0 . 6 I { Z 1 = 1 } + 0 . 4 I { Z 1 = − 1 } − 0 . 2 I { C t Z 2 = 1 } + 0 . 2 I { C t Z 2 = − 1 } be DTR-dependent, The responder status is history-dependent, with logit( p ( R = 1 | H t ∗ )) = − 0 . 62 + e X 1 + ( A t ∗ − 1 − p t ∗ − 1 (1 | H t ∗ − 1 )) + 0 . 5 Z 1 , where e X t = X t − E [ X t | A t − 1 , H t − 1 ] is the centered state. The realized values of the probability of being a responder are about 0.45 and 0.27 for indi viduals with Z 1 = 1 and − 1 , respecti vely . G.2 Baseline Model Setup in Simulations. In WCLS, the numerator probability in the MR T weight was set as e p t (1) = 1 2 . The control variables g t ( H t ) included non-centered X t and X t Z 1 . The mar ginal model of WCLS analysis included DTRs as moderators is speciﬁed as E [ Y t +1  s d, ( ¯ A t − 1 , 1)  − Y t +1  s d, ( ¯ A t − 1 , 0)  ] = f t ( s d ) ⊤ β WCLS . (34) The coef ﬁcient b β WCLS is obtained by solving the estimating equation P n T X t =1 W M t h Y t +1 − g t ( H t ) ⊤ α 0 − ( A t − ρ ) f t ( s d ) ⊤ β WCLS i  g t ( H t ) − µ t, s d ( H t ) ( A t − ρ ) f t  s d   = 0 . (35) Note that in ( 35 ), the control v ariables g t ( H t ) are not centered as in ( 16 ) of the proposed approach. In both simulation scenarios I and II, we let f t ( s d ) = (1 , d 1 , δ t d 2 , δ t d 1 d 2 ) ⊤ , taking the same form as in the hybrid method. The WR method focuses only on the mar ginal mean proximal outcome under a DTR av eraging over all MR T treatment sequences, and does not account for time-v arying control v ariables. The marginal mean model is speciﬁed as E [ Y t +1  s d, ¯ A t  ] = m t ( s d ) ⊤ β WR . (36) The coef ﬁcient b β WR is obtained by solving the estimating equation P n X s d ∈D T X t =1 W S t h Y t +1 − m t ( s d ) ⊤ β WR i m t  s d  = 0 . (37) In scenario I, m t ( s d ) = (1 , d 1 , δ t d 2 , δ t d 1 d 2 ) ⊤ ; in scenario II, m t ( s d ) = (1 − δ t , (1 − δ t ) d 1 , δ t , δ t d 1 , δ t d 2 , δ t d 1 d 2 ) ⊤ . The moderators f t were the same as ( 13 ) in both scenarios I and II, respecti vely . G.3 Expression of T rue Marginal Effects in Simulation. W e consider three types of marginal quantities: (1) the marginal means of the proximal outcome at a ﬁxed DTR and A t = a , (2) the marginal means of the proximal outcome for a giv en A t = a , av eraging over all DTRs, and (3) the marginal means of the proximal outcome for a giv en DTR, av eraging ov er all A t ’ s. G.3.1 Mar ginal Mean At A F ixed DTR And A t = a . W e compute E [ Y ( A t = a, ¯ Z = ¯ z ] for all a ∈ { 0 , 1 } and ¯ z ∈ D . Based on the causal assumptions, we kno w that E [ Y t +1 (( ¯ A t − 1 , a ) , ¯ Z = ¯ z )] = E [ Y t +1 | A t = a, ¯ Z = ¯ z ] . In addition, since E [ e X t ] = 0 for all 1 ≤ t ≤ T in the data generating process ( 24 ), we hav e that E [( A t − p t (1 | H t )) e X t ] = 0 , E [( A t − p t (1 | H t )) e X t Z 1 ] = 0 , and E [ e X t Z 1 ] = 0 . Recall that H M t = H t \ { ¯ Z } denotes the history up to time t e xcept for SMAR T factors. W e now consider Stage 1 and Stage 2 separately . 30 Stage 1. When t < t ∗ , E [ Y t +1  ( ¯ A t − 1 , a ) , D 1 = d 1  ] = X z 1 ∈{− 1 , 1 } I { z 1 = d 1 ( X 0 ) } E [ Y t +1  ( ¯ A t − 1 , a ) , Z 1 = z 1  ] = X z 1 ∈{− 1 , 1 } I { z 1 = d 1 ( X 0 ) } E [ E [ Y | A t = a, Z 1 = z 1 , H M t ]] = X z 1 ∈{− 1 , 1 } I { z 1 = d 1 ( X 0 ) } [( a − p t (1 | z 1 ))( β ∗ 0 + β ∗ 1 z 1 ) + γ ∗ 0 + γ ∗ 1 z 1 ] . (38) Stage 2. When t > t ∗ , E  Y t +1  ¯ d = ( d 1 , d 2 ) , ( ¯ A t − 1 , a )  = X z ∈Z E  I { z 1 = d 1 ( X 0 ) } I { z 2 = d 2 ( H t ∗ ( z 1 )) } Y  ( ¯ A t − 1 , a ) , ¯ Z = ( z 1 , z 2 )  = X z ∈Z 1 X r =0 E [ I { z 1 = d 1 ( X 0 ) } I { z 2 = d 2 ( H t ∗ ( z 1 ) , R ( z 1 ) = r ) } Y t +1  ( ¯ A t − 1 , a ) , ¯ Z = ( z 1 , z 2 ) , R ( z 1 ) = r  P ( R ( z 1 ) = r )  = X z ∈Z 1 X r =0 E [ I { z 1 = d 1 ( X 0 ) } I { z 2 = d 2 ( Z 1 = z 1 , R = r ) } E  Y t +1 | A t = a, ¯ Z = ( z 1 , z 2 ) , R = r, H M t  P ( R = r | Z 1 = z 1 )  = X z ∈Z I { z 1 = d 1 ( X 0 ) } I { z 2 = d 2 ( Z 1 = z 1 , R = r ) }  ( A t − p t (1 | ¯ Z = ( z 1 , z 2 ) , R = 0))( β ∗ 0 + β ∗ 1 z 1 + β ∗ 2 z 2 + β ∗ 3 z 1 z 2 ) + γ ∗ 0 + γ ∗ 1 z 1 + γ ∗ 2 z 2 + γ ∗ 3 z 1 z 2 ] · P ( R = 0 | Z 1 = z 1 )+  ( A t − p t (1 | ¯ Z = ( z 1 , 0) , R = 1))( β ∗ 0 + β ∗ 1 z 1 ) + γ ∗ 0 + γ ∗ 1 z 1  · P ( R = 1 | Z 1 = z 1 )  . (39) G.3.2 Mar ginal Mean at A F ixed A t = a A veraging o ver DTRs. Stage 1. When t < t ∗ , since we hav e equal probabilities of assigning two SMAR T factors, the marginal means a veraging ov er all DTRs can be expressed as X d 1 {− 1 , 1 } E [ Y t +1  ( ¯ A t − 1 , a ) , d 1  ] P ( D 1 = d 1 ) = 1 2 E [ Y t +1  ( ¯ A t − 1 , a ) , 1  ] + 1 2 E [ Y t +1  ( ¯ A t − 1 , a ) , − 1  ] = 1 2 [( a − p t (1 | z 1 = 1))( β ∗ 0 + β ∗ 1 ) + γ ∗ 0 + γ ∗ 1 ] + 1 2 [( a − p t (1 | z 1 = − 1))( β ∗ 0 − β ∗ 1 ) + γ ∗ 0 − γ ∗ 1 ] = 1 2 [( a − p t (1 | z 1 = 1))( β ∗ 0 + β ∗ 1 ) + ( a − p t (1 | z 1 = − 1))( β ∗ 0 − β ∗ 1 )] + γ ∗ 0 . HYBRID SMAR T -MR T DESIGN 31 As a result, the marginal effect of the MR T treatment on the proximal outcome a veraging ov er DTRs is P d 1 {− 1 , 1 } E [ Y t +1  ( ¯ A t − 1 , 1) , d 1  − Y t +1  ( ¯ A t − 1 , 0) , d 1  ] P ( D 1 = d 1 ) = β ∗ 0 . Stage 2. When t > t ∗ , the marginal means a veraging ov er all DTRs can be expressed as X ¯ d ∈D E [ Y t +1  ( ¯ A t − 1 , a ) , ¯ D = ( d 1 , d 2 )  ] P ( D 1 = d 1 , D 2 = d 2 ) = X ¯ d ∈{− 1 , 1 } 2 E [ Y t +1  ( ¯ A t − 1 , a ) , ¯ D = ( d 1 , d 2 )  ] · 1 2 · 1 2 = 1 4  ( a − p t (1 | ¯ Z = (1 , 1) , R = 0))( β ∗ 0 + β ∗ 1 + β ∗ 2 + β ∗ 3 ) + γ ∗ 0 + γ ∗ 1 + γ ∗ 2 + γ ∗ 3  P ( R = 0 | Z 1 = 1) +  ( a − p t (1 | ¯ Z = (1 , 0) , R = 1))( β ∗ 0 + β ∗ 1 ) + γ ∗ 0 + γ ∗ 1  P ( R = 1 | Z 1 = 1) +  ( a − p t (1 | ¯ Z = (1 , − 1) , R = 0))( β ∗ 0 + β ∗ 1 − β ∗ 2 − β ∗ 3 ) + γ ∗ 0 + γ ∗ 1 − γ ∗ 2 − γ ∗ 3  P ( R = 0 | Z 1 = 1) +  ( a − p t (1 | ¯ Z = (1 , 0) , R = 1))( β ∗ 0 + β ∗ 1 ) + γ ∗ 0 + γ ∗ 1  P ( R = 1 | Z 1 = 1) +  ( a − p t (1 | ¯ Z = ( − 1 , 1) , R = 0))( β ∗ 0 − β ∗ 1 + β ∗ 2 − β ∗ 3 ) + γ ∗ 0 − γ ∗ 1 + γ ∗ 2 − γ ∗ 3  P ( R = 0 | Z 1 = − 1) +  ( a − p t (1 | ¯ Z = ( − 1 , 0) , R = 1))( β ∗ 0 − β ∗ 1 ) + γ ∗ 0 − γ ∗ 1  P ( R = 1 | Z 1 = − 1) +  ( a − p t (1 | ¯ Z = ( − 1 , − 1) , R = 0))( β ∗ 0 − β ∗ 1 − β ∗ 2 + β ∗ 3 ) + γ ∗ 0 − γ ∗ 1 − γ ∗ 2 + γ ∗ 3  P ( R = 0 | Z 1 = − 1) +  ( a − p t (1 | ¯ Z = ( − 1 , 0) , R = 1))( β ∗ 0 − β ∗ 1 ) + γ ∗ 0 − γ ∗ 1  P ( R = 1 | Z 1 = − 1)  = 1 4  ( a − p t (1 | ¯ Z = (1 , 1) , R = 0))( β ∗ 0 + β ∗ 1 + β ∗ 2 + β ∗ 3 ) P ( R = 0 | Z 1 = 1) + ( a − p t (1 | ¯ Z = (1 , − 1) , R = 0))( β ∗ 0 + β ∗ 1 − β ∗ 2 − β ∗ 3 ) P ( R = 0 | Z 1 = 1) + ( a − p t (1 | ¯ Z = ( − 1 , 1) , R = 0))( β ∗ 0 − β ∗ 1 + β ∗ 2 − β ∗ 3 ) P ( R = 0 | Z 1 = − 1) + ( a − p t (1 | ¯ Z = ( − 1 , − 1) , R = 0))( β ∗ 0 − β ∗ 1 − β ∗ 2 + β ∗ 3 ) P ( R = 0 | Z 1 = − 1) + 2( a − p t (1 | ¯ Z = (1 , 0) , R = 1))( β ∗ 0 + β ∗ 1 ) P ( R = 1 | Z 1 = 1) + 2( a − p t (1 | ¯ Z = ( − 1 , 0) , R = 1)( β ∗ 0 − β ∗ 1 ) P ( R = 1 | Z 1 = − 1) + 4 γ ∗ 0 } . As a result, the marginal ef fect of the MR T treatment on the proximal outcome a veraging ov er DTRs is P ¯ d ∈D E [ Y t +1  ( ¯ A t − 1 , 1) , ¯ D = ( d 1 , d 2 )  − Y t +1  ( ¯ A t − 1 , 0) , ¯ D = ( d 1 , d 2 )  ] P ( D 1 = d 1 , D 2 = d 2 ) = β ∗ 0 . G.3.3 Mar ginal Mean at A F ixed DTR A veraging over A t . Stage 1. When t < t ∗ , the marginal mean of the proximal outcome at a ﬁx ed DTR a veraging ov er MR T treatments is E  Y t +1  ¯ A t , D 1 = d 1  = X z 1 ∈{− 1 , 1 } 1 X a =0 E  I { z 1 = d 1 ( X 0 ) } Y t +1  ( ¯ A t − 1 , a ) , Z 1 = z 1  P ( A t ( z 1 ) = a ) = X z 1 ∈{− 1 , 1 } 1 X a =0 E  I { z 1 = d 1 ( X 0 ) } E  Y t +1 | A t = a, Z 1 = z 1 | H M t  P ( A t = a | Z 1 = z 1 ) 32 = X z 1 ∈{− 1 , 1 } I { z 1 = d 1 ( X 0 ) } { [(1 − p t (1 | z 1 ))( β ∗ 0 + β ∗ 1 z 1 ) + γ ∗ 0 + γ ∗ 1 z 1 ] p t (1 | z 1 ) + [ − p t (1 | z 1 )( β ∗ 0 + β ∗ 1 z 1 ) + γ ∗ 0 + γ ∗ 1 z 1 ] (1 − p t (1 | z 1 )) } = X z 1 ∈{− 1 , 1 } I { z 1 = d 1 ( X 0 ) } ( γ ∗ 0 + γ ∗ 1 z 1 ) . Stage 2. When t > t ∗ , the marginal mean of the proximal outcome at a ﬁx ed DTR a veraging ov er MR T treatments is E  Y t +1  ¯ Z = ( z 1 , z 2 ) , ¯ A t  = X z ∈Z 1 X a =0 1 X r =0 E  I { z 1 = d 1 ( X 0 ) } I { z 2 = d 2 ( H t ∗ ( z 1 ) , R ( z 1 ) = r ) } Y  ( ¯ A t − 1 , a ) , ¯ Z = ( z 1 , z 2 )  · P ( A t ( z 1 , r , z 2 ) = a ) P ( R ( z 1 ) = r ) = X z ∈Z 1 X a =0 1 X r =0 E [ I { z 1 = d 1 ( X 0 ) } I { z 2 = d 2 ( Z 1 = z 1 , R = r ) } E  Y | A t = a, ¯ Z = ( z 1 , z 2 ) , R = r, | H M t  · P ( A t = a | z 1 , r , z 2 ) P ( R = r | Z 1 = z 1 ) =  (1 − p t (1 | ¯ Z = ( z 1 , z 2 ) , R = 0))( β ∗ 0 + β ∗ 1 z 1 + β ∗ 2 z 2 + β ∗ 3 z 1 z 2 ) + γ ∗ 0 + γ ∗ 1 z 1 + γ ∗ 2 z 2 + γ ∗ 3 z 1 z 2 ] · p t (1 | ¯ Z = ( z 1 , z 2 ) , R = 0) P ( R = 0 | Z 1 = z 1 ) +  − p t (1 | ¯ Z = ( z 1 , z 2 ) , R = 0)( β ∗ 0 + β ∗ 1 z 1 + β ∗ 2 z 2 + β ∗ 3 z 1 z 2 ) + γ ∗ 0 + γ ∗ 1 z 1 + γ ∗ 2 z 2 + γ ∗ 3 z 1 z 2  · (1 − p t (1 | ¯ Z = ( z 1 , z 2 ) , R = 0)) P ( R = 0 | Z 1 = z 1 ) +  (1 − p t (1 | ¯ Z = ( z 1 , 0) , R = 1))( β ∗ 0 + β ∗ 1 z 1 ) + γ ∗ 0 + γ ∗ 1 z 1  · p t (1 | ¯ Z = ( z 1 , 0) , R = 1) + γ ∗ 3 z 1 z 2 ] · p t (1 | ¯ Z = ( z 1 , z 2 ) , R = 0) P ( R = 1 | Z 1 = z 1 ) +  − p t (1 | ¯ Z = ( z 1 , 0) , R = 1)( β ∗ 0 + β ∗ 1 z 1 ) + γ ∗ 0 + γ ∗ 1 z 1  · (1 − p t (1 | ¯ Z = ( z 1 , 0) , R = 1)) · P ( R = 1 | Z 1 = z 1 ) = [ γ ∗ 0 + γ ∗ 1 z 1 + γ ∗ 2 z 2 + γ ∗ 3 z 1 z 2 ] P ( R = 0 | Z 1 = z 1 ) + [ γ ∗ 0 + γ ∗ 1 z 1 ] P ( R = 1 | Z 1 = z 1 ) . HYBRID SMAR T -MR T DESIGN 33 G.4 Additional Simulation Results. T ables 7 and 8 display results from 500 replica- tions of simulation scenarios I and II, respecti vely , under sample size N = 400 . T A B L E 7 Mar ginal effect estimation comparisons among thr ee methods in simulation scenario I, where the MRT randomization pr obability . Sample size N = 400 . (a) Comparison of effects between A t = 1 and 0, for a ﬁx ed DTR or averaging o ver DTRs. Hybrid WCLS Stage Condition T rue Bias SE CP Bias SE CP 1) 1 Fix d 1 = 1 0.1 0 0.03 0.98 -0.03 0.03 1 2) 1 Fix d 1 = − 1 0.7 0 0.02 0.98 0.06 0.03 0.3 3) 2 Fix s d = (1 , 1) 0.14 0 0.03 0.98 -0.01 0.03 0.98 4) 2 Fix s d = (1 , − 1) 0.06 0 0.03 0.98 -0.04 0.06 0.99 5) 2 Fix s d = ( − 1 , 1) 0.86 0 0.03 0.98 0.09 0.05 0.53 6) 2 Fix s d = ( − 1 , − 1) 0.54 0 0.03 0.97 0.04 0.03 0.81 7) 1 A veraging DTR 0.4 0 0.02 0.98 0.02 0.02 0.85 8) 2 A veraging DTR 0.4 0 0.02 0.98 0.02 0.02 0.85 (b) Comparison of effects on the proximal outcome between DTRs a veraging over MR T treatments. Hybrid WR Stage Contrast T rue Bias SE CP Bias SE CP mRE sdRE 1) 1 d 1 = 1 vs -1 0.4 0 0.03 0.98 0 0.03 0.99 1.20 0.06 2) 2 s d = (1 , 1) vs (1, -1) -0.16 0 0.04 0.96 0 0.04 0.96 1.03 0.11 3) 2 s d = (1 , 1) vs (-1, 1) 0.32 0 0.04 0.97 0 0.04 0.98 1.05 0.06 4) 2 s d = (1 , 1) vs (-1, -1) 0.32 0 0.04 0.98 0 0.04 0.99 1.09 0.07 5) 2 s d = (1 , − 1) vs (-1, 1) 0.48 0 0.04 0.97 0 0.04 0.98 1.18 0.08 6) 2 s d = (1 , − 1) vs (-1, -1) 0.48 0 0.03 0.97 0 0.04 0.98 1.24 0.08 7) 2 s d = ( − 1 , 1) vs (-1, -1) 0 0 0.03 0.96 0 0.03 0.96 1.05 0.06 (c) Comparison of effects on the proximal outcome between DTRs for a ﬁx ed ﬁxed MR T treatment. Hybrid Stage Contrast T rue Bias SE CP 1) 1 d 1 = 1 vs -1 0.7 0 0.03 0.98 2) 2 s d = (1 , 1) vs (1, -1) -0.2 0 0.04 0.98 3) 2 s d = (1 , 1) vs (-1, 1) 0.68 0 0.05 0.97 4) 2 s d = (1 , 1) vs (-1, -1) 0.52 0 0.05 0.98 5) 2 s d = (1 , − 1) vs (-1, 1) 0.88 0 0.04 0.98 6) 2 s d = (1 , − 1) vs (-1, -1) 0.72 0 0.04 0.97 7) 2 s d = ( − 1 , 1) vs (-1, -1) -0.16 0 0.04 0.97 8) 1 d 1 = 1 vs -1 0.1 0 0.03 0.96 9) 2 s d = (1 , 1) vs (1, -1) -0.12 0 0.04 0.94 10) 2 s d = (1 , 1) vs (-1, 1) -0.04 0 0.04 0.98 11) 2 s d = (1 , 1) vs (-1, -1) 0.12 0 0.05 0.97 12) 2 s d = (1 , − 1) vs (-1, 1) 0.08 0 0.04 0.96 13) 2 s d = (1 , − 1) vs (-1, -1) 0.24 0 0.04 0.95 14) 2 s d = ( − 1 , 1) vs (-1, -1) 0.16 0 0.04 0.95 34 T A B L E 8 Mar ginal effect estimation comparisons among thr ee methods in simulation scenario II, where the MRT randomization pr obability depends on the DTR assignment. Sample size N = 400 . (a) Comparison of effects on the proximal outcome between A t = 1 and 0, for a ﬁxed DTR or averaging over DTRs. Hybrid WCLS Stage Condition T rue Bias SE CP Bias SE CP 1) 1 Fix d 1 = 1 0.1 0 0.03 0.94 -0.02 0.03 0.9 2) 1 Fix d 1 = − 1 0.7 0 0.02 0.96 0.06 0.03 0.48 3) 2 Fix s d = (1 , 1) 0.15 0 0.04 0.93 -0.01 0.03 0.93 4) 2 Fix s d = (1 , − 1) 0.05 0 0.04 0.96 -0.03 0.07 0.92 5) 2 Fix s d = ( − 1 , 1) 0.89 0 0.04 0.96 0.08 0.05 0.69 6) 2 Fix s d = ( − 1 , − 1) 0.51 0 0.03 0.96 0.03 0.03 0.83 7) 1 A veraging DTR 0.4 0 0.02 0.94 0.02 0.02 0.88 8) 2 A veraging DTR 0.4 0 0.02 0.94 0.02 0.02 0.88 (b) Comparison of effects on the proximal outcome between DTRs a veraging over MR T treatments. Hybrid WR Stage Contrast T rue Bias SE CP Bias SE CP mRE sdRE 1) 1 d 1 = 1 vs -1 0.4 0 0.03 0.94 0 0.03 0.94 1 0 2) 2 s d = (1 , 1) vs (1, -1) -0.2 0 0.04 0.96 0 0.04 0.96 1 0 3) 2 s d = (1 , 1) vs (-1, 1) 0.3 0 0.04 0.96 0 0.04 0.96 1 0 4) 2 s d = (1 , 1) vs (-1, -1) 0.3 0 0.04 0.95 0 0.04 0.95 1 0 5) 2 s d = (1 , − 1) vs (-1, 1) 0.5 0 0.04 0.95 0 0.04 0.95 1 0 6) 2 s d = (1 , − 1) vs (-1, -1) 0.5 0 0.04 0.94 0 0.04 0.94 1 0 7) 2 s d = ( − 1 , 1) vs (-1, -1) 0 0 0.04 0.96 0 0.04 0.96 1 0 (c) Comparison of effects on the proximal outcome between DTRs for a ﬁx ed MR T treatment. Hybrid Stage Contrast T rue Bias SE CP 1) 1 d 1 = 1 vs -1 0.62 0 0.05 0.93 2) 2 s d = (1 , 1) vs (1, -1) -0.24 0 0.06 0.96 3) 2 s d = (1 , 1) vs (-1, 1) 0.46 0 0.05 0.97 4) 2 s d = (1 , 1) vs (-1, -1) 0.48 0 0.05 0.96 5) 2 s d = (1 , − 1) vs (-1, 1) 0.7 0 0.05 0.95 6) 2 s d = (1 , − 1) vs (-1, -1) 0.73 0 0.06 0.94 7) 2 s d = ( − 1 , 1) vs (-1, -1) 0.03 0 0.04 0.96 8) 1 d 1 = 1 vs -1 0.02 0 0.05 0.91 9) 2 s d = (1 , 1) vs (1, -1) -0.14 0 0.05 0.95 10) 2 s d = (1 , 1) vs (-1, 1) -0.28 0 0.06 0.94 11) 2 s d = (1 , 1) vs (-1, -1) 0.12 0 0.05 0.91 12) 2 s d = (1 , − 1) vs (-1, 1) -0.14 0 0.05 0.93 13) 2 s d = (1 , − 1) vs (-1, -1) 0.27 0 0.04 0.93 14) 2 s d = ( − 1 , 1) vs (-1, -1) 0.41 0 0.05 0.95 HYBRID SMAR T -MR T DESIGN 35 H. M-BRIDGE STUD Y D A T A CONSTR UCTION The study cohort analyzed in Section 6 was part of a larger trial to inform the de vel- opment of an API for reducing binge drinking among ﬁrst-year under graduates in a large Midwestern Univ ersity for the 2019–2020 academic year ( N = 891 , 62.4% female, 76.8% White; see Patrick et al. (2021 ) and Carpenter et al. (2023 ), for baseline characteristics). T wo-thirds ( N = 591 ) of the students in volved in the lar ger trial were randomized to a Stage 1 intervention group at the start of the fall semester, whereas the remaining students were randomized to a control group. Among the 591 students, 295 of them were assigned to early Stage 1 intervention and 296 to late Stage 1 intervention, re garding the timing of recei v- ing personalized normati ve feedback (PNF). These 591 students were subsequently in volv ed in an MR T design, where they were randomized biweekly to recei ve one of the two types of self-relev ant prompts (self-interest or pro-social) that encouraged completion of four bi- weekly self-monitoring (SM) surve ys of alcohol use. Not all participants were randomized to a prompt at every assessment time. As part of the API, whenev er a student was classiﬁed as a heavy drinker based on the SM survey response, the in vitation to self-rele vant prompts stopped and the student was sent a link to an indicated Stage 2 intervention. A total of 158 students e ventually entered Stage 2 interventions, depending on when the y became hea vy drinkers. Consequently , the proximal outcome, maximum number of alcohol drinks consumed in any 24-hour period between two assessment time points, could be unobserved for two rea- sons: (1) a student receiv ed an MR T prompt b ut did not complete a SM survey , or (2) a student w as ﬂagged as a hea vy drinker at a previous assessment time and stopped receiv- ing in vitations to SM surve ys. For the unobserved proximal outcomes of these individuals, we use their responses to the end-of-semester follow-up (FU) surve y as proxy outcomes. In December 2019, the FU survey was distributed to all students, regardless of whether they entered Stage 2, to gather additional drinking information. The FU surve y collected timeline follo wback (TLFB) data about the number of drinks a student drank on each typical day of a week (Sunday through Monday) in the past 30 days. W e took the maximum of these number of drinks ov er sev en days as the proxy response for an unobserved proximal outcome at the second, third, and fourth assessment time. Ho wever , since the ﬁrst assessment time preceded the FU surve y for more than 30 days, the TLFB data might be an inappropriate proxy for un- observed proximal outcome at the ﬁrst assessment time. F or illustrating our proposed method in a typical hybrid design scenario, we excluded students who did not complete the ﬁrst SM surve y ( N = 117 ). W e further excluded N = 38 students whose unobserved proximal out- comes could not be imputed by TLFB data due to missingness. The ﬁnal cohort consisted of N = 428 students. T able 9 sho ws the number of students randomized at each assessment time during the MR T design. 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Evaluating time-varying treatment effects in hybrid SMART-MRT designs

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