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# Data integration methods for micro-randomized trials

E. Huch1,∗<EMAIL_ADDRESS>I. Nahum–Shani1,∗∗<EMAIL_ADDRESS>L. Potter2 C.
Lam2 D.W. Wetter2 and W. Dempsey1,∗∗∗
1d3 Center<EMAIL_ADDRESS>University of Michigan  Ann Arbor  Michigan  U.S.A.
2Department of Population Health Sciences and Huntsman Cancer Institute
University of Utah  Salt Lake City  Utah  U.S.A

(Received March 2024.; 2024)

###### Abstract

Existing statistical methods for the analysis of micro-randomized trials
(MRTs) are designed to estimate causal excursion effects using data from a
single MRT. In practice, however, researchers can often find previous MRTs
that employ similar interventions. In this paper, we develop data integration
methods that capitalize on this additional information, leading to statistical
efficiency gains. To further increase efficiency, we demonstrate how to
combine these approaches according to a generalization of multivariate
precision weighting that allows for correlation between estimates, and we show
that the resulting meta-estimator possesses an asymptotic optimality property.
We illustrate our methods in simulation and in a case study involving two MRTs
in the area of smoking cessation.

###### keywords:

Causal inference; Dynamic treatment regimes; Meta-analysis; mHealth.

††volume: Volume

##  1 Introduction

Micro-randomized trials (MRTs; Liao et al., 2016; Qian et al., 2022) are
becoming an increasingly popular tool for developing just-in-time adaptive
interventions (JITAIs) in mobile health (mHealth). JITAIs leverage digital
technologies to adapt the delivery of interventions to the rapidly changing
needs of individuals (Nahum-Shani et al., 2015, 2018). To empirically inform
these interventions, MRTs involve rapid sequential randomizations, meaning
that each participant is randomized many times during the trial with short
time intervals (e.g., a few hours or minutes) between subsequent
randomizations. The goal of an MRT is to collect data that will enable
investigators to estimate the causal effect of delivering just-in-time
interventions on a proximal health-related outcome; e.g., the effect of
delivering (vs. not delivering) a mobile-based supportive message on tobacco
use in the next hour.

MRT data is typically analyzed with the weighted and centered least squares
method (WCLS; Boruvka et al., 2018). WCLS is a semi-parametric method that
enables unbiased estimation of time-varying treatment effects even when the
model includes time-varying endogenous variables. WCLS has been extended in
various directions, including for use with different estimands (Dempsey et
al., 2020), binary outcomes (Qian et al., 2021), clustered data structure (Shi
et al., 2023a), and greater statistical efficiency via auxiliary variables and
supervised machine learning methods (Shi et al., 2023b; Shi and Dempsey,
2023). However, WCLS has not yet been extended to allow for data integration
across multiple studies.

The benefit of data integration methods is that they can result in substantial
increases in statistical efficiency relative to a single-study analysis. They
are particularly useful in causal inference because experimental data sets are
often small, which makes statistical efficiency a key consideration (Shi et
al., 2023). The potential for efficiency gains depends largely on the size of
the additional data set and its relevance to the target estimand.
Investigators seeking to develop a JITAI often use previous MRTs and/or
observational intervention studies as a starting point for new studies.
Consequently, previous studies often contain data that are relevant to
estimating the target estimand in the study under consideration, indicating
the possibility of efficiency gains via data integration.

The key challenge in realizing these efficiency gains is that the causal
effects may differ between studies, so naively pooling the data sets together
will, in general, result in biased estimates. Here, we propose an approach
that achieves asymptotic unbiasedness under a weaker assumption—that the
causal effects must be equal only in a conditional sense. We show that this
assumption leads to several estimation strategies. Because the resulting
estimators are correlated with one another, the classical method of precision
weighting (Konstantopoulos and Hedges, 2019; Hedges and Olkin, 2014, Chapter
6) is not optimal for combining estimates. Instead, we introduce a meta-
estimator that generalizes precision weighting to allow for correlation both
_within_ and _between_ vector-valued estimates, and we show that it possesses
an asymptotic optimality property.

The paper proceeds as follows. Section 2 formally introduces the problem setup
and WCLS. Sections 3, 4, and 5 develop our proposed data integration methods.
Sections 6 and 7 present results from applying our methods in a simulation and
case study, respectively. Section 8 closes with a discussion.

##  2 Preliminaries

This section introduces the problem setup, including the target estimand and
basic causal assumptions. We then provide the main technical details for WCLS
and discuss the extension of our problem setup to multiple studies.

###  2.1 Single-study problem setup

For a given participant, we observe a sequence of covariates ($X_{t}$),
treatment assignments ($A_{t}$), and outcomes ($Y_{t}$) for $T$ time points.
To simplify the exposition, we assume that $A_{t}$ is binary (we discuss the
extension to three or more treatment levels in Section 8). We denote the full
treatment history up to time point $t$ as

$H_{t}=(X_{1},A_{1},Y_{2},X_{2},A_{2},Y_{3},...,X_{t}).$

We adopt the potential outcomes framework of Rubin (1974) and posit the
existence of potential values for all variables from $Y_{2}$ onward in the
above sequence. For $Y_{2}$, $X_{2}$, and $A_{2}$ the potential values are
$Y_{2}(a_{1})$, $X_{2}(a_{1})$, and $A_{2}(a_{1})$, respectively, with
$a_{1}\in\\{0,1\\}$. After $A_{2}$, the potential values depend on multiple
treatment assignments. For instance, the potential value for $Y_{3}$ is
$Y_{3}(a_{1},a_{2})$. To make the notation more compact, we use an overbar to
represent the history of a given variable up to a particular time point; e.g.,
$\bar{a}_{3}=(a_{1},a_{2},a_{3})$. Using this notation, the potential values
can then be written as $Y_{t}(\bar{a}_{t-1})$, $X_{t}(\bar{a}_{t-1})$, and
$A_{t}(\bar{a}_{t-1})$. We employ the following standard causal assumptions
for dynamic treatment regimes:

(_Consistency_) The observed variables are equal to the potential variables as
follows: $Y_{t}=Y_{t}(\bar{A}_{t-1})$, $X_{t}=X_{t}(\bar{A}_{t-1})$, and
$A_{t}=A_{t}(\bar{A}_{t-1})$.

(_Strict Positivity_) The treatment assignment probabilities are bounded away
from zero and one: $\exists\epsilon\in(0,0.5]$ such that
$\epsilon<\Pr(A_{t}=1\,|\,H_{t})<1-\epsilon$ for all $t\in[T]$.

(_Sequential Unconfoundedness_) The treatment assignment at time $t$ is
independent of all future potential values; i.e., for all $t\in[T]$ and all
$(a_{t},a_{t+1},\dots,a_{T})\in\\{0,1\\}^{T-t+1}$, we have

$A_{t}\perp\\!\\!\\!\perp\big{\\{}Y_{t+1}(\bar{A}_{t-1},a_{t}),X_{t+1}(\bar{A}_{t-1},a_{t}),A_{t+1}(\bar{A}_{t-1},a_{t}),\dots,\phantom{\\{}Y_{T+1}(\bar{A}_{t-1},a_{t},a_{t+1},\dots,a_{T})\big{\\}}\big{|}H_{t}.$

Because $A_{t}$ is randomized in an MRT, Assumptions 2.1 and 2.1 typically
hold by design. The scientific goal is to learn how a set of (potentially)
time-varying covariates, $R_{t}\subset H_{t}$, moderates the causal effect of
$A_{t}$ on the proximal outcome, $Y_{t+1}$:

$\mathbb{E}\left\\{Y_{t+1}(\bar{A}_{t-1},1)-Y_{t+1}(\bar{A}_{t-1},0)\,|\,R_{t}(\bar{A}_{t-1})=r^{*}\right\\}.$
(1)

We denote these causal effects as ${\beta_{r}}(t,r^{*})$. Boruvka et al.
(2018) refer to ${\beta_{r}}(t,r^{*})$ as ‘causal excursion effects’ because
they represent the average effect of following the (stochastic) treatment
policy up until $t-1$ and then taking an ‘excursion’ from the policy. We use
$r^{*}$ to denote a generic value of $r^{*}$ from the support of $R_{t}$. The
definition given in Boruvka et al. (2018) allows for lags of more than one
time point; e.g., the effect of $A_{t}$ on $Y_{t+2}$. The extension to
additional lags is straightforward, so we use a single lag to simplify the
exposition. Boruvka et al. (2018) show that ${\beta_{r}}(t,r^{*})$ can be
identified from the data as

$\mathbb{E}\left[\mathbb{E}\left\\{Y_{t+1}\,|\,H_{t},A_{t}=1\right\\}-\mathbb{E}\left\\{Y_{t+1}\,|\,H_{t},A_{t}=0\right\\}\,|\,R_{t}=r^{*}\right].$

Following Boruvka et al. (2018), we assume that ${\beta_{r}}(t,r^{*})$ is a
linear function:

(_Linear $R_{t}$-moderated Effects_)
${\beta_{r}}(t,r^{*})={f_{r}}(r^{*})^{\intercal}{\beta_{r}}$ for some known
function, ${f_{r}}$, determining feature transformations of $r^{*}$ and some
unknown parameter vector, ${\beta_{r}}$.

In a slight abuse of notation, we use ${\beta_{r}}$ to refer to both (1) the
nonparametrically defined estimand and (2) the vector of coefficients that
determine it according to Assumption 2.1. In principle, $f_{r}$ and other
feature transformation functions we introduce later could depend on $t$;
however, we suppress the dependence in our notation for brevity.

###  2.2 Weighted, centered least squares (WCLS)

This section introduces the WCLS method as a means of faciliating the
development of our methods in subsequent sections. To simplify notation, we
use $\mathbb{P}_{n}$ to denote the empirical expectation operator:
$\mathbb{P}_{n}Z=\frac{1}{n}\sum_{i=1}^{n}Z_{i}$. Using this notation, the
WCLS estimator is defined as the solution to the estimating equation
$0=\mathbb{P}_{n}U({\alpha},{\beta_{r}})$ with $U({\alpha},{\beta_{r}})$ equal
to

$\sum_{t=1}^{T}W_{t}\left[Y_{t+1}-{g}(H_{t})^{\intercal}{\alpha}-\left\\{A_{t}-p_{r}(1\,|\,R_{t})\right\\}{f_{r}}(R_{t})^{\intercal}{\beta_{r}}\right]\begin{bmatrix}{g}(H_{t})\\\
\left\\{A_{t}-p_{r}(1\,|\,R_{t})\right\\}{f_{r}}(R_{t})\end{bmatrix}.$ (2)

In Equation (2), $W_{t}=p_{r}(A_{t}\,|\,R_{t})/p_{h}(A_{t}\,|\,H_{t})$ is a
ratio of conditional treatment assignment probabilities. ${g}(H_{t})$ is a
transformation of $H_{t}$ thought to explain variation in the expected value
of the outcome. ${\alpha}$ is a corresponding nuisance parameter that is
included as a means of increasing statistical efficiency by ‘de-noising’ the
outcome. Crucially, ${g}(H_{t})$ need not correspond with the true conditional
expectation of $Y_{t+1}$ given $H_{t}$. Boruvka et al. (2018) show that the
WCLS estimator of ${\beta_{r}}$ is consistent and asymptotically Gaussian with
a covariance matrix that can be consistently estimated via a sandwich
estimator. These sandwich estimators take the form
$\mathbf{B}^{-1}\mathbf{M}\mathbf{B}^{-\intercal}$, where
$\mathbf{B}=\mathbb{P}_{n}\,\partial U({\alpha},{\beta_{r}})$ and
$\mathbf{M}=\mathbb{P}_{n}\,U({\alpha},{\beta_{r}})U({\alpha},{\beta_{r}})^{\intercal}$
(Stefanski and Boos, 2002).

We can still apply WCLS when $p_{r}$, $p_{h}$, or both are unknown, provided
we know the parametric form (e.g., logistic regression) for the unknown
probabilities. In this case, we estimate the corresponding parameters
according to their own estimating equations and plug the solutions into the
WCLS estimating equation. We can propagate the uncertainty due to estimating
these parameters by ‘stacking’ all of the estimating equations vertically and
forming a joint sandwich estimator (See background material in Carroll et al.,
2006). Although the parametric form of $p_{h}$ must be correctly specified,
$p_{r}$ may be assumed constant without loss of consistency or asymptotic
normality.

###  2.3 Extension to multiple studies

We now extend the problem setup above to multiple studies. We assume that
${\beta_{r}}(t,r^{*})$ is defined in a single study population, and we refer
to this study as the “internal study.” We allow access to one or more
additional “external studies.” Mathematically, we distinguish between these
studies using the variable $I$ (for “internal”), which takes on a value of one
if an observation is sampled from the internal study population and zero
otherwise. Similarly, we denote the number of participants as $n_{1}$ and
$n_{0}$ in the internal and external studies, respectively. We assume the
following relationship between the internal and external studies:

(_Shared $S_{t}$-moderated Effects_) There exists a set of (potentially) time-
varying moderators $S_{t}\supseteq R_{t}$ such that the $S_{t}$-conditional
causal excursion effects, ${\beta_{s}}(t,s^{*})$, are equal between studies
for all $s^{*}$:

$\displaystyle\
\mathbb{E}\left\\{Y_{t+1}(\bar{A}_{t-1},1)-Y_{t+1}(\bar{A}_{t-1},0)\,|\,S_{t}(\bar{A}_{t-1})=s^{*},I=1\right\\}$
$\displaystyle=$ $\displaystyle\
\mathbb{E}\left\\{Y_{t+1}(\bar{A}_{t-1},1)-Y_{t+1}(\bar{A}_{t-1},0)\,|\,S_{t}(\bar{A}_{t-1})=s^{*},I=0\right\\}.$

Assumption 2.3 effectively assumes that the average causal effects are equal
between studies within appropriately defined strata. The $R_{t}$-moderated
causal excursion effects may still differ between studies; however, Assumption
2.3 implies that these differences must be due to differences in the
conditional distribution of $S_{t}$ given $R_{t}$.

##  3 Methods based on conditional mean models

Beginning in this section, we develop five data integration methods that
extend WCLS to the multi-study setting. Table 1 compares the methods and
offers recommendations on when each method is most applicable. This section
introduces the first two methods, A-WCLS and P-WCLS, which rely on models for
certain conditional means.

|  $\mathbb{E}({f_{s}}(S_{t})|R_{t})$ model |  ${\beta_{s}}(t,s^{*})$ model |  $\omega(S_{t})$ model |  Combination method |
---|---|---|---|---|---
| Recommended use case
A-WCLS | ✓ |  |  |  |  Known structure/constraints for $\mathbb{E}({f_{s}}(S_{t})|R_{t})$
P-WCLS |  | ✓ |  |  |  High confidence in ${\beta_{s}}(t,s^{*})$; low confidence in $\omega(S_{t})$
ET-WCLS |  |  | ✓ |  | Subcomponent in PET-WCLS
DR-WCLS |  | ✓ | ✓ | ✓ |  Medium confidence in both ${\beta_{s}}(t,s^{*})$ and $\omega(S_{t})$
PET-WCLS |  | ✓ | ✓ | ✓ |  High confidence in both ${\beta_{s}}(t,s^{*})$ and $\omega(S_{t})$
Table 1: Summary of proposed methods. The first three columns indicate which
models are required for each method. “Combination method” indicates which
methods combine several individual estimators. “Recommended use case” presents
recommendations on when each method is particularly appropriate; see Section 8
for further discussion.

###  3.1 A-WCLS

Our first method is based on the following smoothing identity:

${\beta_{r}}(t,r^{*})=\mathbb{E}\left\\{{\beta_{s}}(t,S_{t})\,|\,R_{t}=r^{*}\right\\}.$
(3)

This identity implies that one strategy to estimate ${\beta_{r}}(t,r^{*})$ is
to first estimate ${\beta_{s}}(t,S_{t})$ and then appropriately average its
value conditional on $R_{t}$. To that end, we introduce an additional
linearity assumption:

(_Linear $S_{t}$-moderated Effects_)
${\beta_{s}}(t,s^{*})={f_{s}}(s^{*})^{\intercal}{\beta_{s}}$ for some known
function, ${f_{s}}$, determining feature transformations of $s^{*}$ and some
unknown parameter vector, ${\beta_{s}}$.

Assumption 3.1 is identical to Assumption 2.1, replacing $R$’s with $S$’s.
Given Assumption 3.1, Equation (3) implies that

${\beta_{r}}(t,r^{*})=\mathbb{E}\left\\{{f_{s}}(S_{t})\,|\,R_{t}=r^{*}\right\\}^{\intercal}{\beta_{s}}.$
(4)

From this identity, we observe that it would be sufficient to estimate the
expected value of each component of ${f_{s}}(S_{t})$ conditional on
$R_{t}=r^{*}$ and multiply the resulting vector by ${\beta_{s}}$. For
concreteness, we now assume a simple parametric form for this expectation:

(_Linear $\mathbb{E}\left\\{{f_{s}}(S_{t})\,|\,R_{t}\right\\}$_)
$\mathbb{E}\left\\{{f_{s}}(S_{t})\,|\,R_{t}=r^{*}\right\\}=\boldsymbol{\Gamma}^{\intercal}{f_{r}}(r^{*})$
for some known function, ${f_{r}}$, determining feature transformations of
$r^{*}$ and some partially known parameter matrix, $\boldsymbol{\Gamma}$. We
further assume that each column of $\boldsymbol{\Gamma}$ is either fully known
or fully unknown.

Applying Assumption 3.1 to Equation (4), we see that
${\beta_{r}}(t,r^{*})={f_{r}}(r^{*})^{\intercal}\boldsymbol{\Gamma}{\beta_{s}}$
and, consequently, we have ${\beta_{r}}=\boldsymbol{\Gamma}{\beta_{s}}$. We
name this method Apportioned-WCLS (A-WCLS) because it apportions the
$S_{t}$-conditional effects among the elements of ${f_{r}}(R_{t})$ according
to the corresponding conditional expectations. We make the following
clarifying remarks regarding Assumption 3.1.

###### Remark 3.1

The expressions above indicate that Assumptions 3.1 and 3.1 actually imply
Assumption 2.1. We list Assumption 2.1 as a separate assumption because it is
required for the other methods. A key insight from the above derivation is
that the analyst must choose feature transformations of $R_{t}$ (encoded by
${f_{r}}$) that allow the expectation of ${f_{s}}(S_{t})$ to be expressed as a
linear function. This assumption can be checked using standard regression
diagnostics and plots.

###### Remark 3.2

Assumption 3.1 is not required for valid inference of ${\beta_{r}}(t,r^{*})$;
rather, it is a simplifying assumption. We could, for instance, assume a GLM-
style link function for the various components of
$\mathbb{E}\left\\{{f_{s}}(S_{t})\,|\,R_{t}=r^{*}\right\\}$, and the methods
described below would generalize naturally (though, Assumption 2.1 would no
longer hold).

###### Remark 3.3

Because $R_{t}\subseteq S_{t}$, some columns of $\boldsymbol{\Gamma}$ are
known. In particular, for each common element of ${f_{r}}(R_{t})$ and
${f_{s}}(S_{t})$, there will be a corresponding column in
$\boldsymbol{\Gamma}$ with a single nonzero entry equal to one. In a similar
fashion, we could assume that other elements in $\boldsymbol{\Gamma}$ are zero
and modify the method below accordingly.

With the parametric identification formula
${\beta_{r}}=\boldsymbol{\Gamma}{\beta_{s}}$, estimation proceeds as follows:

  1. 1.

Estimate the unknown treatment assignment probabilities $p_{h}$, $p_{s}$, and
$p_{r}$ by solving a set of estimating equations (e.g., score functions for
logistic regression).

  2. 2.

Estimate ${\beta_{s}}$ by applying WCLS to the combined data set.

  3. 3.

Set $\sigma_{r}^{2}(R_{t})=p_{r}(1|R_{t})\left\\{1-p_{r}(1|R_{t})\right\\}$
and solve the following estimating equation for all $j$ corresponding to non-
common elements of ${f_{r}}(R_{t})$ and ${f_{s}}(S_{t})$:

$0=\mathbb{P}_{n}\,I\sum_{t=1}^{T}\sigma_{r}^{2}(R_{t})\Big{[}\big{\\{}{f_{s}}(S_{t})\big{\\}}_{j}-{f_{r}}(R_{t})^{\intercal}\gamma_{j}\Big{]}{f_{r}}(R_{t}),$

The final step is a weighted least-squares regression within the internal-
study population. The weights are not required for consistency or asymptotic
normality, but they can improve the efficiency of the resulting estimator.

To simplify the inference procedures, we order the elements of
${f_{s}}(S_{t})$ and ${f_{r}}(R_{t})$ such that the common elements appear
first and in the same positions. We denote the number of common elements as
$c$ and the dimensions of ${f_{r}}(R_{t})$ and ${f_{s}}(S_{t})$ as $d_{r}$ and
$d_{s}$, respectively. We form $\boldsymbol{\hat{\Gamma}}$ as

$\boldsymbol{\hat{\Gamma}}=\left[\begin{array}[]{c:c:c:c:c}\begin{array}[]{c}\mathbf{I}_{c}\\\
\hdashline\mathbf{0}_{(d_{r}-c),c}\end{array}&\hat{\gamma}_{d_{s}-c+1}&\hat{\gamma}_{d_{s}-c+2}&...&\hat{\gamma}_{d_{s}}\end{array}\right].$

We then calculate our estimate of ${\beta_{r}}$ as
$\hat{\beta}_{r}=\boldsymbol{\hat{\Gamma}}\hat{\beta}_{s}$. Letting
$\phi=({\beta_{s}}\ \gamma_{d_{s}-c+1}\ ...\gamma_{d_{s}})^{\intercal}$ with
$\hat{\phi}$ defined similarly, standard results in M-estimation then
guarantee (under mild regularity conditions) that
$\sqrt{n}(\hat{\phi}-\phi)\overset{d}{\longrightarrow}N(0,\boldsymbol{\Sigma}_{\phi})$
with $\boldsymbol{\Sigma}_{\phi}$ consistently estimated via a sandwich
estimator. The delta method gives the asymptotic distribution of
$\hat{\beta_{r}}$:
$\sqrt{n}(\hat{\beta_{r}}-{\beta_{r}})\overset{d}{\longrightarrow}N(0,\mathbf{D}_{\phi}\boldsymbol{\Sigma}_{\phi}\mathbf{D}_{\phi}^{\intercal}),$
where $\mathbf{D}_{\phi}$ is the total derivative of
${\beta_{r}}=\boldsymbol{\Gamma}{\beta_{s}}$ with respect to $\phi$:

$\mathbf{D}_{\phi}=\frac{\partial}{\partial\phi^{\intercal}}{\beta_{r}}=\big{[}\begin{array}[]{c:c}\boldsymbol{\Gamma}&\beta^{\intercal}_{s,(d_{s}-c+1):d_{s}}\otimes\mathbf{I}_{d_{r}}\end{array}\big{]}.$

In practice, we plug in estimates for $\mathbf{D}_{\phi}$ and
$\boldsymbol{\Sigma}_{\phi}$, and inference proceeds as usual.

###  3.2 P-WCLS

This section introduces Projected-WCLS (P-WCLS). P-WCLS requires Assumptions
2.1–3.1, but it does not require Assumption 5; hence, the required assumptions
are somewhat weaker than those of A-WCLS. P-WCLS estimates ${\beta_{r}}$ by
projecting ${f_{s}}(S_{t})^{\intercal}\hat{\beta}_{s}$ onto ${f_{r}}(R_{t})$.
In terms of implementation, we estimate $p_{h}$, $p_{s}$, $p_{r}$, and
${\beta_{s}}$ as detailed in Section 3.1, but we replace the estimation of
$\boldsymbol{\Gamma}$ with direct estimation of ${\beta_{r}}$ according to the
following estimating equation:

$U_{{\beta_{r}}}=\mathbb{P}_{n}\sum_{t=1}^{T}I\,\sigma_{r}^{2}(R_{t})\big{\\{}{f_{s}}(S_{t})^{\intercal}\hat{\beta}_{s}-{f_{r}}(R_{t})^{\intercal}{\beta_{r}}\big{\\}}{f_{r}}(R_{t}).$
(5)

Solving this equation for ${\beta_{r}}$ amounts to performing a regression of
the estimated $S_{t}$-moderated causal excursion effects,
${f_{s}}(S_{t})^{\intercal}\hat{\beta}_{s}$, on ${f_{r}}(R_{t})$ weighted by
$\sigma_{r}^{2}(R_{t})$. A natural follow-up question is whether and how this
estimator differs from A-WCLS. Theorem 3.4 provides a partial answer.

###### Theorem 3.4

Under Assumption 3.1, A-WCLS and P-WCLS are equivalent.

The proof relies on straightforward linear algebra and is provided in Web
Appendix A. As pointed out in Stefanski and Boos (2002), the estimating
equations defining a given estimator are not unique; however, provided two
sets of estimating equations produce the same estimator, they also have the
same asymptotic variance. Consequently, the asymptotic for variances for
A-WCLS and P-WCLS are also equivalent under Assumption 3.1—a fact that we
confirmed empirically while running our simulation study. We introduce A-WCLS
as a separate method because (a) it provides guidance on how to select
${f_{r}}$ (see Remark 3.1), (b) it is distinct from P-WCLS under modifications
of Assumption 3.1 (see Remarks 3.2 and 3.3), and (c) the interpretation of
${\beta_{r}}$ as a linear combination of ${\beta_{s}}$ allows us to connect
and interpret multiple levels of moderation analysis. In cases where A-WCLS
and P-WCLS are equivalent, we recommend performing inference via P-WCLS
because it allows for a simpler implementation—the delta method is not needed.

##  4 Methods based on density ratio models

This section introduces Exponentially Tilted WCLS (ET-WCLS). Rather than
assuming a model for shared conditional expectations, ET-WCLS relates the
internal and external studies via an exponential density ratio model. We also
show how to combine the resulting estimator with WCLS to produce a meta-
estimator that is asymptotically superior to WCLS.

###  4.1 ET-WCLS

Similar to A-WCLS and P-WCLS, ET-WCLS assumes that the $S_{t}$-conditional
causal excursion effects are equal between studies (Assumption 2.3). In
contrast to A-WCLS and P-WCLS, however, ET-WCLS does not require them to
assume a particular parametric form (i.e., Assumptions 3.1 and 3.1 do not
apply). Instead, we assume the studies can be related according to a density
ratio as follows:

(_Exponential Tilt Density Ratio_) $p(S_{t}\,|\,I=0)>0$ if
$p(S_{t}\,|\,I=1)>0$ except, perhaps, on sets of measure zero. Further, there
exists a set of feature transformations, $d(S_{t})$, and corresponding
parameter vector, $\omega$, such that the conditional densities of $S_{t}$ can
be related as follows:

$\omega(S_{t})\coloneqq\frac{p(S_{t}\,|\,I=1)}{p(S_{t}\,|\,I=0)}=\exp\left\\{d(S_{t})^{\intercal}\omega\right\\}.$

Assumption 4.1 allows us to view the external data as an unevenly sampled
representation of the internal study population. This relationship suggests a
strategy in which we estimate $\omega(S_{t})$ and use it to counterbalance the
under- and over-representation of each data point, similar to the methods
provided in Graham et al. (2012, 2016) and Tan (2006). Qin (1998) details how
to estimate density ratios of this form via a slight modification of logistic
regression (due to case-control parity, the non-intercept coefficients are
identical). The full estimation procedure for ET-WCLS is given below:

  1. 1.

If needed, estimate $p_{h}$ and $p_{r}$ as explained in Section 3.1

  2. 2.

Maximize the log-likelihood
$\ell(\omega)=\mathbb{P}_{n}\sum_{t=1}^{T}\left(Id(S_{t})^{\intercal}\omega-\log\left[1+\rho\exp\left\\{d(S_{t})^{\intercal}\omega\right\\}\right]\right)$,
where $\rho\coloneqq n_{1}/n_{0}$ to obtain an estimate of $\omega$

  3. 3.

Estimate ${\beta_{r}}$ as the solution to Equation (2), replacing $W_{t}$ with
$\left[(1-I)\,\exp\left\\{d(S_{t})^{\intercal}\omega\right\\}\,W_{t}\right]$

In Web Appendix A, we prove the following result about ET-WCLS:

###### Theorem 4.1

Assuming the existence of certain moments, the ET-WCLS estimator is a
consistent, asymptotically normal estimator of ${\beta_{r}}$.

We prove Theorem 4.1 using classical results in M-estimation. The regularity
conditions require the reweighted WCLS equation to have bounded fourth
moments. From a practical standpoint, this requirement means that there must
be sufficient overlap between studies in the distributions of $S_{t}$;
otherwise, ET-WCLS may suffer from instability in a manner similar to other
weighting methods, such as inverse probability weighting (Robins et al., 2007;
Seaman and White, 2013).

###  4.2 Meta-estimator

A common practical challenge with weighting-based estimators is that they can
exhibit high variance under insufficient overlap, even if the theoretical
requirements are satisfied. For ET-WCLS in particular, this challenge results
in ET-WCLS being less efficient than WCLS-Internal (the single-study analysis)
unless the external study is large relative to the internal study. This
dilemma begs the question: Is it possible to pool the WCLS-Internal and ET-
WCLS estimates such that the resulting estimator is more efficient than either
estimate individually? This section introduces an asymptotically optimal meta-
estimator that provides an affirmative answer to this question.

The central challenge in developing the meta-estimator is that WCLS-Internal
and ET-WCLS are, in general, correlated due to estimation of $\omega$, which
involves both the internal and external studies. Consequently, the classical
solution of precision weighting is not guaranteed to be optimal. We note that
closely related problems have been considered in the meta-analysis literature.
Valentine (2019) summarizes multivariate methods that are applicable to
vector-valued effect-size estimates provided there is no correlation between
estimates. In contrast, Hedges and Olkin (2014, Chapter 10) provide a meta-
analysis method for dependent estimates of a scalar-valued parameter. However,
our setting involves both of these challenges: vector-valued estimates with
correlation both within _and_ between estimates. We were unable to locate a
method that addresses both of these challenges.

The first step toward a solution is to jointly stack all relevant estimating
equations. For full generality, we consider the case with $J\geq 2$ distinct
estimators of ${\beta_{r}}$. We stack these estimators in a single vector as
follows:

$\hat{\theta}=\left(\hat{\beta}_{r1}^{\intercal},\
\hat{\beta}_{r2}^{\intercal},\dots,\hat{\beta}_{rJ}^{\intercal}\right)^{\intercal}.$

We then form a sandwich estimator, $\hat{\Sigma}$, of
$\Sigma\coloneqq\text{Var}\left(\sqrt{n}\hat{\theta}\right)$ that captures the
dependence structure both within and between the $\hat{\beta}_{rj}$’s. Letting
$\hat{\Lambda}\coloneqq\hat{\Sigma}^{-1}$, the meta-estimator is then defined
as

$\hat{\beta}_{r}=\left(\sum_{j=1}^{J}\sum_{k=1}^{J}\hat{\Lambda}_{j,k}\right)^{-1}\sum_{j=1}^{J}\hat{\Lambda}_{j.}\hat{\theta}=(1_{J}^{\intercal}\otimes\hat{\Omega})\,\hat{\Lambda}\,\hat{\theta}$

with
$\hat{\Omega}=\left(\sum_{j=1}^{J}\sum_{k=1}^{J}\hat{\Lambda}_{j,k}\right)^{-1}$.
We denote the population-level analogs of $\hat{\Lambda}$ and $\hat{\Omega}$
as $\Lambda$ and $\Omega$, respectively. $\hat{\beta}_{r}$ reduces to standard
precision weighting when the off-diagonal blocks of $\hat{\Lambda}$ (or,
equivalently, $\hat{\Sigma}$) are zero. Similarly, it reduces to the estimator
given in Hedges and Olkin (2014, Chapter 10) when ${\beta_{r}}$ is a scalar.
$\hat{\Omega}$ is guaranteed to exist and, consequently, ${\beta_{r}}$ is
well-defined provided $\hat{\Sigma}$ is positive definite. This property
follows from the following proposition, the proof of which is given in Web
Appendix A.

###### Proposition 4.2

Let $A$ be a positive definite square matrix. Then for any partition of this
matrix into equal-sized blocks, the sum of these blocks is also positive
definite and, hence, invertible.

We now turn our attention to the asymptotic properties of $\hat{\beta}_{r}$.
In Web Appendix A, we prove the following result:

###### Theorem 4.3

Consider the class of unbiased estimators of ${\beta_{r}}$ that can be written
in the form $B\hat{\theta}$ for some matrix $B$ that may depend on
$\hat{\Sigma}$. Among this class of estimators, $\hat{\beta}_{r}$ achieves the
minimum asymptotic variance,
$\frac{1}{n}(1_{J}^{\intercal}\otimes\Omega)\Lambda(1_{J}\otimes\Omega)$,
which can be consistently estimated as
$\frac{1}{n}(1_{J}^{\intercal}\otimes\hat{\Omega})\hat{\Lambda}(1_{J}\otimes\hat{\Omega})$.

The proof relies on showing that $\hat{\beta}_{r}$ is asymptotically
equivalent to an oracle estimator having the same mathematical formulation as
${\beta_{r}}$, but employing $\Sigma$ and $\Lambda$ instead of $\hat{\Sigma}$
and $\hat{\Lambda}$. The optimality of this oracle estimator follows from an
application of the Gauss–Markov Theorem for generalized least squares. Theorem
4.3 immediately implies that pooling the WCLS-Internal and ET-WCLS estimators
is asymptotically superior to simply selecting one of the two estimators. More
generally, the meta-estimator provides an asymptotically optimal formula for
combining correlated estimates that is applicable in broader settings than
those considered here.

One drawback of the meta-estimator is that it relies on accurate estimation of
a large, full-rank covariance matrix. Consequently, we expect its performance
to degrade in small samples when this covariance matrix is poorly estimated.
In these settings, one path forward is to assume that $\Sigma$ takes on a
Kronecker structure, a common assumption in high-dimensional covariance
estimation (Srivastava et al., 2008). Web Appendix B provides the
methodological details.

##  5 Combination methods

Having developed two general data integration strategies, we now turn our
attention to developing methods that combine them for increased robustness and
efficiency.

###  5.1 DR-WCLS

The first combination method we develop is Doubly Robust WCLS (DR-WCLS): a
combination of P-WCLS and ET-WCLS that produces valid inferences provided that
the assumptions for at least one method hold. We develop methods separately
for the internal and external study, then discuss how to combine them using
the meta-estimator of Section 4.2. We discuss the internal study first. Our
proposed method is based on a ‘pseudo-outcome,’ $\tilde{Y}_{t+1}$, similar to
those employed in Bang and Robins (2005) and Kennedy (2023):

$\displaystyle\,\frac{Y_{t+1}-m(H_{t},A_{t})}{A_{t}-p_{h}(0|H_{t})}+m(H_{t},1)-m(H_{1},0)$
$\displaystyle=$
$\displaystyle\frac{W_{t}\\{A_{t}-p_{r}(1|R_{t})\\}\\{Y_{t+1}-m(H_{t},A_{t})\\}}{\sigma_{r}^{2}(R_{t})}+m(H_{t},1)-m(H_{1},0),$

where $m(H_{t},A_{t})$ is a model having
$\mathbb{E}\left\\{m(H_{t},1)-m(H_{t},0)|S_{t}\right\\}={f_{s}}(S_{t})^{\intercal}{\beta_{s}}$,
such as

$m(H_{t},A_{t})={g}(H_{t})^{\intercal}{\alpha}+\\{A_{t}-p_{s}(1|S_{t})\\}{f_{s}}(S_{t})^{\intercal}{\beta_{s}}.$

We then have $E\left[\tilde{Y}_{t+1}|S_{t},I=1\right]={\beta_{s}}(t,S_{t})$
_even if_ ${f_{s}}(S_{t})^{\intercal}{\beta_{s}}$ is misspecified. Thus, based
on our smoothing identity (Equation (3)), we can estimate ${\beta_{r}}$ by
regressing $\tilde{Y}_{t+1}$ on ${f_{r}}(R_{t})$ in the internal study
according to the following estimating equation:

$0=\mathbb{P}_{n}I\sum_{t=1}^{T}\sigma_{r}^{2}(R_{t})\big{[}\tilde{Y}_{t+1}-{f_{r}}(R_{t})^{\intercal}{\beta_{r}}\big{]}{f_{r}}(R_{t}).$
(6)

The resulting estimator is fully robust in the sense that it does not require
any of Assumptions 3.1, 3.1, or 4.1 to hold. However, the estimator above uses
the external study only to produce a shared model, $m(H_{t},A_{t})$, and as
such produces limited efficiency gains relative to standard WCLS.

To attain meaningful efficiency gains, we now turn our attention toward
developing a doubly-robust estimator that uses the external study more
directly. This estimator cannot be represented using a single pseudo-outcome
as above. Instead, we work directly from an expanded version of Equation (6).
The estimating equation defining the external-study is
$0=\mathbb{P}_{n}\sum_{t=1}^{T}\tilde{U}({\beta_{r}})$, where
$\tilde{U}({\beta_{r}})$ is defined as

$\displaystyle\frac{1-I}{1-\pi}\,\omega(S_{t})W_{t}\\{A_{t}-p_{r}(1|R_{t})\\}\\{Y_{t+1}-m(H_{t},A_{t})\\}{f_{r}}(R_{t})$
(7)
$\displaystyle\quad\quad+\frac{I}{\pi}\,\sigma_{r}^{2}(R_{t})\\{m(H_{t},1)-m(H_{1},0)\\}{f_{r}}(R_{t})$
$\displaystyle\quad\quad-\frac{I}{\pi}\,\sigma_{r}^{2}(R_{t})\,{f_{r}}(R_{t})^{\intercal}{\beta_{r}}{f_{r}}(R_{t}),$

with $\pi$ representing the population proportion of individuals belonging to
the internal study. Note that this nuisance parameter requires its own
estimating equation of the form $0=\mathbb{P}_{n}(I-\pi)$. On the one hand, if
the model-based assumptions (Assumptions 3.1 and 3.1) hold, then the first
term has expectation zero and the second term (equal to ${\beta_{s}}(t,S_{t})$
by Assumption 3.1) is projected onto ${f_{r}}(R_{t})$ in a manner similar to
P-WCLS. On the other hand, if Assumption 4.1 holds, then

$\mathbb{E}\sum_{t=1}^{T}\frac{1-I}{1-\pi}\,\omega(S_{t})W_{t}[A_{t}-p_{r}(1|R_{t})]\,m(H_{t},A_{t})=\mathbb{E}\sum_{t=1}^{T}\frac{I}{\pi}\sigma_{r}^{2}(R_{t})[m(H_{t},1)-m(H_{t},0)],$

so the corresponding terms in Equation (7) cancel in expectation, leaving only
the term involving $Y_{t+1}$ and the last line. Under Assumption 4.1, the
expectation of the former is
$\sigma_{r}^{2}(R_{t}){\beta_{r}}(t,R_{t})=\sigma_{r}^{2}(R_{t}){f_{r}}(R_{t})^{\intercal}{\beta_{r}}$.
Consequently, ${\beta_{r}}$ solves the estimating equation in expectation,
which implies that the solution to Equation (7) is a consistent estimator of
${\beta_{r}}$.

Combining these observations, we see that the external study estimator
possesses the desired double-robustness property. After forming these two
estimates and the corresponding joint sandwich estimator, we can then combine
them via the meta-estimator of Section 4.2. The meta-estimator preserves the
doubly-robust nature of the individual estimators and, by Theorem 4.3, results
in an estimator with lower asymptotic variance than either individual
estimator.

###  5.2 PET-WCLS

In this section, we briefly discuss our second combination method, PET-WCLS,
which combines P-WCLS and ET-WCLS, effectively requiring Assumptions 3.1, 3.1,
and 4.1 to hold. We form the estimator by combining WCLS-Internal, P-WCLS, and
ET-WCLS according to our meta-estimator. When Assumptions 3.1, 3.1, and 4.1
all hold, PET-WCLS is guaranteed to be asymptotically superior to any other
linear combination of its constituent methods by Theorem 4.3.

##  6 Simulation

This section details a simulation study we performed to empirically evaluate
the performance of our proposed estimators.

###  6.1 Simulation setup

Throughout the simulation study, we use $T=20$ time points. The simulation
includes two simulated studies: one internal, one external. We vary the number
of participants in each study, $n_{1}$ and $n_{0}$, to investigate the impact
of sample size on the efficiency of our estimators. The simulation employs
three simulated covariates: $X_{t1}$, $X_{t2}$, and $X_{t3}$. We set
$R_{t}=X_{t1}$, $S_{t}=\\{X_{t1},X_{t2}\\}$,
${\beta_{s}}(t,S_{t})=1+2X_{t1}-3X_{t2}$, and
${\beta_{r}}(t,R_{t})=-2+5X_{t1}$. Web Appendix E provides additional details
on the simulation setup and a link to the computer code.

In the results below, we label methods that use only the internal data set
with the suffix -Internal. The other methods use both the internal and
external data sets. WCLS-Pooled naively applies WCLS to the pooled data set,
which generally results in biased estimates.

###  6.2 Simulation results

Coefficient name |  True value |  Method |  Avg estimate |  Relative efficiency |  rMSE |  Coverage
---|---|---|---|---|---|---
Intercept | -2 | WCLS-Internal | -1.94 | 100.0% | 1.57 | 94.5%
-2 | WCLS-Pooled | -0.48 | N/A | 1.77 | 61.0%
-2 | P-WCLS-Internal | -1.94 | 99.9% | 1.57 | 95.3%
-2 | P-WCLS-Pooled | -1.96 | 119.0% | 1.31 | 95.8%
-2 | ET-WCLS | -2.32 | 113.1% | 1.42 | 92.5%
-2 | DR-WCLS | -2.27 | 116.6% | 1.37 | 95.3%
-2 | PET-WCLS | -2.27 | 136.3% | 1.18 | 94.3%
Slope | 5 | WCLS-Internal | 4.86 | 100.0% | 1.77 | 96.0%
5 | WCLS-Pooled | 3.44 | N/A | 1.86 | 58.8%
5 | P-WCLS-Internal | 4.86 | 99.9% | 1.77 | 96.0%
5 | P-WCLS-Pooled | 4.92 | 130.9% | 1.35 | 95.8%
5 | ET-WCLS | 5.02 | 116.7% | 1.51 | 93.8%
5 | DR-WCLS | 5.10 | 124.9% | 1.42 | 96.3%
5 | PET-WCLS | 5.14 | 153.9% | 1.16 | 94.0%
Table 2: Results from the simulation with 400 individuals in both the internal
and external studies. For the “Avg estimate” and “Coverage” columns, the
boldface indicates values within Monte Carlo error ($3\sigma$) of the truth.
For the “Relative efficiency” and “rMSE” columns, the boldface indicates the
best performance for each coefficient (PET-WCLS in both cases).

Table 2 summarizes the performance of each method with $n_{1}=n_{0}=400$. As
expected, every method except WCLS-Pooled exhibited minimal bias and near-
nominal confidence interval coverage. We show in Web Appendix F that the
bias—albeit minimal—apparent in the density ratio methods (ET-WCLS, DR-WCLS,
and PET-WCLS) diminishes with larger sample sizes.

P-WCLS, ET-WCLS, DR-WCLS, and PET-WCLS all outperform WCLS-Internal in terms
of statistical efficiency and rMSE. PET-WCLS performs best—likely because it
is asymptotically superior to WCLS, P-WCLS, and ET-WCLS—achieving efficiency
improvements of 36.3% and 53.9% relative to WCLS-Internal. P-WCLS outperforms
WCLS-Internal only when it uses the external data; otherwise, its performance
is quite similar to WCLS-Internal. Web Appendix F shows additional
comparisons, including different sample sizes and method variations.

##  7 Case study

In this section, we apply P-WCLS to the MARS and Affective Science studies.

###  7.1 Study design

In both studies, the treatment is to send push notifications that prompt
participants to engage in evidence-based self-regulatory strategies. The
stated goal of these notifications is to “improve smokers’ ability to resist
craving and build self-regulatory skills” (Nahum-Shani et al., 2021).

Six times per day for ten days, the mHealth app prompted participants to
answer a two-question (2-Q) survey about cigarette availability and current
affect. Immediately upon answering the survey (or about two minutes later in
the case of non-response), the app randomly assigned participants to receive a
prompt to engage in a self-regulatory strategy with probability 0.5.
Approximately one hour after randomization, the app prompted participants to
answer additional questions in an ecological momentary assessment (EMA;
Shiffman et al., 2008).

###  7.2 Data analysis

At the time of writing, the Affective Science study is still enrolling
participants, but the MARS study has completed enrollment and the results are
currently being analyzed. Consequently, we have access only to a subset of
preliminary data ($n_{1}=68$, $n_{0}=97$) that will be further refined to
assess the studies’ primary aims. Although the MARS study study recruited 114
participants, we excluded 17 due to withdrawn consent (3), lack of
microrandomizations (1), insufficient EMA adherence (11), or missing baseline
tobacco dependence (2). The main hypotheses for the MARS study are based on
self-reported (via EMA) engagement with self-regulatory strategies. Because
our primary goal is to use these studies to demonstrate our methods, we
restrict attention to a separate outcome: The amount of time (in seconds)
spent in the app activities in the hour following micro-randomization.

To apply our methods, we must select two sets of moderators, $S_{t}$ and
$R_{t}$. The goal is to estimate how $R_{t}$ moderates treatment efficacy. In
contrast, $S_{t}\supseteq R_{t}$ is a set of moderators such that Assumption
2.3 is plausible; i.e., conditional on $S_{t}$, the causal excursion effects
are equal between studies. We include two variables in $R_{t}$: an indicator
for whether the participant completed the last 2-Q survey and the smoker’s
baseline tobacco dependence (measured by the participant’s response to the
question “How many cigarettes do you smoke per day?”). $S_{t}$ includes two
additional variables: an indicator for whether the participant identifies as
male and the participant’s self-reported race/ethnicity. The race/ethnicity
variable is divided into three categories: Latino, non-Latino white, and
other, which includes non-Latino black and other minority races.

Based on these choices, Assumption 2.3 states that participants with the same
2-Q survey completion status, baseline tobacco dependence, gender, and
race/ethnicity have the same causal excursion effects regardless of which
study they participated in (MARS or Affective Science). Although we cannot
verify this assumption, we can attempt to falsify it via a $\chi^{2}$ model
comparison test. The resulting p-value is 0.98, indicating essentially no
evidence against Assumption 2.3.

We apply P-WCLS because (1) it performed well in our simulation study and (2)
it is less sensitive to misspecification of its required sub-model
($\beta_{s}(t,s^{*})$) than the methods relying on density ratios. Table 3
displays the estimates of ${\beta_{r}}$ via WCLS-Internal, P-WCLS-Internal,
and P-WCLS-Pooled. As expected, P-WCLS-Pooled produces the smallest standard
errors because it leverages the MARS data to improve statistical efficiency.
For completeness, we also show results for our other methods in Web Appendix
C.

Figure 1 provides a visual depiction of the estimated effects. We have strong
evidence of a positive effect across most values of $R_{t}$. The estimated
coefficients suggest that the effects are more than twice as large among those
who responded to the previous 2-Q survey compared to those who did not. In
contrast, we see limited evidence that a smoker’s baseline tobacco dependence
(mean: 13.1, SD: 7.6) moderates these effects.

Coefficient Name | WCLS-Internal | P-WCLS-Internal | P-WCLS-Pooled
---|---|---|---
Intercept | *25.15 (9.50) | *25.16 (10.04) | *28.77 (6.61)
Tobacco Dependence | 0.09 (0.58) | 0.09 (0.61) | -0.04 (0.43)
2-Q Survey Completion | *39.47 (7.35) | *39.45 (7.76) | *40.89 (6.13)
Table 3: Estimates of $\beta_{r}$ (standard errors) from applying WCLS-
Internal, P-WCLS-Internal, and P-WCLS-Pooled in the case study; asterisks
denote statistical significance at the 0.05 level. P-WCLS-Pooled produces the
smallest standard errors. Figure 1: Estimated effect of prompts on time spent
in the mobile app within the Affective Science study. Dashed lines denote
pointwise 95% confidence bands.

##  8 Discussion

This paper presents five methods for pooling data across multiple MRTs for the
purpose of estimating causal excursion effects. The simulations demonstrate
that the methods outperform the standard single-study analysis (WCLS-Internal)
without losing the ability to perform calibrated inference on ${\beta_{r}}$.
The case study provides an illustrative example of (1) the mechanics of
applying the method to real MRTs and (2) the potential for efficiency gains.

In practice, researchers must decide which method and corresponding
assumptions to employ for a given analysis. While implementing the case study
and simulation, we found that the exponential tilt models were more difficult
to specify accurately than the conditional mean models, in part because common
regression diagnostics can be used to check the latter. Moreover, when the
exponential tilt models are misspecified, we observed that the resulting
estimators—including the robust estimators—can be highly unstable. Our
recommendations in Table 1 largely reflect this finding.

All five data integration methods rely on Assumption 2.3—that the
$S_{t}$-moderated causal excursion effects are equal between studies. Future
work might consider how to relax this assumption, perhaps assuming bounds on
discrepancies between the $S_{t}$-moderated causal excursion effects in each
study. Under this modified assumption, a natural approach would be to penalize
the internal-study estimates toward those of the external study. This approach
would require careful consideration of the resulting bias–variance tradeoff.

In addition to the data integration methods, we also developed an
asymptotically optimal meta-estimator that generalizes the classical method of
precision weighting. We suspect this meta-estimator could be applied in other
settings outside of mHealth, especially in the areas of meta-analysis and data
integration.

An interesting direction for future research would be to combine some of the
ideas here with other extensions of WCLS; e.g., the extensions to binary
outcomes or clustered MRTs. This direction would allow the application of
these methods in broader mHealth contexts. Boruvka et al. (2018) briefly
mention that it is possible to extend WCLS to the setting of three or more
treatment levels by introducing multiple centered treatment indicators. Web
Appendix D provides a proof that this strategy does, in fact, result in
consistent estimates and discusses how to apply this strategy to our methods.

As these methods are applied to future MRTs, one potential challenge is
identifying how to relate measurements and treatments across different study
designs. At a minimum, our proposed methods require (1) treatments with
similar effects, (2) a shared set of moderators, and (3) a similar outcome
between studies. Moreover, these variables must be selected so that Assumption
2.3 is plausible. Relaxations of Assumption 2.3 could help address this
challenge, but the practical relevance of external studies will remain a
central consideration in whether and how to integrate data across MRTs.

## Acknowledgements

The authors thank Jamie Yap and Qinggang Yu (d3c team members) for their
assistance in curating the data for the case study. This work was supported in
part by grant P50DA054039 provided through the NIH and NIDDK. The MARS and
Affective Science studies were funded by grants U01CA229437 and R01CA224537,
respectively, provided through the NIH and NCI.

## Data availability statement

The data underlying this article cannot be shared publicly due to privacy of
the participants involved in the MARS and Affective Science studies.

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## Supplementary Materials

Web Appendices referenced in Sections 3, 4, 6, and 7 are available at the end
of this document. The code for the simulation study is publicly available at
https://github.com/eastonhuch/mrt-data-integration.git.