# Paradoxes and resolutions for semiparametric fusion of individual and
summary data

Wenjie Hu  Ruoyu Wang  Wei Li  Wang Miao00footnotetext: Co-corresponding
authors: Wei Li and Wang Miao

###### Abstract

Suppose we have available individual data from an internal study and various
types of summary statistics from relevant external studies. External summary
statistics have been used as constraints on the internal data distribution,
which promised to improve the statistical inference in the internal data;
however, the additional use of external summary data may lead to paradoxical
results: efficiency loss may occur if the uncertainty of summary statistics is
not negligible and a large estimation bias can emerge even if the bias of
external summary statistics is small. We investigate these paradoxical results
in a semiparametric framework. We establish the semiparametric efficiency
bound for estimating a general functional of the internal data distribution,
which is shown to be no larger than that using only internal data. We propose
a data-fused efficient estimator that achieves this bound so that the
efficiency paradox is resolved. Besides, we propose a debiased estimator that
can achieve the same asymptotic distribution as the oracle estimator as if one
knew whether the summary statistics were biased or not. Simulations and
application to a Helicobacter pylori infection dataset are used to illustrate
the proposed methods.

###### keywords:

Causal inference; Data fusion; Integrative data analysis; Semiparametric
efficiency bound.

##  section Introduction

Suppose we have available individual data from an internal study for the
investigation of a particular scientific purpose. It is appealing to fuse
together external datasets from different sources with the internal data to
improve statistical inference. Methods for data fusion with external
individual data have grown in popularity in recent years (e.g., Yang & Ding,
2020; Li et al., 2023; Li & Luedtke, 2021; Sun & Miao, 2022; Chen et al.,
2021). However, sometimes it is impossible to access the individual data due
to ethics and privacy concerns and one can only have certain summary
statistics from the external studies. Meta-analysis has been widely applied to
integrate summary statistics on a common parameter from multiple studies
(e.g., Singh et al., 2005; Lin & Zeng, 2010; Kundu et al., 2019; Li et al.,
2020), but it becomes challenging if participating studies are analyzed with
different statistical models and are concerned with different parameters. In
order to assimilate various types of summary statistics from multiple sources,
previous authors have developed a suite of methods which essentially view
external summary statistics as certain constraints on the internal data
distribution. For example, Bickel et al. (1993, Section 3.2 Example 3)
established the semiparametric theory for using external summary statistics as
moment equation constraints; Qin (2000) proposed an empirical likelihood
method and Chatterjee et al. (2016) proposed a constrained maximum likelihood
approach, which leverage summary statistics obtained from a large external
dataset to improve estimation efficiency for a parametric model for the
internal dataset. In situations where the uncertainty of external summary
statistics is negligible, these methods can achieve higher efficiency than the
maximum likelihood estimate based solely on the internal individual data.
Nonetheless, Zhang et al. (2020) cautioned that if the uncertainty of external
summary statistics is not negligible, the efficiency gain of the constrained
maximum likelihood estimate is not guaranteed, and paradoxically, it can be
even less efficient. Such an efficiency paradox may arise when the sample size
of external studies is comparable or smaller than that of the internal study.
Zhang et al. (2020) developed a generalized data integration method (GIM) for
inference about the parameter of a parametric model for the internal data,
which is able to account for uncertainty of external summary statistics and to
avoid the efficiency paradox. However, in many empirical applications the
parameter of interest is a functional of the internal data distribution and it
is desirable to only assume a semiparametric or nonparametric model rather
than a parametric model. Familiar examples include the average treatment
effect in causal inference and the outcome mean in missing data analysis. It
is so far not clear whether or how the efficiency paradox can be avoided in
this situation.

In addition to possible efficiency loss, the integration of external summary
statistics can also introduce estimation bias if they are biased, and
moreover, the estimation bias can be large even if the bias of external
summary statistics is small. This bias paradox frequently occurs in the
presence of population heterogeneity, biased sampling, and model
misspecification. It has been noted in conventional meta-analysis and previous
authors have proposed a variety of robust methods for meta-analysis (Singh et
al., 2005; Shen et al., 2020; Wang et al., 2021), which can still deliver
consistent and asymptotically normal estimation in the presence of biased
summary statistics. For the fusion of individual level data, Chen et al.
(2021), Kallus et al. (2018), Yang et al. (2020a) and Yang et al. (2020b)
considered combining randomized trial data and observational data to remove
unmeasured confounding bias. But these methods require the availability of
individual data in multiple data sources. For integration of individual data
and summary statistics, Zhai & Han (2022) proposed a penalized constrained
maximum likelihood method that extends the empirical likelihood methods of Qin
(2000) and Chatterjee et al. (2016) to accommodating possibly biased external
summary statistics. But they have focused on parameters in a parametric model
and assumed the uncertainty of summary statistics is negligible.

In this article, we further investigate these two paradoxes in a
semiparametric framework and propose efficient and robust data fusion methods.
We establish the semiparametric theory for the integration of internal
individual data and external summary statistics. In contrast to previous
parametric approaches, our framework admits semiparametric and nonparametric
models. We derive the semiparametric efficiency bound for inference about a
general functional of the internal data distribution in the presence of
external summary statistics, which is shown to be no larger than that using
only internal data. We construct a data-fused efficient estimator that attains
the efficiency bound so that the efficiency paradox can be avoided. The data-
fused efficient estimator has a closed form, which greatly reduces the
computational complexity compared to the empirical likelihood methods proposed
by Zhang et al. (2020, 2022). Our results provide a formal proof for the
conjecture raised by Zhang et al. (2020) on improving the inference about a
generalized linear model with external summary statistics from a nested
working model using only a subset of covariates. To promote robustness against
biased summary statistics, we employ an adaptive lasso penalty based procedure
to obtain a debiased estimator. The debiased estimator is consistent and
asymptotically normal even if some external summary statistics are biased, and
moreover, it is asymptotically equivalent to the oracle estimator that uses
only unbiased summary statistics. We also discuss the situation when
unbiasedness of a subset of the summary statistics only holds asymptotically
at a suitable rate in the supplementary material. We evaluate performance of
the proposed estimation methods with simulations and apply them to test the
causal effect of a combined therapy on Helicobacter pylori infection.

##  section The classic semiparametric theory

Suppose we have $n$ i.i.d. individual-level observations
$(Z_{1},\ldots,Z_{n})$ from the internal distribution/study
$P_{0}\in\mathcal{P}_{0}$ and a $q$-dimensional summary statistic
$\tilde{\beta}=(\tilde{\beta}_{1},\ldots,\tilde{\beta}_{q})^{\mathrm{\scriptscriptstyle
T}}$ based on individual observations $(W_{1},\ldots,W_{m})$ from the external
distribution/study $P_{1}\in\mathcal{P}_{1}$, where $\mathcal{P}_{0}$ and
$\mathcal{P}_{1}$ are collections of distributions for internal data and
external data respectively. The external sample size $m$ is known, the
external individual data are not available, and $\tilde{\beta}$ is assumed to
be an estimator of some functional $\beta(P_{1})$. The parameter of interest
is a $p$-dimensional functional of the internal data distribution,
$\tau=\tau(P_{0})$, which may differ from $\beta$. Throughout the paper, we
let $E(\cdot)$ denote the expectation with respect to $P_{0}$ and
$\hat{E}(\cdot)$ the empirical mean in the internal data.

We briefly review the classical semiparametric theory when only the internal
i.i.d. individual data are used for estimating $\tau$. Most reasonable
estimators in a range of statistical inference problems are regular and
asymptotically linear (RAL). Following Tsiatis (2006, chapter 3), RAL
estimators in a parametric model indexed by a finite-dimensional parameter
$\theta$, say $\\{P_{0}(Z;\theta);\theta\in\mathbb{R}^{k}\\}$, are described
as follows.

######  Definition section.theorem.

An estimator $T_{n}=T_{n}(Z_{1},\ldots,Z_{n})$ of $\tau$ is said to be regular
if $n^{1/2}\\{T_{n}(Z_{1}^{(n)},\ldots,Z_{n}^{(n)})-\tau(\theta_{n})\\}$ has a
limiting distribution that does not depend on the local data generating
process where for each $n$ the data $\\{Z_{1}^{(n)},\ldots,Z_{n}^{(n)}\\}$ are
i.i.d. distributed according to $P_{0}(Z;\theta_{n})$ with
$n^{1/2}(\theta_{n}-\theta)$ converging to a constant; an estimator $T_{n}$ is
said to be asymptotically linear if
$T_{n}=\tau+n^{-1}\sum_{i=1}^{n}\phi(Z_{i})+o_{p}(n^{-1/2})$ for some vector
function $\phi$ with $E\\{\phi(Z)\\}=0$ and
$E\\{\phi(Z)\phi^{\mathrm{\scriptscriptstyle T}}(Z)\\}$ finite and
nonsingular; and an estimator $T_{n}$ is said to be RAL if it is both regular
and asymptotically linear.

The function $\phi$ is called an influence function for $\tau$, describing the
influence of each observation on the estimation of $\tau$. Regularity is often
desirable, which rules out pathological estimators such as the superefficient
estimator of Hodges and estimators that invoke more information than is
contained in the model. Moreover, it can be shown that the most efficient
regular estimator is asymptotically linear (Hájek, 1970); hence, it is
reasonable to restrict attention to RAL estimators. For a differentiable
parameter $\tau(\theta)$ in the parametric model $P_{0}(Z;\theta)$, letting
$S_{\theta}$ denote the score for $\theta$, then the Cramer–Rao bound,
$V_{\theta}=\\{\partial\tau(\theta)/\partial\theta\\}\\{E(S_{\theta}S^{\mathrm{\scriptscriptstyle
T}}_{\theta})\\}^{-1}\\{\partial\tau(\theta)/\partial\theta^{\mathrm{\scriptscriptstyle
T}}\\}$, describes the smallest possible asymptotic variance for RAL
estimators of $\tau$. However, lack of flexibility and thus potential
misspecification of parametric models incur biased inferences, and in many
situations one is only interested in a finite-dimensional parameter rather
than the full data distribution. This leads to the adoption of semiparametric
or nonparametric models that admit infinite-dimensional parameters embodying
less restrictive assumptions beyond the parameter of interest. Bickel et al.
(1993) described the efficiency theory for inference in semiparametric and
nonparametric models. One can view a semiparametric model as the collection of
many parametric submodels that satisfy the semiparametric assumptions and
contain the true data generating process but impose no additional
restrictions. A (pathwise) differentiable functional $\tau$ on a
semiparametric model needs to be differentiable on all parametric submodels
and satisfy $\partial\tau(\theta)/\partial\theta=E\\{\phi S_{\theta}\\}$ for
some squared integrable function $\phi$ and score function $S_{\theta}$ of an
arbitrary parametric submodel. An estimator is said to be regular on a
semiparametric model if it is regular on all parametric submodels. A key
concept in the semiparametric theory is the semiparametric efficiency bound,
which is the supremum of the Cramer-Rao bounds for all parametric submodels.
The semiparametric efficiency bound is the lower bound for the asymptotic
variance of any RAL estimator. The influence function attaining the
semiparametric efficiency bound is called the efficient influence function,
and the corresponding estimator is the efficient estimator.

In the rest of this paper, we let $\phi_{\mathrm{eff}}$ denote the efficient
influence function,
$E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle T}})$
the efficiency bound, $\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$ the efficient
estimator of $\tau$ based on the internal data in the class of semiparametric
or nonparametric model under consideration. We illustrate these concepts with
an influential causal inference example; see Bang & Robins (2005) and Hahn
(1998) for details.

######  Example section.theorem.

Suppose we have internal individual data on $Z=(T,X,Y)$ from an observational
study $P_{0}$ about the effect of a binary treatment $T$ on the outcome $Y$
with covariates $X$. Let $Y_{t}$ denote the potential outcome if the treatment
were set to $T=t$ for $t=0,1$ and $\tau=E(Y_{1}-Y_{0})$ the average treatment
effect. Let $p(X)=\text{pr}(T=1\mid X)$ be the treatment propensity score and
$\mu_{t}(X)=E(Y\mid T=t,X)$ ($t=0,1$) the outcome regression function. Under
the ignorability assumption ($Y_{t}\perp\\!\\!\\!\perp T\mid X$ and
$0<p(X)<1$), $\tau$ is identified from the observed data with
$\tau=E\\{\mu_{1}(X)-\mu_{0}(X)\\}$. The efficient influence function for
$\tau$ in the nonparametric model that imposes no other restrictions than the
ignorability is

$\phi_{\mathrm{eff}}(Z;\tau)=\frac{T}{p(X)}\\{Y-\mu_{1}(X)\\}-\frac{1-T}{1-p(X)}\\{Y-\mu_{0}(X)\\}+\mu_{1}(X)-\mu_{0}(X)-\tau.$

The efficiency bound for $\tau$ is $E(\phi_{\mathrm{eff}}^{2})$. The efficient
estimator $\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$ can be obtained by first
estimating $\\{p(X),\mu_{t}(X)\\}$ and then solving
$\hat{E}\\{\phi_{\mathrm{eff}}(Z;\tau)\\}=0$ with these nuisance estimators
plugged in. For instance, one can specify and fit parametric working models
$p(X;\hat{\alpha})$ and $\mu_{t}(X;\hat{\psi}_{t})$, then

$\displaystyle\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\biggl{[}\frac{T_{i}}{p(X_{i};\hat{\alpha})}\\{Y_{i}-\mu_{1}(X_{i};\hat{\psi}_{1})\\}-\frac{1-T_{i}}{1-p(X_{i};\hat{\alpha})}\\{Y_{i}-\mu_{0}(X_{i};\hat{\psi}_{0})\\}$
$\displaystyle+\mu_{1}(X_{i};\hat{\psi}_{1})-\mu_{0}(X_{i};\hat{\psi}_{0})\biggl{]}.$

The semiparametric theory for i.i.d. data has been well established. Applying
the classical semiparametric theory, Bickel et al. (1993, Section 3.2 Example
3) has established a well-known result that the efficient influence function
for $\tau$, when a relevant functional $\beta$ is known, is
$\phi_{\mathrm{eff}}-E\left(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}}\right)\left\\{E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\right\\}^{-1}\eta_{\mathrm{eff}}$, where $\eta_{\mathrm{eff}}$ is the
efficient influence function for $\beta$ based on internal individual data.
This influence function must have variance no larger than
$\phi_{\mathrm{eff}}$, and the corresponding estimator denoted by
$T_{n}^{\mathrm{eff}}(\beta)=T_{n}^{\mathrm{eff}}(Z_{1},\ldots,Z_{n},\beta)$
is more efficient than the estimator
$\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$ solely based on internal data.
Without knowing the true value $\beta$ but given a good (RAL) estimator
$\tilde{\beta}$, a natural estimator of $\tau$ is the plug-in estimator
$T_{n}^{\mathrm{eff}}(\tilde{\beta})$, which is expected to still deliver
better efficiency than $\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$.
Nonetheless, Zhang et al. (2020) cautioned for a paradoxical result that
$T_{n}^{\mathrm{eff}}(\tilde{\beta})$ can have asymptotic variance even larger
than $\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$ and they illustrated about
this in a parametric framework.

It is desirable to further investigate the efficiency paradox in a
semiparametric framework where the parameter of interest is a functional of
internal data distribution with least restrictive assumptions imposed.
However, the classical semiparametric theory for i.i.d. data does not apply
because the external summary data $\tilde{\beta}$ are distributed differently
from the internal data. Although there exists a sparse literature on the
semiparametric theory for non-i.i.d data (Strasser, 1989; McNeney & Wellner,
2000), their theory is not applicable here. In the next section, we extend the
classical semiparametric theory to the data fusion setting we consider.

##  section Semiparametric theory for fusion of individual and summary data

###  section.subsection Assumptions, data-fused RAL estimators and the
efficiency paradox

In order to make use of the external summary statistic, we make the following
assumptions. The external summary statistic $\tilde{\beta}$ is a RAL estimator
of a $q$-dimensional functional $\beta(P_{1})$ of the external data
distribution and $m^{1/2}\\{\tilde{\beta}-\beta(P_{1})\\}\rightarrow
N(0,\Sigma_{1})$; a consistent covariance estimator $\hat{\Sigma}_{1}$ is also
available; and $m/n\rightarrow\rho\in(0,+\infty)$.

$\beta(P_{0})=\beta(P_{1})$. Assumption section.subsection is met with
standard estimation methods under mild regularity conditions and has been
widely adopted in meta-analysis (Singh et al., 2005; Xie et al., 2011; Kundu
et al., 2019; Zhang et al., 2020). Note that the functional $\beta(\cdot)$ is
not necessarily the same as $\tau$, the parameter of interest. Assumption
section.subsection states that the values of $\beta$ are the same across the
internal and external studies. This unbiasedness assumption establishes the
connection between the internal data distribution $P_{0}$ and external data
distribution $P_{1}$, which is essential for efficiency improvement with
external summary statistics. Analogous assumptions such as mean/distribution
exchangeability have been used in previous work (e.g., Dahabreh et al., 2019;
Li et al., 2023). We discuss in Section 4 when Assumption 2 fails and a fixed
bias exists. Furthermore, we consider the scenario when unbiasedness of a
subset of the summary statistics only holds asymptotically at a suitable rate
in the supplementary material. In the rest of the paper, we denote
$\beta=\beta(P_{0})$, and we assume that $\tau$ is pathwise differentiable on
$\mathcal{P}_{0}$ at $P_{0}$ and $\beta$ is pathwise differentiable on
$\mathcal{P}_{0}$ and $\mathcal{P}_{1}$ at $P_{0}$ and $P_{1}$, respectively.
We let $\eta_{\mathrm{eff}}$ denote the efficient influence function for
$\beta$ when only internal individual data are available.

We consider estimation of $\tau$ in the semiparametric model
$\mathcal{P}_{0}\times\mathcal{P}_{1}=\\{P_{0}\times P_{1}:P_{0},P_{1}\text{
satisfy Assumption~{}\ref{well defined}}\\}$. We consider the following class
of estimators that incorporate both internal individual data and the external
summary statistic.

######  Definition section.theorem (Data-fused RAL estimator).

Let $T_{n}(Z_{1},\ldots,Z_{n},\tilde{\beta})$ denote a data-fused estimator of
$\tau$ and we write $T_{n}(\tilde{\beta})$ for shorthand.

  * (i)

$T_{n}(\tilde{\beta})$ is said to be regular if for every parametric submodel
$P_{0}(Z;\theta)\times
P_{1}(W;\theta)\in\mathcal{P}_{0}\times\mathcal{P}_{1}$,
$n^{1/2}\big{\\{}T_{n}(Z_{1}^{(n)},\ldots,Z_{n}^{(n)},\tilde{\beta}^{(m)})-\tau(P_{0}(Z;\theta_{n}))\big{\\}}$
has a limiting distribution that does not depend on the local data generating
process, where the data $\\{Z_{1}^{(n)},\ldots,Z_{n}^{(n)}\\}$ and
$\\{W_{1}^{(m)},\ldots,W_{m}^{(m)}\\}$ are i.i.d. samples from
$P_{0}(Z;\theta_{n})$ and $P_{1}(W;\theta_{n})$ respectively, and
$\tilde{\beta}^{(m)}$ is obtained from $\\{W_{1}^{(m)},\ldots,W_{m}^{(m)}\\}$,
with $m/n\rightarrow\rho\in(0,\infty)$ and $n^{1/2}(\theta_{n}-\theta)$
converging to a constant.

  * (ii)

$T_{n}(\tilde{\beta})$ is said to be asymptotically linear if
$T_{n}(\tilde{\beta})=\tau+n^{-1}\sum_{i=1}^{n}\psi(Z_{i})+\gamma(\tilde{\beta})+o_{p}(n^{-1/2})$
with $E\\{\psi(Z)\\}=0$, $E\\{\psi(Z)\psi^{\mathrm{\scriptscriptstyle
T}}(Z)\\}$ finite and nonsingular, $\gamma(\tilde{\beta})$ continuously
differentiable in $\tilde{\beta}$ and $\gamma(\beta)=0$.

  * (iii)

$T_{n}(\tilde{\beta})$ is said to be regular and asymptotically linear (RAL)
if it satisfies both (i) and (ii).

Analogous to the classical semiparametric theory, Definition section.theorem
(i) characterizes the regularity with respect to both the internal data
distribution and external data distribution. This class of data-fused regular
estimators contain all the regular estimators based only on the internal
individual data. Following the spirit of classical asymptotic linearization,
Definition section.theorem (ii) treats $\tilde{\beta}$ as an additional sample
to the internal data and uses $\psi(Z_{i})$ as well as $\gamma(\tilde{\beta})$
to depict the influence of $(Z_{1},\ldots,Z_{n},\tilde{\beta})$ on the
estimation of $\tau$. The restrictions on $\gamma(\tilde{\beta})$ ensure that
$T_{n}(\tilde{\beta})$ satisfying Definition section.theorem (ii) is
consistent and asymptotically normal. The class of data-fused RAL estimators
in Definition section.theorem (iii) include all the RAL estimators that use
only the internal data.

######  Proposition section.theorem.

Under Assumptions section.subsection–section.subsection, a data-fused RAL
estimator $T_{n}(\tilde{\beta})$ has the following representation,

$T_{n}(\tilde{\beta})=\tau+\frac{1}{n}\sum_{i=1}^{n}\\{\phi(Z_{i})-\xi\eta_{\mathrm{eff}}(Z_{i})\\}+\xi(\tilde{\beta}-\beta)+o_{p}(n^{-1/2}),$
(equation)

and its asymptotic variance is

$E(\phi\phi^{\mathrm{\scriptscriptstyle T}})+\xi
E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\xi^{\mathrm{\scriptscriptstyle T}}-2\xi
E(\eta_{\mathrm{eff}}\phi^{\mathrm{\scriptscriptstyle
T}})+\rho^{-1}\xi\Sigma_{1}\xi^{\mathrm{\scriptscriptstyle T}},$

where $\phi$ is an influence function for $\tau$ based only on the internal
data, $\xi=\xi(P_{0})$ is a $p\times q$ matrix, and the form of $\phi$ and
$\xi$ depends on the estimator $T_{n}(\tilde{\beta})$.

Proposition section.theorem reveals how the estimation of $\beta$ in the
external data affects the efficiency of a data-fused RAL estimator
$T_{n}(\tilde{\beta})$. If the estimation $\tilde{\beta}$ in the external
study is very precise or the true value $\beta$ is known, then
$T_{n}(\tilde{\beta})$ reduces to $T_{n}(\beta)$, which is a RAL estimator
with influence function $\psi=\phi-\xi\eta_{\mathrm{eff}}$. As mentioned in
Section 2, let $T_{n}^{\mathrm{eff}}(\beta)$ be the efficient estimator when
$\beta$ is known and $T_{n}^{\mathrm{eff}}(\tilde{\beta})$ the plug-in
estimator when an estimate $\tilde{\beta}$ is available instead, then
$T_{n}^{\mathrm{eff}}(\tilde{\beta})=\tau+n^{-1}\sum_{i=1}^{n}\\{\phi_{\mathrm{eff}}-A\eta_{\mathrm{eff}}\\}+A(\tilde{\beta}-\beta)+o_{p}(n^{-1/2})$
with
$A=E\left(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}}\right)\left\\{E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\right\\}^{-1}$, and its asymptotic variance is

$E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})+A\\{\Sigma_{1}/\rho-E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\}A^{\mathrm{\scriptscriptstyle T}}.$ (equation)

This result has three meaningful implications: First, when the external data
sample size is much larger than the internal data, i.e.,
$\rho\rightarrow+\infty$, the asymptotic variance of the plug-in estimator
$T_{n}^{\mathrm{eff}}(\tilde{\beta})$ approximates the semiparametric
efficiency bound when $\beta$ is known; this has been noted by Qin (2000) and
Chatterjee et al. (2016) for estimation of parameters in a parametric model
for the internal data; here we extend this result to semiparametric models
where the parameter of interest is a functional of data distribution. Second,
when the external data sample size is much smaller than the internal data,
i.e., $\rho\rightarrow 0$, the asymptotic variance of
$T_{n}^{\mathrm{eff}}(\tilde{\beta})$ diverges, suggesting that it has a
slower convergence rate than $n^{-1/2}$. In this case, the large variability
in external summary statistics will severely damage the estimation efficiency
of the plug-in estimator. Third, for sufficiently small $\rho$ such that
$\Sigma_{1}/\rho-E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})>0$ (positive definite), the asymptotic variance of
$T_{n}^{\mathrm{eff}}(\tilde{\beta})$ is larger than that of the efficient
estimator using only internal data. This explains why the efficiency paradox
arises: the external summary statistic is used as the true value while its
uncertainty is not negligible. The following example illustrates a situation
frequently encountered in practice where the efficiency paradox occurs.

######  Example section.theorem.

Suppose $Z=(X,Y)\sim P_{0}$ in the internal study and $W=X\sim P_{1}$ in the
external study with $P_{1}$ equal to the marginal distribution of $P_{0}$.
This is a common setting in semi-supervised learning or missing data analysis
where $Z$ is the labeled/complete data and $W$ is the unlabeled/incomplete
data. Consider the estimation of $\tau=E(Y)$. Suppose individual samples in
the internal study and $\tilde{\beta}$ the sample mean of $X$ in external data
are available. Using only the internal data, the efficiency bound for $\tau$
is $\mathrm{var}(Y)$. Treating $\tilde{\beta}$ as the true underlying value
$\beta$, then the plug-in estimator is
$T_{n}^{\mathrm{eff}}(\tilde{\beta})=n^{-1}\sum_{i=1}^{n}\\{Y_{i}-\hat{\alpha}(X_{i}-\tilde{\beta})\\}$,
where $\hat{\alpha}$ is the least squares coefficient of $X$ in the linear
regression of $Y$ on $X$. The asymptotic variance of
$T_{n}^{\mathrm{eff}}(\tilde{\beta})$ is
$\mathrm{var}(Y)-(1-1/\rho)\mathrm{cov}^{2}(X,Y)\\{\mathrm{var}(X)\\}^{-1}$.
If $\rho<1$ and $\mathrm{cov}(X,Y)\neq 0$, the efficiency paradox emerges.

###  section.subsection Efficiency bound

To assess how external summary statistics can improve the efficiency for
estimating $\tau$, we establish the semiparametric efficiency bound for the
data-fused RAL estimators. Theorem section.theorem characterizes their
asymptotic distribution.

######  Theorem section.theorem (Convolution theorem).

Under Assumptions section.subsection and section.subsection, for any data-
fused RAL estimator $T_{n}(\tilde{\beta})$ we have

$n^{1/2}\left[\begin{matrix}T_{n}(\tilde{\beta})-\tau-n^{-1}\sum_{i=1}^{n}\left(\phi_{\mathrm{eff}}-M\eta_{\mathrm{eff}}\right)-M(\tilde{\beta}-\beta)\\\
n^{-1}\sum_{i=1}^{n}\left(\phi_{\mathrm{eff}}-M\eta_{\mathrm{eff}}\right)+M(\tilde{\beta}-\beta)\end{matrix}\right]\rightarrow\begin{pmatrix}\Delta_{0}\\\
S_{0}\end{pmatrix},$

where $M=E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\left\\{\Sigma_{1}/\rho+E\left(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}}\right)\right\\}^{-1}$, $S_{0}$ and $\Delta_{0}$ are independent, and
$S_{0}\sim N(0,B)$ with
$B=E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})-E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\left\\{\Sigma_{1}/\rho+E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\right\\}^{-1}E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})^{\mathrm{\scriptscriptstyle T}}$.

In classical semiparametric theory, convolution theorem is a key venue for
establishing the asymptotic bound for the limiting distribution of RAL
estimators. Here we extend it to the data fusion setting with both individual
and summary data. The proof follows from Bickel et al. (1993), but innovation
here is that we incorporate the external summary statistics, which is
different from the i.i.d case considered in classical semiparametric theory.
Theorem section.theorem asserts that the asymptotic distribution of any data-
fused RAL estimator $T_{n}(\tilde{\beta})$ can be decomposed into two
independent parts $\Delta_{0}$ and $S_{0}$, where $S_{0}$ follows a normal
distribution. We have
$n^{1/2}\\{T_{n}(\tilde{\beta})-\tau(P_{0})\\}\to\Delta_{0}+S_{0}$ and
$\mathrm{var}(\Delta_{0}+S_{0})=\mathrm{var}(\Delta_{0})+\mathrm{var}(S_{0})\geq\mathrm{var}(S_{0})=B$,
which is a lower bound for the asymptotic variance of any data-fused RAL
estimator $T_{n}(\tilde{\beta})$.

######  Theorem section.theorem.

Under Assumptions section.subsection and section.subsection, the efficiency
bound for data-fused RAL estimators given in Definition section.theorem is

$\displaystyle
B=E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})-E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\left\\{\Sigma_{1}/\rho+E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\right\\}^{-1}E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})^{\mathrm{\scriptscriptstyle T}}.$ (equation)

Note that the efficiency bound when only internal data are available is
$E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle T}})$,
Theorem section.theorem suggests that the efficiency bound does not increase
in the presence of external summary statistics. But $B$ is larger than
$E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})-E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\left\\{E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\right\\}^{-1}E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})^{\mathrm{\scriptscriptstyle T}}$, the efficiency bound when $\beta$ is
known, suggesting that $\tilde{\beta}$ can provide no more information than
the true value of $\beta$ for estimating $\tau$, and $B$ reduces to the latter
as $\rho\rightarrow+\infty$. The efficiency bound $B$ also depends on the
efficiency of $\tilde{\beta}$, captured by $\Sigma_{1}$; $B$ increases as
$\Sigma_{1}$ increases.

######  Proposition section.theorem.

Under Assumptions section.subsection, if $P_{1}$ is a marginal distribution of
$P_{0}$ and $\tilde{\beta}$ is an efficient estimator of $\beta$ in the
external study, i.e.,
$\Sigma_{1}=E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})$, then the efficiency bound for $\tau$ is
$E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})-\rho(1+\rho)^{-1}E\left(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}}\right)\left\\{E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\right\\}^{-1}E\left(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}}\right)^{\mathrm{\scriptscriptstyle T}}$.

Theorem section.theorem also shows that external summary statistics do not
bring efficiency gain if
$E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle T}})=0$;
that is, knowing $\beta$ has no influence on the estimation of $\tau$. This
happens if $P_{0}$ factorizes as $P_{0}(Z)=f_{1}(Z)f_{2}(Z)$ and
$\tau(P_{0})=\tau(f_{1}),\beta(P_{0})=\beta(f_{2})$. The following is an
example concerning marginal and joint regressions in genome-wide association
studies, etc.

######  Example section.theorem.

Suppose the internal data are random samples of $(X_{1},X_{2},Y)\sim P_{0}$
and the external individual data are random samples of $(X_{1},X_{2},Y)\sim
P_{1}$, with $P_{0}=P_{1}$, $Y=X_{1}\tau_{1}+X_{2}\tau_{2}+\varepsilon$,
$X_{1},X_{2}$ mean zero, $E(\varepsilon\mid X_{1},X_{2})=0$ and
$\mathrm{var}(\varepsilon\mid X_{1},X_{2})=\sigma^{2}$. The external summary
statistics are
$\tilde{\beta}=(\tilde{\beta}_{1},\tilde{\beta}_{2})^{\mathrm{\scriptscriptstyle
T}}$, where $\tilde{\beta}_{1}$ and $\tilde{\beta}_{2}$ are the ordinary least
squares coefficients obtained by regressing $Y$ on $X_{1}$ and $X_{2}$
separately with the external data. This happens in genome-wide association
studies where researchers provide summary statistics of separate univariate
regression coefficients of quantitative trait ($Y$) on each centered genotypes
($X$) (Zhu & Stephens, 2017). The efficient influence function for $\beta$ is
$\eta_{\mathrm{eff}}=(\eta_{\mathrm{eff},1},\eta_{\mathrm{eff},2})^{\mathrm{\scriptscriptstyle
T}}$, where
$\eta_{\mathrm{eff},1}=\\{E(X_{1}^{2})\\}^{-1}X_{1}(Y-X_{1}\beta_{1})$ and
$\eta_{\mathrm{eff},2}=\\{E(X_{2}^{2})\\}^{-1}X_{2}(Y-X_{2}\beta_{2})$.
Denoting $X=(X_{1},X_{2})^{\mathrm{\scriptscriptstyle T}}$, the data-fused
efficiency bound for $\tau=(\tau_{1},\tau_{2})^{\mathrm{\scriptscriptstyle
T}}$ is

$\sigma^{2}\\{E(XX^{\mathrm{\scriptscriptstyle
T}})\\}^{-1}-\frac{\rho\sigma^{4}}{1+\rho}\mathrm{diag}\\{1/E(X_{1}^{2}),1/E(X_{2}^{2})\\}\\{E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\}^{-1}\mathrm{diag}\\{1/E(X_{1}^{2}),1/E(X_{2}^{2})\\}.$

If only $\tilde{\beta}_{1}$ is available and without loss of generality we
assume $X_{1},X_{2}$ have unity variance and correlation coefficient $\kappa$,
the data-fused efficiency bound for $\tau$ is

$\sigma^{2}\begin{pmatrix}\frac{1}{1-\kappa^{2}}&-\kappa\\\
-\kappa&\frac{1}{1-\kappa^{2}}\end{pmatrix}-\frac{\rho}{1+\rho}\begin{pmatrix}\sigma^{4}/\mathrm{var}(\eta_{\mathrm{eff,1}})&0\\\
0&0\\\ \end{pmatrix}.$

There is no efficiency gain for estimating $\tau_{2}$ from external marginal
regression estimate $\tilde{\beta}_{1}$. Zhang et al. (2020) has obtained this
result under a special case where the distribution of $\varepsilon$ is
$N(0,1)$. Nevertheless, we note that the efficiency gain for estimating
$\tau_{2}$ emerges when $\varepsilon$ is heteroscedastic, that is,
$\mathrm{var}(\varepsilon\mid X_{1},X_{2})$ is not a constant. A simulation is
given in the supplementary material for illustration.

Applying Theorem section.theorem to estimation of the generalized linear
model, we have the following result that provides a formal justification of
the conjecture raised by Zhang et al. (2020).

######  Proposition section.theorem.

Suppose $P_{0}=P_{1}$ and $E(Y\mid
X_{1},X_{2})=g^{-1}(X_{1}^{\mathrm{\scriptscriptstyle
T}}\tau+X_{2}^{\mathrm{\scriptscriptstyle T}}\alpha)$ with $g$ being the
canonical link function. Suppose in the external study
$g^{-1}(X_{2}^{\mathrm{\scriptscriptstyle T}}\beta)$ is used as a working
model for $E(Y\mid X_{2})$ and $\beta$ is estimated by solving estimating
equation $E[X_{2}\\{Y-g^{-1}(X_{2}^{\mathrm{\scriptscriptstyle
T}}\beta)\\}]=0$. Then the resultant estimator $\tilde{\beta}$ does not bring
efficiency gain for estimating $\tau$.

###  section.subsection Efficient estimation to avoid the efficiency paradox

Let $\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}$ denote the efficient estimator
of $\beta$ and $\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$ the efficient
estimator of $\tau$ based only on internal individual data. Motivated by
Theorem section.theorem, we propose the following data-fused estimator:

$\displaystyle\hat{\tau}_{\mathrm{eff}}$
$\displaystyle=\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}-\hat{E}(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\left\\{\hat{\Sigma}_{1}/\rho+\hat{E}(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\right\\}^{-1}(\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}-\tilde{\beta}),$
(equation)

where
$\hat{E}(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}}),\hat{E}(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})$ are consistent estimators of
$E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}}),E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})$ based on the internal data respectively, and $\hat{\Sigma}_{1}$ is a
consistent estimator of $\Sigma_{1}$.

######  Theorem section.theorem.

Under Assumptions section.subsection–section.subsection, suppose that
$n^{1/2}(\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}-\tau)\rightarrow
N\\{0,E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\}$ and
$n^{1/2}(\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}-\beta)\rightarrow
N\\{0,E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\}$, then $n^{1/2}(\hat{\tau}_{\mathrm{eff}}-\tau)$ is asymptotically
normal with asymptotic variance equal to $B$ in (equation).

Theorem section.theorem shows that $\hat{\tau}_{\mathrm{eff}}$ attains the
efficiency bound for estimating a general functional in semiparametric or
nonparametric models when both internal individual data and external summary
statistics are available. This generalizes previous efficiency results (Zhang
et al., 2020) on parameter estimation in parametric models. We refer to
$\hat{\tau}_{\mathrm{eff}}$ as the data-fused efficient estimator. This
estimator is more efficient than any RAL estimator using only internal data,
and thus, resolves the efficiency paradox. The estimator
$\hat{\tau}_{\mathrm{eff}}$ has a closed form. This fact benefits the
computation of the estimator in general, although the calculation of efficient
influence functions $\phi_{\text{eff}},\eta_{\text{eff}}$ may not be easy in
complicated semiparametric models. For example, the closed form estimator
$\hat{\tau}_{\mathrm{eff}}$ can greatly reduce the computational burden
compared to the constrained maximum likelihood or empirical likelihood methods
(Chatterjee et al., 2016; Zhang et al., 2020, 2022) when estimating parameters
in parametric models.

Let

$\Sigma=\left\\{\begin{matrix}E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})&E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\\ E(\eta_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})&E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\end{matrix}\right\\}$

denote the asymptotic covariance of
$(\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}},\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}})$.
The estimator $\hat{\tau}_{\mathrm{eff}}$ and its asymptotic variance
(equation) are determined once we obtain
$(\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}},\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}},\hat{\Sigma})$
and $(\tilde{\beta},\hat{\Sigma}_{1})$. The nuisance estimators
$(\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$,
$\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}},\hat{\Sigma})$ can be obtained with
internal individual data, $\tilde{\beta}$ is available from external data, and
$\hat{\Sigma}_{1}$ is routinely reported as summary statistics together with
$\tilde{\beta}$. One can also consistently estimate $\Sigma_{1}$ using
internal data if $P_{1}$ is a marginal distribution of $P_{0}$ and the
estimation method of $\tilde{\beta}$ is known. Otherwise, one can use a
positive-definite working matrix $\Omega$ for $\Sigma_{1}$ without
compromising consistency of $\hat{\tau}_{\mathrm{eff}}$, but in this case
there is no guarantee for efficiency gain from external summary statistics and
the efficiency paradox may still occur. Additional discussion on the choice of
the working covariance matrix and the efficiency is provided in the proof of
Proposition section.theorem.

If $\beta$ is the same functional as $\tau$, then $\hat{\tau}_{\mathrm{eff}}$
reduces to

$\frac{\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}/\hat{\mathrm{var}}(\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}})+\tilde{\beta}/\hat{\mathrm{var}}(\tilde{\beta})}{1/\hat{\mathrm{var}}(\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}})+1/\hat{\mathrm{var}}(\tilde{\beta})},$

which is the well-known inverse variance weighted estimator in meta-analysis
(Lin & Zeng, 2010). The estimator $\hat{\tau}_{\mathrm{eff}}$ can be viewed as
a calibration estimator where the external summary statistic $\tilde{\beta}$
is used to calibrate the internal data-based efficient estimator
$\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$. Calibration is a standard
technique used in survey sampling for efficiency improvement with auxiliary
information. Chen & Chen (2000), Wang & Wang (2015) and Yang & Ding (2020)
considered calibration with validation data in the contexts of measurement
error and confounding adjustment, where the validation dataset contains
individual random samples from the big internal data. In contrast, here we
consider the situation where only summary statistics are available from the
external study and the external data are not necessarily random samples from
the internal population.

The estimator $\hat{\tau}_{\mathrm{eff}}$ can also be viewed as a confidence
density estimator (Liu et al., 2015) in which only summary statistics
$(\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}},\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}},\hat{\Sigma})$
and $(\tilde{\beta},\hat{\Sigma}_{1})$ were used for estimating $\tau$.

######  Proposition section.theorem.

Given
$(\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}},\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}},\hat{\Sigma})$
and $(\tilde{\beta},\hat{\Sigma}_{1})$, then for some $\hat{\beta}$,

$(\hat{\tau}_{\mathrm{eff}},\hat{\beta})=\arg\min_{\tau,\beta}\biggl{\\{}\left(\begin{array}[]{c}\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}-\tau\\\
\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}-\beta\end{array}\right)^{\mathrm{\scriptscriptstyle
T}}\hat{\Sigma}^{-1}\left(\begin{array}[]{c}\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}-\tau\\\
\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}-\beta\end{array}\right)+(\tilde{\beta}-\beta)^{\mathrm{\scriptscriptstyle
T}}\left(\frac{\hat{\Sigma}_{1}}{\rho}\right)^{-1}(\tilde{\beta}-\beta)\biggl{\\}}.$
(equation)

The confidence density approach of Liu et al. (2015) concerns estimation of
parameters in a parametric model with summary statistics whose functional
relationships to the parameters are completely known. Here however, we
consider estimation of a general functional in semiparametric models and the
availability of internal individual data obviates the need to know the
functional relationship between summary statistics and the functional of
interest.

For illustration, we apply our estimation method to causal inference and semi-
supervised learning examples.

######  Example section.theorem (Continuation of Example section.theorem).

Suppose in addition to individual samples on $Z=(T,X,Y)\sim P_{0}$ in the
internal study, we also have the ordinary least squares estimate
$\tilde{\beta}$ of $\beta=\\{E(VV^{\mathrm{\scriptscriptstyle
T}})\\}^{-1}E(VY)$ obtained from the linear regression of $Y$ on
$V=(1,X^{\mathrm{\scriptscriptstyle T}},T)^{\mathrm{\scriptscriptstyle T}}$ in
the external study. Suppose $P_{1}=P_{0}$. The efficient influence function
for $\beta$ using only internal data is
$\eta_{\mathrm{eff}}=\\{E(VV^{\mathrm{\scriptscriptstyle
T}})\\}^{-1}V(Y-V^{\mathrm{\scriptscriptstyle T}}\beta)$. Given
$(\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}},\phi_{\mathrm{eff}})$ described in
Example section.theorem and $\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}$ by
regressing $Y$ on $V$ in the internal data, $\hat{\tau}_{\mathrm{eff}}$ is
equal to

$\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}-\frac{m}{m+n}\left[\hat{E}\big{\\{}\phi_{\mathrm{eff}}(Y-V^{\mathrm{\scriptscriptstyle
T}}\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}})V\big{\\}}\right]^{\mathrm{\scriptscriptstyle
T}}\left[\hat{E}\big{\\{}(Y-V^{\mathrm{\scriptscriptstyle
T}}\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}})^{2}V^{\mathrm{\scriptscriptstyle
T}}V\big{\\}}\right]^{-1}\hat{E}(VV^{\mathrm{\scriptscriptstyle
T}})(\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}-\tilde{\beta}).$

######  Example section.theorem (Continuation of Example section.theorem).

We have $\hat{\tau}_{\text{eff}}^{\mathrm{int}}=\bar{Y}$ and
$\hat{\beta}_{\text{eff}}^{\mathrm{int}}=\bar{X}$, the sample mean of $Y$ and
$X$, respectively, in the internal data. The data-fused efficient estimator is
$\hat{\tau}_{\text{eff}}=\bar{Y}-\rho(1+\rho)^{-1}\hat{\alpha}(\bar{X}-\tilde{\beta})$,
where $\hat{\alpha}$ is the least squares coefficient of $X$ in the linear
regression of $Y$ on $X$. This recovers the semi-supervised least squares
estimator given by Zhang et al. (2019).

##  section Bias paradox and resolution

The external summary statistics may contain biased components, i.e.,
Assumption 2 may not hold for some entries. In this case, the integration of
summary statistics as in (equation) will introduce bias. Letting
$b^{*}=\beta(P_{1})-\beta(P_{0})$ denote the asymptotic bias of
$\tilde{\beta}$, the asymptotic bias of $\hat{\tau}_{\mathrm{eff}}$ is

$E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\{\Sigma_{1}/\rho+E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\}^{-1}b^{\ast},$

which can be large even if $b^{*}$ is small, for instance, when $\rho$ is
large and
$E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle T}})$ is
small. That explains why the bias paradox occurs. To mitigate this problem, we
next construct a robust estimator that can effectively use the unbiased
components of the external summary statistics to improve the efficiency while
keeping invulnerable to biased components.

Let $\mathcal{A}=\\{j:\beta_{j}(P_{0})=\beta_{j}(P_{1}),j=1,\ldots,q\\}$
denote the set of unbiased external summary statistics. If such a set were
known a priori, an oracle estimator could be obtained by incorporating only
this subset of external summary statistics, $\tilde{\beta}_{\mathcal{A}}$,
into the efficient estimation in (equation). Let
$\hat{\tau}_{\mathrm{orc}}=\hat{\tau}_{\mathrm{eff}}^{\mathcal{A}}$ denote
such an oracle estimator and $B_{\mathcal{A}}$ denote its asymptotic variance,
which is the efficiency bound when only unbiased external summary statistics
$\tilde{\beta}_{\mathcal{A}}$ and the internal individual data are used for
estimating $\tau$. In practice, $\mathcal{A}$ is unknown. One can first
conduct $q$ tests with null hypotheses
$\beta_{j}(P_{0})=\beta_{j}(P_{1}),j=1,\ldots,q$, to select unbiased summary
statistics, and then use the selected statistics to construct a data-fused
estimator. However, the statistical inference of this approach involves the
stochastic error inherited from the tests, which is generally difficult to
assess. To resolve the difficulty of statistical inference, we propose a
debiased approach that can consistently select the set $\mathcal{A}$ and
achieve the same asymptotic distribution as the oracle estimator. We estimate
the bias $b^{*}$ by solving the optimization problem

$\displaystyle\hat{b}=$
$\displaystyle\arg\min_{b}\biggl{[}(\tilde{\beta}-\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}-b)^{\mathrm{\scriptscriptstyle
T}}\big{\\{}\hat{\Sigma}_{1}/\rho+\hat{E}(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\big{\\}}^{-1}(\tilde{\beta}-\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}-b)$
(equation)
$\displaystyle\qquad\qquad+\lambda\sum_{j=1}^{q}|\tilde{\beta}_{j}-\hat{\beta}_{\mathrm{eff},j}^{\mathrm{int}}|^{-\alpha}|b_{j}|\biggl{]},$

where an adaptive lasso penalty on $b$ is included to account for the
influence of the bias of summary statistics, $\lambda,\alpha>0$ are tuning
parameters, and $\alpha$ is commonly taken to be $1$ or $2$. Then
$\mathcal{A}$ can be estimated by $\hat{\mathcal{A}}=\\{j:\hat{b}_{j}=0\\}$,
and the debiased estimator is defined as
$\hat{\tau}_{\mathrm{dbs}}=\hat{\tau}_{\mathrm{eff}}^{\hat{\mathcal{A}}}$. For
biased summary statistics, the corresponding $\hat{b}_{j}$ tends to be
nonzero, and for unbiased summary statistics, the adaptive lasso penalty
shrinks the corresponding $\hat{b}_{j}$’s to zero. The adaptive lasso penalty
is convex and enjoys the desirable selection consistency (Zou, 2006) under
certain conditions.

######  Theorem section.theorem.

Under Assumption section.subsection, if $\lambda n^{1/2}\rightarrow 0$,
$\lambda n^{(\alpha+1)/2}\rightarrow+\infty$,
$n^{1/2}(\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}-\tau)\rightarrow
N\\{0,E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\}$ and
$n^{1/2}(\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}-\beta)\rightarrow
N\\{0,E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\}$, then we have

  1. (i)

$\lim_{n\rightarrow\infty}P(\mathcal{A}=\hat{\mathcal{A}})=1$, where
$\hat{\mathcal{A}}=\\{j:\hat{b}_{j}=0\\}$;

  2. (ii)

$\hat{\tau}_{\mathrm{dbs}}$ has the same asymptotic distribution as
$\hat{\tau}_{\mathrm{orc}}$ and
$n^{1/2}(\hat{\tau}_{\mathrm{dbs}}-\tau)\rightarrow N(0,B_{\mathcal{A}})$.

Theorem section.theorem shows the selection consistency for the unbiased
external summary statistics; besides, the debiased estimator
$\hat{\tau}_{\mathrm{dbs}}$ retains consistency and asymptotic normality even
if $\tilde{\beta}$ contains biased components, and it is asymptotically as
efficient as the oracle estimator $\hat{\tau}_{\mathrm{orc}}$. Given
$\hat{\mathcal{A}}$, the efficiency bound $B_{\hat{\mathcal{A}}}$ can be
consistently estimated according to (equation) and the procedure described in
Section 3.3, which results in a consistent estimator of the asymptotic
variance $B_{\mathcal{A}}$. Let
$X=\\{\hat{\Sigma}_{1}/\rho+\hat{E}(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\}^{-1/2}$ and
$Y=\\{\hat{\Sigma}_{1}/\rho+\hat{E}(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\\}^{-1/2}(\tilde{\beta}-\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}})$, then
$\hat{b}$ can be obtained with off-the-shelf R packages such as glmnet by
treating $\hat{b}$ as the coefficient in the linear regression of $Y$ on $X$
with no intercept.

A crucial issue for the debiased estimation is the choice of the tuning
parameter $\lambda$. We can specify $\lambda=Cn^{-w}$ with
$1/2<w<(\alpha+1)/2$, which ensures that $\lambda n^{1/2}\rightarrow 0$,
$\lambda n^{(\alpha+1)/2}\rightarrow+\infty$. To choose the constant loading
term $C$, we adopt the cross validation method. Specifically, we split the
full internal data into $K\geq 2$ balanced parts; the cross validation
consists of $K$ steps, where for step $k$ the $k$’th part of internal data is
treated as the test sample and produces an internal data-based estimate
$\hat{\tau}^{(k)}$, and the remaining $K-1$ parts of internal data and the
external summary statistics are treated as the training sample and produce
estimate $\hat{\tau}_{\mathrm{dbs}}^{(k)}$; the prediction error for step $k$
is the squared difference between these two estimates and the cross validation
prediction error is defined as the average of these $K$ prediction errors; $C$
is chosen by minimizing the cross validation error. In the simulation and real
data analysis, we choose $K=3$ and use the above cross validation procedure to
choose $C$. However, as a common issue for cross validation, when the sample
size of the internal data is too small, the cross validation procedure may
have poor performance.

Zhai & Han (2022) have previously proposed a penalized constrained maximum
likelihood method for the fusion of internal individual data and possibly
biased summary statistics; however, they have focused on inference about
parameters in a parametric model and ignored the uncertainty of external
summary statistics, thus efficiency paradox may still occur. In contrast, our
method applies to a general functional in semiparametric models and accounts
for the uncertainty of external summary statistics, which is able to avoid
both the efficiency and the bias paradoxes. Besides, our estimator enjoys a
closed form, which bypasses sophisticated programming and computation and is
easily implemented using existing software.

We have assumed exact unbiasedness in Section 3 and allowed for fixed bias in
Section 4. However, as raised by anonymous reviewers, to mimic the situation
where the internal and external studies have a minor difference, it is of
interest to investigate scenarios when the unbiasedness of certain summary
statistics only holds asymptotically at a suitable rate. Due to space limit,
we give a formal description of this scenario in the supplementary material
and then investigate the asymptotic properties of our proposed estimators. We
show that when the bias is of order smaller than $n^{-1/2}$, the data-fused
efficient estimator $\hat{\tau}_{\mathrm{eff}}$ can still achieve the
efficiency bound $B$ in Theorem 2. When bias of order $n^{-1/2}$ or greater
than $n^{-1/2}$ also exists, the debiased approach will select summary
statistics with bias of order no greater than $n^{-1/2}$. As a result, when
bias of order $n^{-1/2}$ does not exist, the debiased estimator has the same
limiting distribution as the oracle estimator that uses only summary
statistics with bias of order smaller than $n^{-1/2}$ under suitable
conditions; otherwise, the debiased estimator will have asymptotic bias of
order $n^{-1/2}$, which will lead to confidence intervals with coverage
probability below the nominal level. In the supplementary material, we also
conduct additional simulations to illustrate the finite sample property of our
estimators when summary statistics with bias of order $n^{-1/2}$ are
considered.

##  section Numerical illustrations

###  section.subsection Simulation

We conduct simulation studies to evaluate performance of the proposed
estimators. We consider two scenarios with unbiased and biased external
summary statistics, respectively. In Scenario I, a triplet of treatment $T$,
outcome $Y$ and covariate $X$ are generated as follows in both the internal
and external data:

$\displaystyle X\sim N(0,1),\quad\text{pr}(T=1\mid X)=\text{expit}(1-X),$
$\displaystyle
Y=1+X+TX^{2}+T\varepsilon_{1}+(1-T)\varepsilon_{0},\quad(\varepsilon_{0},\varepsilon_{1})\perp\\!\\!\\!\perp(X,T),$

where $\varepsilon_{1}\sim N(0,4)$, $\varepsilon_{0}\sim N(0,1)$ and
$\text{expit}(x)=1/\\{1+\exp(-x)\\}$. The internal sample size $n$ is $1000$
throughout the simulations and the external sample size $m$ increases from
$200$ to $2000$. The functional of interest $\tau$ is the treatment effect of
$T$ on $Y$. Following Example section.theorem we use the internal individual
data and the ordinary least squares estimate $\tilde{\beta}$ obtained by
regressing $Y$ on $(1,X^{\mathrm{\scriptscriptstyle
T}},T)^{\mathrm{\scriptscriptstyle T}}$ in the external study for estimating
$\tau$. We implement four estimators, (i) INT:
$\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$ using only internal data; (ii) CRD:
the crude estimator ignoring uncertainty of $\tilde{\beta}$; (iii) EFF: the
data-fused efficient estimator $\hat{\tau}_{\mathrm{eff}}$; (iv) KNW: the
efficient estimator knowing the true value of $\beta$. The GIM estimator by
Zhang et al. (2020) does not apply to the causal effect functional, and we do
not implement it.

We replicate 1000 simulations. Figure figure and Table table show bias, root
mean squared error (RMSE), average standard error and coverage probability of
the four estimators under different external sample sizes. The crude estimator
improves efficiency against $\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$ only if
the external sample size is sufficiently large ($m=2000$) relative to that of
internal data, otherwise ($m\leq 1000$) efficiency loss emerges. The data-
fused EFF estimator outperforms the INT and CRD estimators and ensures
efficiency gain under all sample sizes. The EFF estimator is less efficient
than the KNW estimator, but the KNW estimator is not feasible in practice
because one does not know $\beta$. Coverage probabilities of the $95\%$
confidence intervals are close to the nominal level for all four estimators.

Table table:  RMSE, average standard error (ASE) and coverage probability of
95% confidence interval (CP) for Scenario I. All numbers are multiplied by
100.

| $m=200$ | $m=500$ | $m=1000$ | $m=2000$
---|---|---|---|---
| RMSE | ASE | CP | RMSE | ASE | CP | RMSE | ASE | CP | RMSE | ASE | CP
INT | 11.62 | 11.78 | 95.4 | 11.62 | 11.78 | 95.4 | 11.62 | 11.78 | 95.4 | 11.62 | 11.78 | 95.4
CRD | 20.86 | 22.64 | 96.2 | 15.30 | 15.26 | 95.5 | 11.79 | 11.79 | 94.2 | 9.44 | 9.59 | 95.1
EFF | 10.97 | 11.10 | 94.8 | 10.25 | 10.37 | 95.5 | 9.56 | 9.58 | 94.4 | 8.65 | 8.72 | 94.8
KNW | 6.94 | 6.69 | 93.1 | 6.94 | 6.69 | 93.1 | 6.94 | 6.69 | 93.1 | 6.94 | 6.69 | 93.1

Figure figure: Bias of estimators in Scenario I. The horizontal line marks the
true value.

In Scenario II, we consider fusion with biased external summary statistics. We
generate $(Y,X_{1},X_{2})$ in the internal data and $(Y,X_{1},\tilde{X}_{2})$
in the external data as follows:

$\displaystyle Y$
$\displaystyle=X_{1}\tau_{1}+X_{2}\tau_{2}+\varepsilon_{1},\quad(X_{1},X_{2})^{\mathrm{\scriptscriptstyle
T}}\sim N\left\\{\begin{pmatrix}0\\\ 0\end{pmatrix},\begin{pmatrix}1&0.6\\\
0.6&1\end{pmatrix}\right\\},$ $\displaystyle\tilde{X}_{2}$
$\displaystyle=X_{2}+\varepsilon_{2},\quad\varepsilon_{1}\sim
N(0,4),\quad\varepsilon_{2}\sim N(0,1).$

where $\tilde{X}_{2}$ in the external data is viewed as a surrogate of $X_{2}$
with measurement error. The internal sample size is $n=1000$ and the external
sample size is $m=4000$. Let
$\tilde{\beta}=(\tilde{\beta}_{1},\tilde{\beta}_{2})$ be the ordinary least
squares estimate obtained from the external data by regressing $Y$ on $X_{1}$
ad $\tilde{X}_{2}$ separately. We use the internal individual data and
$\tilde{\beta}$ to estimate
$\tau=(\tau_{1},\tau_{2})^{\mathrm{\scriptscriptstyle T}}$. We implement five
estimation methods: (i) INT: $\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}$ using
only internal data; (ii) ORC: the oracle estimator using internal data and
only $\tilde{\beta}_{1}$; (iii) DBS: the debiased estimator
$\hat{\tau}_{\mathrm{dbs}}$; (iv) EFF: the efficient estimator
$\hat{\tau}_{\mathrm{eff}}$ in (equation) using both $\tilde{\beta}_{1}$ and
$\tilde{\beta}_{2}$; (v) GIM: the estimator of Zhang et al. (2020).

We replicate 1000 simulations. Figure figure and Table table show the bias,
root mean squared error, average standard error and coverage probability of
the estimators. The GIM and EFF estimators exhibit non-negligible bias as they
are agnostic to the biasedness of external summary statistics and use both
unbiased and biased ones. In contrast, the DBS estimator can adaptively select
and use unbiased external summary statistics; as a result, it performs
approximately to the oracle estimator, both showing negligible bias. Besides,
the DBS estimator of $\tau_{1}$ has smaller variance than the INT estimator
while the DBS estimator of $\tau_{2}$ does not enjoy the efficiency gain,
which coincides with our analysis in Example section.theorem. Coverage
probabilities of the 95% confidence intervals are close to the nominal level
for INT, ORC and DBS estimators; while for EFF and GIM estimators, the
coverage probabilities are below the nominal level.

We also evaluate performance of these estimators when the external data has
accurate measurement of $X_{2}$, in which case, $\tilde{\beta}_{2}$ is also
unbiased for $\beta_{2}$. Figure figure and Table table show the bias, mean
squared error, average standard error and coverage probability of the five
estimators. In this setting, the ORC, EFF, and GIM estimators behave close to
each other, and DBS estimator has slightly larger mean squared error and
average standard error, but they all have smaller bias and mean squared errors
than the INT estimator. Although, we recommend the DBS estimator because it is
both robust against biased external summary statistics and computationally
convenient.

Table table:  RMSE, ASE and CP for Scenario II with biased summary statistics.
All numbers are multiplied by 100.

| $\tau_{1}$ | $\tau_{2}$
---|---|---
| RMSE | ASE | CP | RMSE | ASE | CP
INT | 7.88 | 7.90 | 94.7 | 7.85 | 7.90 | 94.8
ORC | 5.97 | 5.89 | 94.7 | 7.85 | 7.88 | 95.3
DBS | 6.42 | 6.01 | 94.4 | 7.84 | 7.89 | 95.2
EFF | 43.10 | 3.53 | 0 | 82.90 | 3.56 | 0
GIM | 47.27 | 3.73 | 0 | 73.18 | 4.62 | 0

Table table: RMSE, ASE and CP for Scenario II with unbiased summary
statistics. All numbers are multiplied by 100.

| $\tau_{1}$ | $\tau_{2}$
---|---|---
| RMSE | ASE | CP | RMSE | ASE | CP
INT | 7.75 | 7.90 | 95.9 | 7.49 | 7.91 | 96.8
ORC | 4.77 | 4.75 | 94.9 | 4.71 | 4.74 | 94.4
DBS | 5.82 | 5.01 | 94.2 | 5.76 | 5.03 | 93.0
EFF | 4.77 | 4.75 | 94.9 | 4.71 | 4.74 | 94.4
GIM | 4.78 | 4.76 | 94.8 | 4.70 | 4.76 | 95.1

Figure figure:  Bias of estimators in Scenario II with biased external summary
statistics.

Figure figure:  Bias of estimators in Scenario II with unbiased external
summary statistics.

###  section.subsection Real data analysis

We apply the proposed methods to analyze a Helicobacter pylori infection
dataset described by Li et al. (2023). Helicobacter pylori infection is a
leading world-wide infectious disease. The triple therapy (clarithromycin,
amoxicillin, and omeprazole) is a standard treatment for Helicobacter pylori
infection. The internal study is a two-arm randomized clinical trial conducted
at a traditional Chinese medicine hospital. This trial aims to investigate
whether the additional taking of traditional Chinese medicine ($T=1$) has
better efficacy than the standard triple therapy treatment ($T=0$) on
Helicobacter pylori infection. It contains 362 observations, of which 180
patients are assigned to the triple therapy and the rest are assigned to a
combination treatment including both the triple therapy and traditional
Chinese medicine. The external study is a single-arm study conducted at a
Western-style hospital, where 110 patients are all assigned to the triple
therapy. The outcome $Y$ is the post-treatment infection status assessed with
the C–14 urea breath test and baseline covariates $X$ include age, gender,
height, BMI, occupation, education level, marital status, and information on
patients’ symptoms. The internal and external studies adopt the same inclusion
and exclusion criteria and the same treatment protocols. The parameter of
interest is the average causal effect of the combination treatment against the
standard triple therapy treatment, i.e., $\tau=E(Y_{1}-Y_{0})$. We illustrate
how to use the individual data from the internal study and the outcome sample
mean $(\tilde{\beta})$ from the external study to make inference about $\tau$,
where $\tilde{\beta}$ is a consistent estimator of $\beta=E(Y_{0})$ in the
external study.

We implement four methods INT, CRD, EFF and DBS to estimate $\tau$. Our goal
is to test the null hypothesis $\mathbb{H}_{0}:\tau\leq 0$ against
$\mathbb{H}_{1}:\tau>0$ to investigate whether the combination treatment can
improve the efficacy. We calculate the estimate of $\tau$ and its standard
error, and then use z-test to calculate one-sided p-values. Table
section.subsection presents the analysis results. The four point estimates are
close to each other, all showing a potentially beneficial effect of additional
using of traditional Chinese medicine. The EFF estimate and the DBS estimate
are identical, suggesting unbiasedness of the external summary statistic; this
is because the same inclusion criterion and the same treatment protocols are
adopted in both the internal and external studies, and it is reasonable to
assume that they are from the same population. The INT estimate based solely
on the internal data does not reject the null hypothesis $\mathbb{H}_{0}$ at
level $0.1$, and the CRD estimate has an even larger standard error.
Nonetheless, by appropriate integration of the external summary statistic, the
EFF and DBS estimates achieve smaller standard errors, which reject
$\mathbb{H}_{0}$ at level $0.06$. This may serve as evidence in favor of a
beneficial effect of additional using of traditional Chinese medicine.

Point estimates of $\tau$, standard errors, and $p$-values  point estimate
standard error $p$-value INT 0.0543 0.0442 0.1100 CRD 0.0773 0.0520 0.0684 EFF
0.0628 0.0394 0.0553 DBS 0.0628 0.0394 0.0553

##  section Discussion

The proposed methods can be extended to integrate summary statistics from
multiple external studies. Consider $S$ independent external studies, each
study $s$ $(1\leq s\leq S)$ with sample size $m_{s}$ from population $P_{s}$
providing a summary statistic $\tilde{\beta}_{(s)}$ on a functional
$\beta_{(s)}$.

######  Theorem section.theorem.

Suppose $m_{s}^{1/2}\\{\tilde{\beta}_{(s)}-\beta_{(s)}(P_{s})\\}\rightarrow
N(0,\Sigma_{s})$ for $s=1,\dots,S$ and
$\\{\beta_{(1)}^{\mathrm{\scriptscriptstyle
T}}(P_{1}),\ldots,\beta_{(S)}^{\mathrm{\scriptscriptstyle
T}}(P_{S})\\}^{\mathrm{\scriptscriptstyle
T}}=\\{\beta_{(1)}^{\mathrm{\scriptscriptstyle
T}}(P_{0}),\ldots,\beta_{(S)}^{\mathrm{\scriptscriptstyle
T}}(P_{0})\\}^{\mathrm{\scriptscriptstyle T}}\eqqcolon\beta(P_{0})$. Denote
the efficient influence function for $\beta(P_{0})$ by $\eta_{\mathrm{eff}}$
$=(\eta_{\mathrm{eff},(1)}^{\mathrm{\scriptscriptstyle
T}},\ldots,\eta_{\mathrm{eff},(S)}^{\mathrm{\scriptscriptstyle
T}})^{\mathrm{\scriptscriptstyle T}}$. If
$m_{s}/n\rightarrow\rho_{s}\in(0,\infty)$ for each $s$, then the data-fused
efficiency bound for $\tau$ is
$E(\phi_{\mathrm{eff}}\phi_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})-E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\left\\{\Sigma_{\mathrm{ext}}+E(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\right\\}^{-1}E(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})^{\mathrm{\scriptscriptstyle T}}$, where
$\Sigma_{\mathrm{ext}}=\mathrm{diag}(\Sigma_{1}/\rho_{1},\ldots,\Sigma_{S}/\rho_{S})$.

Theorem section.theorem establishes the efficiency bound for estimating $\tau$
when multiple external summary statistics are available; the proof is
analogous to that of Theorem section.theorem and is omitted. The data-fused
efficient estimator with unbiased summary statistics is
$\hat{\tau}_{\mathrm{eff}}=\hat{\tau}_{\mathrm{eff}}^{\mathrm{int}}-\hat{E}(\phi_{\mathrm{eff}}\eta_{\mathrm{eff}}^{T})\big{\\{}\hat{\Sigma}_{\mathrm{ext}}+\hat{E}(\eta_{\mathrm{eff}}\eta_{\mathrm{eff}}^{\mathrm{\scriptscriptstyle
T}})\big{\\}}^{-1}(\hat{\beta}_{\mathrm{eff}}^{\mathrm{int}}-\tilde{\beta})$.
In the presence of biased summary statistics, the debiased estimator can be
obtained analogously, with $\hat{\Sigma}_{1}/\rho$ replaced by
$\hat{\Sigma}_{\mathrm{ext}}$ in formula (equation).

We have focused on integration of finite-dimensional summary statistics. It is
of both theoretical and practical interest to study how to integrate infinite-
dimensional external summary curves, such as estimates of a density function,
regression curve, or conditional mean. We plan to pursue this extension in the
future.

## Supplementary material

Supplementary material available at Biometrika online includes proofs of
Theorems 1–4 and Propositions 1–4, details of Examples 2 and 3, additional
simulation results and discussion on the scenario when Assumption 2 only holds
asymptotically.

## References

  * Bang & Robins (2005) Bang, H. & Robins, J. M. (2005).  Doubly robust estimation in missing data and causal inference models.  Biometrics 61, 962–973.
  * Bickel et al. (1993) Bickel, P. J., Klaassen, C. A. J., Ritov, Y. & Wellner, J. A. (1993).  Efficient and Adaptive Estimation for Semiparametric Models.  Baltimore: Johns Hopkins University Press.
  * Chatterjee et al. (2016) Chatterjee, N., Chen, Y.-H., Maas, P. & Carroll, R. J. (2016).  Constrained maximum likelihood estimation for model calibration using summary-level information from external big data sources.  Journal of the American Statistical Association 111, 107–117.
  * Chen et al. (2021) Chen, S., Zhang, B. & Ye, T. (2021).  Minimax rates and adaptivity in combining experimental and observational data.  arXiv preprint arXiv:2109.10522 .
  * Chen & Chen (2000) Chen, Y.-H. & Chen, H. (2000).  A unified approach to regression analysis under double-sampling designs.  Journal of the Royal Statistical Society: Series B 62, 449–460.
  * Dahabreh et al. (2019) Dahabreh, I. J., Robertson, S. E., Tchetgen Tchetgen, E., Stuart, E. A. & Hernán, M. A. (2019).  Generalizing causal inferences from individuals in randomized trials to all trial-eligible individuals.  Biometrics 75, 685–694.
  * Hahn (1998) Hahn, J. (1998).  On the role of the propensity score in efficient semiparametric estimation of average treatment effects.  Econometrica 66, 315–331.
  * Hájek (1970) Hájek, J. (1970).  A characterization of limiting distributions of regular estimates.  Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 14, 323–330.
  * Kallus et al. (2018) Kallus, N., Puli, A. M. & Shalit, U. (2018).  Removing hidden confounding by experimental grounding.  Advances in Neural Information Processing Systems 31.
  * Kundu et al. (2019) Kundu, P., Tang, R. & Chatterjee, N. (2019).  Generalized meta-analysis for multiple regression models across studies with disparate covariate information.  Biometrika 106, 567–585.
  * Li et al. (2020) Li, H., Miao, W., Cai, Z., Liu, X., Zhang, T., Xue, F. & Geng, Z. (2020).  Causal data fusion methods using summary-level statistics for a continuous outcome.  Statistics in Medicine 39, 1054–1067.
  * Li & Luedtke (2021) Li, S. & Luedtke, A. (2021).  Efficient estimation under data fusion.  arXiv preprint arXiv:2111.14945 .
  * Li et al. (2023) Li, X., Miao, W., Lu, F. & Zhou, X.-H. (2023).  Improving efficiency of inference in clinical trials with external control data.  Biometrics 79, 394–403.
  * Lin & Zeng (2010) Lin, D.-Y. & Zeng, D. (2010).  On the relative efficiency of using summary statistics versus individual-level data in meta-analysis.  Biometrika 97, 321–332.
  * Liu et al. (2015) Liu, D., Liu, R. Y. & Xie, M. (2015).  Multivariate meta-analysis of heterogeneous studies using only summary statistics: efficiency and robustness.  Journal of the American Statistical Association 110, 326–340.
  * McNeney & Wellner (2000) McNeney, B. & Wellner, J. A. (2000).  Application of convolution theorems in semiparametric models with non-iid data.  Journal of Statistical Planning and Inference 91, 441–480.
  * Qin (2000) Qin, J. (2000).  Combining parametric and empirical likelihoods.  Biometrika 87, 484–490.
  * Shen et al. (2020) Shen, J., Liu, R. Y. & Xie, M.-g. (2020).  ifusion: Individualized fusion learning.  Journal of the American Statistical Association 115, 1251–1267.
  * Singh et al. (2005) Singh, K., Xie, M. & Strawderman, W. E. (2005).  Combining information from independent sources through confidence distributions.  The Annals of Statistics 33, 159–183.
  * Strasser (1989) Strasser, H. (1989).  Tangent vectors for models with independent but not identically distributed observations.  Statistics & Risk Modeling 7, 127–152.
  * Sun & Miao (2022) Sun, B. & Miao, W. (2022).  On semiparametric instrumental variable estimation of average treatment effects through data fusion.  Statistical Sinica 32, 569–590.
  * Tsiatis (2006) Tsiatis, A. (2006).  Semiparametric Theory and Missing Data.  New York: Springer.
  * Wang et al. (2021) Wang, R., Wang, Q. & Miao, W. (2021).  A robust fusion-extraction procedure with summary statistics in the presence of biased sources.  arXiv preprint arXiv:2108.12600 .
  * Wang & Wang (2015) Wang, X. & Wang, Q. (2015).  Semiparametric linear transformation model with differential measurement error and validation sampling.  Journal of Multivariate Analysis 141, 67–80.
  * Xie et al. (2011) Xie, M., Singh, K. & Strawderman, W. E. (2011).  Confidence distributions and a unifying framework for meta-analysis.  Journal of the American Statistical Association 106, 320–333.
  * Yang & Ding (2020) Yang, S. & Ding, P. (2020).  Combining multiple observational data sources to estimate causal effects.  Journal of the American Statistical Association 115, 1540–1554.
  * Yang et al. (2020a) Yang, S., Zeng, D. & Wang, X. (2020a).  Elastic integrative analysis of randomized trial and real-world data for treatment heterogeneity estimation.  arXiv preprint arXiv:2005.10579 .
  * Yang et al. (2020b) Yang, S., Zeng, D. & Wang, X. (2020b).  Improved inference for heterogeneous treatment effects using real-world data subject to hidden confounding.  arXiv preprint arXiv:2007.12922 .
  * Zhai & Han (2022) Zhai, Y. & Han, P. (2022).  Data integration with oracle use of external information from heterogeneous populations.  Journal of Computational and Graphical Statistics 31, 1001–1012.
  * Zhang et al. (2019) Zhang, A., Brown, L. D. & Cai, T. T. (2019).  Semi-supervised inference: General theory and estimation of means.  Annals of Statistics 47, 2538–2566.
  * Zhang et al. (2020) Zhang, H., Deng, L., Schiffman, M., Qin, J. & Yu, K. (2020).  Generalized integration model for improved statistical inference by leveraging external summary data.  Biometrika 107, 689–703.
  * Zhang et al. (2022) Zhang, H., Deng, L., Wheeler, W., Qin, J. & Yu, K. (2022).  Integrative analysis of multiple case-control studies.  Biometrics 78, 1080–1091.
  * Zhu & Stephens (2017) Zhu, X. & Stephens, M. (2017).  Bayesian large-scale multiple regression with summary statistics from genome-wide association studies.  Annals of Applied Statistics 11, 1561.
  * Zou (2006) Zou, H. (2006).  The adaptive lasso and its oracle properties.  Journal of the American Statistical Association 101, 1418–1429.