A new information integration framework for complex models with applications to real-world data

Abstract

Over the past few decades, various advanced methods have been developed to facilitate information integration. These methods leverage summary statistics (e.g. point estimates) from multiple sites or studies, which can be readily extracted from existing publications or efficiently shared via correspondence, without requiring the sharing of raw individual-level data. Despite these advancements, existing methods may not be directly applicable to the model that allows certain covariate effects to vary with the values of another covariate. This model is widely used in biomedical research, particularly in scenarios where assuming linear effects across all covariates is inappropriate. This article addresses this gap by introducing a new information integration framework. This new framework (a) enables computationally efficient integration of information from a different external model type (e.g. generalized linear models), (b) does not assume homogeneous data distributions across sites or studies, and (c) supports variable selection. Extensive simulations validate the proposed method, demonstrating substantial variance reduction with minimal estimation bias in various cases. Finally, we apply this method to two distinct datasets to identify (a) risk factors influencing blood pressure using data from the UK Biobank and ARIC studies, and (b) risk factors associated with prolonged hospital length of stay among older fracture patients with ADRD using Medicare claims.

1. Introduction

Sharing individual-level raw data from independent studies, administrative datasets, and survey data can present substantial challenges, primarily due to the need to protect sensitive participant information and the administrative burden associated with data sharing.¹ This is particularly true in biomedical research, where the data often include detailed medical histories and other personally identifiable details, all governed by strict privacy regulations and ethical guidelines.² Balancing the demand for large dataset research innovations with ethical consideration remains a complex challenge with few established statistical best practices.

Several solutions have been proposed to facilitate research innovation without sharing raw individual data, including synthetic data methods,^3,4 distributed learning,^5,6 and approaches that share summary statistics between studies.^7,8 These methods employ different data-sharing and communication strategies, each suited to specific contexts. In this article, we focus on the last proposed solution: sharing summary statistics between studies, a widely adopted and easily implementable approach in modern biomedical research. One of the most established methods for sharing summary statistics is meta-analysis,⁹ which pools published results from multiple similar studies to produce an enhanced estimate without requiring access to raw individual data. Recently, numerous methods have been developed to integrate summary statistics in the context of more complex data structures (e.g. differing covariate sets) and modeling approaches (e.g. borrowing information across diverse models). Examples include constrained optimization-based estimators,^10–17 generalized method of moments,¹⁸ generalized meta-estimators,¹⁹ and Bayesian approaches that incorporate external information into informative priors.^20,21 These methods have been applied across various statistical contexts, including generalized linear models (GLMs),¹⁰ survival models,²² partial linear models,¹⁵ kernel regression models,¹² among others.

Despite advancements, the application of the aforementioned methods remains limited: it is still unclear whether and how information integration could be implemented for the varying coefficient model (VCM),^23,24,27,32 a semi-parametric model that allows some covariate effects (not all) to vary depending on the values of another covariate. This model has been extensively studied in theoretical statistics^23,25–27 and is particularly relevant to our real-world data application. In analyzing UK Biobank data to examine potential risk factors associated with blood pressure (Figure 3), we observed that (a) glucose level was non-linearly associated with systolic blood pressure (SBP), and (b) the association between sex and SBP potentially varied due to different glucose levels. Consequently, to study risk factors associated with elevated SBP (i.e. hypertension), it may be more appropriate to incorporate these varying effects in the modeling strategy. Besides, this application is also interested in identifying other biomarkers from neuroimaging and blood biochemistry predictive to SBP. Thus, a VCM with variable selection is well-suited for our primary analysis. We highlight here that the need for variable selection extends beyond this example; it is commonly encountered in multiple applied contexts. For instance, in healthcare research (our second data application), there is interest in identifying baseline chronic conditions that associated with longer hospital length of stay for older fracture patients with Alzheimer’s disease and related dementias (ADRDs). Identifying subcohorts with specific influential chronic conditions can help target these groups for more tailored interventions. Given the importance of applying advanced models with additional need of variable selection as the target analysis, it is highly valuable to explore how summary statistics from external studies can be integrated to enhance the primary analysis.

Figure 3.

Estimation of varying coefficient effects due to glucose levels (i.e. for intercept and Sex). IB stands for Information Borrowed, where the two graphs on the left are results from analysis including ARIC data, while the graph on the right does not. The colored areas are confidence regions.

This article aims to explore the potential for improving estimation efficiency, particularly for linear components, in complex settings where some covariates in the internal model may have varying coefficients, by introducing a novel, general, and computationally efficient framework. This approach integrates summary statistics from a single distinct model (e.g. GLMs) to enhance estimation performance for the more complex model through the adaptation of profile least squares (PLS) and empirical likelihood weighting.^15,28,29 Integrating information from a distinct external model is not only theoretically intriguing but also highly practical in real-world applications. For instance, summary statistics available in existing scientific literature are often derived from GLMs, while internal analyses may rely on more advanced modeling strategies, such as VCMs, which are better suited to internal data and tailored analysis plans. Additionally, GLMs are more straightforward to implement in distributed settings, where external study teams may lack access to sophisticated statistical expertise. Thus, developing a method that facilitates the integration of information from distinct models is both desirable and of significant practical value. It is important to note that our proposed method conceptually differs from existing empirical likelihood-based data integration approaches,^{8,10,11,17,30} which typically require specifying a functional relationship between the internal and external models and necessitate complex computational strategies to jointly estimate all parameters. Such a setting is difficult to formulate when the internal analysis is based on a VCM, as it includes nonparametric components that cannot be readily incorporated into existing frameworks. As a result, current empirical likelihood-based framework cannot be directly applicable in our setting. In contrast, our approach avoids the need to identify an explicit internal–external model relationship, enabling more convenient and feasible integration of external information to improve efficiency in the primary analysis.

Moreover, inspired by real-world applications described above, the proposed approach incorporates variable selection by integrating penalized regression³¹ into the framework to facilitate the selection of biomarkers (application 1) and individual chronic conditions (application 2). Our theoretical development accounts for heterogeneous covariate distributions and adopts a selective-borrowing scheme to accommodate differences in conditional outcome means between datasets, enhancing its applicability to practical settings. This flexibility has the potential to reduce estimation bias while maintaining high efficiency, even when external summary statistics deviate significantly from the target data source. Finally, the proposed method is easy to implement and well suited for achieving all of the aforementioned tasks. Extensive simulation studies demonstrate significant variance reduction for the estimates corresponding to parametric terms with minimal estimation bias after information integration, compared to existing methods. We applied our method to analyze two real datasets to identify (a) risk factors (e.g. demographics, brain metrics, and blood biomarkers) influencing SBP using the UK Biobank and ARIC datasets, and (b) risk factors (e.g. demographics and individual chronic conditions) associated with prolonged hospital length of stay for older patients with ADRD experiencing a hip fracture using Medicare claims.

The remainder of the article is organized as follows: Section 2 presents the mathematical notations and foundational setting for information integration. Section 3 details the proposed methods. Section 4 provides extensive numerical evaluations through computer simulations. Section 5 illustrates two data applications to demonstrate the use of the method. Section 6 concludes the article. Detailed proofs, additional simulations, and real data results, and further discussions are available in the Supplementary Material. In the rest of the article, we use the terms internal/external data, study, or analysis interchangeably in different contexts, all essentially referring to the same concept.

2. Preliminaries

We use our first data application to illustrate the notation. For $n$ independent observations in the internal (target) study, let $\left\{Y_i,U_i,{\mathbf{X}}_i,{\mathbf{Z}}_i,{\mathbf{W}}_i\right\}$ be the collection of individual-level data from the $i$th subject, where $Y_i$ is the continuous response (e.g. SBP); $U_i$ is a covariate (e.g. glucose levels) associated with varying coefficients in the outcome model; ${\mathbf{X}}_i\in R^d$ is a vector of covariates, including the intercept as the first element $(d\ge 1)$, with varying effects on the outcome (e.g. gender); and ${\mathbf{Z}}_i\in R^q$ contains covariates with constant impact on the outcome (e.g. age and BMI). Additionally, ${\mathbf{W}}_i\in R^s$ is a vector only available in the internal data (e.g. 39 regions of Fractional Anisotropy (FA) scans of the brain white matter and 25 whole blood test results). Using these variables, the conditional mean model of interest can be summarized as $E_{\left(1\right)}\left(Y_i\mid U_i,{\mathbf{X}}_i,{\mathbf{Z}}_i\right)={\mathbf{Z}}_i^T{\mathit{\beta }}_0+{\mathbf{X}}_i^T{\mathit{\alpha }}_0\left(U_i\right)+{\mathbf{W}}_i^T{\gamma }_0$, where ${\mathit{\alpha }}_0\left(U_i\right)\in R^d$ represents the vector of covariate $U_i$-associated varying effects from covariates in ${\mathbf{X}}_i$, and $E_{\left(1\right)}$ represents the expectation based on the population of the internal data. We assume that the functions in ${\mathit{\alpha }}_0\left(U_i\right)$ are unknown, subject to continuous smoothing, and have a bounded second-order derivative. We also assume that only a finite number of non-zero values are in ${\mathit{\gamma }}_0$ (sparsity). For method illustration in the later context, we define the stacked vectors $\mathbf{Y}={\left(Y_1,\dots ,Y_n\right)}^T$, $\mathbf{U}={\left(U_1,\dots ,U_n\right)}^T$, $\mathbf{X}={\left({\mathbf{X}}_1^T,\dots ,{\mathbf{X}}_n^T\right)}^T$, $\mathbf{Z}={\left({\mathbf{Z}}_1^T,\dots ,{\mathbf{Z}}_n^T\right)}^T$, and $\mathbf{W}={\left({\mathbf{W}}_1^T,\dots ,{\mathbf{W}}_n^T\right)}^T$.

On the other hand, the external study includes $N$ subjects targeting a population similar to that in the internal study. We assume the external study measures only a limited set of covariates, summarized in the vector ${\tilde{\mathbf{X}}}_j={\left(U_j,{\mathbf{X}}_j^T,{\mathbf{Z}}_j^T\right)}^T$, for $j=1,\dots ,N$. Unlike the internal study, the variables in ${\mathbf{W}}_j$ are not recorded in the external study. This setup aligns with our first real data application, where the ARIC study, serving as the external data source, does not collect neuroimaging data and certain blood biomarkers. This is also a common scenario in real-world applications, where certain variables are cohort-specific or require additional effort to obtain. To account for potential population heterogeneity between the internal and external data, we allow different marginal covariate distributions, that is $f_{\left(1\right)}\left(\tilde{\mathbf{X}}\right)\ne f_{\left(2\right)}\left(\tilde{\mathbf{X}}\right)$. Here, $f_{\left(1\right)}(\cdot )$ and $f_{\left(2\right)}(\cdot )$ represent the density functions of covariates from the internal and external studies, respectively. The presence of heterogeneous populations is common in practice. For example, in our first case study, age, sex, and BMI showed different distributions in the UK Biobank and ARIC data (Table S6).

Furthermore, we assume that the external study fits a mean model for $Y_j$ conditional on covariates ${\tilde{\mathbf{X}}}_j$ by imposing a linear structure (which may be mis-specified) as ${\tilde{\mathbf{X}}}_j^T\mathit{\theta }$, with some parameter vector $\mathit{\theta }$. The estimates of $\mathit{\theta }$ are obtained by solving general estimating equations $\sum _{j=1}^N\mathbf{H}\left(Y_j,{\tilde{\mathbf{X}}}_j,\mathit{\theta }\right)=\mathbf{0}$, where $\mathbf{H}(\cdot )$ can be the derivative of the least square loss or generalized estimating equations. We denote the estimates as ${\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\left(2\right)}$ and the estimated variance–covariance matrix of $\sqrt{N}({\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\left(2\right)}-{\mathit{\theta }}_0)$ as ${\stackrel{\mathbf{ˆ}}{\mathbf{V}}}_{\left(2\right)}$, with ${\mathit{\theta }}_0$ being the limiting values of ${\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\left(2\right)}$. We assume that only the summary statistics ${\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\left(2\right)}$ and ${\stackrel{\mathbf{ˆ}}{\mathbf{V}}}_{\left(2\right)}$, instead of the raw data, is available from the external study. To ensure an unbiased estimator when integrating summary statistics into the internal data analysis, we assume that there exists a uniform ${\mathit{\theta }}_0$ satisfying $E_{\left(1\right)}\left\{\mathbf{H}\left(Y_i,{\tilde{\mathbf{X}}}_i,{\mathit{\theta }}_0\right)\right\}=E_{\left(2\right)}\left\{\mathbf{H}\left(Y_j,{\tilde{\mathbf{X}}}_j,{\mathit{\theta }}_0\right)\right\}=\mathbf{0}$, where $E_{\left(2\right)}$ denotes the expectation based on the population of the external data. This assumption holds if, for instance, two reduced models from internal and external analyses that maximize Kullback–Leibler divergence are the same, i.e. $f_{\left(1\right)}\left(Y\mid \tilde{\mathbf{X}};{\mathit{\theta }}_0\right)=f_{\left(2\right)}\left(Y\mid \tilde{\mathbf{X}};{\mathit{\theta }}_0\right)$, a milder condition compared to existing literature.^10,13,30 The assumption of a uniform ${\mathit{\theta }}_0$ can be tested in practice by comparing two estimates of $\mathit{\theta }$: one derived from the internal data, denoted by ${\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\left(1\right)}$, and the other from the summary statistics of the external data. A significant difference between these two estimates would indicate a potential issue. In Sections 3.2 and 3.3, we focus on the setting where the uniform ${\mathit{\theta }}_0$ assumption holds, and in Section 3.4 introduces a two-step approach to address the violation of the uniform ${\mathit{\theta }}_0$ assumption.

By incorporating summary statistics from the external source, this article aims to provide a valid and more efficient estimate of the conditional mean for the internal population, in particular for the linear component ${\mathit{\beta }}_0$, and conduct variable selection in $\mathbf{W}$.

3. The proposed method

3.1. Varying coefficient model with variable selection

Before introducing our method, we first describe the existing framework of the semi-parametric VCM with variable selection. Following Hastie and Tibshirani,³² Fan et al.,²⁴ the model for the subject $i=1,\dots ,n$ from the internal data can be written as

$$Y_i={\mathbf{Z}}_i^T\mathit{\beta }+{\mathbf{X}}_i^T\mathit{\alpha }\left(U_i\right)+{\mathbf{W}}_i^T\mathit{\gamma }+{\epsilon }_i,$$ (3.1)

where the residual term ${\epsilon }_i$ has a mean of zero and finite variance, without any distributional assumptions imposed on it. In addition, this model allows interaction between covariates $U_i$ and ${\mathbf{X}}_i$. Other notations are described in Section 2. The VCM in (3.1) is a semi-parametric model that provides a better explanation of heterogeneous covariate effects and improves prediction accuracy while maintaining reasonable interpretability.²⁷ The identification of ${\mathbf{X}}_i$ and ${\mathbf{Z}}_i$ in model (3.1) can be determined either by experts using domain-specific prior knowledge or by constructing statistical tests to assess the presence of varying coefficients.^23,26 To efficiently estimate the parameters in (3.1), we employ the PLS approach.^23,24,33 Specifically, we define a pseudo outcome $Y_i^{\mathrm{*}}:=Y_i-{\mathbf{Z}}_i^T\mathit{\beta }-{\mathbf{W}}_i^T\mathit{\gamma }$ and re-write the model in (3.1) as

$$Y_i^{\mathrm{*}}=\sum _{l=1}^d{\alpha }_l\left(U_i\right)X_{il}+{\epsilon }_i.$$ (3.2)

Therefore, this transforms the semi-parametric model into the VCM. Assuming that $U$ has the same degree of smoothness on all $\mathbf{X}$, we use the local linear approach to estimate $\mathit{\alpha }(\cdot )$ in a small neighborhood of $u_0$, which leads to ${\alpha }_k\left(U\right)={\alpha }_k\left(u_0\right)+{\alpha }_k^{\prime }\left(U-u_0\right)+o_p\left\{\left(U-u_0\right)\right\}$, for $k=1,\dots ,d$, with ${\alpha }_k^{\prime }\left(\cdot \right):=b(\cdot )$. As a result, the estimator of ${\alpha }_k$ will be the estimator of $\mathit{\alpha }(\cdot )$ at the value $u_0$.^24,34 To obtain the estimator, we minimize the weighted quadratic loss with respect to $a_k$ and $b_k$

$$\sum _{i=1}^n{\left[Y_i^{\mathrm{*}}-\sum _{k=1}^d\left\{a_k+b_k\left(U_i-u_0\right)\right\}{X}_{ik}\right]}^2{K}_h\left(U_i-u_0\right),$$ (3.3)

where $K_h(\cdot )=h^{-1}K(\cdot /h)$ is a smooth kernel function with a bandwidth $h$ and some kernel $K(\cdot )$.³⁵ Using the model in (3.2) and following the routine kernel smoothing calculation,^23,24 we have $\mathbf{X}\stackrel{\mathit{ˆ}}{\mathit{\alpha }}\left(\mathbf{U}\right)=\mathbf{S}(\mathbf{Y}-\mathbf{Z}\mathit{\beta }-\mathbf{W}\mathit{\gamma })$. The form of $\mathbf{S}$, which contains covariates $\mathbf{U}$ and $\mathbf{X}$, is explicitly described in equations (2.3) and (2.4) of Fan and Huang,²³ which is free of parameters $\mathit{\beta }$ and $\mathit{\gamma }$. The explicit form of $\mathbf{S}$ is described in the proof of Theorem 3 in the supplementary materials.

Next, to facilitate variable selection in $\mathbf{W}$, we consider the following penalized least squares approach,³³ which minimizes the following loss function

$$\begin{array}{l}\frac{1}{2}(\mathbf{Y}-\mathbf{X}\stackrel{\mathit{ˆ}}{\mathit{\alpha }}(\mathbf{U})-\mathbf{Z}\mathit{\beta }-\mathbf{W}\mathit{\gamma }{)}^T(\mathbf{Y}-\mathbf{X}\stackrel{\mathit{ˆ}}{\mathit{\alpha }}(\mathbf{U})-\mathbf{Z}\mathit{\beta }-\mathbf{W}\mathit{\gamma })+n\sum _{l=1}^s{\lambda }_l{p}_l\left(\left|{\gamma }_l\right|\right)\\ =\frac{1}{2}(\mathbf{Y}-\mathbf{Z}\mathit{\beta }-\mathbf{W}\mathit{\gamma }{)}^T(\mathbf{I}-\mathbf{S}{)}^T(\mathbf{I}-\mathbf{S})(\mathbf{Y}-\mathbf{Z}\mathit{\beta }-\mathbf{W}\mathit{\gamma })+n\sum _{l=1}^s{\lambda }_l{p}_l\left(\left|{\gamma }_l\right|\right).\end{array}$$ (3.4)

There are many choices for the penalty function $p(|\cdot |)$. Commonly chosen ones, such as Lasso, SCAD, and MCP, fit the optimization in (3.4).

3.2. Information integration in varying coefficient semiparametric model with transportable information

In this section, we describe the proposed information integration scheme by using summary statistics ${\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)}$ and ${\stackrel{\mathbf{ˆ}}{\mathbf{V}}}_{(2)}$. The new method allows the marginal covariate distributions between two datasets to be different, i.e. $f_1(\tilde{\mathbf{X}})\ne f_2(\tilde{\mathbf{X}})$. We assume that there exists a uniform ${\mathit{\theta }}_0$ satisfying $E_1\left\{\mathbf{H}\left(Y_i,{\tilde{\mathbf{X}}}_i,{\mathit{\theta }}_0\right)\right\}=E_2\left\{\mathbf{H}\left(Y_j,{\tilde{\mathbf{X}}}_j,{\mathit{\theta }}_0\right)\right\}=0$. The extension to accommodate the violation of the uniform ${\mathit{\theta }}_0$ assumption is described in Section 3.4.

To simultaneously achieve information integration and variable selection, we utilize the weighted and penalized PLS estimation by re-weighting the loss function in (3.4) using a diagonal weight matrix $\stackrel{\mathbf{ˆ}}{\mathbf{P}}$. The estimators of $\mathit{\beta }$ and $\mathit{\gamma }$, denoted by $\stackrel{\mathit{ˆ}}{\mathit{\beta }}$ and $\stackrel{\mathit{ˆ}}{\mathit{\gamma }}$, can be obtained by minimizing the following weighted loss function,

$$\frac{1}{2}(\mathbf{Y}-\tilde{\mathbf{Z}}\mathit{\eta }{)}^T(\mathbf{I}-\mathbf{S}{)}^T\stackrel{\mathbf{ˆ}}{\mathbf{P}}(\mathbf{I}-\mathbf{S})(\mathbf{Y}-\tilde{\mathbf{Z}}\mathit{\eta })+n\sum _{l=1}^{s+q}{c}_l{\lambda }_l{p}_l\left(\left|{\gamma }_l\right|\right).$$ (3.5)

Here, $\mathit{\eta }={\left({\mathit{\beta }}^T,{\mathit{\gamma }}^T\right)}^T$, $\tilde{\mathbf{Z}}=(\mathbf{Z},\mathbf{W})$, and $c_l$ for $l=1,\dots ,s+q$ represents a user-specified sequence. In our data applications, we set $c_l=1$ for variables in $\mathbf{W}$ and $c_l=0$ otherwise.

To improve the estimation efficiency, the weight matrix $\stackrel{\mathbf{ˆ}}{\mathbf{P}}=Diag{\left\{{\hat{p}}_i\right\}}_{i=1}^n$ should be informative and contain the information that is deliverable from the external data to the internal data. To proceed, we adapt the idea from the constrained optimization that could be used to improve estimation efficiency.^15,28,36 In particular, we obtain the weights ${\hat{p}}_i$ by maximizing the following semi-parametric likelihood of $Y_i$, ${\tilde{\mathbf{X}}}_i$, and $\stackrel{\mathit{ˆ}}{\mathit{\theta }}$

$$l=\sum _{i=1}^{n}log(p_i)-N{(\mathit{\theta }-{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\left(2\right)})}^T{({\stackrel{\mathbf{ˆ}}{\mathbf{V}}}_{\left(2\right)})}^{-1}(\mathit{\theta }-{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\left(2\right)})/2,$$ (3.6)

with respect to $p_i$ and $\mathit{\theta }$, and subject to the following three constraints:

$$p_i>0,\sum _{i=1}^n{p}_i=1,\sum _{i=1}^n{p}_i\mathbf{H}(Y_i,{\tilde{\mathbf{X}}}_i,\mathit{\theta })=\mathbf{0}.$$ (3.7)

The proposed loss function in (3.6) is analogous to maximizing the logarithm of the joint distribution $F(\mathbf{Y},\tilde{\mathbf{X}}\mid {\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)})=F(\mathbf{Y},\tilde{\mathbf{X}}\mid {\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)})F({\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)})$, where ${\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)}$ is estimated by the external data and expected to be normally distributed when the ample size of the external data is large. Moreover, the first two constraints in (3.7) guarantee the property of the likelihood and the third constraint together with the quadratic loss in (3.6) integrates information by incorporating the complete inference solutions of $\stackrel{\mathit{ˆ}}{\mathit{\theta }}$ obtained from the external data. The intuition of improvement of efficiency is that the parameter $\mathit{\theta }$ is estimated by using both internal and external data. Therefore, the function $\mathbf{H}(\cdot )$ in (3.7) becomes “over-identified” with additional degrees of freedom borrowed from the external data, leading more efficient estimate of empirical probability mass ${\hat{p}}_i$ based on the empirical likelihood theory.^28,29 Using informative score ${\hat{p}}_i$ as weights in the optimization (3.5), we expect more efficient estimates of $\mathit{\eta }$. Noted that the proposed integration approach does not require joint estimation of parameters across the internal and external models, nor does it require specifying an explicit internal–external model relationship. This makes the method particularly feasible for improving efficiency in the primary VCM-based analysis.

After obtaining the estimates of $\mathit{\beta }$ and $\mathit{\gamma }$, denoted as $\stackrel{\mathit{ˆ}}{\mathit{\eta }}:={({\stackrel{\mathit{ˆ}}{\mathit{\beta }}}^T,{\stackrel{\mathit{ˆ}}{\mathit{\gamma }}}^T)}^T$, we substitute these estimates into $\mathbf{X}\stackrel{\mathit{ˆ}}{\mathit{\alpha }}(\mathbf{U};\stackrel{\mathit{ˆ}}{\mathit{\eta }})=\mathbf{S}(\mathbf{Y}-\mathbf{Z}\stackrel{\mathit{ˆ}}{\mathit{\beta }}-\mathbf{W}\stackrel{\mathit{ˆ}}{\mathit{\gamma }})$ to derive the estimate of $\mathbf{X}\mathit{\alpha }(\mathbf{U})$. Additionally, we can obtain $\stackrel{\mathit{ˆ}}{\mathit{\alpha }}\left(u_0;\stackrel{\mathit{ˆ}}{\mathit{\eta }}\right)$ and $\mathbf{X}\stackrel{\mathit{ˆ}}{\mathit{\alpha }}\left(u_0;\stackrel{\mathit{ˆ}}{\mathit{\eta }}\right)$ evaluated at a specific value $u_0$ by following the standard procedure described in Fan and Huang.²³

3.3. Asymptotics theory with transportable information

In this section, we describe the asymptotic properties of $\stackrel{\mathit{ˆ}}{\mathit{\eta }}$ and $\stackrel{\mathit{ˆ}}{\mathit{\alpha }}\left(u_0;\stackrel{\mathit{ˆ}}{\mathit{\eta }}\right)$. Let ${\mathit{\eta }}_{\mathrm{*}}={\left({\mathit{\beta }}^T,{\mathit{\gamma }}_{\mathrm{*}}^T\right)}^T$, here, ${\mathit{\gamma }}_{\mathrm{*}}$ correspond to the non-zero elements of $\mathit{\gamma }$ with a fixed length of $s_1<s$, and ${\mathit{\eta }}_{\mathrm{*}\mathrm{*}}$ represent remaining zero elements in $\mathit{\gamma }$. For simplicity, we assume constants in the penalty term $c_l=1$ for all $l$. In practice, certain $c_l$ can be zero if the $l$th covariate should always stay with the model. Let $a_n$ and $b_n$ be the maximum value of the absolute value of first and second-order derivatives of the penalty function $p_{{\lambda }_l}\left(\left|{\gamma }_l\right|\right)$ on the ${\mathit{\eta }}_{\mathrm{*}}$, respectively. Moreover, we denote ${\mathbf{\Sigma }}_b$ as the sub-matrix of $\mathbf{\Sigma }=E_{(1)}[\{{\tilde{\mathbf{Z}}}_i-E_{(1)}\left({\tilde{\mathbf{Z}}}_i\mid {\mathbf{X}}_i,U_i\right)\}{\left\{{\tilde{\mathbf{Z}}}_i-E_{(1)}\left({\tilde{\mathbf{Z}}}_i\mid {\mathbf{X}}_i,U_i\right)\right\}}^T]$ corresponding to ${\mathit{\eta }}_{\mathrm{*}}$. Similarly, let ${\mathbf{B}}_b$ be the corresponding sub-matrix of $\mathbf{B}=\mathbf{G}-\mathbf{M}\left(\mathbf{\Omega }-\mathbf{\Omega }\mathbf{Q}{\mathbf{V}}_T^{-1}{\mathbf{Q}}^T\mathbf{\Omega }\right){\mathbf{M}}^T$, where $\mathbf{Q}=E_{(1)}\left(\partial {\mathbf{H}}_0/\partial \mathit{\theta }\right)$ and ${\mathbf{V}}_T=\{{\mathbf{V}}_{(1)}^{-1}+\rho {\mathbf{V}}_{(2)}^{-1}\}$ with $\rho ={\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}N/n$ being a finite constant value larger than zero. ${\mathbf{V}}_{(1)}$ and ${\mathbf{V}}_{(2)}$ are the variance–covariance matrices of the internal estimates $\sqrt{n}({\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(1)}-{\mathit{\theta }}_0)$ and external estimates $\sqrt{N}({\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)}-{\mathit{\theta }}_0)$, respectively. Besides, we have $\mathbf{G}=E_{(1)}[\left\{{\tilde{\mathbf{Z}}}_i-E_{(1)}\left({\tilde{\mathbf{Z}}}_i\mid {\mathbf{X}}_i,U_i\right)\right\}{\left\{{\tilde{\mathbf{Z}}}_i-E_{(1)}\left({\tilde{\mathbf{Z}}}_i\mid {\mathbf{X}}_i,U_i\right)\right\}}^T{\epsilon }_i^2],\mathbf{\Omega }=E_{(1)}^{-1}\left({\mathbf{H}}_i{\mathbf{H}}_i^T\right)$, and $\mathbf{M}=E_{(1)}\left[\left\{{\tilde{\mathbf{Z}}}_i-E_{(1)}\left({\tilde{\mathbf{Z}}}_i\mid {\mathbf{X}}_i,U_i\right)\right\}{\epsilon }_i{\mathbf{H}}_i^T\right]$. In addition, let ${\mathbf{D}}_b$ be $diag{\{p_{{\lambda }_l}^{\prime \prime }\left({\eta }_l\right)\}}_{l=1}^{s_1+q}$ and $\mathbf{b}$ be ${(p_{{\lambda }_1}^{\prime }\left(\left|{\eta }_{\mathrm{*}}\right|\right),\dots ,p_{{\lambda }_{s_1+q}}^{\prime }\left(\left|{\eta }_{\mathrm{*}}\right|\right))}^T$.

The following theorem outlines the asymptotic properties of $\stackrel{\mathit{ˆ}}{\mathit{\eta }}$, including estimation consistency, oracle selection for ${\stackrel{\mathit{ˆ}}{\mathit{\eta }}}_{\mathrm{*}\mathrm{*}}$, and asymptotic normality for ${\stackrel{\mathit{ˆ}}{\mathit{\eta }}}_{\mathrm{*}}$. The proof is provided under the assumption that the number of variables in $\mathbf{W}$ remains fixed, which aligns with our applications where there is a fixed number of FA scans and whole blood measures based on the UKB study design in our first data application and fixed number of common chronic conditions (> 5% rate) in our second data application.

Theorem 1.

Under the regularity conditions (category B and S) listed in the Supplementary material, the estimator $\stackrel{\mathit{ˆ}}{\mathit{\eta }}$ obtained by minimizing equation (3.5), with informative score obtained by the constrained optimization over (3.6) and (3.7), is a consistent estimator of the underlying truth ${\mathit{\eta }}_0={\left({\mathit{\beta }}_0^T,{\mathit{\gamma }}_0^T\right)}^T$. In addition, for each $l$ from $1,\dots ,s+q$, ${\lambda }_l\to 0$, we assume $\sqrt{n}{\lambda }_l\to \mathrm{\infty }$, and the penalty function satisfies

$$\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}}\underset{{\eta }_l\to 0^{+}}{\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}}\frac{p_{{\lambda }_l}^{\prime }\left({\eta }_l\right)}{{\lambda }_l}>0.$$ (3.8)

With $a_n=O_p\left(n^{-1/2}\right)$ and $b_n=o_p(1)$, we have the oracle property ${\stackrel{\mathit{ˆ}}{\mathit{\eta }}}_{\mathrm{*}\mathrm{*}}=\mathbf{0}$ with probability tending to one, and asymptotic normality

$$\sqrt{n}\left({\mathbf{\Sigma }}_b+{\mathbf{D}}_b\right)\left\{{\stackrel{\mathit{ˆ}}{\mathit{\eta }}}_{\mathrm{*}}-{\mathit{\eta }}_{\mathrm{*},0}+{\left({\mathbf{\Sigma }}_b+{\mathbf{D}}_b\right)}^{-1}\mathbf{b}\right\}\to \mathbf{N}\left(0,{\mathbf{B}}_b\right)$$ (3.9)

in distribution.

There are three key insights from Theorem 1. First, the convergence rate of the non-zero elements in ${\stackrel{\mathit{ˆ}}{\mathit{\eta }}}_{ib}$ follows the oracle rate, i.e. $n^{-0.5}$, a desired property that is fulfilled by applying PLS estimation. Second, the term $\mathbf{M}\left(\mathbf{\Omega }-\mathbf{\Omega }\mathbf{Q}{\mathbf{V}}_T^{-1}{\mathbf{Q}}^T\mathbf{\Omega }\right){\mathbf{M}}^T$ in ${\mathbf{B}}_b$ is generally positive definite, ensuring variance reduction by incorporating summary statistics from the external study. Lastly, if the covariate distributions between the internal and external datasets are identical, no variable selection is applied, and the VCM is reduced to the partial linear model described in Liang et al.,¹⁵ the resulting variance from (3.9) will reduce to the variance in Theorem 2 of Liang et al.¹⁵ Thus, our theorem is more general, accommodating differences in covariate distributions between datasets and the need for variable selection, with the method in Liang et al.¹⁵ as a special case.

Moreover, the following Corollary provides insight for an alternative estimation algorithm that is computationally more efficient. This new algorithm makes use of a plug-in estimate of ${\mathit{\theta }}_1$ using the information from both internal and external data, and the resulting estimator is asymptotically equivalent to the estimator obtained by iteratively solving the constrained optimization (3.10) with constraints in (3.7).

Corollary 2.

The empirical likelihood estimator of $\mathit{\theta }$ obtained from the constrained optimization (3.6) and (3.7) is asymptotically equivalent to the following meta-estimator from two studies, i.e. ${\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\mathit{meta}}=\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}\{n{(\mathit{\theta }-{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(1)})}^T{\stackrel{\mathit{ˆ}}{\mathit{V}}}_{(1)}^{-1}(\mathit{\theta }-{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(1)})+N(\mathit{\theta }-{({\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)}}^T{\stackrel{\mathit{ˆ}}{\mathit{V}}}_{(2)}^{-1}(\mathit{\theta }-{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)})\}$, where ${\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(1)}$ is the estimator obtained by solving the estimating equation $\sum _{i=1}^n\mathit{H}\left(Y_i,{\tilde{\mathit{X}}}_i,\mathit{\theta }\right)=\mathbf{0}$ using the internal data, and ${\stackrel{\mathit{ˆ}}{\mathit{V}}}_{(1)}$ is the estimated variance–covariance matrix of $\sqrt{n}({\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(1)}-{\mathit{\theta }}_0)$.

After obtaining the meta-analysis estimator described in Corollary 2, we can directly calculate the informative score ${\hat{p}}_i$ by applying the algorithm described in Section 2.2 of the Supplementary Material and then solve the weighted loss function in (3.5) to estimate $\mathit{\eta }$. Using the meta-analysis estimator of $\mathit{\theta }$ offers the advantage of speeding up computation without needing to iterate through the empirical likelihood estimator.

On the other hand, given oracle selection in ${\mathit{\eta }}_{\mathrm{*}\mathrm{*}}$ and oracle convergence rate in ${\mathit{\eta }}_{\mathrm{*}}$, the asymptotic distributions of $\stackrel{\mathit{ˆ}}{\mathit{\alpha }}\left(u_0;\stackrel{\mathit{ˆ}}{\mathit{\eta }}\right)$ turn out to be not affected by the estimation of $\mathit{\eta }$ with a high probability.^15,23 As a result, following the routine derivation^25,26 of the local linear approximation on $\mathit{\alpha }\left(u_0\right)$, we have

$$\sqrt{nh}\left[\stackrel{\mathit{ˆ}}{\mathit{\alpha }}\left(u_0\right)-\mathit{\alpha }\left(u_0\right)-\frac{1}{2}{h}^2{\mu }_2{\mathit{\alpha }}^{\prime \prime }\left(u_0\right)+o_p\left(h^2\right)\right]\to N\left(0,\mathbf{\Sigma }\left(u_0\right)\right)$$

in distribution. Here, ${\mu }_2=\int u^{2}K(u)du$, and $\mathbf{\Sigma }\left(u_0\right)$ equals ${\sigma }^2\int K^2(u)du{\mathbf{\Gamma }}^{-1}\left(u_0\right)/f_U\left(u_0\right)$, with $\mathbf{\Gamma }(u)=E\left({\mathbf{X}\mathbf{X}}^T\mid U=u\right)$.

3.4. When information is partially transportable

The methods developed in the beginning of Section 3 are based on the assumption of a uniform ${\mathit{\theta }}_0$, meaning there exists a common ${\mathit{\theta }}_0$ such that $E_{(1)}\left\{\mathbf{H}\left(Y_i,{\tilde{\mathbf{X}}}_i,{\mathit{\theta }}_0\right)\right\}=E_{(2)}\left\{\mathbf{H}\left(Y_j,{\tilde{\mathbf{X}}}_j,{\mathit{\theta }}_0\right)\right\}=\mathbf{0}$. A sufficient condition for this assumption is that the reduced models maximizing the Kullback–Leibler divergence are identical for both the internal and external data, that is $f_{(1)}\left(Y\mid \tilde{\mathbf{X}};{\mathit{\theta }}_0\right)=f_{(2)}\left(Y\mid \tilde{\mathbf{X}};{\mathit{\theta }}_0\right)$. While this is a weaker assumption than that of homogeneous data distribution, this assumption is still not guaranteed in real applications. In our first case study, we found that the BMI effects were significantly different between the internal cohort (UK Biobank) and the external cohort (ARIC data). The violation of the uniform ${\mathit{\theta }}_0$ would result in biased estimates when integrating information from the external source. This leads us to consider the situation where only part of the information in $\mathit{\theta }$ is transportable. To address this, we propose the following selective-borrowing method extension, which has also been similarly adopted by existing literature.^37,38

Without loss of generality, we first rewrite $\mathit{\theta }={\left({\mathit{\theta }}_1^T,{\mathit{\theta }}_2^T\right)}^T$, where ${\mathit{\theta }}_1$ represents the uniform parameters for both the internal and external conditional expectations, while ${\mathit{\theta }}_2$ is the vector of parameters that differ between internal and external studies. In practice, we can perform a hypothesis test by evaluating the difference between the estimated $\mathit{\theta }$ from the internal and external data to identify ${\mathit{\theta }}_2$. Given this separation in $\mathit{\theta }$, our proposed method integrates information only from the summary statistics for ${\mathit{\theta }}_1$. The estimation procedure remains the same as described in Section 3.2, with the only change being the modification of the loss function:

$$l=\sum _{i=1}^n\mathrm{l}\mathrm{o}\mathrm{g}\left(p_i\right)-N{({\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\left(2\right)1}-{\mathit{\theta }}_1)}^T{({\stackrel{\mathbf{ˆ}}{\mathbf{V}}}_{\left(2\right)1})}^{-1}({\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\left(2\right)1}-{\mathit{\theta }}_1)/2,$$ (3.10)

where ${\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)}={({\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)1}^T,{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)2}^T)}^T$ is the parameter used in the external study. The matrix ${\stackrel{\mathbf{ˆ}}{\mathbf{V}}}_{(2)1}$ is the estimated variance–covariance matrix of $\sqrt{N}({\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)1}-{\mathit{\theta }}_{01})$ Note that the proposed selective-borrowing strategy differs from approaches that explicitly model heterogeneity between two datasets. The latter typically requires a correctly specified heterogeneity model, which may be difficult to achieve, especially when individual-level data cannot be merged. We summarize the asymptotic properties of the resulting estimator in Theorem 3.

Theorem 3.

Under the regularity conditions listed in the Supplementary material (category $B$ and $S$), the estimator $\stackrel{\mathit{ˆ}}{\mathit{\eta }}$ obtained by minimizing equation (3.5) with informative score obtained by the constrained optimization over (3.10), is a consistent estimator of ${\mathit{\eta }}_0$. In addition, with the same conditions about ${\lambda }_l$, $a_n$ and $b_n$ listed in Theorem 1, we have the oracle property ${\stackrel{\mathit{ˆ}}{\mathit{\eta }}}_{\mathrm{*}\mathrm{*}}=0$ with probability tending to one and the asymptotic normality of ${\stackrel{\mathit{ˆ}}{\mathit{\eta }}}_{\mathrm{*}}$

$$\sqrt{n}\left({\mathbf{\Sigma }}_b+{\mathbf{D}}_b\right)\left({\stackrel{\mathit{ˆ}}{\mathit{\eta }}}_{\mathrm{*}}-{\mathit{\eta }}_{\mathrm{*},0}+{\left({\mathbf{\Sigma }}_b+{\mathbf{D}}_b\right)}^{-1}\mathbf{b}\right)\to \mathbf{N}\left(0,{\mathbf{B}}_{11}\right)$$ (3.11)

in distribution.

The detailed formula for ${\mathbf{B}}_{11}$ can be found in Section S2 of the Supplementary Material. It is worth noting that if there is no uniform ${\mathit{\theta }}_0$, i.e. $\mathit{\theta }={\mathit{\theta }}_2$, by simple calculation the variance in (3.11) will reduce to that of the PLS estimator in Fan et al.²⁴ Conversely, if $\mathit{\theta }={\mathit{\theta }}_1$, the estimator simplifies to the one where information from all covariates is borrowed (Theorem 1). Thus, the above theorem presents a more general framework and provides an alternative approach for handling violations of the uniform ${\mathit{\theta }}_0$ assumption.

Additionally, similar to Corollary 2, we identify a more computationally efficient estimator for this extended method based on an asymptotic equivalent estimator. The results are summarized in the following corollary. For detailed algorithm, we refer interested readers to Section 2.2 of the Supplementary Material.

Corollary 4.

The empirical likelihood estimator of $\mathit{\theta }$ obtained from the constrained optimization (3.10) and (3.7) is asymptotically equivalent to the following meta-analysis estimator, that is ${\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{\text{meta}}=\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}\{n{(\mathit{\theta }-{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(1)})}^T{\stackrel{\mathbf{ˆ}}{\mathbf{V}}}_{(1)}^{-1}(\mathit{\theta }-{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(1)})+N{({\mathit{\theta }}_1-{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)1})}^T{\stackrel{\mathbf{ˆ}}{\mathbf{V}}}_{(2)1}^{-1}({\mathit{\theta }}_1-{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}_{(2)1})\}$.

4. Simulation

In this section, we evaluated the proposed method based on two simulation scenarios. In the first scenario, we considered a setting where the conditional outcome distribution was the same between internal and external data, but allow for differences in the marginal distributions of certain covariates. In the second scenario, we considered the violation of homogeneous conditional distributions (i.e. $Y\mid \tilde{X}$) in two datasets. In both simulations, the internal sample size $(n)$ was set to be 200 and 500, while the ratio of external to internal sample sizes $(N/n)$ was set to be 1, 3, and 10. For nonparametric smoothing of the varying coefficients, we used the Epanechnikov kernel $(K(u)=0.75{\left(1-u^2\right)}_{+})$. The bandwidth formula $h=a\cdot n^{-1/5}$ was employed to evaluate the sensitivity of the estimates with respect to bandwidth, using different values of $a=0.8,1,1.25$ (results for $a=0.8$ and 1.25 are provided in the Supplementary Materials). For the real data application, bandwidth was selected via 10-fold cross-validation.

4.1. Data generation

4.1.1. Scenario I: Different marginal covariate distributions.

Following the notations from Section 2, we considered the following model to generate outcomes for both studies

$$Y_i={\alpha }_0\left(U_i\right)+{\alpha }_1\left(U_i\right)X_i+{\mathit{\beta }}^T{\mathbf{Z}}_i+{\gamma }^T{\mathbf{W}}_i+{\epsilon }_i,$$ (4.1)

where $\mathit{\beta }=(1,1,1{)}^T$ and $\mathit{\gamma }=(0.5,0.5,0,\dots ,0{)}^T$ (only first twos are non-zeros). For the internal data, the variable $U_i$ followed a uniform distribution $Unif[0,1]$ with ${\alpha }_0\left(U_i\right)=cos\left(5U_i\right)$ and ${\alpha }_1\left(U_i\right)=sin\left(3U_i\right)$. We generated $X_i$ by another uniform distribution $Unif[1,3]$ and considered three covariates in ${\mathbf{Z}}_i=\left(Z_{i1},Z_{i2},Z_{i3}\right)$, where $Z_{i1}$ followed a normal distribution with mean $U_i$ and variance one, $Z_{i2}$ followed a Binomial distribution with probabilities equal to $\mathrm{e}\mathrm{x}\mathrm{p}\left(U_i\right)/\left\{1+\mathrm{e}\mathrm{x}\mathrm{p}\left(U_i\right)\right\}$, and $Z_{i3}$ followed a uniform distribution $Unif[-\mathrm{1,1}]$. Moreover, we considered 100 covariates in $\mathbf{W}$ as candidates for variable selection, which followed a multivariate normal distribution with mean zeros, variance ones, and a constant correlation coefficient 0.2. The residual $\epsilon$ followed a standard normal distribution. For the external data, the same data generation mechanism was applied, except that $X_j$ followed a normal distribution with mean two and variance one, and $Z_{j3}$ followed a standard normal distribution. Thus, heterogeneous covariate distributions were considered here. As described in Section 2, we mimicked the real data situation where the covariates in $\mathbf{W}$ were not observed. Therefore, the external study considered the following model $Y_j={\tilde{\mathbf{X}}}_j^T\mathit{\theta }+{\tilde{\epsilon }}_j$, where the vector ${\tilde{\mathbf{X}}}_j$ contained covariates $U_j$, $X_j$, and ${\mathbf{Z}}_j$. Thus, in this case the conditional distributions $Y\mid \tilde{X}$ were the same between the internal and external datasets.

4.1.2. Scenario II: violation of homogeneous conditional distributions.

We followed the same data generation mechanism described above for both the internal and external data, except that we changed the parameter values to $\mathit{\beta }=(2,1,1{)}^T$ for generating the internal data. Everything else remained the same. In this scenario, the conditional distributions $Y\mid \tilde{X}$ differed between the two datasets, with the difference occurring only in the coefficient of the covariate $Z_1$. Therefore, our method would NOT borrow information from the external study for the coefficient of $Z_1$. In practice, we can perform a hypothesis test to identify the borrowed effects. We refer readers to Sections 2 and 3.4 for more details.

4.2. Evaluation metrics and results

4.2.1. Evaluation of $\stackrel{\mathit{ˆ}}{\mathit{\beta }}$ and $\stackrel{\mathit{ˆ}}{\mathit{\gamma }}$.

We performed 1000 Monte Carlo simulations for each scenario to assess the performance of $\stackrel{\mathit{ˆ}}{\mathit{\beta }}$ and $\left\{{\hat{\gamma }}_1,{\hat{\gamma }}_2\right\}$ using the following metrics: bias, Monte Carlo Standard Deviation (MCSD), Asymptotic Standard Error (ASE), 95% Coverage Probability (CP), and Relative Efficiency (RE). True positive rate and false positive rate were used for evaluating variable selection in $\left\{{\hat{\gamma }}_1,{\hat{\gamma }}_2\right\}$ and $\left\{{\hat{\gamma }}_3,\dots ,{\hat{\gamma }}_{100}\right\}$, respectively.

First, we observed that true positive rates for selecting ${\gamma }_1$ or ${\gamma }_2$ was 100% across 1, 000 Monte Carlo runs, and the false positive rates among the 98 nuisance parameters were well below 0.05 when the internal sample size was 500 (Table S5 in the Supplementary Material). This implies the validity of the variable selection procedure. Second, the evaluations for $\stackrel{\mathit{ˆ}}{\mathit{\beta }},{\hat{\gamma }}_1$ and ${\hat{\gamma }}_2$ estimates are summarized in Tables 1 (scenario I) and 2 (scenario II). In general, we observed the following: (a) the estimation bias was negligible regardless of sample size; (b) as the internal data sample size increased, the asymptotic standard error (ASE) became closer to the Monte Carlo standard deviation (MCSD), and the empirical coverage probability (CP) based on the 95% confidence interval converged to the nominal 95% level; (c) the relative efficiency (RE PLS/IB), defined as the ratio of the Monte Carlo mean squared error of the parameters $({\hat{\beta }}_1,{\hat{\beta }}_2,{\hat{\beta }}_3,{\hat{\gamma }}_1,{\hat{\gamma }}_2)$ estimated by the profile least squares (PLS) using the internal data only, over the estimates after information borrowing (PLS+IB), indicated substantial efficiency gains, with some estimated coefficients showing over two-fold efficiency improvement, underscoring the benefits of integrating information from external data. Moreover, we found that bandwidth variation had little impact on bias control and variance reduction for parametric effect estimates. However, bandwidth selection remains crucial for varying coefficient estimates, as it often involves the “variance-bias tradeoff” problem in real data applications. Therefore, as recommended by Fan and Huang,²³ we used cross-validation to select the optimal bandwidth in Section 5.

Table 1.

Evaluation of the estimator $\stackrel{\mathit{ˆ}}{\mathit{\beta }}$ under scenario I, with various combination of internal and external sample sizes and bandwidth equal to $n^{-1/5}$. All results are based on 1000 Monte Carlo runs.

	Ratio	${\beta }_1$	${\beta }_2$	${\beta }_3$	${\gamma }_1$	${\gamma }_2$	${\beta }_1$	${\beta }_2$	${\beta }_3$	${\gamma }_1$	${\gamma }_2$
		Internal sample size: n = 200					Internal sample size: n = 500
BIAS	1	−0.004	−0.001	−0.006	−0.051	−0.048	0.001	0.001	0.001	−0.002	−0.002
MCSD	1	0.063	0.134	0.097	0.137	0.135	0.040	0.081	0.062	0.055	0.054
ASE	1	0.048	0.099	0.076	0.099	0.098	0.037	0.077	0.057	0.051	0.052
CP	1	0.906	0.896	0.880	0.817	0.807	0.928	0.937	0.929	0.943	0.936
RE(PLS/IB)	1	1.378	1.409	1.884	0.932	0.958	1.382	1.370	1.690	0.976	0.971
BIAS	3	0.001	−0.002	0.001	−0.053	−0.046	0.001	−0.002	−0.003	0.003	0.002
MCSD	3	0.058	0.117	0.093	0.142	0.139	0.033	0.073	0.055	0.055	0.053
ASE	3	0.048	0.100	0.076	0.102	0.101	0.033	0.069	0.052	0.050	0.050
CP	3	0.901	0.904	0.901	0.800	0.817	0.942	0.931	0.931	0.925	0.933
RE(PLS/IB)	3	1.665	1.886	2.090	0.961	0.918	1.751	1.677	1.987	0.956	0.981
BIAS	10	0.001	−0.000	−0.002	−0.033	−0.049	0.001	−0.002	0.000	0.000	−0.001
MCSD	10	0.051	0.105	0.088	0.138	0.139	0.033	0.065	0.053	0.054	0.057
ASE	10	0.048	0.099	0.075	0.102	0.102	0.030	0.062	0.050	0.051	0.051
CP	10	0.894	0.915	0.893	0.812	0.797	0.927	0.937	0.940	0.934	0.926
RE(PLS/IB)	10	2.179	2.088	2.400	0.923	0.939	1.933	1.967	2.327	0.975	0.917

The Ratio is N/n, external sample size over internal sample size; MCSD stands for Monte Carlo Standard Deviation; ASE is Asymptotic Standard Error; RE is Relative Efficiency, i.e. the mean squared error of ${\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_{\mathit{pls}}$ divided by the mean squared error of ${\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_{\mathit{ib}}$; CP is 95% Coverage Probability.

In addition to the common observations across both scenarios, each scenario exhibited unique features. In Scenario I, our derived ASE closely matched the MCSD (Table 1), whereas the formula from the existing method¹⁵ produced less accurate ASE estimates (results not shown). This underscores the critical importance of accounting for heterogeneous covariate distributions in information integration. In Scenario II, we obtained unbiased estimates for all coefficients, even when the conditional distributions differed. However, we observed that ${\hat{\beta }}_1$ exhibited little efficiency gain (Table 2), which is expected, as our method does not borrow information from the external study for the coefficient of $Z_1$. The benefit of excluding partial information from the external study is that it helps avoid introducing bias when heterogeneous conditional distributions are a concern. Noted that the relative efficiency values for ${\hat{\gamma }}_1$ and ${\hat{\gamma }}_2$ were slightly lower than 1, as the external summary statistics did not contain useful information for estimating these parameters. Thus, in finite sample settings, the increased model complexity introduced by data integration may yield a slightly increase in estimation variance. From Table S6 in the Supplementary Material, we observed significant estimation bias when all information was integrated into the internal data analysis. These findings underscore the validity and flexibility of our method in addressing various practical scenarios.

Table 2.

Evaluation of the estimator $\stackrel{\mathit{ˆ}}{\mathit{\beta }}$ under scenario II, with various combination of internal and external sample sizes and bandwidth equal to $n^{-1/5}$. All results are based on 1000 Monte Carlo runs.

	N/n	${\beta }_1$	${\beta }_2$	${\beta }_3$	${\gamma }_1$	${\gamma }_2$	${\beta }_1$	${\beta }_2$	${\beta }_3$	${\gamma }_1$	${\gamma }_2$
		Internal sample size: n = 200					Internal sample size: n = 500
BIAS	1	0.002	0.007	0.003	−0.009	−0.013	0.001	0.000	0.000	0.004	0.002
MCSD	1	0.074	0.134	0.099	0.105	0.104	0.044	0.082	0.059	0.048	0.048
ASE	1	0.066	0.120	0.089	0.091	0.090	0.043	0.078	0.057	0.046	0.046
CP	1	0.909	0.918	0.924	0.918	0.919	0.941	0.937	0.948	0.941	0.939
RE(PLS/IB)	1	1.051	1.381	1.744	0.950	0.960	1.116	1.338	1.877	0.976	0.968
BIAS	3	0.000	0.005	0.001	−0.008	−0.008	0.001	0.001	0.001	0.003	0.004
MCSD	3	0.070	0.117	0.091	0.110	0.106	0.044	0.072	0.055	0.048	0.048
ASE	3	0.064	0.108	0.083	0.091	0.092	0.042	0.069	0.053	0.046	0.046
CP	3	0.925	0.935	0.921	0.897	0.909	0.929	0.940	0.937	0.935	0.942
RE(PLS/IB)	3	1.131	1.680	1.992	0.947	0.902	1.078	1.721	2.107	0.958	0.939
BIAS	10	−0.001	−0.006	0.003	−0.010	−0.012	−0.001	−0.002	0.001	0.006	0.002
MCSD	10	0.069	0.104	0.083	0.107	0.110	0.042	0.066	0.052	0.048	0.050
ASE	10	0.063	0.099	0.079	0.090	0.090	0.041	0.063	0.050	0.046	0.046
CP	10	0.910	0.940	0.938	0.908	0.895	0.939	0.936	0.948	0.924	0.934
RE(PLS/IB)	10	1.078	2.103	2.288	0.921	0.875	1.158	2.283	2.291	0.965	0.959

The Ratio is N/n, external sample size over internal sample size; MCSD stands for Monte Carlo Standard Deviation; ASE is Asymptotic Standard Error; RE is Relative Efficiency, i.e. the mean squared error of ${\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_{\mathit{pls}}$ over the mean squared error of ${\stackrel{\mathit{ˆ}}{\mathit{\beta }}}_{\mathit{ib}}$; CP is 95% Coverage Probability.

4.2.2. Evaluation of varying coefficients.

From Figure 1, we observed that in both scenarios, the estimated varying coefficients ${\hat{\alpha }}_0(u)$ and ${\hat{\alpha }}_1(u)$ closely matched the underlying true values. Additionally, we noted little efficiency gains for these estimates compared to those obtained without information integration. This observation aligns with our theoretical study, which suggests that the asymptotic distributions of the estimated varying coefficients may be unaffected by more efficient estimates of $\mathit{\eta }$.

Figure 1.

Evaluation of two varying coefficient estimators under Scenario I (Upper Panel) and II (Lower Panel). The internal sample size is 500, the external sample size is 1500, and bandwidth is selected to be 500^−1/5. The red line represents the average predicted value over 1000 Monte Carlo simulations, and the blue line is the true value. The shaded region represents the 95% empirical confidence interval.

4.2.3. Sample size recommendation.

To further evaluate the impact of sample size and inform practical recommendations, we considered the same simulation setting as Scenario II and examined multiple sample sizes. The results for ${\beta }_2$ are summarized in Figure 2, while the results for other parameter estimates are summarized in Figure S.4, S.5, S.6, and S.7 in the Supplementary material. We observed that >200 internal sample size yielded more stabilized efficiency gain and desired coverage probability close to its nominal level. In contrast, the estimation bias appeared relatively robust to small sample sizes, remaining low even when the internal sample size was as small as 100.

Figure 2.

Evaluation of bias, relative efficiency, and coverage probability of ${\beta }_2$ estimation in Scenario II (Heterogeneous Inference) across incremental sample sizes.

4.2.4. Other evaluations.

In addition, we conducted supplementary simulation studies to assess robustness across various settings, including the use of a non-linear link function instead of a linear model (Figure S.3, Table S.7), alternative parameter tuning strategies (Table S.8). Similar patterns were observed, and the main conclusions remained consistent across these evaluations.

5. Data application

5.1. Study on systolic blood pressure

In this section, we examined several risk factors influencing SBP using the UK Biobank data as the internal study while incorporating additional summary statistics from a statistical model based on the ARIC data as the external study. Directly sharing raw data among all collaborators can be challenging. Therefore, one author performed a linear regression of glucose, age, BMI, and sex on SBP using the ARIC data and provided the summary statistics (i.e. estimates with the variance–covariance matrix) to other authors, who then conducted varying coefficient modeling with the UK Biobank data, leveraging summary statistics from the ARIC study.

After data cleaning, we had 5,250 patients from the ARIC database and 965 patients from the UK Biobank, with demographic details in Table S.9 of the Supplementary Material. Beyond the covariates used in the ARIC analysis, we focused on the selection of brain imaging markers (39 fractional anisotropy regions) and 21 blood biomarkers available only in the UK Biobank (as $\mathbf{W}$) to identify influential biomarkers predictive to SBP. Recognizing the established association between diabetes and hypertension,³⁹ we treated glucose level as a potential varying factor (as $U$) that could also be associated with effects of other risk factors, including age, BMI, and gender. Using the generalized likelihood ratio test by Fan et al.,²⁶ we found that sex effects varied with glucose levels ($P=0.019$), leading us to model BMI and age as linear terms (as $\mathbf{Z}$) and sex as a varying coefficient term (as $\mathbf{X}$). Additionally, comparing internal estimates ${\mathit{\theta }}_{(1)}$ and external estimates ${\mathit{\theta }}_{(2)}$ using a Wald-type test revealed significant differences in BMI effects between the ARIC and UK Biobank datasets. As a result, we borrowed external information only from the Age effect, as suggested in Section 3.4. We emphasize that investigating the underlying mechanisms that create heterogeneity between datasets is beyond the scope of this article. Our primary goal is to integrate useful information to enhance estimation efficiency.

The parametric estimates and variable selection (using SCAD penalty) results are summarized in Table 3. Below the horizontal line, we reported only the covariates selected by the SCAD penalty under each proposed method. We found that the standard error for the estimated age effect was substantially reduced with our proposed method (Profile Least Squares plus Information Borrow (abbreviated as PLS+IB) vs. Profile Least Squares (abbreviated as PLS)), highlighting the advantage of information integration. Additionally, the proposed method selected more imaging and blood biomarkers compared to the PLS only method (lower half of the table, blank space indicates that the covariate is not selected in the corresponding model). It was interesting to observe that borrowing external information led to more covariates being retained during the variable selection. Notably, blood total protein, blood apolipoprotein A, and blood albumin were significantly associated with higher SBP, while blood phosphate and RLIC-L (brain) showed a negative association. These findings were consistent across different models, with the results regarding total protein and phosphate aligning well with the existing literature.⁴⁰ In addition, prior studies reported significant positive correlations for total protein and albumin.^41,42 Serum phosphate was also reported to have a negative association with SBP among middle-aged men.⁴³ These findings are consistent with our results. Apolipoprotein A, in contrast, has primarily been studied in relation to cardiovascular outcomes and mortality, including myocardial infarction and stroke.⁴⁴ Our findings provide additional evidence suggesting an association between apolipoprotein A and systolic blood pressure. In contrast, the linear model (covariate effects were linear) with SCAD penalty (LM+SCAD) produced very different effect sizes compared to the PLS-based method. Given the flexibility of the PLS method in capturing more complex effects, we place greater confidence in the PLS-based results. In addition, it is interesting to observe that standard errors for certain biomarkers that were not available in the external data became larger after information borrowing. This is not unexpected in theory as the proposed integration method does not guarantee efficiency gain for those variables when sample size is small.

Table 3.

Estimation result on the study of systolic blood pressure, using the UK Biobank data with summary statistics obtained from the ARIC data.

	PLS			PLS+IB			LM+SCAD
	Effect	SE	P value	Effect	SE	P value	Effect	SE	P value
Age *	0.730	0.104	<0.001	0.688	0.0616	<0.001	0.736	0.106	<0.001
BMI	1.320	0.165	<0.001	1.302	0.1647	<0.001	1.317	0.167	<0.001
Glucose							0.077	0.043	0.075
Sex (M vs. F)**							7.021	0.893	<0.001
Total Protein	1.851	0.371	<0.001	2.174	0.434	<0.001	0.416	0.082	<0.001
Phosphate	−1.087	0.281	<0.001	−1.582	0.404	<0.001	−5.214	1.364	<0.001
Apolipoprotein A	0.617	0.206	0.003	1.104	0.353	0.002	1.664	0.560	0.003
Albumin	0.551	0.188	0.003	0.558	0.210	0.008	0.242	0.071	<0.001
RLIC-L	−0.294	0.124	0.018	−0.515	0.215	0.017	−4.121	1.828	0.024
EX-L	−0.293	0.124	0.018	−0.292	0.155	0.060	−12.703	4.839	0.008
SHBG				−0.315	0.175	0.072
A. A.**				−0.048	0.037	0.189
IGF–1				0.089	0.063	0.159
Vitamin D	−0.023	0.013	0.082	−0.212	0.125	0.090
UN-R				−0.162	0.103	0.115

Bolded names indicate covariates that were not included in the variable selection procedure.

^*: only summary statistics for the age effect was integrated into the internal data analysis;

^**: A.A. stands for Aspartate Aminotransferase. (M vs. F) stands for Male versus Female.

The results for the varying coefficient estimates are shown in Figure 3. From the bottom panels, we observed a positive association between glucose levels and SBP, which became negative when glucose levels exceeded 100 mg/dL. This negative association pattern might be partially explained by more prevalent use of diabetes medications, such as SGLT2 inhibitors, or anti-hypertensive medications, which have blood pressure-lowering effects.⁴⁵ Additionally, the sex effect varied with glucose levels (upper panels): males had significantly higher SBP than females when glucose levels were between 60 and 90 mg/dL. However, this difference diminished and became non-significant as glucose levels exceeded 90 mg/dL. These findings indicate potential heterogeneity in sex difference due to different glucose levels.

5.2. Study on hospital length of stay

In this section, we examined patient-level factors influencing hospital length of stay (HLOS) among older hip fracture patients living with ADRD, integrating information between two distinct States (Texas and California) identified in the Medicare dataset that treated hip fracture patients with ADRD. Beyond understanding the influence of basic demographic factors, i.e. sex (male and female), age, race (white vs. non-white), and pre-fracture days at home over half of a year prior to fracture, another interest was to identify specific chronic conditions (variables in $\mathbf{W}$, 16 in total) associated with prolonged HLOS. We refer readers to Table S.10 for a detailed data summary. For method illustration, we designated the State of Texas as the internal data source, where detailed individual chronic condition records (variables in $\mathbf{W}$) were available for each patient, and the State of California as the external data source, where only an aggregated score—the Elixhauser Comorbidity Index (ECI)—was available. Notably, the ECI is widely used and is often reported in healthcare literature⁴⁶ instead of individual comorbidity information. Thus, this application mimics a realistic scenario where individual chronic conditions are not analyzed or available in the external study.

We also considered a potential non-linear effect for the covariate “months since ADRD” (variable $U$) as a proxy for ADRD severity. After testing for varying coefficients with respect to “months since ADRD,” only race is shown to improve fit with varying coefficients (P-value equals 0.0205). Thus, we modeled pre-fracture days at home, age, and sex as linear terms in the internal model. We treated “months since ADRD” as the varying intercept while race has varying coefficient in the internal model. In the model based on the external data, we performed linear regression with HLOS as the outcome and included sex, pre-fracture days spent healthy at home, age, race, months since Alzheimer’s disease, and the Elixhauser Comorbidity Index as covariates. All summary statistics (estimates and the variance–covariance matrix) shared from the external analysis were used to enhance the internal analysis. All other technical settings remained consistent with those in Section 5.1.

Estimates for the linear covariate effects are summarized in Table 4. We reported only the covariates selected by the SCAD penalty below the horizontal line. Compared with inferences from PLS and a linear model using only the internal data, our proposed method (PLS+IB) achieved a substantial reduction in standard errors and identified a significant sex effect, which was undetected by PLS and linear model. Though not passing the 0.05 threshold, we also observed stronger evidence for an age effect (P-value= 0.087). These results suggest that male and elderly patients might be subgroups at higher risk for prolonged HLOS, highlighting potential healthcare disparities. Additionally, several chronic conditions were found to be associated with prolonged HLOS, indicating that older fracture patients with ADRD exhibit heterogeneous HLOS outcomes. Patients with congestive heart failure, renal failure, and fluid and electrolyte disorders were among the top three groups more likely to experience prolonged HLOS. These findings are consistent with prior literature linking congestive heart failure,⁴⁷ fluid and electrolyte disorders,⁴⁸ and renal failure⁴⁹ to poorer post-fracture recovery outcomes. We also observed a non-linear effect of “months since ADRD” and slight varying coefficients for “Race” (Figure 4). These findings suggest that non-white patients with approximately 110 months into ADRD may experience the highest HLOS, and those patients may have significantly longer HLOS compared to white patients with around 100 to 140 months into ADRD.

Table 4.

Estimation result on the study of length of stay in hospital for older fracture patients with ADRD, using the raw data from the State of Texas with summary statistics from State of California.

	PLS			PLS+IB			LM+SCAD
	Effect	SE	P value	Effect	SE	P value	Effect	SE	P value
Month since ADRD							−0.006	0.003	0.060
Pre days at home *	−0.006	0.018	0.729	−0.01	0.015	0.503	−0.014	0.023	0.546
Sex* (Female vs. Male)	−0.083	0.259	0.748	−0.472	0.177	0.008	−0.110	0.334	0.743
Age *	0.246	0.18	0.172	0.203	0.119	0.087	0.269	0.216	0.215
Race (W vs. Oth)**							−0.227	0.451	0.615
CHF **	0.596	0.162	<0.001	0.818	0.242	0.001	0.518	0.147	<0.001
Renal Failure	0.507	0.148	0.001	0.596	0.201	0.003	0.493	0.143	0.001
FED **	0.341	0.113	0.003	0.658	0.211	0.002	0.323	0.107	0.002
Diabetes Complicated				0.291	0.153	0.058
Cardiac Arrhythmia				0.003	0.002	0.157
HU **				−0.057	0.036	0.107
Hypothyroidism				−0.051	0.032	0.116

Bolded names indicate covariates that were not included in the variable selection procedure.

^*: summary statistics for pre-days at home, sex, and age were integrated into the internal data analysis;

^**: (W vs. Oth) stands for White versus Others. CHF stands for Congestive Heart Failure. FED stands for Fluid and Electrolyte Disorders. HU stands for Hypertension Uncomplicated.

Figure 4.

Estimation of the varying intercept due to months since ADRD. PLS+IB stands for Information Borrowed, and PLS stands for estimation using only internal data. The colored areas are confidence regions.

6. Discussion

We have developed a versatile and computationally efficient method to integrate external summary statistics from an external source to enhance the estimation in the internal analysis. This method can be flexibly applied to the case where the internal analysis relies on a model where some covariates have varying effects, with additional function of variable selection, while the summary statistics from the external source is based on a distinct model, such as GLM. We have developed theory that relaxes the assumption of homogeneous covariate distributions between studies and proposed an extended framework to alleviate estimation bias when the summary statistics substantially deviate from the internal source. Extensive simulations have demonstrated the validity this new method, and method applications successfully detected several key risk factors that cannot be detected based on the model without information integration.

The proposed information integration not only improves the efficiency of estimating target parameters but also has measurable effects on variable selection in numerical studies. One possible explanation is that incorporating external information enhances estimation precision for key baseline covariates (e.g. age in both studies; sex and pre-days at home in the ADRD study), leading to more stable adjustment for these factors. This stabilization may strengthen the signal for associations with additional biomarkers, allowing more covariates to be retained during the variable selection stage.

Despite the substantial advancements, we acknowledge several limitations in the proposed methodology. First, the framework relies on cross-sectional internal data and is restricted to continuous outcome analysis; extending it to handle longitudinal/survival data and categorical outcomes is of substantial interest. Second, the current integration method primarily contributes to the effects associated with the linear terms in the VCM, which is expected when the external summary statistics are from a linear-based model. How to leverage summary statistics of non-linear components from external studies, if exist, to ensure the efficiency gain of varying coefficients is unknown and a theoretically interesting avenue for future research, although such settings (i.e. external studies providing summary statistics for nonparametric components) is likely uncommon in practice. Lastly, integrating summary statistics from multiple external sources—each with different covariates or modeling strategies—remains underexplored. These extensions fall outside the scope of this article and warrant further investigation.

Supplementary Material

Supplemental material for this article is available online.

Data availability statement

The UK Biobank data are publicly available to approved researchers through the UK Biobank Access Management System (https://www.ukbiobank.ac.uk). The ARIC data are publicly accessible through the NHLBI BioData Catalyst platform (https://biodatacatalyst.nhlbi.nih.gov/) and dbGaP (https://www.ncbi.nlm.nih.gov/gap) for approved researchers. Medicare data are not publicly available and require restricted access through the Centers for Medicare & Medicaid Services.