Estimating Optimal Site‐Specific Individualized Treatment Rules With Limited Data Sharing

ABSTRACT

Estimation of optimal individualized treatment rules (ITRs) tailored to patient characteristics is a crucial component of precision medicine. When such problems arise in multicenter randomized trials with known treatment assignment probabilities, the optimal ITR may vary across sites because of site‐level heterogeneity, and directly pooling data across centers can lead to biased ITR estimation. Meanwhile, privacy constraints often prevent investigators from accessing individual‐level data from all centers. We therefore propose Heterogeneous Distributed Learning (HD‐learning) to estimate optimal site‐specific ITRs using only summary‐level data from each site in a single communication round. The method uses a distributed mixed‐effects model to accommodate common fixed effects and site‐specific random effects. We further develop Shrinkage HD‐learning (SHD‐learning) for moderate‐ or high‐dimensional covariates. Under suitable conditions, we establish asymptotic properties of the proposed estimators. Simulations show that our methods are lossless relative to the corresponding pooled estimator and outperform methods that ignore heterogeneity. We illustrate the proposed methods using the acute upper respiratory tract infections (AURTIs) study and the opioid withdrawal study.

1. Introduction

Precision medicine aims to tailor medical treatments by maximizing clinical benefits according to individual patient characteristics, and it has received increasing attention in modern biomedical research. One of the primary goals of precision medicine is to estimate the optimal individualized treatment rules (ITRs), which map observed patient‐level information to a recommended treatment [1]. With the increasing volume, accessibility, and quality of patient‐level data, statistical methods play a crucial role in estimating and evaluating individualized treatment rules.

Many existing methods have been proposed to estimate optimal ITRs [2, 3, 4, 5, 6, 7], but many of them are developed under relatively homogeneous settings. In multicenter studies, clinical data often exhibit between‐site heterogeneity. Such heterogeneity may arise from differences in patient composition, treatment quality, disease severity, geographical regions, and other site‐level factors, and may lead to site‐specific optimal ITRs. Figure 1 provides descriptive evidence of between‐site variation in the response distributions in the AURTIs study, suggesting that site‐specific ITRs may be needed.

FIGURE 1.

Response distributions across sites and treatment groups in the AURTIs study.

Most related studies require individual‐level data for estimation and inference. In practice, however, because of privacy concerns, data‐sharing costs, and institutional constraints on data sharing, investigators may only have access to site‐level summary statistics rather than individual‐level records, as in settings such as GWAS consortia [8] or privacy‐preserving analysis platforms such as DataSHIELD [9]. Recently, various methods have been developed to analyze multicenter data while respecting privacy regulations and reducing operational burden [10, 11, 12]. For example, Danieli and Moodie [13] developed a weighted ordinary least squares method for multicenter ITR estimation, which implicitly assumes homogeneous data distributions across centers. This assumption may be violated in practical scenarios, making the model vulnerable in estimation and inference. Although their method respects data‐sharing constraints, it has limited utility when dealing with heterogeneous data. Cheng et al. [14] proposed an individual‐level meta‐analysis of ITRs, which jointly learns site‐specific ITRs by enforcing sign coherence of the covariate effects across sites. This approach assumes complete heterogeneity in parameters across sites. However, that approach effectively treats the site‐specific parameters as fully heterogeneous, which may be overly flexible in practice because some population‐level effects may still be shared across sites. In this paper, “privacy‐preservation” refers to the fact that only aggregated site‐level summaries, rather than individual‐level records, are transmitted across sites.

In this study, we focus on multicenter randomized trials with known treatment assignment probabilities and aim to estimate optimal site‐specific ITRs using only summary‐level statistics from each site. Our HD‐learning method models site‐specific treatment decision functions through a distributed mixed‐effects framework. The key technical contributions are threefold. First, HD‐learning incorporates site‐specific random effects in addition to common fixed effects in the decision function, allowing the method to account for site‐level heterogeneity. Second, the resulting estimator is lossless relative to the corresponding pooled linear mixed‐model estimator based on the full individual‐level data, while requiring only a single round of communication of site‐level summary statistics. Third, in practice, treatment decisions may depend on only a subset of covariates, so we further develop a distributed penalized extension, Shrinkage HD‐learning (SHD‐learning), to obtain sparse solutions in moderate‐ or high‐dimensional settings. We establish consistency and asymptotic theory for these penalized estimators.

The rest of the paper is organized as follows. In Section 2, we first present HD‐learning, a distributed mixed‐effects framework for estimating site‐specific ITRs, and then develop its shrinkage extension, SHD‐learning, for sparse estimation in moderate‐ or high‐dimensional settings. In Section 3, we evaluate the finite‐sample performance of the proposed methods through simulation studies. In Section 4, we illustrate the proposed methods using the AURTIs study. Theoretical results are presented in the Appendix A. Additional simulation results, analysis of opioid withdraw study and proofs are provided in the Supporting Information.

2. Methodology

2.1. Heterogeneous Distributed Learning for Individualized Treatment Rules

We consider a multicenter study with $K$ sites, indexed by $k=1,\dots ,K$. At site $k$, we observe ${𝒟}_k=\{(R_{ki},A_{ki},{\mathit{X}}_{ki}):i=1,\dots ,n_k\}$, where $R_{ki}\in ℝ$ is the observed outcome, $A_{ki}\in 𝒜=\{-1,1\}$ is the binary treatment, and ${\mathit{X}}_{ki}\in 𝒳\subset {ℝ}^p$ is the covariate vector for subject $i$ at site $k$. Let $n_k$ denote the sample size at site $k$, and $N={\sum }_{k=1}^K{n}_k$ be the total sample size. Without loss of generality, we assume that larger values of $R_{ki}$ correspond to more desirable clinical outcomes. Let $R_k(1)$ and $R_k(-1)$ denote the potential outcomes at site $k$ under treatments 1 and $-1$, respectively. To account for between‐site heterogeneity, we define a site‐specific individualized treatment rule (ITR) $d_k:𝒳\to 𝒜$, and the corresponding value function is $V_k(d_k)=E\{R_k(d_k({\mathit{X}}_k))\}$. Our goal is to estimate the optimal site‐specific ITR

$$\begin{array}{l}\hfill d_k^{\star }({\mathit{X}}_k)=\arg \underset{d_k}{\max }{V}_k(d_k),k=1,\dots ,K.\end{array}$$

In this paper, we focus on multicenter randomized trials with known treatment assignment probabilities. Let $S=k$ denote the site indicator for observations from site $k$, and define the treatment assignment probability by ${\pi }_k(a)=P(A_k=a|{\mathit{X}}_k,S=k)$ for $a\in \{-1,1\}$. Similar to Qi and Liu [15] and Chen et al. [16], we make the following assumptions: (i) consistency, $R_k=R_k(A_k)$; (ii) no unmeasured confounding, $A_k\perp \{R_k(-1),R_k(1)\}|({\mathit{X}}_k,S=k)$; and (iii) positivity, ${\pi }_k(a)\in (c,1-c)$ for all $a\in 𝒜$ and $\mathit{x}\in 𝒳$, where $c>0$ is a constant. Under these assumptions, the optimal site‐specific ITR can be written as

$$\begin{array}{ll}\hfill d_k^{\star }({\mathit{X}}_k)& =\text{sign}\left(E[R_k|{\mathit{X}}_k,S=k,A_k=1]\right.\\ \hfill & \left.-E[R_k|{\mathit{X}}_k,S=k,A_k=-1]\right)\\ \hfill & =\text{sign}\{f_k^{\star }({\mathit{X}}_k)\},\end{array}$$ (1)

where the site‐specific treatment decision function is

$$\begin{array}{l}\hfill f_k^{\star }({\mathit{X}}_k)=E\left\{\frac{R_k{A}_k}{{\pi }_k(A_k)}|{\mathit{X}}_k,S=k\right\}.\end{array}$$ (2)

To model $R_k$, we decompose it into two components for each site: a main effect for both treatments $m_k({\mathit{X}}_k)$ and a treatment‐covariate interaction ${\delta }_k({\mathit{X}}_k)$:

$$\begin{array}{l}\hfill R_k=m_k({\mathit{X}}_k)+{\delta }_k({\mathit{X}}_k)A_k+{\eta }_k,k=1,\dots ,K,\end{array}$$ (3)

where the residual satisfies $E({\eta }_k|{\mathit{X}}_k,A_k,S=k)=0.$.

Remark 1

By (2) and (3), we have $f_k^{\star }({\mathit{X}}_k)=2{\delta }_k({\mathit{X}}_k)$. Therefore, the optimal ITR can be estimated by directly modeling $f_k^{\star }({\mathit{X}}_k)$ without specifying the form of the main effect $m_k({\mathit{X}}_k)$.

For simplicity, and to match the balanced randomized‐trial setting considered in the main development, we assume ${\pi }_k(A_k)=1/2$ in the subsequent derivations. Then $f_k^{\star }({\mathit{X}}_k)=E(2R_k{A}_k|{\mathit{X}}_k,S=k)$, which motivates the modified outcome ${\tilde{Y}}_{ki}=2R_{ki}{A}_{ki}$ for $i=1,\dots ,n_k$ and $k=1,\dots ,K$. We treat ${\pi }_k(A_k)=1/2$ as a special case in the main text, and provide the extension to more general settings in Supporting Information: Section S1.

While heterogeneity exists across sites, there are also common aspects. Estimating each site separately might lead to overfitting and fail to take full advantage of the data. To borrow strength across sites while allowing residual site‐level heterogeneity, we adopt the following working linear mixed model for the modified outcome:

$$\begin{array}{l}\hfill {\tilde{Y}}_{ki}={\mathit{X}}_{ki}^{\top }\mathit{\beta }+{\alpha }_k+{ϵ}_{ki},k=1,\dots ,K,i=1,\dots ,n_k,\end{array}$$ (4)

where ${\alpha }_k$ is a site‐specific random intercept capturing residual heterogeneity across sites. Equation (4) implies the distributed linear mixed model (LMM). Similar to Peng and Lu [17], we make the following distribution assumptions:

$$\begin{array}{l}\hfill {ϵ}_{ki}\sim N(0,{\sigma }^2),{\alpha }_k\sim N(0,{\tau }^2).\end{array}$$ (5)

Then ${\tilde{\mathit{Y}}}_k\sim N({\mathbf{X}}_k\mathit{\beta },{\mathbf{\Sigma }}_k)$, where ${\mathbf{\Sigma }}_k={\tau }^2{U}_k{U}_k^{\top }+{\sigma }^2{\mathrm{I}}_{n_k}$. Here ${\tau }^2$ is a positive constant, ${\tilde{\mathit{Y}}}_k={({\tilde{Y}}_{k1},\dots ,{\tilde{Y}}_{k{n}_k})}^{\top }$, ${\mathbf{X}}_k={({\mathit{X}}_{k1},\dots ,{\mathit{X}}_{k{n}_k})}^{\top }$, and $U_k={(1,\dots ,1)}^{\top }\in {ℝ}^{n_k}$. Under the normality assumption (5), we can estimate $\mathit{\beta }$, ${\sigma }^2$ and ${\tau }^2$ based on full samples by maximizing the following log‐likelihood function

$$\begin{array}{ll}\hfill L\left(\mathit{\beta },{\sigma }^2,{\tau }^2\right)& =-\frac{1}{2}\sum _{k=1}^K\left\{{\left({\tilde{\mathit{Y}}}_k-{\mathbf{X}}_k\mathit{\beta }\right)}^{\top }{\mathbf{\Sigma }}_k^{-1}\left({\tilde{\mathit{Y}}}_k-{\mathbf{X}}_k\mathit{\beta }\right)\right.\\ \hfill & +\log \left|{\mathbf{\Sigma }}_k\right|\}+c_1,\end{array}$$ (6)

where $|·|$ denotes the matrix determinant and $c_1$ does not depend on $\mathit{\beta },{\sigma }^2,{\tau }^2$. To simplify estimation, we adopt the Hartley–Rao variance‐component parameterization [18]. Specifically, define $\theta =({\sigma }^2,\gamma )$ with $\gamma ={\tau }^2/{\sigma }^2$, and the covariance matrix is then reparameterized as:

$$\begin{array}{l}\hfill {\mathbf{\Sigma }}_k(\gamma )={\sigma }^2({\mathrm{I}}_{n_k}+\gamma U_k{U}_k^{\top })\triangleq {\sigma }^2{\mathrm{H}}_k(\gamma ),\end{array}$$ (7)

where ${\mathrm{H}}_k(\gamma )\triangleq {\mathrm{I}}_{n_k}+\gamma U_k{U}_k^{\top }$. This decomposition separates ${\mathbf{\Sigma }}_k$ into a scale parameter ${\sigma }^2$ and a structural matrix ${\mathrm{H}}_k(\gamma )$. It is especially convenient in our distributed setting, because it allows $\mathit{\beta }$ and ${\sigma }^2$ to be profiled out in closed form. By differentiating the joint log‐likelihood function (6) with respect to $\mathit{\beta }$ and ${\sigma }^2$, we obtain the conditional Maximum Likelihood Estimates (MLEs):

$$\begin{array}{ll}\hfill \mathit{\beta }(\gamma )& ={\left(\sum _{k=1}^K{\mathbf{X}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\mathbf{X}}_k\right)}^{-1}\left(\sum _{k=1}^K{\mathbf{X}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\tilde{\mathit{Y}}}_k\right),\end{array}$$ (8)

$$\begin{array}{ll}\hfill {\sigma }^2(\gamma )& =\frac{1}{N}\left[\sum _{k=1}^K{\tilde{\mathit{Y}}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\tilde{\mathit{Y}}}_k-{\left(\sum _{k=1}^K{\mathbf{X}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\tilde{\mathit{Y}}}_k\right)}^{\top }\right.\\ \hfill & \left.{\left(\sum _{k=1}^K{\mathbf{X}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\mathbf{X}}_k\right)}^{-1}\left(\sum _{k=1}^K{\mathbf{X}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\tilde{\mathit{Y}}}_k\right)\right].\end{array}$$ (9)

where $N={\sum }_{k=1}^K{n}_k$ is the total sample size. Substituting $\mathit{\beta }(\gamma )$ and ${\sigma }^2(\gamma )$ into the original log‐likelihood function (6) yields the profile log‐likelihood function with respect to $\gamma$:

$$\begin{array}{ll}\hfill L(\gamma )& =-\frac{1}{2}\sum _{k=1}^K\left\{{\sigma }^{-2}(\gamma ){\left({\tilde{\mathit{Y}}}_k-{\mathbf{X}}_k\mathit{\beta }(\gamma )\right)}^{\top }{\mathrm{H}}_k^{-1}(\gamma )\left({\tilde{\mathit{Y}}}_k-{\mathbf{X}}_k\mathit{\beta }(\gamma )\right)\right.\\ \hfill & +n_k\log {\sigma }^2(\gamma )+\left.\log |{\mathrm{H}}_k(\gamma )|\right\}.\end{array}$$ (10)

Since $\gamma$ lacks a closed‐form solution, numerical optimization (e.g., Newton–Raphson) is required to maximize $L(\gamma )$. However, direct computation involves sharing individual‐level data across sites, violating distributed constraints. To address this, we leverage the Woodbury matrix identity [19] and the matrix determinant lemma [20], then ${\mathrm{H}}_k^{-1}(\gamma )={\mathrm{I}}_{n_k}-U_k{\left({\gamma }^{-1}+U_k^{\top }{U}_k\right)}^{-1}{U}_k^{\top }={\mathrm{I}}_{n_k}-\gamma {\left(1+n_k\gamma \right)}^{-1}{U}_k{U}_k^{\top }$, and $\left|{\mathrm{H}}_k(\gamma )\right|=1+n_k\times \gamma$. We reconstruct the profile log‐likelihood (10) using aggregated statistics across sites, thereby avoiding the need to share individual‐level patient data:

$$\begin{array}{ll}\hfill L(\gamma )& =-\frac{1}{2}\left\{N\log \left[\frac{1}{N}\left(\sum _{k=1}^K{\tilde{\mathit{Y}}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\tilde{\mathit{Y}}}_k\right.\right.\right.\\ & -{\left(\sum _{k=1}^K{\mathbf{X}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\tilde{\mathit{Y}}}_k\right)}^{\top }{\left(\sum _{k=1}^K{\mathbf{X}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\mathbf{X}}_k\right)}^{-1}\\ & \left.\left.\left.\left(\sum _{k=1}^K{\mathbf{X}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\tilde{\mathit{Y}}}_k\right)\right)\right]+\sum _{k=1}^K\log (1+n_k\gamma )\right\}+c,\end{array}$$ (11)

where $c$ is the constant. Note that the individual components can be decomposed as

$$\begin{array}{ll}& {\mathbf{X}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\mathbf{X}}_k={\mathbf{X}}_k^{\top }{\mathbf{X}}_k-{\mathbf{X}}_k^{\top }{U}_k{\left({\gamma }^{-1}+n_k\right)}^{-1}{U}_k^{\top }{\mathbf{X}}_k,\\ & {\mathbf{X}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\tilde{\mathit{Y}}}_k={\mathbf{X}}_k^{\top }{\tilde{\mathit{Y}}}_k-{\mathbf{X}}_k^{\top }{U}_k{\left({\gamma }^{-1}+n_k\right)}^{-1}{U}_k^{\top }{\tilde{\mathit{Y}}}_k,\\ & {\tilde{\mathit{Y}}}_k^{\top }{\mathrm{H}}_k^{-1}(\gamma ){\tilde{\mathit{Y}}}_k={\tilde{\mathit{Y}}}_k^{\top }{\tilde{\mathit{Y}}}_k-{\tilde{\mathit{Y}}}_k^{\top }{U}_k{\left({\gamma }^{-1}+n_k\right)}^{-1}{U}_k^{\top }{\tilde{\mathit{Y}}}_k.\end{array}$$

It is worth mentioning that function (11) can be easily constructed and computed at the master site on a distributed system. For notational and computational convenience, define the augmented matrix ${\tilde{\mathbf{X}}}_k=(U_k,{\mathbf{X}}_k)\in {ℝ}^{n_k\times (p+1)}$, which is used only for transmitting summary statistics. Then each local site sends the $(p+1)\times (p+1)$ matrix $S_k^X={\tilde{\mathbf{X}}}_k^{\top }{\tilde{\mathbf{X}}}_k$, the $(p+1)\times 1$ vector $S_k^{XY}={\tilde{\mathbf{X}}}_k^{\top }{\tilde{\mathit{Y}}}_k$, the scalar $S_k^Y={\tilde{\mathit{Y}}}_k^{\top }{\tilde{\mathit{Y}}}_k$, and the local sample size $n_k$ to the master site. From the block structure of $S_k^X$ and $S_k^{XY}$, the master site can recover ${\mathbf{X}}_k^{\top }{\mathbf{X}}_k$, ${\mathbf{X}}_k^{\top }{U}_k$, $U_k^{\top }{\tilde{\mathit{Y}}}_k$, and ${\mathbf{X}}_k^{\top }{\tilde{\mathit{Y}}}_k$, and hence reconstruct the profile log‐likelihood without sharing individual‐level data.

For simplicity, we omit the $\gamma$ of the ${\mathrm{H}}_k(\gamma )$ unless clarification is needed. The master site can then obtain $\widehat{\gamma }$ as the maximizer of (11) and then ${\widehat{\mathrm{H}}}_k$, $\widehat{\mathit{\beta }}$ and ${\widehat{\sigma }}^2$ can be obtained from Equations (8) and (9). Finally, the best linear unbiased predictor (BLUP) [21] for ${\alpha }_k$ at the $k$th site is given as

$$\begin{array}{l}\hfill {\widehat{\alpha }}_k=\widehat{\gamma }{U}_k^{\top }{\widehat{\mathrm{H}}}_k^{-1}({\tilde{\mathit{Y}}}_k-{\mathbf{X}}_k\widehat{\mathit{\beta }}).\end{array}$$ (12)

Based on (8) and (12), we can construct the estimation of site‐specific ITRs of individual $i$ as ${\widehat{d}}_k({\mathit{X}}_{ki})=\text{sign}\{{\widehat{f}}_k({\mathit{X}}_{ki})\}$, and ${\widehat{f}}_k({\mathit{X}}_{ki})={\mathit{X}}_{ki}^{\top }\widehat{\mathit{\beta }}+{\widehat{\alpha }}_k$, $i=1,\dots ,n_k$, $k=1,\dots ,K$. Note that the proposed estimator is statistically optimal, similar to the global pooled LMM estimator. Simulation results further support this conclusion. We also emphasize that we do not choose the restricted MLE in this case, as we focus on the scenario where $p$ is fixed. In a distributed data setting, where the number of observations $N$ is significantly larger than the number of parameters $p$, the difference between MLE and REML estimates becomes negligible.

We also provide some theoretical results about $\widehat{\mathit{\beta }}$ in Appendix A that the proposed distributed estimator is lossless relative to the corresponding centralized pooled estimator. Specifically, Theorem A1 shows the consistency of the HD‐learning estimator $\widehat{\mathit{\beta }}$. Theorem A2 establishes the asymptotic normality of $\widehat{\mathit{\beta }}$ and shows its asymptotic distribution is identical to that of the pooled linear mixed‐model estimator (${\widehat{\mathit{\beta }}}_{\text{LMM}}$), obtained from the full, pooled dataset.

Remark 2

When the data are assumed to arise from a common population, so that the sample is homogeneous across sites, the site‐specific random effect is absent, and we set ${\alpha }_k=0$. In this case, model (4) degenerates to the linear regression model as mentioned in the modified outcome method [22] and D‐learning [15].

Remark 3

The random‐intercept specification provides a simple and interpretable version of our distributed framework while capturing a main source of between‐site heterogeneity. More general covariate‐dependent random effects can also be accommodated, and we present a random‐coefficient extension in Section S2 of the Supporting Information.

2.2. Shrinkage Estimation for Heterogeneous Distributed Learning

In this subsection, we introduce SHD‐learning for estimating individualized treatment rules. Variable selection is important in clinical studies, because in practice only a subset of covariates may contribute substantially to treatment decision‐making. Although many clinical covariates may be available, retaining irrelevant variables can reduce estimation efficiency and complicate interpretation. However, to the best of our knowledge, variable selection for decision functions in distributed heterogeneous settings has received limited attention. In addition, simultaneous estimation and tuning‐parameter selection in distributed systems can be computationally demanding [23].

Considering communication efficiency and estimation accuracy, we focus on sparse estimation of the fixed‐ effects when $p<n_k$. Since the variance components have already been estimated in Section 2.1, we construct the shrinkage estimator by penalizing the quadratic loss induced by the fitted linear mixed model. Specifically, let ${\widehat{\mathbf{\Sigma }}}_k={\widehat{\sigma }}^2{\widehat{\mathrm{H}}}_k$, and consider the penalized objective

$$\begin{array}{ll}{Q}_L(\mathit{\beta })& =L(\mathit{\beta })+\sum _{j=1}^p{p}_{{\lambda }_j}(|{\beta }_j|),\\ L(\mathit{\beta })& =\frac{1}{2N}\sum _{k=1}^K{({\tilde{\mathit{Y}}}_k-{\mathbf{X}}_k\mathit{\beta })}^{\top }{\widehat{\mathbf{\Sigma }}}_k^{-1}({\tilde{\mathit{Y}}}_k-{\mathbf{X}}_k\mathit{\beta }),\end{array}$$ (13)

where $p_{{\lambda }_j}(·)$ is a penalty function with regularization parameter ${\lambda }_j>0$.

Directly minimizing (13) would typically require iterative computation based on the full data, which is undesirable in distributed settings because of communication costs. To avoid this, we adopt the least‐squares approximation (LSA) [23, 24, 25] together with the adaptive Lasso penalty [26] on the master site. We first compute the HD‐learning estimator $\widehat{\mathit{\beta }}$ in Section 2.1 as a pilot estimator, using the first‐round summary statistics already transmitted from the local sites. We then approximate $L(\mathit{\beta })$ by a second‐order Taylor expansion at $\widehat{\mathit{\beta }}$. This gives $L(\mathit{\beta })\approx \tilde{L}(\mathit{\beta })+c_2$, where $c_2$ does not depend on $\mathit{\beta }$ and

$$\begin{array}{cc}& \tilde{L}(\mathit{\beta })=\frac{1}{2N}\sum _{k=1}^K{(\mathit{\beta }-\widehat{\mathit{\beta }})}^{\top }{\mathrm{M}}_k(\mathit{\beta }-\widehat{\mathit{\beta }}),\hfill \\ \hfill & {\mathrm{M}}_k={\mathbf{X}}_k^{\top }{\widehat{\mathbf{\Sigma }}}_k^{-1}{\mathbf{X}}_k={\widehat{\sigma }}^{-2}{\mathbf{X}}_k^{\top }{\widehat{\mathrm{H}}}_k^{-1}{\mathbf{X}}_k.\hfill \end{array}$$

We then specialize the penalty in (13) to the adaptive Lasso, that is, $p_{{\lambda }_j}(|{\beta }_j|)={\lambda }_j|{\beta }_j|$, and obtain the SHD‐learning objective

$$\begin{array}{l}\hfill Q_{\tilde{L}}(\mathit{\beta })=\tilde{L}(\mathit{\beta })+\sum _{j=1}^p{\lambda }_j|{\beta }_j|,{\lambda }_j={\lambda }_0|{\widehat{\beta }}_j{|}^{-\rho },\end{array}$$ (14)

where ${\lambda }_0>0$ is a global tuning parameter, ${\widehat{\beta }}_j$ is the $j$th component of the pilot estimator $\widehat{\mathit{\beta }}$, and $\rho >0$ is fixed. Here, $\rho =1$ is used. Thus, coefficients with larger pilot estimates receive smaller penalties, whereas coefficients with smaller pilot estimates receive larger penalties. Compared with the ordinary Lasso, the adaptive Lasso therefore applies data‐dependent shrinkage across coefficients.

Let $\tilde{\mathit{\beta }}=\arg {\min }_{\mathit{\beta }}{Q}_{\tilde{L}}(\mathit{\beta })$. In the Appendix A, we establish several theoretical properties of $\tilde{\mathit{\beta }}$. Specifically, Theorem A3 establishes the $\sqrt{N}$‐consistency and selection consistency of the penalized estimator, Theorem A4 establishes its oracle property, and Theorem A5 establishes the consistency of tuning‐parameter selection based on distributed bayesian information criterion (DBIC).

Because $Q_{\tilde{L}}(\mathit{\beta })$ depends only on the pilot estimator $\widehat{\mathit{\beta }}$ and the matrices ${\{{\mathrm{M}}_k\}}_{k=1}^K$, all of which can be reconstructed at the master site from the first‐round summary statistics in Section 2.1, the sparse estimator $\tilde{\mathit{\beta }}$ can be computed without any additional communication with local sites. Consequently, only a single communication round is required throughout the entire procedure, including variable selection. The SHD‐learning procedure is summarized in Algorithm 1.

ALGORITHM 1. Communication‐efficient shrinkage estimation for heterogeneous distributed learning.

2.3. Tuning Parameter Selection

To implement the variable selection procedure in Section 2.2, we need to select the tuning parameter. Let $a_{\lambda }=\max \{{\lambda }_j:1\le j\le p_0\}$ and $b_{\lambda }=\min \{{\lambda }_j:p_0<j\le p\}$. Theoretically, for consistent selection of the fixed effects, the tuning parameter should satisfy $a_{\lambda }\sqrt{N}\stackrel{P}{\to }0$ and $b_{\lambda }\sqrt{N}\stackrel{P}{\to }\infty$ as $N\to \infty$. However, traditional tuning‐parameter selection methods are difficult to implement in distributed settings, because they typically require repeated model fitting and additional communication. We therefore adopt a distributed BIC criterion. Recall that ${\mathrm{M}}_k={\mathbf{X}}_k^{\top }{\widehat{\mathbf{\Sigma }}}_k^{-1}{\mathbf{X}}_k={\widehat{\sigma }}^{-2}{\mathbf{X}}_k^{\top }{\widehat{\mathrm{H}}}_k^{-1}{\mathbf{X}}_k$, and let $\mathrm{M}={\sum }_{k=1}^K{\mathrm{M}}_k$. We define

$$\begin{array}{l}\hfill {\text{DBIC}}_{\lambda }=\frac{1}{N}{(\tilde{\mathit{\beta }}-\widehat{\mathit{\beta }})}^{\top }\mathrm{M}(\tilde{\mathit{\beta }}-\widehat{\mathit{\beta }})+\frac{\log (N)}{N}{\text{df}}_{\lambda },\end{array}$$ (15)

where ${\text{df}}_{\lambda }$ is the number of nonzero components in $\tilde{\mathit{\beta }}$.

The criterion in (15) is motivated by Wang and Leng [24] and Zhu et al. [23]. In our setting, the difference is that the criterion is constructed from the pilot estimator $\widehat{\mathit{\beta }}$ obtained in Section 2.1 and the aggregated quadratic matrix $\mathrm{M}$ arising from the distributed linear mixed‐model framework. Since $\mathrm{M}$ can be reconstructed on the master site from the first‐round transmitted summaries, tuning‐parameter selection can be carried out without any additional communication.

3. Simulation Studies

We conduct extensive simulation studies to investigate the empirical performance of the proposed methods and evaluate their robustness and flexibility. Specifically, refer to Shah et al. [27], we choose three generalized criteria:

• Correct classification rate (CCR): the percent of correct best treatment assignments;

• Empirical value: $\frac{{\sum }_{k=1}^K{n}_k{P}_{n_k}^{(k)}[R_k·𝕀\{A_k=d_k({\mathit{X}}_k)\}]}{{\sum }_{k=1}^K{n}_k{P}_{n_k}^{(k)}[𝕀\{A_k=d_k({\mathit{X}}_k)\}]}$, where $P_{n_k}^{(k)}g=\frac{1}{n_k}{\sum }_{i=1}^{n_k}g(O_{ki})$ represents the empirical average at $k$th site, with $O_{ki}=(R_{ki},\hspace{0.17em}{A}_{ki},\hspace{0.17em}{\mathit{X}}_{ki})$;

• Average prediction error (APE): Coefficient accuracy based on MSE of true versus predicted decision functions, $\text{APE}=N^{-1}{\sum }_{k=1}^K{\sum }_{i=1}^{n_k}{({\mathit{X}}_{ki}^{\top }{\mathit{\beta }}_0+{\alpha }_k-{\mathit{X}}_{ki}^{\top }\widehat{\mathit{\beta }}-{\widehat{\alpha }}_k)}^2$, where ${\mathit{\beta }}_0$ and ${\alpha }_k$ represent the true parameters.

Better performance is indicated by the higher CCR and empirical value, along with the lower APE. All simulations evaluate these criteria using a large test dataset comprised of 10 000 observations across $K$ sites. For reliable assessment, each simulation setting is based on 100 replications. We consider different scenarios to demonstrate the efficacy of our methods, evaluating both estimation efficiency and variable selection accuracy. Because our main interest is treatment‐effect heterogeneity across sites, we generate ${\mathit{X}}_k$, $A_k$, and the main‐effect function from the same distribution across sites, and introduce heterogeneity through the site‐specific treatment‐decision function. To simplify notation, we omit the subscript $k$ for the homogeneous components whenever there is no ambiguity. Throughout, $\mathit{X}={(X_1,\dots ,X_p)}^{\top }$, with each component independently generated from $U[-1,1]$, and the treatment $A$ is randomly assigned to $\pm 1$ with equal probability.

3.1. Case 1

Case 1 presents simulations to emphasize the advantages of HD‐learning estimates under a heterogeneous decision function. We compare the following methods:

• Our proposed HD‐learning;

• DWOLS: Distributed weighted ordinary least square proposed in Danieli and Moodie [13] with a modified outcome, which ignores the site heterogeneity;

• Average: Averaging the parameter estimates of the decision function obtained from each site;

• Pooled D‐learning: A pooled version of D‐learning [27] that uses the entire dataset. This method ignores the site heterogeneity and privacy constraints;

• Pooled LMM: Pooled linear mixed model, which can be used to validate the lossless property of our proposed method.

We consider four scenarios as follows

$$\begin{array}{cc}\hfill (1)m(\mathit{X})& =1+X_1+X_2+2X_3+0.5X_4,f_k(\mathit{X})=1.8(X_1+X_2-0.3).\hfill \\ \hfill (2)m(\mathit{X})& =1+X_1+X_2+2X_3+0.5X_4,f_k(\mathit{X})=-X_1-X_2+{\alpha }_k.\hfill \\ \hfill (3)m(\mathit{X})& =1+2X_1+X_2+0.5X_4,f_k(\mathit{X})=1.8(X_1+X_2+X_3+X_4)+{\alpha }_k.\hfill \\ \hfill (4)m(\mathit{X})& =1+X_1^2+X_2^2,f_k(\mathit{X})=0.3X_1+{\alpha }_k.\hfill \end{array}$$

Scenario (1) is considered in Qi and Liu [15], without heterogeneity in the decision function. Scenarios (2) through (4) illustrate varying levels of treatment decision variation across sites via the random effect term ${\alpha }_k$. Specifically, Scenario (2) presents a situation with the inverse effect of covariates on the decision. In Scenario (3), the interaction effect considers linear combinations containing all four covariates. Finally, Scenario (4) represents the situation where covariates have weak impacts on decision‐making, yet the random effect ${\alpha }_k$ is relatively vital. In all instances, the random error $\eta$ follows a standard normal distribution, $\eta \sim N(0,1)$, and $\delta (\mathit{X})=0.5f(\mathit{X})$. The total sample size is set at $N\in \{10^3,2\times 10^3,3\times 10^3\}$, with the number of sites $K$ fixed at 10. Conversely, with a fixed $N=3\times 10^3$, the number of sites varies as $K\in \{10,30,50\}$.

Figure 2 displays the APE results for all four scenarios as $K$ or $N$ varies, and Tables S1, S2 in the Supporting Information report the corresponding CCRs and empirical values on the test dataset. In Scenario (1), where the decision function is homogeneous across sites, methods that do not account for site heterogeneity perform comparably to HD‐learning. In Scenarios (2) and (3), HD‐learning outperforms the competing methods, as reflected by lower APE and higher CCR and empirical value, because it explicitly accounts for between‐site heterogeneity in the treatment decision function. In Scenario (4), where the random effect ${\alpha }_k$ plays a dominant role, the advantage of HD‐learning becomes even more pronounced. Across all four scenarios, the results of HD‐learning closely match those of the pooled linear mixed model, confirming the lossless property of the proposed distributed estimator. At the same time, HD‐learning requires only one round of communication, making it particularly attractive for large multicenter datasets with limited data sharing.

FIGURE 2.

Simulation results for Case1: APE values calculated of the four scenarios. (a): APE for different site numbers $K$. (b) APE for a varied total sample size $N$. Empirical Average prediction errors are evaluated on testing data over 100 replications. Error bars represent the 95% confidence intervals.

We further investigate different signal‐to‐noise ratios (SNRs), defined by $\text{SNR}={\tau }^2/{\sigma }^2\in \{0.1^2,1^2,1.5^2,2^2\}$ [28]. A small SNR indicates that the random effect contributes little relative to the within‐site random error, whereas a large SNR indicates substantial between‐site heterogeneity. Figure 3 presents boxplots of the CCR values under the four scenarios for different SNR levels. Note that Scenario (1) is the homogeneous special case in which ${\alpha }_k$ is absent, so the heterogeneity‐related SNR parameter has no impact on its results. In contrast, in the heterogeneous scenarios, the advantage of HD‐learning becomes more pronounced as the SNR increases, especially in Scenario (4), where the random effect plays a dominant role.

FIGURE 3.

Simulation results for Case 1: Boxplots of the CCR values from 100 replications for 5 methods under four scenarios under a heterogeneous decision function with $SNR={\tau }^2/{\sigma }^2=({0.1}^2,1^2,{1.5}^2,2^2)$. (a) Scenario (1), (b) Scenario (2), (c) Scenario (3), (d) Scenario (4).

3.2. Case 2

In this subsection, we compare SHD‐learning and shrinkage pooled D‐learning under the data‐generating mechanisms of Section 3.1, with $p$ ranging from 10 to 50. For shrinkage pooled D‐learning, we employ adaptive Lasso to estimate decision function parameters as follows:

$$\begin{array}{ll}\hfill {\widehat{\mathit{\beta }}}_N^{\text{SPD}}& =\underset{\mathit{\beta }}{\text{arg min}}\frac{1}{N}\sum _{k=1}^K{({\tilde{\mathit{Y}}}_k-{\mathbf{X}}_k\mathit{\beta })}^{\top }({\tilde{\mathit{Y}}}_k-{\mathbf{X}}_k\mathit{\beta })+\sum _{j=1}^p{\lambda }_j|{\beta }_j|,\\ \hfill {\lambda }_j& ={\lambda }_0|{\widehat{\beta }}_j{|}^{-\rho }.\end{array}$$

Figure 4 shows the APE under high‐dimensional settings. In Scenario (1), the APE curves of the two estimators nearly overlap, indicating comparable performance under a homogeneous decision function. In contrast, in Scenarios (2–4), where heterogeneity is present, the shrinkage pooled D‐learning estimator performs substantially worse than SHD‐learning, and the gap widens as the dimension increases. This deterioration is driven by the bias induced by ignoring between‐site heterogeneity. Table S3 in the Supporting Information reports the corresponding CCRs and empirical values on the test dataset, showing that SHD‐learning remains stable and advantageous even as the dimension increases. Table S4 further reports the numbers of correctly selected nonzero coefficients and incorrectly selected zero coefficients, indicating satisfactory variable‐selection performance.

FIGURE 4.

Simulation results for Case 2: APE values calculated under the four scenarios for $p=\{10,20,30,40,50\}$ in the high‐dimensional setting. SHD‐learning represents shrinkage heterogeneous distributed learning. Average prediction errors are evaluated on testing data at different dimensionalities of covariates over 100 replications, with $N=1000,K=10$. Error bars represent the 95% confidence intervals.

4. Application to AURTIs Study

In this section, we illustrate the HD‐learning and SHD‐learning methods using data from a multicenter, double‐blind, randomized, placebo‐controlled trial on acute upper respiratory tract infections (AURTIs). Our goal is to identify the optimal individualized treatment rule for nonantibiotic anti‐inflammatory medication in the treatment of AURTIs. AURTIs are commonly caused by various viruses and bacteria, and typical symptoms include cough, phlegm, sore throat, headache, mucosal hyperemia, dry throat, and low‐grade fever. The use of nonantibiotic medication may help avoid unnecessary side effects associated with antibiotics [29]. The treatment period lasted seven days, during which antiviral, hormonal, antibacterial, and other related therapies were not allowed. Physical cooling measures were used for patients whose body temperature was below $38.5^{\circ }$C after enrollment. Treatment was coded as $1$ for the experimental group and $-1$ for the control group. The dataset contains 580 observations, including 290 subjects in the experimental group and 290 in the control group, collected from five research centers across the country. The site‐specific sample sizes are 128, 120, 120, 92, and 120. Each center contributes equal numbers of subjects to the experimental and control groups, thereby maintaining a balanced $1:1$ allocation within the site.

We include 12 covariates: Two binary variables, sex ($1=$ male, $0=$ female) and past medical history ($1=$ yes, $0=$ no), together with 10 continuous variables, namely age, course of disease, and eight baseline symptom scores (health status, expectoration, thirst, cough, halitosis, headache, mucosal hyperemia, and pharyngalgia visual analogue scale). The clinical outcome is defined as the change in the total symptom score from baseline to day 7, with a larger change indicating greater therapeutic benefit. Before constructing the modified outcome, we center the original outcome to avoid cancellation induced by the symmetric treatment coding. Under balanced randomization, this centering does not change the population treatment contrast, but it may improve numerical stability and finite‐sample precision.

We apply SHD‐learning to the data from the five centers without sharing individual‐level records, and compare its performance with that of shrinkage‐pooled D‐learning based on the full pooled data. We use a proportionate stratified bootstrap with optimism correction to estimate the mean empirical value and its standard error (SE). This approach avoids further data splitting and reduces the potential optimistic bias caused by evaluating a treatment rule on the same data used for model fitting [30, 31]. More details of the bootstrap procedure can be found in Section S6 of the Supporting Information. The bootstrap method is used to provide an uncertainty summary for empirical value comparison, and is not used to construct confidence intervals or conduct inference for model coefficients. SHD‐learning outperforms shrinkage pooled D‐learning, with a mean empirical value of 1.216 (SE = 0.190), compared with 1.136 (SE = 0.202) for shrinkage pooled D‐learning. These results suggest that SHD‐learning provides better treatment decisions in this application.

To further quantify uncertainty for the selected active set and enhance clinical interpretability, we conduct post‐selection inference based on the asymptotic variance in Theorem A2. Specifically, after selecting the active set, we refit the corresponding unpenalized model and report model‐based SEs, 95% Wald confidence intervals, and $p$ values. The resulting estimates are reported in Table 1. Features such as course of disease ($p<0.001$), sex ($p=0.002$), mucosal hyperemia ($p=0.002$), and past medical history ($p=0.002$) are strongly associated with the optimal individualized treatment rule, while age ($p=0.007$) also shows clear statistical significance. Note that the inferential results above rely on Assumption C2. To assess the plausibility of this assumption in practice, investigators can examine site‐wise covariate comparability using summary balance diagnostics such as site‐specific means and standard deviations or graphical checks. When individual‐level data are available, formal distributional diagnostics, such as Kolmogorov‐Smirnov tests for key continuous covariates, chi‐square tests for categorical covariates, and energy‐distance‐based multivariate tests [32], may also be used. When substantial cross‐site covariate differences are detected, the model‐based $p$‐values and confidence intervals should be interpreted with caution and complemented by sensitivity analyses.

TABLE 1.

Post‐selection refit inference of the clinical features for the optimal individualized treatment rule in the AURTIs study.

Clinical feature	Unpenalized estimate ($\widehat{\mathit{\beta }}$)	Standard error	Statistics	$p$	95% Confidence interval
Age	−0.064	0.024	−2.702	$0.007^{\ast \ast }$	$[-0.111,-0.017]$
Sex	2.311	0.751	3.076	$0.002^{\ast \ast }$	[0.839, 3.783]
Cough	−0.956	0.536	−1.783	$0.075^{·}$	$[-2.007,0.095]$
Mucosal hyperemia	−2.021	0.633	−3.193	$0.002^{\ast \ast }$	$[-3.262,-0.780]$
Past medical history	0.342	0.107	3.179	$0.002^{\ast \ast }$	[0.132, 0.552]
Course of disease	3.113	0.890	3.498	$<0.001^{\ast \ast \ast }$	[1.369, 4.857]

Note: To conduct rigorous post‐selection inference, the estimates presented in this table are unpenalized maximum likelihood estimates fit on the selected active set. They differ slightly from the penalized coefficients in the final SHD‐learning decision rule. The reported standard errors, confidence intervals, and $p$‐values are based on the model‐based asymptotic variance in Theorem A2 and should be interpreted under the stated regularity conditions. Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘$·$’ 0.1 ‘ ’ 1.

Based on Table 1, the estimated site‐specific individualized treatment rule can be written as ${\widehat{d}}_k({\mathit{X}}_k)=\text{sign}(-0.064\times \text{Age}+2.311\times \text{Sex}-0.956\times \text{Cough}-2.021\times \text{Mucosal hyperemia}+0.342\times \text{Past medical history}+3.113\times \text{Course of disease}+{\widehat{\alpha }}_k),k=1,\dots ,5,$where ${\alpha }_k$ is a site‐specific intercept capturing residual differences across centers, with estimated values ${\widehat{\alpha }}_k=(0.303,1.280,-0.569,0.145,-1.547)$ and SE $=(0.598,0.768,0.721,0.674,0.657)$. The estimated site effects differ across centers, with Site 2 having the largest value and Site 5 the smallest. Given the limited sample size and the small number of centers, these differences should be interpreted as preliminary exploratory evidence of possible site‐level heterogeneity. This pattern is broadly consistent with Figure 1, where the treatment group appears to respond more favorably than the control group at Site 2, whereas at Site 5 the treatment group does not appear to outperform the control group.

5. Discussion

This paper addresses the estimation of site‐specific optimal individualized treatment rules in multicenter clinical trials with heterogeneous treatment effects across sites. We propose HD‐learning and SHD‐learning, which incorporate common fixed effects and site‐specific random effects into ITR estimation requiring only summary‐level information from each site. The proposed framework achieves the same asymptotic efficiency as the corresponding pooled mixed‐effects estimator without sharing individual‐level data and requires only a single round of communication. We further develop a distributed variable selection procedure to identify covariates that contribute to the treatment decision function. Theoretical results establish asymptotic normality for HD‐learning and oracle‐type properties for SHD‐learning under suitable regularity conditions. Simulation studies and real‐data analyses demonstrate that the proposed methods can improve correct classification rates and empirical values when treatment‐effect heterogeneity is present across sites.

Several limitations of the proposed framework should be noted. First, HD‐learning is built upon a working linear mixed‐effects model for the site‐specific treatment decision function. Although this formulation is simple, interpretable, and communication‐efficient, it may be restrictive when the true treatment effects are highly nonlinear or when more complex forms of site‐level heterogeneity are present. Extending the proposed distributed framework to nonlinear mixed‐effects models or more flexible machine‐learning‐based decision functions is an important direction for future work. Second, the current development assumes that the treatment assignment probabilities are known, as is natural in randomized trials. Extending the method to observational studies, where propensity scores must be estimated, would require additional theoretical investigation. Third, the formal Wald‐type inference established in this paper relies on Assumption C2. Although the proposed estimator may still be used as a working estimation procedure under violation of Assumption C2 and can retain satisfactory predictive performance, the current theoretical results no longer guarantee the validity of the associated inference; in particular, the standard errors may be underestimated, leading to lower‐than‐nominal coverage probabilities. The score‐based bootstrap method may serve as a practical supplementary sensitivity analysis [33, 34], while the formal validity of Wald‐type inference under violation of Assumption C2 remains an important topic for future work.

Several future extensions are also worth pursuing. The proposed distributed framework could be generalized to other types of outcomes, such as binary, ordinal, or survival outcomes, by adopting appropriate loss functions. More flexible decision‐function models, including nonlinear mixed‐effects models and machine‐learning‐based methods, could be incorporated to capture complex treatment‐effect heterogeneity. It would also be useful to develop extensions that allow missing or partially collected covariates across centers, possibly through distributed imputation strategies. In addition, important variables for ITRs may themselves vary across sites, and accounting for such heterogeneity in variable selection is an interesting direction for future research. Finally, this paper focuses on single‐stage treatment decisions; extending HD‐learning to dynamic treatment regimes for multistage decision‐making with longitudinal data would further broaden its practical applicability. Strengthening the method under formal differential privacy frameworks is an important topic for future work.

Funding

This work was supported by MOE Project of Key Research Institute of Humanities and Social Sciences (22JJD910001) and Guangdong Basic and Applied Basic Research Foundation (2026A1515010197).

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Table S1. Simulation results under unbalanced treatment assignment with ${\pi }_{ki}(A_{ki}=1)=0.7.$ Table S2. Simulation results under the random‐coefficient model (random intercept and random slope). All five methods were evaluated on an independent test set of 10 000 subjects. The true random slope variance is denoted by ${\tau }_1^2$. Figure S1. Simulation results for Case 1: The correct classification rates (CCRs) and mean empirical values for heterogeneous decision function simulations with varying numbers of sites $K\in \{10,30,50\}(N=3\times 10^3$). Error bars represent the 95% confidence intervals. Note: The error bars are sometimes not clearly visible, likely because they are very small and partially obscured by the points. Figure S2. Simulation results for Case 1: The correct classification rates (CCRs) and mean empirical values for heterogeneous decision function simulations with varying total sample sizes $N\in \{10^3,2\times 10^3,3\times 10^3\}(K=10)$. Error bars represent the 95% confidence intervals. Note: The error bars are sometimes not clearly visible, likely because they are very small and partially obscured by the points. Figure S3. Simulation results for Case 2: The CCRs and mean empirical values comparing shrinkage pooled D‐learning (SPD learning) versus shrinkage HD‐learning (SHD‐learning) with varying covariate dimensions $p\in \{10,30,50\}$. The sample size is fixed at $N=1000$, and the number of sites is $K=10$. Error bars represent the 95% confidence intervals. Figure S4. Simulation results for Case 2: Variable selection results of Scenario (2), where the true parameter vector contains exactly 2 nonzero values. The plots show the average number of correctly and incorrectly selected variables across different covariate dimensions $p\in \{10,20,30,40,50\}$ with $N=1000,K=10$. Table S3. Simulation results for small sample sizes across different methods. Table S4. Simulation results for Case 1: The CCRs and mean empirical value, along with standard error of the mean, for heterogeneous decision function simulations with $K\in \{10,30,50\}(N=3\times 10^3)$. Table S5. Simulation results for Case 1: The CCRs and mean empirical value, along with standard error of the mean, for heterogeneous decision function simulations with $N\in \{10^3,2\times 10^3,3\times 10^3\}(K=10)$. Table S6. Simulation results for Case 2: The CCRs and mean empirical value, along with standard error of the mean, for heterogeneous decision function simulations for shrinkage pooled D‐learning (SPD‐learning) versus shrinkage HD‐learning (SHD‐learning) with $p\in \{10,30,50\}(N=1000,K=10)$. Table S7. Simulation results for Case 2: Variable selection results of Scenario (2), where the true parameter with nonzero values equals 2,in case of $K=10$ and $N=1000$. Table S8. Simulation results for statistical inference of the fixed effects ($\beta$) and variance component (${\tau }^2$) across Scenarios 1‐4. Results are based on $N=1000,K=10$, and 100 Monte Carlo replications. Inference evaluation contains Bias, Empirical Standard Errors (ESE), Asymptotic Standard Errors (ASE), and 95% Confidence Interval Coverage Probabilities (CP) of the fixed effects. Note: The ${\tau }^2$ estimation for Scenario 1 is omitted (—) as it represents a strictly homogeneous setting (${\tau }^2=0$). Table S9. Simulation results with correlated covariates generated via the Gaussian Copula method. The covariates exhibit an AR(1) correlation structure with $\rho ==0.5$. The sample size is $N=1000$ and $K=10$ across all cases, and all methods were evaluated on an independent test set of 10 000 subjects. Table S10. A toy example with violation of Assumption C2 under four scenarios. Results are reported as mean (SE). Table S11. Summary statistics for characteristics of the selected subjects.

Appendix A.

In this section, we derive theoretical results of the proposed HD‐learning and SHD‐learning methods under a fixed dimensionality setting, denoted as $p$. For simplicity, we assume that ${\pi }_k(A_k)=1/2$ for any $A_k\in 𝒜$ and ${\mathit{X}}_k\in 𝒳$, where $𝒜=\{-1,1\}$. We can demonstrate that both the HD‐learning estimator $\widehat{\mathit{\beta }}$ and the shrinkage estimator $\tilde{\mathit{\beta }}$, given by Equations (8) and (14), are consistent and possess asymptotic normality or the oracle property under some mild conditions. All proofs are provided in Supporting Information.

We then establish the exact finite‐sample equivalence of the HD‐learning estimator in Theorem A1. The asymptotic normality is demonstrated in Theorem A2.

Theorem A1

(Exact finite‐sample equivalence) For every finite sample, $\widehat{\mathit{\beta }}={\widehat{\mathit{\beta }}}_{\mathrm{LMM}}$.

Theorem A2

(Asymptotic normality) Under the regularity conditions (C1–C3) in Supporting Information, for the non‐sparse true parameter ${\mathit{\beta }}_0$, $\sqrt{N}\left(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0\right)\stackrel{d}{\to }N(0,\mathbf{\Sigma })$, where $\stackrel{d}{\to }$ denotes convergence in distribution, $\mathrm{\Omega }={\lim }_{N\to \infty }\frac{1}{N}{\sum }_{k=1}^K{\mathbf{X}}_k^{\top }{\mathbf{\Sigma }}_k^{-1}{\mathbf{X}}_k$ and $\mathbf{\Sigma }={\mathrm{\Omega }}^{-1}$. The sample plug‐in estimator is $\widehat{\mathbf{\Sigma }}={\left(\frac{1}{N}{\sum }_{k=1}^K{\mathbf{X}}_k^{\top }{\widehat{\mathbf{\Sigma }}}_k^{-1}{\mathbf{X}}_k\right)}^{-1}={\widehat{\sigma }}^2{\left(\frac{1}{N}{\sum }_{k=1}^K{\mathbf{X}}_k^{\top }{\widehat{\mathrm{H}}}_k^{-1}{\mathbf{X}}_k\right)}^{-1}.$

By Theorem A2, we know that the HD‐learning estimator $\widehat{\mathit{\beta }}$ exhibits the same asymptotic distribution as the global full‐data estimator ${\widehat{\mathit{\beta }}}_{\text{LMM}}$. This implies that $\widehat{\mathit{\beta }}$ shares the same asymptotic efficiency as ${\widehat{\mathit{\beta }}}_{\text{LMM}}$. For valid statistical inference, we need to consistently estimate $\mathbf{\Sigma }$. Given a $\sqrt{N}$‐consistent estimation $\widehat{\gamma }$, $\widehat{\mathbf{\Sigma }}={\widehat{\sigma }}^2{(1/N{\sum }_{k=1}^K{\mathbf{X}}_k^{\top }{\widehat{\mathrm{H}}}_k^{-1}{\mathbf{X}}_k)}^{-1}$. It is remarkable that both $\widehat{\mathbf{\Sigma }}$ and ${\widehat{\mathrm{H}}}_k$ can be estimated easily on our distributed system.

Moreover, we can establish the $\sqrt{N}$‐consistency as well as the selection consistency result of the sparse estimator $\tilde{\mathit{\beta }}$ under certain conditions of $\lambda ={\left({\lambda }_1,\dots ,{\lambda }_p\right)}^{\top }$. We first define some notations. Without loss of generality, we assume the first $p_0\left(0<p_0<p\right)$ parameters to be nonzero, that is, ${\beta }_j\ne 0$ for $1\le j\le p_0$ and ${\beta }_j=0$ for $j>p_0$. Correspondingly, ${𝒮}_T=\left\{1,\dots ,p_0\right\}$ denotes the true support set. In addition, let $𝒮=\left\{i_1,\dots ,i_d\right\}$ be an arbitrary candidate model with size $d$. In addition, for an arbitrary vector $V={\left(V_j:1\le j\le p\right)}^{\top }$, define $V^{(𝒮)}={\left(V_i:i\in 𝒮\right)}^{\top }\in {ℝ}^d$ and $V^{(-𝒮)}={\left(V_i:i\notin 𝒮\right)}^{\top }\in {ℝ}^{p-d}$.

Theorem A3

Under the regularity conditions (C1–C3) in Supporting Information, for the sparse true parameter ${\mathit{\beta }}_0$, let $a_{\lambda }=\max \{{\lambda }_j:1\le j\le p_0\}$ and $b_{\lambda }=\min \{{\lambda }_j:p_0<j\le p\}$ . Then, as $N\to \infty$,

• 1.
  If $a_{\lambda }\sqrt{N}\stackrel{P}{\to }0$, then

  $$\tilde{\mathit{\beta }}-{\mathit{\beta }}_0=O_p(N^{-1/2}).$$
• 2.
  If $a_{\lambda }\sqrt{N}\stackrel{P}{\to }0$ and $b_{\lambda }\sqrt{N}\stackrel{P}{\to }\infty$, then

  $$P\left({\tilde{\mathit{\beta }}}^{(-{𝒮}_T)}=0\right)\to 1.$$

Theorem A4

(Oracle property) Under the regularity conditions (C1–C4) in Supporting Information, for the sparse true parameter ${\mathit{\beta }}_0$, if $a_{\lambda }\sqrt{N}\stackrel{P}{\to }0$ and $b_{\lambda }\sqrt{N}\stackrel{P}{\to }\infty$, then

$$\sqrt{N}\left({\tilde{\mathit{\beta }}}^{({𝒮}_T)}-{\mathit{\beta }}_0^{({𝒮}_T)}\right)\stackrel{d}{\to }N\left(0,V_{{𝒮}_T}\right),$$

where $V_{{𝒮}_T}={\mathrm{\Omega }}_{{𝒮}_T}^{-1}$.

For a given $\lambda$, let ${𝒮}_{\lambda }$ denote the selected model. Define ${ℝ}^{-}=\{\lambda :{𝒮}_{\lambda }\not\supset {𝒮}_T\}$, ${ℝ}^0=\{\lambda :{𝒮}_{\lambda }={𝒮}_T\}$, ${ℝ}^{+}=\{\lambda :{𝒮}_{\lambda }⊋{𝒮}_T\}.$

Theorem A5

Under the regularity conditions (C1–C4) in Supporting Information, for the sparse true parameter ${\mathit{\beta }}_0$ and $n_k/N^{1/2}\to \infty$, define a reference tuning sequence ${\{{\lambda }_N\}}_{N=1}^{\infty }$ by ${\lambda }_{N,j}=N^{-1}$, $1\le j\le p_0$, and ${\lambda }_{N,j}=\log N/\sqrt{N}$, $p_0<j\le p.$ Then

$$\begin{array}{l}\hfill P\left(\underset{\lambda \in {ℝ}^{-}\cup {ℝ}^{+}}{\inf }{\text{DBIC}}_{\lambda }>{\text{DBIC}}_{{\lambda }_N}\right)\to 1.\end{array}$$

Data Availability Statement

To ensure reproducibility, all source codes used in this study have been made publicly available on the project GitHub repository at https://github.com/qiaonan0915/shrinkageHDlearning. Opioid withdrawal study data are publicly available through the National Institute on Drug Abuse (NIDA) (https://datashare.nida.nih.gov/study/). The AURTIs study data used in this study are not publicly available because they contain sensitive personal health information and are subject to institutional and data‐use restrictions.