Integration of Individual Participant and Aggregate Data Under Dataset Shift: Summary Statistic Comparison and Scalable Computation

TL;DR

It is found that AD derived from outcome-stratified subgroups consistently yield greater efficiency gains than those based on covariate-stratified subgroups, and these findings support the routine inclusion of outcome-stratified summaries for continuous endpoints in trial reports and public data repositories to facilitate more efficient evidence synthesis.

Abstract

Integrated IPD-AD analysis, which combines individual participant data (IPD) with aggregate data (AD), is increasingly recognized as an effective strategy for generating more reliable and generalizable inferences from heterogeneous studies. While most existing work has focused on algorithmic approaches, this paper investigates a complementary yet underexplored question: how different forms of AD influence the efficiency of data integration. Working within a constrained maximum likelihood estimation framework, we compare commonly reported summary statistics and show that subgroup-specific summaries can substantially improve estimation efficiency. In particular, we find that AD derived from outcome-stratified subgroups (e.g., cases and controls) consistently yield greater efficiency gains than those based on covariate-stratified subgroups (e.g., age or exposure categories), especially when the outcome is continuous. Although outcome-stratified summaries are commonly reported for discrete outcomes, they are rarely provided when the outcome is continuous. Our findings therefore support the routine inclusion of outcome-stratified summaries for continuous endpoints in trial reports and public data repositories to facilitate more efficient evidence synthesis. We further extend the constrained maximum likelihood framework to accommodate dataset shift and develop a fast, non-iterative estimation procedure to improve numerical stability and scalability. We illustrate the proposed methodology with two applications: an analysis of income data under covariate shift and an analysis of housing data under prior probability shift.

Figures (4)

• Figure 1: The biases (top panel) and relative efficiencies (bottom panel) of the constrained maximum likelihood estimator for $\beta_{00}$ (left), $\beta_{01}$ (center), and $\beta_{02}$ (right), with various AD: $\widetilde{\pmb{\phi}}^Y$ (solid line with $\circ$), $\widetilde{\pmb{\phi}}^{\mathbf{X}|Y}_{\mathrm{median}}$ (solid line with $\triangle$), $\widetilde{\pmb{\phi}}^{\mathbf{X}|Y}_{\mathrm{quartile}}$ (dashed line with $\triangle$), $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{1}$ (solid line with $+$), $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{2}$ (dashed line with $+$), and $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{3}$ (dotted line with $+$) under AD sample size $N=1000$.
• Figure 2: The relative efficiencies of the constrained maximum likelihood estimator for $\beta_{00}$ (top row), $\beta_{01}$ (center row), and $\beta_{02}$ (bottom row) under IPD sample sizes $n=100$ (left column), $n=200$ (center column), and $n=400$ (right column), with various AD: $\widetilde{\pmb{\phi}}^Y$ (solid line with $\circ$), $\widetilde{\pmb{\phi}}^{\mathbf{X}|Y}_{\mathrm{median}}$ (solid line with $\triangle$), $\widetilde{\pmb{\phi}}^{\mathbf{X}|Y}_{\mathrm{quartile}}$ (dashed line with $\triangle$), $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{1}$ (solid line with $+$), $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{2}$ (dashed line with $+$), and $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{3}$ (dotted line with $+$).
• Figure 3: The biases (top panel) and relative efficiencies (bottom panel) of the constrained maximum likelihood estimator for $\beta_{00}$ (left), $\beta_{01}$ (center), and $\beta_{02}$ (right) under covariate shift, with various AD: $\widetilde{\pmb{\phi}}^Y$ (solid line with $\circ$), $\widetilde{\pmb{\phi}}^{\mathbf{X}|Y}_{\mathrm{median}}$ (solid line with $\triangle$), $\widetilde{\pmb{\phi}}^{\mathbf{X}|Y}_{\mathrm{quartile}}$ (dashed line with $\triangle$), $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{1}$ (solid line with $+$), $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{2}$ (dashed line with $+$), and $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{3}$ (dotted line with $+$) under AD sample size $N=1000$.
• Figure 4: The biases (top panel) and relative efficiencies (bottom panel) of the constrained maximum likelihood estimator for $\beta_{00}$ (left), $\beta_{01}$ (center), and $\beta_{02}$ (right) under prior probability shift, with various AD: $\widetilde{\pmb{\phi}}^Y$ (solid line with $\circ$), $\widetilde{\pmb{\phi}}^{\mathbf{X}|Y}_{\mathrm{median}}$ (solid line with $\triangle$), $\widetilde{\pmb{\phi}}^{\mathbf{X}|Y}_{\mathrm{quartile}}$ (dashed line with $\triangle$), $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{1}$ (solid line with $+$), $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{2}$ (dashed line with $+$), and $\widetilde{\pmb{\phi}}^{Y|\mathbf{X}}_{3}$ (dotted line with $+$) under AD sample size $N=1000$.

Theorems & Definitions (1)

• Theorem 1