Subgroup Identification with Latent Factor Structure
Subgroup analysis has garnered increasing attention for its ability to identify meaningful subgroups within heterogeneous populations, thereby enhancing predictive power. However, in many fields such as social science and biology, covariates are often highly correlated due to common factors. This correlation poses significant challenges for subgroup identification, an issue that is often overlooked in existing literature. In this paper, we aim to address this gap in the ``diverging dimension" regime by proposing a center-augmented subgroup identification method within the Factor Augmented (sparse) Linear Model framework. This method bridges dimension reduction and sparse regression. Our proposed approach is adaptable to the high cross-sectional dependence among covariates and offers computational advantages with a complexity of $O(nK)$, compared to the $O(n^2)$ complexity of the conventional pairwise fusion penalty method in the literature, where $n$ is the sample size and $K$ is the number of subgroups. We also investigate the asymptotic properties of the oracle estimators under conditions on the minimal distance between group centroids. To implement the proposed approach, we introduce a Difference of Convex functions-based Alternating Direction Method of Multipliers (DC-ADMM) algorithm and demonstrate its convergence to a local minimizer in a finite number of steps. We illustrate the superiority of the proposed method through extensive numerical experiments and a real macroeconomic data example. An \texttt{R} package, \texttt{SILFS}, implementing the method is also available on CRAN.