Generalized partially linear single-index models (GPLSIMs) provide a flexible and interpretable semiparametric framework for longitudinal outcomes by combining a low-dimensional parametric component with a nonparametric index component. For repeated measurements, valid inference is challenging because within-subject correlation induces nuisance parameters and variance estimation can be unstable in semiparametric settings. We propose a profile estimating-equation approach based on spline approximation of the unknown link function and construct a subject-level block empirical likelihood (BEL) for joint inference on the parametric coefficients and the single-index direction. The resulting BEL ratio statistic enjoys a Wilks-type chi-square limit, yielding likelihood-free confidence regions without explicit sandwich variance estimation. We also discuss practical implementation, including constrained optimization for the index parameter, working-correlation choices, and bootstrap-based confidence bands for the nonparametric component. Simulation studies and an application to the epilepsy longitudinal study illustrate the finite-sample performance.

翻译：广义部分线性单指标模型（GPLSIMs）通过结合低维参数分量与非参数指标分量，为纵向结局提供了灵活且可解释的半参数框架。对于重复测量数据，由于受试者内部相关性会引入冗余参数，且半参数设定下的方差估计可能不稳定，因此有效的统计推断具有挑战性。我们提出了一种基于未知连接函数样条近似的轮廓估计方程方法，并构建了受试者层面的块经验似然（BEL），用于对参数系数和单指标方向进行联合推断。所得的BEL比统计量具有Wilks型卡方极限分布，可在无需显式三明治方差估计的情况下获得无似然置信域。我们还讨论了实际实施细节，包括指标参数的约束优化、工作相关性选择，以及非参数分量的基于自助法的置信带。模拟研究和癫痫纵向数据应用展示了该方法的有限样本性能。