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，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，并构建了面向参数系数与单指标方向的联合推断的受试者级块经验似然（BEL）。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方案，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。