Rathouz and Gao [2] and Luo and Tsai [3] proposed valuable extensions to the generalized linear model for modeling a nonlinear monotonic relationship between the mean response and a set of covariates. In their extensions for discrete data the baseline response distribution is unspecified and is estimated from the data. We propose to extend this model for the analysis of longitudinal data by incorporating random effects into the linear predictor, and using maximum likelihood for estimation and inference. Motivated in particular by longitudinal studies of clinical scale outcomes, we developed an estimation procedure for a finite‐support response using a generalized expectation‐maximization algorithm where Gauss‐Hermite quadrature is employed to approximate the integrals in the E step of the algorithm. Upon convergence, the observed information matrix is estimated through second‐order numerical differentiation of the log‐likelihood function. Asymptotic properties of the maximum likelihood estimates follow under the usual regularity conditions. Simulation studies are conducted to assess its finite‐sample properties and compare the proposed model to the generalized linear mixed model. The proposed method is illustrated in an analysis of data from a longitudinal study of Huntington's disease.

中文翻译：

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离散纵向数据的比例似然比混合模型

Rathouz和Gao [2]以及Luo和Tsai [3]提出了对广义线性模型的有价值的扩展，用于对平均响应和一组协变量之间的非线性单调关系进行建模。在离散数据的扩展中，基线响应分布是不确定的，是根据数据估算的。我们建议通过将随机效应合并到线性预测变量中，并使用最大似然进行估计和推断，来扩展此模型以用于纵向数据分析。特别是由于对临床规模结果的纵向研究的推动，我们使用广义期望最大化算法开发了有限支持响应的估计程序，该算法采用Gauss-Hermite正交来近似算法E步中的积分。收敛后 通过对数似然函数的二阶数值微分估计观察到的信息矩阵。在通常的规律性条件下，最大似然估计的渐近性质遵循。进行仿真研究以评估其有限样本属性，并将拟议的模型与广义线性混合模型进行比较。在对亨廷顿氏病进行纵向研究的数据分析中说明了所提出的方法。