Associated longitudinal response variables are faced with variations caused by repeated measurements over time along with the association between the responses. To model a longitudinal ordinal outcome using generalized linear mixed models, integrating over a normally distributed random intercept in the proportional odds ordinal logistic regression does not yield a closed form. In this paper, we combined a longitudinal count and an ordinal response variable with Bridge distribution for the random intercept in the ordinal logistic regression submodel. We compared the results to that of a normal distribution. The two associated response variables are combined using correlated random intercepts. The random intercept in the count outcome submodel follows a normal distribution. The random intercept in the ordinal outcome submodel follows Bridge distribution. The estimations were carried out using a likelihood-based approach in direct and conditional joint modelling approaches. To illustrate the performance of the model, a simulation study was conducted. Based on the simulation results, assuming a Bridge distribution for the random intercept of ordinal logistic regression results in accurate estimation even if the random intercept is normally distributed. Moreover, considering the association between longitudinal count and ordinal responses resulted in estimation with lower standard error in comparison to univariate analysis. In addition to the same interpretation for the parameter in marginal and conditional estimates thanks to the assumption of a Bridge distribution for the random intercept of ordinal logistic regression, more efficient estimates were found compared to that of normal distribution.

中文翻译：

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序数和过度分散计数结果的纵向联合建模：序数随机截距的桥梁分布

相关的纵向响应变量面临着随着时间的推移重复测量以及响应之间的关联而引起的变化。为了使用广义线性混合模型对纵向序数结果进行建模，在比例优势序数逻辑回归中对正态分布随机截距进行积分不会产生封闭形式。在本文中，我们将纵向计数和有序响应变量与桥分布相结合，用于有序逻辑回归子模型中的随机截距。我们将结果与正态分布的结果进行了比较。使用相关随机截距组合两个关联的响应变量。计数结果子模型中的随机截距遵循正态分布。序数结果子模型中的随机截距遵循 Bridge 分布。在直接和条件联合建模方法中使用基于似然的方法进行估计。为了说明模型的性能，进行了仿真研究。根据仿真结果，假设序数逻辑回归的随机截距为桥分布，即使随机截距是正态分布的，也能得到准确的估计。此外，考虑到纵向计数和有序响应之间的关联，与单变量分析相比，估计的标准误差较低。