评估牙周病(PD)状态和进展的研究/试验通常集中在量化聚集(受试者内的牙齿)双变量终点之间的关系。如探孔深度(PPD),和临床依恋水平(CAL)与协变量。尽管可以在线性混合模型(LMM)框架下对随机项(随机效应和误差)调用多元正态的假设,违反这些假设可能会导致不精确的推断。此外,响应-协变量关系可能不是线性的,如在LMM拟合下假设的那样,其中获得的回归估计并没有提供PD风险的总体摘要,从协变量获得。受PD对讲古拉的非裔美国人2型糖尿病患者的研究的启发,我们将非对称聚类双变量(PPD和CAL)响应转换为非线性混合模型框架,其中两个随机项都遵循多元非对称拉普拉斯分布(ALD)。为了提供一个单一的风险摘要,通过单指数模型对关系中可能的非线性进行建模,由索引函数的多项式样条逼近提供动力,和ALD的正常混合物表达式。要进行最大似然推理设置,我们设计了一个优雅的EM型算法。此外,在一些温和条件下建立了大样本的理论性质。使用在各种情况下生成的合成数据进行模拟研究,以研究我们的估计量的有限样本属性,并证明了我们提出的模型和估计算法可以有效地处理非对称,重尾数据,与异常值。最后,我们通过应用于激励PD研究来说明我们提出的方法。
Studies/trials assessing status and progression of periodontal disease (PD) usually focus on quantifying the relationship between the clustered (tooth within subjects) bivariate endpoints, such as probed pocket depth (PPD), and clinical attachment level (CAL) with the covariates. Although assumptions of multivariate normality can be invoked for the random terms (random effects and errors) under a linear mixed model (LMM) framework, violations of those assumptions may lead to imprecise inference. Furthermore, the response-covariate relationship may not be linear, as assumed under a LMM fit, and the regression estimates obtained therein do not provide an overall summary of the risk of PD, as obtained from the covariates. Motivated by a PD study on Gullah-speaking African-American Type-2 diabetics, we cast the asymmetric clustered bivariate (PPD and CAL) responses into a non-linear mixed model framework, where both random terms follow the multivariate asymmetric Laplace distribution (ALD). In order to provide a one-number risk summary, the possible non-linearity in the relationship is modeled via a single-index model, powered by polynomial spline approximations for index functions, and the normal mixture expression for ALD. To proceed with a maximum-likelihood inferential setup, we devise an elegant EM-type algorithm. Moreover, the large sample theoretical properties are established under some mild conditions. Simulation studies using synthetic data generated under a variety of scenarios were used to study the finite-sample properties of our estimators, and demonstrate that our proposed model and estimation algorithm can efficiently handle asymmetric, heavy-tailed data, with outliers. Finally, we illustrate our proposed methodology via application to the motivating PD study.