Motivated by the analysis of data from the UK Millennium Cohort Study on emotional and behavioural disorders, we develop an M‐quantile regression model for multivariate longitudinal responses. M‐quantile regression is an appealing alternative to standard regression models; it combines features of quantile and expectile regression and it may produce a detailed picture of the conditional response variable distribution, while ensuring robustness to outlying data. As we deal with multivariate data, we need to specify what it is meant by M‐quantile in this context, and how the structure of dependence between univariate profiles may be accounted for. Here, we consider univariate (conditional) M‐quantile regression models with outcome‐specific random effects for each outcome. Dependence between outcomes is introduced by assuming that the random effects in the univariate models are dependent. The multivariate distribution of the random effects is left unspecified and estimated from the observed data. Adopting this approach, we are able to model dependence both within and between outcomes. We further discuss a suitable model parameterisation to account for potential endogeneity of the observed covariates. An extended EM algorithm is defined to derive estimates under a maximum likelihood approach.

中文翻译：

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多元纵向数据的M分位数回归及其在Millennium Cohort Study中的应用

通过对来自英国千禧一代队列研究的情绪和行为障碍的数据进行分析，我们开发了用于多变量纵向反应的M分位数回归模型。M分位数回归是标准回归模型的吸引人的替代方法；它结合了分位数和预期回归的特征，并且可以生成条件响应变量分布的详细图片，同时确保对外围数据的鲁棒性。在处理多元数据时，我们需要指定在这种情况下M位数的含义，以及如何解释单变量配置文件之间的依存关系。在这里，我们考虑对每个结果具有特定于结果的随机效应的单变量（条件）M分位数回归模型。通过假设单变量模型中的随机效应是相关的来引入结果之间的相关性。随机效应的多元分布不明确，可以从观察到的数据中估计出来。采用这种方法，我们能够对结果内部和结果之间的依赖性进行建模。我们进一步讨论了合适的模型参数化，以解决观察到的协变量的潜在内生性。定义了扩展的EM算法，以根据最大似然法得出估计。我们进一步讨论了合适的模型参数化，以解决观察到的协变量的潜在内生性。定义了扩展的EM算法，以根据最大似然法得出估计。我们进一步讨论了合适的模型参数化，以解决观察到的协变量的潜在内生性。定义了扩展的EM算法，以根据最大似然法得出估计。