Linear Mixed Effects (LME) models have been widely applied in clustered data analysis in many areas including marketing research, clinical trials, and biomedical studies. Inference can be conducted using maximum likelihood approach if assuming Normal distributions on the random effects. However, in many applications of economy, business and medicine, it is often essential to impose constraints on the regression parameters after taking their real-world interpretations into account. Therefore, in this paper we extend the classical (unconstrained) LME models to allow for sign constraints on its overall coefficients. We propose to assume a symmetric doubly truncated Normal (SDTN) distribution on the random effects instead of the unconstrained Normal distribution which is often found in classical literature. With the aforementioned change, difficulty has dramatically increased as the exact distribution of the dependent variable becomes analytically intractable. We then develop likelihood-based approaches to estimate the unknown model parameters utilizing the approximation of its exact distribution. Simulation studies have shown that the proposed constrained model not only improves real-world interpretations of results, but also achieves satisfactory performance on model fits as compared to the existing model.

翻译：在许多领域,包括营销研究、临床试验和生物医学研究,在集群数据分析中广泛采用线性混合效应模型(LME)在许多领域广泛应用。如果假设正常分布随机效应,可以使用最大可能性方法进行推论。然而,在许多经济、商业和医学应用中,往往有必要考虑到对真实世界的解释,对回归参数施加限制。因此,在本文件中,我们扩展传统(不受限制的)LME模型,以便对其总系数进行标志性限制。我们提议假设对随机效应进行对称性双轨正常(SDTN)分布,而不是古典文献中经常发现的不受限制的正常分布。随着上述变化,随着依赖变量的确切分布变得难以分析,难度急剧增大。我们随后制定基于可能性的方法,利用精确分布的近似值估计未知模型参数。模拟研究表明,拟议的受限模型不仅改善了对结果的真实世界解释,而且与现有模型相比,还取得了令人满意的效果。