标题： 具有形状限制样条的异方差增长曲线建模 显示英文标题

标题： Heteroscedastic Growth Curve Modeling with Shape-Restricted Splines

摘要： 生长曲线分析（GCA）在需要建模增长轨迹的各个领域有着广泛的应用。 误差项通常存在异方差性，这不能通过标准线性固定效应或混合效应模型以足够的灵活性来处理。 已解决的一种情况是误差方差由某些协变量的线性预测器表征的情况。 然而，在GCA中经常遇到的情况是方差是均值的平滑函数，并且具有已知的形状限制。 对标准线性混合效应模型的直接应用会低估固定效应估计量的方差，从而导致对估计增长曲线的不确定性估计不足。 我们建议使用形状受限的样条函数，将响应变量的方差建模为边际或条件均值的（单调递增/递减；凸/凹）函数。 开发了一种简单的迭代重加权拟合算法，该算法利用了现有的线性混合效应模型软件。 对于推断，推荐使用参数化自助法。 我们的模拟研究表明，所提出的方法在适度样本量下给出了令人满意的结果。 通过两个真实世界的应用展示了该方法的实用性。 显示英文摘要

摘要： Growth curve analysis (GCA) has a wide range of applications in various fields where growth trajectories need to be modeled. Heteroscedasticity is often present in the error term, which can not be handled with sufficient flexibility by standard linear fixed or mixed-effects models. One situation that has been addressed is where the error variance is characterized by a linear predictor with certain covariates. A frequently encountered scenario in GCA, however, is one in which the variance is a smooth function of the mean with known shape restrictions. A naive application of standard linear mixed-effects models would underestimate the variance of the fixed effects estimators and, consequently, the uncertainty of the estimated growth curve. We propose to model the variance of the response variable as a shape-restricted (increasing/decreasing; convex/concave) function of the marginal or conditional mean using shape-restricted splines. A simple iteratively reweighted fitting algorithm that takes advantage of existing software for linear mixed-effects models is developed. For inference, a parametric bootstrap procedure is recommended. Our simulation study shows that the proposed method gives satisfactory inference with moderate sample sizes. The utility of the method is demonstrated using two real-world applications.