Fang Hong-Bin, Tian Guo-Liang, Xiong Xiaoping, Tan Ming
Division of Biostatistics, University of Maryland Greenebaum Cancer Center, 22 South Greene Street, Baltimore, MD 21201, USA.
Stat Med. 2006 Jun 15;25(11):1948-59. doi: 10.1002/sim.2364.
In clinical studies, multiple endpoints are often measured for each patient longitudinally. The multivariate random-effects or random coefficient model has been a useful method for analysis. However, medical research problems may impose restrictions on the model parameters of interests. For example, in a paediatric brain tumour study on radiation therapy, there is a natural ordering in the white matter relaxation time of brain tissues among different regions surrounding the primary tumour, i.e. the closer a specific region of brain tissues is to the centre of primary tumour, the shorter is the relaxation time. Such parameter constraints should be accounted for in the analysis. This article proposes a class of multivariate random coefficient models with restricted parameters and derives its maximum likelihood estimates (MLE). We propose a modified EM algorithm for the quadratic optimalization with linear inequality constraints necessary in deriving the MLE. The method is applied to analysing the paediatric brain tumour study.
在临床研究中，通常会对每位患者纵向测量多个终点指标。多元随机效应或随机系数模型一直是一种有用的分析方法。然而，医学研究问题可能会对感兴趣的模型参数施加限制。例如，在一项关于小儿脑肿瘤放射治疗的研究中，原发性肿瘤周围不同区域的脑组织白质弛豫时间存在自然顺序，即特定区域的脑组织离原发性肿瘤中心越近，弛豫时间越短。在分析中应考虑这种参数约束。本文提出了一类参数受限的多元随机系数模型，并推导了其最大似然估计（MLE）。我们提出了一种改进的期望最大化（EM）算法，用于在推导MLE时进行具有线性不等式约束的二次优化。该方法应用于小儿脑肿瘤研究的分析。
