Department of Statistics, University of Haifa, Mount Carmel, Haifa, Israel.
Biostatistics. 2011 Apr;12(2):327-40. doi: 10.1093/biostatistics/kxq051. Epub 2010 Aug 18.
In medical studies, endpoints are often measured for each patient longitudinally. The mixed-effects model has been a useful tool for the analysis of such data. There are situations in which the parameters of the model are subject to some restrictions or constraints. For example, in hearing loss studies, we expect hearing to deteriorate with time. This means that hearing thresholds which reflect hearing acuity will, on average, increase over time. Therefore, the regression coefficients associated with the mean effect of time on hearing ability will be constrained. Such constraints should be accounted for in the analysis. We propose maximum likelihood estimation procedures, based on the expectation-conditional maximization either algorithm, to estimate the parameters of the model while accounting for the constraints on them. The proposed methods improve, in terms of mean square error, on the unconstrained estimators. In some settings, the improvement may be substantial. Hypotheses testing procedures that incorporate the constraints are developed. Specifically, likelihood ratio, Wald, and score tests are proposed and investigated. Their empirical significance levels and power are studied using simulations. It is shown that incorporating the constraints improves the mean squared error of the estimates and the power of the tests. These improvements may be substantial. The methodology is used to analyze a hearing loss study.
在医学研究中，通常会对每个患者进行纵向测量终点。混合效应模型是分析此类数据的有用工具。在某些情况下，模型的参数受到某些限制或约束。例如，在听力损失研究中，我们预计听力会随时间恶化。这意味着反映听力敏锐度的听力阈值平均而言会随时间增加。因此，与时间对听力能力的平均效应相关的回归系数将受到限制。在分析中应考虑到这些约束。我们提出了基于期望条件最大化算法的最大似然估计程序，以在考虑到对参数的约束的情况下估计模型的参数。所提出的方法在均方误差方面优于无约束估计量。在某些情况下，改进可能很大。提出并研究了包含约束的假设检验程序。具体来说，提出了似然比、 Wald 和得分检验，并对其进行了研究。使用模拟研究了它们的经验显著性水平和功效。结果表明，包含约束可以提高估计量的均方误差和检验的功效。这些改进可能很大。该方法学用于分析听力损失研究。
