Department of Statistics, University of Haifa, Mount Carmel, Haifa 31905, Israel.
Stat Med. 2012 Jul 20;31(16):1761-73. doi: 10.1002/sim.5335.
Multivariate outcomes are often measured longitudinally. For example, in hearing loss studies, hearing thresholds for each subject are measured repeatedly over time at several frequencies. Thus, each patient is associated with a multivariate longitudinal outcome. The multivariate mixed-effects model is 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, it is known that hearing thresholds, at every frequency, increase with age. Moreover, this age-related threshold elevation is monotone in frequency, that is, the higher the frequency, the higher, on average, is the rate of threshold elevation. This means that there is a natural ordering among the different frequencies in the rate of hearing loss. In practice, this amounts to imposing a set of constraints on the different frequencies' regression coefficients modeling the mean effect of time and age at entry to the study on hearing thresholds. The aforementioned constraints should be accounted for in the analysis. The result is a multivariate longitudinal model with restricted parameters. We propose estimation and testing procedures for such models. We show that ignoring the constraints may lead to misleading inferences regarding the direction and the magnitude of various effects. Moreover, simulations show that incorporating the constraints substantially improves the mean squared error of the estimates and the power of the tests. We used this methodology to analyze a real hearing loss study.
多变量结果通常是纵向测量的。例如，在听力损失研究中，每个受试者的听力阈值在多个频率上随时间反复测量。因此，每个患者都与一个多变量纵向结果相关联。多变量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，已知每个频率的听力阈值随年龄增长而增加。此外，这种与年龄相关的阈值升高在频率上是单调的，也就是说，频率越高，平均而言，阈值升高的速度就越高。这意味着在听力损失的速度方面，不同频率之间存在自然的顺序。在实践中，这相当于对不同频率的回归系数施加一组约束，这些回归系数用于建模进入研究时时间和年龄对听力阈值的平均影响。应在分析中考虑到上述约束。结果是一个具有约束参数的多变量纵向模型。我们提出了用于此类模型的估计和检验程序。我们表明，忽略约束可能会导致关于各种效应的方向和幅度的误导性推断。此外，模拟表明，纳入约束可以大大提高估计的均方误差和检验的功效。我们使用这种方法分析了一项真实的听力损失研究。
