The popularity of the classical general linear model (CGLM) is attributable mostly to its ease of fitting and validating; however, the CGLM is inappropriate for correlated observations. In this paper we explore linear models for correlated observations with an exchangeable structure (Arnold in J Am Stat Assoc 74:194–199, 1979). For the case of \(N>1\) repeated measures observations having site-dependent or site-independent covariates, the maximum likelihood estimates (MLEs) of the model’s parameters are derived, likelihood ratio tests are obtained for relevant model building hypotheses, and some Monte Carlo simulation studies are performed to illuminate important aspects of the models and their tests of hypotheses. For the case of site-independent covariates, closed-form solutions exist for the MLEs and exact tests can be constructed for the model building hypotheses. Simulations revealed that these exact tests remain robust in the presence of moderate skewness or outliers. However, these fortuitous closed-form occurrences vanish for the case of site-dependent covariates. In order to ameliorate this deficiency, some Monte Carlo simulations are performed to estimate the bias of these MLEs, the probability of a multimodal likelihood, and the suitability of the limiting chi-squared approximation to the model building hypotheses. These simulations reveal that the estimated biases of the slope parameters are negligible for sample size combinations (n, N) as small as (2, 6). Likewise, this sample size combination resulted in only an approximate 1% estimated probability of a multimodal likelihood, which drastically decreased with the increase of either n or N. Moreover, the limiting \(\chi ^2\) distributional assumption appears to hold reasonably well for a sample size of \(N=100\), regardless of the value of n. Finally, we provide examples of fitting our model and conducting tests of hypotheses using two medical datasets.

经典通用线性模型（CGLM）的普及主要归因于其易于拟合和验证。然而，CGLM不适用于相关观测。在本文中，我们探索了具有可交换结构的相关观测值的线性模型（Arnold in J Am Stat Assoc 74：194-199，1979）。对于\（N> 1 \）重复测量观测值具有与地点相关或与地点无关的协变量，得出模型参数的最大似然估计（MLE），获得相关模型建立假设的似然比检验，并进行了一些Monte Carlo模拟研究以阐明重要方面模型及其假设检验。对于与地点无关的协变量，存在MLE的闭式解，并且可以为模型构建假设构建精确检验。仿真显示，在存在中等偏斜或异常值时，这些精确的测试仍然很可靠。但是，这些偶然的封闭形式出现在依赖于站点的协变量的情况下消失了。为了缓解这种不足，我们进行了一些蒙特卡洛模拟，以估算这些MLE的偏差，多模似然的概率，以及极限卡方近似对模型构建假设的适用性。这些模拟表明，对于样本大小组合，斜率参数的估计偏差可以忽略不计（n，  N）小至（2，6）。同样，此样本大小组合仅导致多模似然估计概率约为1％，随着n或N的增加而急剧下降。此外，对于\（N = 100 \）的样本大小，无论n的值如何，限制\（\ chi ^ 2 \）分布假设似乎都可以很好地保持。最后，我们提供了使用两个医学数据集拟合模型并进行假设检验的示例。