在 170 名医生中，患者被随机分配接受 e-assist，这是一个旨在加强结直肠癌筛查 (CRCS) 或控制的在线计划。部分依从性：78.34 % $$ 78.34\% $$ 的实验患者访问了电子协助，而没有向对照组提供访问权限。令人感兴趣的是分配治疗的平均因果效应和遵守者的平均因果效应以及这些因果效应在不同医生之间的差异。每个医生都会生成筛选实验依从者（访问电子协助的实验患者）、对照依从者（如果被分配到电子协助就会访问电子协助的对照者）和从不接受者（本来会避免电子协助的患者）的概率。 -无论如何都要协助）。共同估计医生特定的概率提出了新的挑战。我们通过最大似然来应对这些挑战，在给定随机效应和随机效应分布的情况下，将“完整数据似然”独特地纳入筛选的条件分布和部分观察到的依从性中。我们使用自适应高斯-埃尔米特求积法来边缘化这种可能性。该方法是双重迭代的，因为条件分布无法进行分析评估。由于每位医生的小样本量限制了多个随机效应的可估计性，因此我们使用具有因子分析结构的共享随机效应模型来降低其维度。我们评估估计者并推荐样本量，以通过模拟产生相当准确和精确的估计，并分析来自 CRCS 干预试验的数据。© 2024 作者。约翰·威利出版的《医学统计》

Within each of 170 physicians, patients were randomized to access e-assist, an online program that aimed to increase colorectal cancer screening (CRCS), or control. Compliance was partial: 78.34 % $$ 78.34\% $$ of the experimental patients accessed e-assist while no controls were provided the access. Of interest are the average causal effect of assignment to treatment and the complier average causal effect as well as the variation of these causal effects across physicians. Each physician generates probabilities of screening for experimental compliers (experimental patients who accessed e-assist), control compliers (controls who would have accessed e-assist had they been assigned to e-assist), and never takers (patients who would have avoided e-assist no matter what). Estimating physician-specific probabilities jointly over physicians poses novel challenges. We address these challenges by maximum likelihood, factoring a "complete-data likelihood" uniquely into the conditional distribution of screening and partially observed compliance given random effects and the distribution of random effects. We marginalize this likelihood using adaptive Gauss-Hermite quadrature. The approach is doubly iterative in that the conditional distribution defies analytic evaluation. Because the small sample size per physician constrains estimability of multiple random effects, we reduce their dimensionality using a shared random effects model having a factor analytic structure. We assess estimators and recommend sample sizes to produce reasonably accurate and precise estimates by simulation, and analyze data from a trial of a CRCS intervention.© 2024 The Author(s). Statistics in Medicine published by John Wiley & Sons Ltd.