Hybrid type 2 studies are gaining popularity for their ability to assess both implementation and health outcomes as co-primary endpoints. Often conducted as cluster-randomized trials (CRTs), five design methods can validly power these studies: p-value adjustment methods, combined outcomes approach, single weighted 1-DF test, disjunctive 2-DF test, and conjunctive test. We compared all of the methods theoretically and numerically. Theoretical comparisons of the power equations allowed us to identify if any method globally had more or less power than other methods. It was shown that the p-value adjustment methods are always less powerful than the combined outcomes approach and the single 1-DF test. We also identified the conditions under which the disjunctive 2-DF test is less powerful than the single 1-DF test. Because our theoretical comparison showed that some methods could be more powerful than others under certain conditions, and less powerful under others, we conducted a numerical study to understand these differences. The crt2power R package was created to calculate the power or sample size for CRTs with two continuous co-primary endpoints. Using this package, we conducted a numerical evaluation across 30,000 input scenarios to compare statistical power. Specific patterns were identified where a certain method consistently achieved the highest power. When the treatment effects are unequal, the disjunctive 2-DF test tends to have higher power. When the treatment effect sizes are the same, the single 1-DF test tends to have higher power. Together, these comparisons provide clearer insights to guide method selection for powering hybrid type 2 studies.

翻译：混合型2类研究因其能够同时评估实施效果和健康结局作为共同主要终点而日益受到青睐。这类研究通常以整群随机试验（CRT）形式开展，现有五种设计方法可有效为此类研究提供统计功效：p值调整法、联合结局法、单自由度加权检验、析取双自由度检验以及合取检验。我们从理论和数值模拟两方面对所有方法进行了比较。通过对功效方程的理论比较，我们得以判断是否存在某种方法在全局上始终优于或劣于其他方法。结果表明，p值调整法的功效始终低于联合结局法和单自由度检验。我们还明确了析取双自由度检验在何种条件下功效会低于单自由度检验。由于理论比较显示某些方法在特定条件下可能更优，而在其他条件下可能更劣，我们通过数值研究来深入理解这些差异。为此开发了crt2power R软件包，用于计算具有两个连续共同主要终点的CRT的功效或样本量。利用该软件包，我们在30,000个输入场景下进行了数值评估以比较统计功效。研究发现特定方法在某些场景下始终能获得最高功效：当处理效应不等时，析取双自由度检验往往具有更高功效；当处理效应相同时，单自由度检验通常更具优势。这些比较结果为混合型2类研究的功效计算方法选择提供了更清晰的指导依据。