Our motivation here is to calculate the power of 3 statistical tests used when there are genetic traits that operate under a pleiotropic mode of inheritance and when qualitative phenotypes are defined by use of thresholds for the multiple quantitative phenotypes. Specifically, we formulate a multivariate function that provides the probability that an individual has a vector of specific quantitative trait values conditional on having a risk locus genotype, and we apply thresholds to define qualitative phenotypes (affected, unaffected) and compute penetrances and conditional genotype frequencies based on the multivariate function. We extend the analytic power and minimum-sample-size-necessary (MSSN) formulas for 2 categorical data-based tests (genotype, linear trend test [LTT]) of genetic association to the pleiotropic model. We further compare the MSSN of the genotype test and the LTT with that of a multivariate ANOVA (Pillai). We approximate the MSSN for statistics by linear models using a factorial design and ANOVA. With ANOVA decomposition, we determine which factors most significantly change the power/MSSN for all statistics. Finally, we determine which test statistics have the smallest MSSN. In this work, MSSN calculations are for 2 traits (bivariate distributions) only (for illustrative purposes). We note that the calculations may be extended to address any number of traits. Our key findings are that the genotype test usually has lower MSSN requirements than the LTT. More inclusive thresholds (top/bottom 25% vs. top/bottom 10%) have higher sample size requirements. The Pillai test has a much larger MSSN than both the genotype test and the LTT, as a result of sample selection. With these formulas, researchers can specify how many subjects they must collect to localize genes for pleiotropic phenotypes.

中文翻译：

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多亲遗传模式下遗传协会研究的功效和样本量计算的解析解决方案。

我们的动机是要计算当遗传特征在多效遗传模式下运行并且通过使用多个定量表型的阈值定义定性表型时，使用三种统计检验的功效。具体来说，我们制定了一个多元函数，该函数提供了一个个体具有特定定量性状值的向量（取决于具有风险位点基因型）的可能性，并且我们应用阈值来定义定性表型（受影响，不受影响），并计算外显率和条件基因型频率基于多元函数。我们将遗传关联的2种基于分类数据的检验（基因型，线性趋势检验[LTT]）的分析能力和所需的最小样本大小（MSSN）公式扩展到了多效模型。我们进一步比较了基因型测试的MSSN和LTT与多元ANOVA（Pillai）的MSSN。我们使用因子设计和ANOVA通过线性模型对MSSN进行统计估算。通过ANOVA分解，我们可以确定哪些因素最明显地改变了所有统计数据的功效/ MSSN。最后，我们确定哪些测试统计信息具有最小的MSSN。在这项工作中，MSSN计算仅针对2个特征（双变量分布）（出于说明目的）。我们注意到，该计算可以扩展为处理任何数量的特征。我们的主要发现是，基因型测试通常比LTT具有更低的MSSN要求。更具包容性的阈值（最高/最低25％与最高/最低10％）对样本量的要求更高。与基因型测试和LTT测试相比，Pillai测试具有更大的MSSN，作为样本选择的结果。利用这些公式，研究人员可以指定必须收集多少个对象才能定位多效性表型的基因。