ABSTRACT

Follmann developed a multivariate test, when $\mathit{X} \sim \text{MVN}(\mathit{\mu },\mathbf{\Sigma })$, to test H₀ versus $H_1 - H_0$ where $H_0:$ $\mathit{\mu }=0$ and $H_1:\mathit{\mu }\ge 0$. Follmann provided strict lower bounds on the power function when an orthogonal mapping requirement was satisfied, the use of which requires knowledge about the unknown population covariance matrix. In this article, we show that the orthogonal mapping requirement for his theorem is equivalent to and can be replaced with $1^{\prime }\mathit{\mu }\ge 0$, which does not require knowledge about the population covariance matrix. Using the lower bound on power, we are able to develop conservative sample sizes for this test. The conservative sample sizes are upper bounds on the actual sample size needed to achieve at least the desired power. Results from a simulation study are provided illustrating that the sample sizes are indeed upper bounds. Also, a simple R program to calculate sample size is provided.

摘要

Follmann 开发了一个多变量测试，当$\mathit{X} ～ \text{MVN}(\mathit{\mu },\mathbf{\Sigma })$, 测试H ₀与$H_1 - H_0$在哪里$H_0：$ $\mathit{\mu }=0$和$H_1：\mathit{\mu }\ge 0$. 当满足正交映射要求时，Follmann 为幂函数提供了严格的下界，其使用需要了解未知总体协方差矩阵。在本文中，我们证明了他的定理的正交映射要求等价于并且可以替换为$1^{\text{'}}\mathit{\mu }\ge 0$，这不需要关于总体协方差矩阵的知识。使用功率的下限，我们能够为此测试开发保守的样本量。保守样本量是至少达到所需功效所需的实际样本量的上限。提供了模拟研究的结果，说明样本量确实是上限。此外，还提供了一个简单的 R 程序来计算样本量。