Calculating Sample Size for Follmann’s Simple Multivariate Test for One-Sided Alternatives

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.

KEYWORDS:

• Follmann’s test
• Multivariate tests with ordered alternatives
• Sample size

Acknowledgments

I would like to express gratitude toward Deidre Kile for her critical review of the article. Also, the anonymous reviewers provided comments that improved the article.