7 - Introduction to Multivariate Methods

Summary

Many scientific questions lead to hypotheses about random vectors. For instance, the question of whether global warming has occurred over a geographic region is a question about whether temperature has changed at each spatial location within the region. One approach to addressing such a question is to apply a univariate test to each location separately and then use the results collectively to make a decision. This approach is called multiple testing or multiple comparisons and is common in genomics for analyzing gene expressions. The disadvantage of this approach is that it does not fully account for correlation between variables. Multivariate techniques provide a framework for hypothesis testing that takes into account correlations between variables. Although multivariate tests are more comprehensive, they require estimating more parameters and therefore have low power when the number of variables is large. Multivariate statistical analysis draws heavily on linear algebra and includes a generalization of the normal distribution, called the multivariate normal distribution, whose population parameters are the mean vector and the covariance matrix.