17 - Analysis of Variance and Predictability

Summary

The correlation test is a standard procedure for deciding if two variables are linearly related. This chapter discusses a test for independence that avoids the linearity assumption. The basic idea is the following. If two variables are dependent, then changing the value of one them, say c, changes the distribution of the other. Therefore, if samples are collected for fixed value of c, and additional samples are collected for a different value of c, and so on for different values of c, then a dependence implies that the distributions for different c’s should differ. It follows that deciding that some aspect of the distributions depend on c is equivalent to deciding that the variables are dependent. A special case of this approach is the t-test, which tests if two populations have identical means. Generalizing this test to more than two populations leads to Analysis of Variance (ANOVA), which is the topic of this chapter. ANOVA is a method for testing if two or more populations have the same means. In weather and climate studies, ANOVA is used most often to quantify the predictability of an ensemble forecast, hence this framing is discussed extensively in this chapter.