10 - Single-factor analysis of variance

Summary

Introduction

So far, this book has only covered tests for one and two samples. Often, however, you are likely to have univariate data from three or more samples, from different localities (or experimental groups), and wish to test the hypothesis that “The means of the populations from which these samples have come from are not significantly different to each other,” or “μ₁ = μ₂ = μ₃ = μ₄ = μ₄ etc…”

For example, you might have data for the percentage of tourmaline in granitic rocks from five different outcrops, and wish to test the hypothesis that these have come from populations with the same mean percentage of tourmaline, or perhaps even the same pluton.

Here you could test this hypothesis by doing a lot of two-sample t tests that compare all of the possible pairs of means (e.g. mean 1 compared to mean 2, mean 1 compared to mean 3, mean 2 compared to mean 3 etc.). The problem with this approach is that every time you do a two-sample test and the null hypothesis applies you run a 5% risk of a Type 1 error. So as you do more and more tests on the same set of data, the risk of a Type 1 error rises rapidly.

Put simply, if you do two or more two-sample tests on the same data set it is like having more than one ticket in a lottery where the chances of winning are 5% – the more tickets you have, the more likely you are to win.