## Anova

Analysis of variance (ANOVA) is a collection of models and procedures in which the observed variance of a particular variable is partitioned into components attributable to different sources. In its simplest form ANOVA provides a test of whether or not the means of several groups are all equal (so that the null hypothesis isfor alland the null hypothesis isfor some)and therefore generalizes the t-test to more than two samples. ANOVA is helpful because it possesses an advantage over a two-sample t-test of being faster when comparing many samples, and reducing the probability of committing a type I error when performing multiple tests.

The samples are independent.

The distributions of the residuals are normal.

The variance of each population from which the samples are taken are the same.

ANOVA involves partitioning of the total sum of squares SST into components (treatment sum of square, SSTr and error sum of squares SSE) related to the effects used in the model. For example, we show the model for a simplified ANOVA with one type of treatment at different levels

SST=SSTr +SSE with

withwhere I is the number of samples and J is the number in each sample.

Then withwith the distribution of the test statistic we can find if there is a difference in the sample means.

Example. Perform an ANOVA test on the three treatments below.

Treatment 1 | 0.56 | 1.12 | 0.9 | 1.07 | 0.94 |

Treatment 2 | 0.72 | 0.69 | 0.87 | 0.78 | 0.91 |

Treatment 3 | 0.62 | 1.08 | 1.07 | 0.99 | 0.93 |

Comparing withcauses us not to reject the null hypothesis.