# ANOVA (Analysis of variance)

**ANOVA (Analysis of variance) is part of:**

### Principles of the Analysis of Variance

Analysis of variance (ANOVA) is a tool used to partition the observed variance in a particular variable into components attributable to different sources of variation.

Analysis of variance (ANOVA) uses the same conceptual framework as linear regression. The main difference comes from the nature of the explanatory variables: instead of quantitative, here they are qualitative. In ANOVA, explanatory variables are often called factors.

### ANOVA model

If p is the number of factors, the ANOVA model is written as follows:

**y _{i} = β_{0} + ∑_{j=1...q} β_{k(i,j),j} + ε_{i}**

where y_{i} is the value observed for the dependent variable for observation i, k_{(i,j)} is the index of the category of factor j for observation i and ε_{i} is the error of the model.

The hypotheses used in ANOVA are identical to those used in linear regression: the errors ε_{i} follow the same normal distribution N(0,s) and are independent. It is recommended to check retrospectively that the underlying hypotheses have been correctly verified. The normality of the residues can be checked by analyzing certain charts or by using a normality test. The independence of the residues can be checked by analyzing certain charts or by using the Durbin Watson test.

### Options for ANOVA in XLSTAT

XLSTAT enables you to perform one and multiple way ANOVA (MANOVA). Interactions up to order 4 can be included in the model as well as nested and random effects.

XLSTAT can handle both balanced and unbalanced ANOVA.

### Results for the analysis of variance in XLSTAT

The results given are a residuals analysis, parameters of the models, the model equation, the standardized coefficients, Type I SS, Type III SS, and predictions are displayed.

In addition several multiple comparison methods can optionally be performed: Bonferroni's and Dunn-Sidak corrected t test, Tukey's HSD test, Fisher's LSD test, Duncan's test, Newman-Keuls' (SNK) method and the REGWQ method. Also the Dunnett's test is available to allow users to perform multiple comparisons with control (MCC) and Multiple comparison with the best (MCB).

### Charts for the analysis of variance in XLSTAT

**Regression chart:**The chart shows the observed values, the regression line and both types of confidence interval around the predictions.**Standardized residuals as a function of the explanatory variable:**This chart shows the standardized residuals as a function of the explanatory variable. In principle, the residuals should be distributed randomly around the X-axis. If there is a trend or a shape, this shows a problem with the model.**The evolution of the standardized residuals as a function of the dependent variable.****The distance between the predictions and the observations:**For an ideal model, the points would all be on the bisector.**The standardized residuals on a bar chart:**The last chart quickly shows if an abnormal number of values are outside the interval]-2, 2[ given that the latter, assuming that the sample is normally distributed, should contain about 95% of the data.

## Tutorials

- Running a one-way ANOVA followed by multiple comparisons tests with XLSTAT
- Running a two-way unbalanced ANOVA with interactions
- Running contrast analysis following a one-way ANOVA
- Running pairwise multiple comparisons after a multi-factor ANOVA