# Analysis of Variance Package

This package provides functions for performing a univariate Analysis of Variance (ANOVA) to examine the differences between groups of means. The function ANOVA can handle models with any number of fixed factors in a crossed design. It can handle both balanced and unbalanced data with or without missing elements. All results are given as type I sums of squares. ANOVA also provides a number of posthoc tests for comparisons.

 ANOVA[data] perform a one‐way analysis of variance ANOVA[data,model,vars] perform a general ANOVA specified by the given model and vars

The ANOVA function.

The data must be of the form {{α1,β1,,y1},{α2,β2,,y2},} where αi, βi, and so on are the values of the categorical variables vars associated with the ith response, yi.

The vars argument is a list of symbols representing the categorical variables in the model.

The model argument is a list of main effects and interactions that together specify the model. The interaction terms are given as the product of variables. For example, the full factorial model for a threeway analysis of variance can be written as {α,β,γ,α β,α γ,β γ,α β γ}, where α, β, γ are the main effects, α β, α γ, β γ are the twoway interactions, and α β γ is the threeway interaction. Models can also be written using All to represent all main effects and interactions between the specified categorical variables. The full factorial model for a threeway analysis of variance can therefore also be written as {α,β,γ,All}.

This defines data of one categorical variable.
This performs a oneway ANOVA on the data.
This defines data of a categorical variable with two levels and a categorical variable with three levels.
This performs a full factorial twoway ANOVA.
Dropping the point {2,2,28.7} gives an unbalanced twoway ANOVA with an empty cell.
Here is a balanced threeway dataset.
Here is a threeway ANOVA with main effects and twoway interactions.
 option name default value CellMeans True whether to give a list of means for each group WorkingPrecision MachinePrecision precision of the arithmetic to use in calculations PostTests {} list of post‐hoc tests to perform SignificanceLevel 0.05 significance level criteria for post‐hoc tests

Options for ANOVA.

 Bonferroni mean comparison test based on the Student t distribution with modified α based on the number of groups Duncan liberal range test based on the Studentized range distribution StudentNewmanKeuls conservative range test based on the Studentized range distribution Tukey mean comparison test based on the Studentized range distribution Dunnett comparison test of group means against a control, taken to be the first group

Available tests for the PostTests option.

Tukey's test finds groups 1 and 4 significantly different from group 3 at the 5% level.
Bonferroni and Tukey's tests find groups 3 and 4 significantly different at the 1% level.