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### ANOVA

INTRODUCTION
ANOVA is a method of analyzing data from designed experiments whose objective is to compare two or more group means.
Factorial ANOVA has two independent variables which are crossed with each other. That means each value of one variable is paired with every value of other variable.
ANOVA is used to compare means between three or more groups, so why is it called Analysis of VARIANCE?
The ANOVA F-test is a comparison of the average variability between groups to the average variability within groups.
The variability within each group is a measure of the spread of the data within each of the groups.
The variability between groups is a measure of the spread of the group means around the overall mean for all groups combined.
If you have three groups, could plausibly do pairwise comparisons. But if you have 10 groups? Too many pairwise comparisons: You would get too many false positives!
You would really like to compare a null hypothesis of all equal, against some difference
ANOVA is  ANalysis Of VAriance
Comparing more than two groups
Up to now we have studied situations with
One observation per object
One group
Two groups
Two or more observations per object
We will now study situations with one observation per object, and three or more groups of objects
The most important question is as usual: Do the numbers in the groups come from the same population, or from different populations?
Types of ANOVA
One-way ANOVA
Ø  Variance around the mean matters for comparison. We must compare the variance within the groups to the variance between the group means.
Ø  It is the simplest type of ANOVA, in which only one source of variation, or factor, is investigated.
Ø  It is an extension to three or more samples of the t test procedure for use with two independent samples
Ø  In another way t test for use with two independent samples is a special case of one-way analysis of variance.
•         Two-way ANOVA (without interaction)
Ø  In two-way ANOVA, data fall into categories in two different ways: Each observation can be placed in a table.
Ø  Example: Both doctor and type of treatment should influence outcome.
Ø  Sometimes we are interested in studying both categories, sometimes the second category is used only to reduce unexplained variance. Then it is called a blocking variable .
•         Two-way ANOVA (with interaction)
Ø  The setup above assumes that the blocking variable influences outcomes in the same way in all categories (and vice versa)
Ø  We can check if there is interaction between the blocking variable and the categories by extending the model with an interaction term

Important points  on ANOVA
•         All analysis of variance (ANOVA) methods are based on the assumptions of normally distributed and independent errors
•         The same problems can be described using the regression framework. We get exactly the same tests and results!