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**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.

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ANOVA is used to compare means between three or more groups, so why is it
called Analysis of VARIANCE?

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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.

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The variability between groups is a measure of the spread of the group
means around the overall mean for all groups combined.

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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!

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You would really like to
compare a null hypothesis of all equal, against some difference

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ANOVA is ANalysis Of VAriance

**Comparing more than two groups**

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Up to now we have studied
situations with

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One observation per
object

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One group

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Two groups

–
Two or more observations
per object

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We will now study
situations with one observation per object, and three or more groups of objects

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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**

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**One-way ANOVA**
Ã˜ Variance around the mean matters for comparison. We
must compare the variance within the groups to the variance between the group
means.

Ã˜ It is the simplest type of ANOVA, in which
only one source of variation, or factor, is investigated.

Ã˜ It is an extension to three or more samples of
the t test procedure for use with two independent samples

Ã˜ In another way t test for use with two
independent samples is a special case of one-way analysis of variance.

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**Two-way ANOVA (without interaction)**
Ã˜ In two-way ANOVA, data fall into categories in two
different ways: Each observation can be placed in a table.

Ã˜ Example: Both doctor and type of treatment should
influence outcome.

Ã˜ 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 .

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**Two-way ANOVA (with interaction)**
Ã˜ The setup above assumes that the blocking variable
influences outcomes in the same way in all categories (and vice versa)

Ã˜ 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**

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All analysis of
variance (ANOVA) methods are based on the assumptions of normally distributed
and independent errors

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The same problems can
be described using the regression framework. We get exactly the same tests and
results!

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