Introduction to ANOVA

Questions

• What null hypothesis is tested by ANOVA?
• What are the uses of ANOVA?

What is ANOVA?

• Analysis of Variance (ANOVA) is a statistical method used to compare two or more means
• Inferences about means are made by analyzing variance (therefore it is not called analysis of mean)
• ANOVA is used to test general rather than specific differences among means

Smiles and Leniency Example

• Let us investigate types of smiles on the leniency
• Types of smiles: neutral, false, felt, miserable

The results from the Tukey hsd test (Six Pairwise Comparisons):

Comparison	Mi-Mj	Q	p
False - Felt	0.46	1.65	0.649
False - Miserable	0.46	1.65	0.649
False - Neutral	1.25	4.48	0.010
Felt - Miserable	0.00	0.00	1.000
Felt - Neutral	0.79	2.83	0.193
Miserable - Neutral	0.79	2.83	0.193

• Notice that the only significant difference is between the False and Neutral conditions.
• ANOVA tests the non-specific null hypothesis that all four populations means are equal

μfalse = μfelt = μmiserable = μneutral

• This non-specific null hypothesis is sometimes called the omnibus null hypothesis
• When the omnibus null hypothesis is rejected, the conclusion is that at least one population mean is different from at least one other mean
• ANOVA does not reveal which means are different from which
• It offers less specific information than the Tukey hsd test
• The Tukey hsd is therefore preferable to ANOVA in this situation

Why to use ANOVA instead of HSD Tukey

• There are complex types of analyses that can be done with ANOVA and not with the Tukey test
• ANOVA is by far the most commonly-used technique for comparing means
• Is important to understand ANOVA in order to understand research reports.

Questions

1 The omnibus null hypothesis when performing an analysis of variance is there are differences between group means, however, no prediction is made concerning where the differences lie.

	True
	False

Answer >>

False, the omnibus null is that all group means are the same.

2 Unlike t-tests an ANOVA may be used to test for differences among more than 2 groups.

	True
	False

Answer >>

True, An analysis of variance (ANOVA) is most often used to determine if there are differences among 3 or more group means. However, if there are more than 2 groups an ANOVA does not provide information regarding where the differences lie.

3 Unlike t-tests, an ANOVA uses both differences between group means and differences within groups to determine whether the difference are significant.

	True
	False

Answer >>

False, both t-tests and ANOVAs use both. In a t test the difference between means is in the numerator. In an ANOVA, the variance of the grouop means (multiplied by n) is in the numerator.

4 It is valid to do the Tukey HSD test without first finding a signficant effect with an ANOVA.

	True
	False

Answer >>

True, the Tukey HSD controls the Type I error rate and is valid without first running an ANOVA.