University Maths Notes: Probability and Statistics – ANOVA
Analysis of variance (ANOVA) is a collection of models and procedures
in which the observed variance of a particular variable is
partitioned into components attributable to different sources. In its
simplest form ANOVA provides a test of whether or not the means of
several groups are all equal (so that the null hypothesis is
for
all
and
the null hypothesis is
for
some
)and
therefore generalizes the t-test to more than two samples. ANOVA is
helpful because it possesses an advantage over a two-sample t-test of
being faster when comparing many samples, and reducing the
probability of committing a type I error when performing multiple
tests.
The samples are independent.
The distributions of the residuals are normal.
The variance of each population from which the samples are taken are the same.
ANOVA involves partitioning of the total sum of squares SST into components (treatment sum of square, SSTr and error sum of squares SSE) related to the effects used in the model. For example, we show the model for a simplified ANOVA with one type of treatment at different levels
SST=SSTr +SSE with
with
where
I is the number of samples and J is the number in each sample.
Then with
with
the distribution of the test statistic we can find if there is a
difference in the sample means.
Example. Perform an ANOVA test on the three treatments below.
|
Treatment 1 |
0.56 |
1.12 |
0.9 |
1.07 |
0.94 |
|
Treatment 2 |
0.72 |
0.69 |
0.87 |
0.78 |
0.91 |
|
Treatment 3 |
0.62 |
1.08 |
1.07 |
0.99 |
0.93 |
Comparing
with
causes
us not to reject the null hypothesis.