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the data points
must be independent from each other

the distributions must be normal (though small departures can be accommodated) K-S or Shapiro-Wilke

the variances of the samples are
not different (though some departures are accommodated) (F-test
or homogeneity
of variance test)

all individuals
must be selected at random from the population

all individuals
must have equal chance of being selected

sample sizes
should be as equal as possible but some differences are allowed

if you do not find a significant difference in your data,
you cannot say that the samples are the same

information covered by this page is only a fraction of
what ANOVA can offer

ANOVA will only indicate a difference between groups,
not which group(s) are different. For the latter you will need
to use a multiple
comparison test.

Two factors
can also be tested with ANOVA. Hormones may affect male animals
to a greater or lesser extent than females. By selecting samples
of females and samples of males for each treatment, sex then becomes
"factor 2". Comparisons could then be made between the
different hormones __and__ between the different sexes. __Additionally__,
ANOVA will test for an interaction between the two i.e. if the
hormone had a positive effect but no interaction then the growth
would be the same for males and females. However, if the males
grew by 50% more than the controls and the females only 10% there
is an interaction.

ANOVA is
incomplete on its own if you have more than two samples. If a
significant difference is found you will only know that the samples
with the highest and lowest means are different. You will not
know with any certainty whether the intermediates are significantly
different from either of the extremes. One of the multiple comparison tests must
be used for this reason. The multiple comparison tests need to
be associated with ANOVA since the latter is more reliable for
detecting differences.

Once you
have established that your data conforms to using an ANOVA, your
data must be arranged thus for use in one of the statistical packages
(SPSS, UNISTAT):

Degrees of
freedom

between groups:
No of samples/treatments - 1

within groups:
*n* - No of samples/treatments,

e.g. Where
there are four treatments with 17 data points per treatment (total
n = 56), between group DF = 4-1=3, and within group DF = 56-4=52.

On completion
of the 1-way ANOVA the results will look something like this:

Although
it looks a bit daunting do not be worried. There are 5 values
that concern the selection of one of the hypotheses. The *F*
value (34.48005) is the calculated statistic from the mathematics
behind ANOVA. The *F crit* (3.354131) is the critical value
as extracted from the f-distribution in statistical tables based
on two values of degrees of freedom *df* of 2 and 27 (No
samples-1 and [n_{1}-1]+[n_{2}-1]+[n_{3}-1]).

However,
this conclusion can be modified based on the *P-value* (3.67E-08).
This is the probability, 0.00000367% in this case, of the differences
occurring purely by chance. It is below the very highly significant
cut-off level of 0.1% therefore the H_{O} is rejected
in favour of the H_{A} and we can say "the differences
between one or more of the samples is/are very highly significant".
In this instance, you would have to go on to perform a multiple
comparison test.

On completion
of the 2-way ANOVA you will have results similar to this:

Daunting
but not much more complicated than the 1-way results. At this
stage there are only three sets of values you need to consider.
If we take the values of *F*, *P-value* and *F crit*
along the "Sample" "Columns" and "Interaction"
rows. The values are used in exactly the same way as the 1-way
ANOVA but "Sample" relates to the first factor (hormone),
"Columns" relates to the second factor (sex) and "Interaction"
relates to the interaction between the two factors.

The H_{O}
for hormone should be rejected: A very high significant difference
exists between two or more of the samples.

The H_{O}
for sex should be rejected: A highly significant difference exists
between the mean weights of the sexes.

The H_{O}
for the interaction between hormone and sex should be accepted:
There is no evidence to suggest that an interaction exists between
the hormone and sex of the animals.

Descriptive
Stats |
Diversity
Indices |
Comparisons |
Correlations |
Regression |

Ted Gaten Department of Biology gat@le.ac.uk Entry approved by the Head of Department. Last Updated: May 2000

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