## Analysis of Variance (ANOVA)

Data types that can be analysed with ANOVA

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

Limitations of the test

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.

Introduction to ANOVA

ANOVA compares between the means of two or more samples. E.g. several manufacturers claim that their growth hormone is best at increasing productivity in livestock. By treating three groups (samples) of animals separately with hormone 1, hormone 2 and saline solution (control), we can assess whether there are differences in the resultant growth. The "factor" concerned here is hormone and therefore is a "one way" or "one factor" ANOVA.

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.

Hypotheses

Data arrangement

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):

Results and interpretation

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 [n1-1]+[n2-1]+[n3-1]).

If the F-value is greater than the F crit value you must reject the HO at the level of confidence you chose during selection of the test (normally 5% or 0.05). In this instance the HO must be rejected.

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 HO is rejected in favour of the HA 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 HO for hormone should be rejected: A very high significant difference exists between two or more of the samples.

The HO for sex should be rejected: A highly significant difference exists between the mean weights of the sexes.

The HO 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.

Graphical representation

 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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