Permutational multivariate analysis of variance (PERMANOVA), is a
non-parametric
Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distri ...
multivariate
Multivariate may refer to:
In mathematics
* Multivariable calculus
* Multivariate function
* Multivariate polynomial
In computing
* Multivariate cryptography
* Multivariate division algorithm
* Multivariate interpolation
* Multivariate optical c ...
statistical
permutation test
A permutation test (also called re-randomization test) is an exact statistical hypothesis test making use of the proof by contradiction.
A permutation test involves two or more samples. The null hypothesis is that all samples come from the same dis ...
. PERMANOVA is used to compare groups of objects and test the
null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
that the
centroids
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
and
dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
of the groups as defined by measure space are equivalent for all groups. A rejection of the null hypothesis means that either the centroid and/or the spread of the objects is different between the groups. Hence the test is based on the prior calculation of the distance between any two objects included in the experiment.
PERMANOVA shares some resemblance to
ANOVA
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...
where they both measure the
sum-of-squares within and between group and make use of
F test
An ''F''-test is any statistical test in which the test statistic has an ''F''-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model th ...
to compare within-group to between-group variance. However, while ANOVA bases the
significance of the result on assumption of normality, PERMANOVA draws tests for significance by comparing the actual F test result to that gained from random permutations of the objects between the groups. Moreover, whilst PERMANOVA tests for similarity based on a chosen distance measure, ANOVA tests for similarity of the group
averages
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
.
Calculation of the statistic
In the simple case of a single factor with ''p'' groups and ''n'' objects in each group,the total sum-of-squares is determined as:
:
where ''N'' is the total number of objects, and
is the squared distance between objects ''i'' and ''j''.
Similarly, the within groups sum-of-squares is determined:
:
where
takes the value of 1 if observation ''i'' and observation ''j'' are in the same group, otherwise it takes the value of zero.
Then, the between groups sum-of-squares (
) can be calculated as the difference between the overall and the within groups sum-of-squares:
:
Finally, a pseudo F-statistic is calculated:
:
where ''p'' is the number of groups.
Drawing significance
Finally, the PERMANOVA procedure draws significance for the actual F statistic by performing multiple permutations of the data. In each such the items are shuffled between groups. For each such permutation of the data the permutation F statistic is calculated. The p-value is then calculated by:
:
Where
is the F statistic obtained from the original data and
is a permutation F statistic.
Implementation and use
PERMANOVA is widely used in the field of ecology and is implemented in several software packages including PERMANOVA software,
PRIMER
Primer may refer to:
Arts, entertainment, and media Films
* ''Primer'' (film), a 2004 feature film written and directed by Shane Carruth
* ''Primer'' (video), a documentary about the funk band Living Colour
Literature
* Primer (textbook), a t ...
and
R (programming language)
R is a programming language for statistical computing and graphics supported by the R Core Team and the R Foundation for Statistical Computing. Created by statisticians Ross Ihaka and Robert Gentleman, R is used among data miners, bioinform ...
Vegan and lmPerm
packages.
References
{{Reflist
Statistical hypothesis testing
Ecology