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Overview The Mann-Whitney U test was described by and . Here, interest lies in comparing two groups of independent samples. This is a non-parametric analogue to a classical two-sample (unpaired) t-test. The null hypothesis Suppose we have independent response...

Overview The Kruskal-Wallis test was described by and . Its purpose is to compare two or more independent groups of samples and it is an extenstion of the Mann-Whitney U test. It operates on ranked values and will indeed yield an equivalent result to the Mann...

Overview The Kolmogorov-Smirnov test is a non-parametric test for comparing two distributions of a continuous variable. Rejection of the null hypothesis indicates that the two distributions differ from one another in some way (location, dispersion, skewness, e...

Overview PRIMER 8 offers several options to achieve a non-parametric bivariate test of association. Here, there are two variables sampled from the same set of sampling units and interest lies in examining the extent to which they co-vary. Do values of the two ...

An example of a paired design with two groups is provided by , who described a study by to investigate the total catch of five different groups of plankton (hence, five variables, named using Roman numerals I, II, III, IV and V) by 2 nets hauled horizontally ...

As an example of the Mann-Whitney U test, we will look at a dataset consisting of counts of the snapper (Chrysophrys auratus) sampled using baited remote underwater videos (BRUVs) from multiple areas inside vs outside several marine reserves along the north-ea...

We will use the Kruskal-Wallis test to compare counts of a bivalve species, Abra prismatica, occurring at sites classified into groups according to their proximity to the Ekofisk oilfield in the North Sea (). Macrofauna were sampled from each of 29 sites that ...

To demonstrate the Kolmogorov-Smirnov test in PRIMER, we shall return to the dataset consisting of length measurements (in mm) of the Sydney rock oyster (Saccostrea commercialis) settling on four different types of surfaces in intertidal estuarine environments...

To demonstrate the test of association, we shall re-visit a dataset of macrofauna assemblages collected from sites near an oilfield in the North Sea. In this study by , macrofauna were sampled from 39 sites in an approximately 5-spoke radial design, at increas...

It is instructive to consider some additional examples of the test of association where the variables are not evenly distributed. Specifically, we wish to cater for situations where the variables of interest are occurrences, densities or counts of species' abu...

Overview What is an empirical distribution? The empirical distribution of a variable is able to be characterised by considering each unique numerical value observed for that variable in a given sample of size $n$. If certain values are repeated, then we simply...

To demonstrate the empirical distribution tool in PRIMER, we shall examine a dataset consisting of length measurements (in mm) of the Sydney rock oyster (Saccostrea commercialis) settling on various surfaces in Quibray Bay, New South Wales, Australia (,). Sett...

Suppose we have measured a given variable in each of several groups. To visualise the distributional shape of each collective set of sample values, we might consider creating several histograms - one for each group - but it then might be difficult to compare t...

Let's re-visit the data on oyster sizes (,). We have already seen some variation in the cumulative distributions of sizes of oysters settling on different types of substrata (see section 2.2). To compare these different distributions as densities, side-by-side...

studied organisms colonising holdfasts of the kelp, Ecklonia radiata, sampled from four different locations along the northeastern coast of New Zealand. One would expect that invertebrate communities colonising holdfasts (which include a wide range of taxa su...

Re-vamped interface To run a PERMANOVA in PRIMER 8, there are two essential steps. From a resemblance matrix of your choice (with associated factors) you: specify the design (click PERMANOVA+ > Create PERMANOVA Design...); then run the PERMANOVA analysis (cli...

The one-way ANOVA model In one-way univariate analysis of variance (ANOVA), interest lies in comparing the means among several groups. More formally, ANOVA tests the null hypothesis of no differences in the population means among groups. Let $y_{ij}$ be the $j...

Overview The Behrens-Fisher problem (BFP) is one of the oldest puzzles in statistics (; ; ). The essence of this problem is how validly to compare the means of two or more populations (groups) when their variances differ. It is clear how the assumption of comm...

Overview In a multivariate context, there are many ways that groups of sampling units can differ from one another. For example, let's consider conceptually just three important ways that groups (i.e., sets of sampling units in a multivariate space) can differ ...

Let's look now at an example where there is a single factor in the study design, the number of replicates per group is unequal and there is clear heterogeneity in multivariate dispersions among the groups. studied the biodiversity of soft-sediment macrobenthi...