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Sediment particle size data from 6 size ranges at 3 sites (A, B, C), at 2 depths (2m and 5m), and 5 replicate samples from each site/depth combination are in C:\Examples v7\Denmark PSA\ Denmark PSA histogram. The data are already standardised to % composition ...

The smoothing effect can best be seen by Plots>Surface Plot of both Data1 and Data2 sheets – a further graph type new to PRIMER 7, but which should be reserved only for cases such as this, where there is genuine ordering of the variables. Symbols can be added ...

Fundamental to the operation of PRIMER and (explicitly or implicitly) any fully multivariate analysis, is an appropriate definition of resemblance between every pair of samples, based on whether the suite of recorded variables (species, environmental variables...

A detailed discussion of the competing properties of different resemblance matrices is outside this manual’s scope (see L&L, CiMC Chapters 2 & 16, or Clarke KR, Somerfield PJ, Chapman MG 2006, J Exp Mar Biol Ecol 330: 55-80). Novice users are recommended to ta...

The most commonly-used similarity coefficient for biological community analysis, because it obeys many of the ‘natural’ biological guidelines in a way that most other coefficients do not (see CiMC), is the Bray-Curtis similarity, defined between samples 1 and ...

A simple modification to the Bray-Curtis coefficient adjusts its behaviour as samples become vanishingly sparse. Standard Bray-Curtis is undefined for two samples containing no species, and can fluctuate wildly for near-blank samples – two samples containing j...

Data on coral communities at a site in Tikus Island, Thousand Islands, Indonesia, over the years 1981, 83, 84, 85, 87 and 88, were reported by Warwick RM, Clarke KR, Suharsono 1990, Coral Reefs 8: 171-179. Ten replicate transects were examined each year, and t...

Euclidean distance, an appropriate measure for environmental (and other) data types, is defined as: $$ D_1 = \sqrt{ \sum_i \left( y _ {i1} - y_{i2} \right) ^ 2 } $$ where the $y_{i1}$ & $y_{i2}$ result from pre-treatment by transformation (sometimes) and subse...

The remaining of the three choices in the initial list, Measure•Index of Association, is essentially Whittaker’s index of association, which when calculated on samples (the default is always Analyse between•Samples)is simply just Bray-Curtis similarity on a s...

PRIMER 7 allows the user choice of 44 other resemblances, firstly divided into two (mutually exclusive) types: Similarities (including L&L’s S numbers) or Dissimilarities/Distances (including L&L’s D numbers); then most of the same coefficients split instead i...

The distance measures defined by L&L and calculated by PRIMER 7 (in addition to $D_1$) are: $ D_2 = \sqrt{ \frac{1}{p} \sum_i \left( y_{i1} - y_{i2} \right) ^ 2 } \text{ \hspace{25mm} average distance,} $ where the number of species p is fixed for all pairs...

Anderson MJ, Ellingsen KE, McArdle BH 2006, Ecol Lett 9: 683-693 used Czekanowski’s mean character difference (above) as their preferred distance measure after a specific transformation of the original counts, advocated for its interpretable properties, namely...

L&L also assign $D_{14}$ to Bray-Curtis dissimilarity, the complement of $S_{17}$, defined earlier. This is also provided in the Dissimilarity list since it is (very occasionally) useful to specify a dissimilarity rather than its complementary similarity – tho...

In addition to Bray-Curtis $S_{17}$, and its zero-adjusted modification, PRIMER 7 also calculates: $$ S_{15} = 100 \frac{1}{p} \sum_i \left[ 1 - \frac{ \left| y_{i1} - y_{i2} \right| }{ R_i} \right] \text{, where } R_i=\max_j \left\{ y_{ij} \right\} - \min_j ...

There are numerous similarity measures defined for simple species lists, i.e. when the data consist only of presence (1) or absence (0) of each species in each sample. Any similarity defined between samples 1 and 2 must then be a combination of only four numbe...

It is instructive to draw the other links between quantitative coefficients and the presence/absence measures they reduce to, when calculating them on a P/A matrix. Pure distance measures such as $D_1$, $D_6$, $D_7$ and $D_{10}$, which are not averaged in some...

Returning to the quantitative resemblance coefficients in the •Others list, five further measures given under the ✓Distance/dissimilarity heading are (loosely) based on likelihood-ratio tests. All are motivated by the (usually unrealistic) model in which the i...

Another useful application of multivariate methods was touched on at the end of Section 4, namely the analysis of structured sets of curves or (pseudo-)frequency distributions, generically referred to as sample profiles. These include particle- or body-size an...

An example of a particle-size analysis (PSA) matrix has already been seen for Danish sediments at the end of Section 4, for which the histogram was smoothed by cumulating the size-classes. Here we examine instead an already smooth frequency distribution from C...

A later section (15) discusses univariate diversity indices that can be computed from each sample, including biodiversity measures that are based on the relatedness of the species making up a simple species list (P/A data), see Chapter 17 of CiMC. Though the s...