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# A3 Index to mathematical notation and symbols
#### Matrices and vectors
**A** = matrix containing elements $a _ {ij} = - \frac{1}{2} d _ {ij} ^ 2 $
**B** = matrix of variables (*N* × *s*) that are linear combinations of normalised **X** variables having maximum correlation with CAP axes
**C** = matrix of CAP axes (*N* × *s*), standardised by the square root of their respective eigenvalues
**D** = matrix containing elements $d _ {ij}$ corresponding to distances or dissimilarities
**G** = Gower’s centred matrix, consisting of elements $g _ {ij} = a_ {ij} - \bar{a} _ {i.} - \bar{a} _ {j.} + \bar{a} _ {\..}$
**H** = ‘hat’ matrix = **X[X′X]$^ {-1}$X′**, used as a projection matrix for regression models
**I** = identity matrix, with 1’s along the diagonal and 0’s elsewhere
**Q** = matrix of PCO axes, standardised by the square root of their respective eigenvalues
**Q**$^0$ = matrix of PCO axes, orthonormalised to SSCP = **I** (‘sphericised’)
**U** = matrix whose columns contain the left singular vectors from a singular value decomposition (SVD) of a matrix (e.g., **X** = **UWV′**); if **X** is (*N* × *q*) and *q* < *N*, then **U** is (*N* × *q*)
**V** = matrix whose columns contain the right singular vectors from a singular value decomposition (SVD) of a matrix (e.g., **X** = **UWV′**); if **X** is (*N* × *q*) and *q* < *N*, then **V** is (*q* × *N*)
**W** = diagonal matrix of eigenvalues from a singular value decomposition (SVD) of a matrix (e.g., **X** = **UWV′**); if **X** is (*N* × *q*) and *q* < *N*, then **W** is (*q* × *q*)
**X** = matrix of predictor variables (*N* × *q*) (often a set of environmental variables)
**X**$^0$ = matrix of **X** variables, orthonormalised to SSCP = **I** (‘sphericised’)
**Y** = matrix of response variables (*N* × *p*) (often a set of species variables)
**Y**$^0$ = matrix of **Y** variables, orthonormalised to SSCP = **I** (‘sphericised’)
$\hat{ {\bf Y}}$ = **HY** = matrix of fitted values (*N* × *p*)
**y**$_ {ij} $ = vector of *p* response variables for the *j*th observation in the *i*th group
$\bar{ {\bf y}}$ = the centroid vector of *p* response variables for group *i*
**Z** = matrix of dbRDA canonical axes (*N* × *s*)
#### Letters
*a*, *b*, *c*, etc… = number of levels of factor A, B, C, etc… in an ANOVA experimental design
*AIC* = multivariate analogue to Akaike’s 'An information criterion'
*AIC*$_c$ = multivariate analogue to the small-sample-size corrected version of *AIC*
*B*$_l$ = the $l$th variable in the space of normalised **X** variables that has maximum correlation with the $l$th coordinate axis (*C*$_l$) from a CAP analysis
*BIC* = multivariate analogue to Schwarz’s ‘Bayesian information criterion’
*C*$ _l $ = the $l$th coordinate axis scores from a CAP analysis
*d*$ _ {ij} $ = distance or dissimilarity between sample *i* and sample *j*
*df* = degrees of freedom
*F* = pseudo-*F* statistic for testing hypotheses in PERMANOVA or DISTLM
*i* = index used for samples (i.e., *i* = 1, …, *N*) or index used for groups (*i* = 1, …, *a*)
*j* = second index used for samples (i.e., *j* = 1, …, *N*) **or** index used for replicates within a group (*j* = 1,…, *n*)
*k* = index used for variables (i.e., *k* = 1, …, *p* or else *k* = 1, …, *q*)
$l$ = index used for canonical axes or eigenvalues for either dbRDA **or** CAP (i.e., $l$ = 1, …, *s*) **or** either the abbreviation for ‘log-likelihood’ or the ‘length’ of a vector (depending on context).
*m* = number of PCO axes chosen as a subset for analysis by CAP
*MC* = Monte Carlo
*MS* = mean square
*N* = total number of samples
*n* = number of samples (replicates) within a group or cell in an experimental design
*P* = *P*-value associated with the test of a null hypothesis
*p* = number of multivariate response variables in matrix **Y**
*q* = total number of predictor variables in matrix **X**
*r* = Pearson correlation coefficient
*R* = the ANOSIM *R* statistic (see {{@324#bkmrk-clarke1993a}})
*R*$^2$ = proportion of explained variation from a model
*s* = number of canonical eigenvalues and associated canonical axes obtained from either a dbRDA **or** a CAP analysis
*SS* = sum of squares
*SSCP* = sum of squares and cross products
*SVD* = singular value decomposition
*t* = pseudo-*t* statistic = $\sqrt{}$pseudo- *F*
*tr* = ‘trace’ of a matrix = the sum of the diagonal elements
*X*$ _ k $ = the *k*th predictor variable
*Y*$ _ k $ = the *k*th response variable
*z*$ _ {ij} $ = distance to group centroid for the *j*th replicate within the *i*th group.
#### Greek symbols and matrices
$ \alpha$ = significance level chosen for a test (usually $\alpha$ = 0.05).
$ \delta _ l ^ 2$ = the $l$th eigenvalue from a CAP analysis, a squared canonical correlation
$ \Delta$ = diagonal matrix containing the square roots of the eigenvalues from a CAP analysis (a capital delta)
$ \gamma _ l ^ 2$ = the $l$th eigenvalue from a dbRDA analysis, a portion of the explained (regression) sum of squares from a dbRDA model.
$ \Gamma$ = diagonal matrix containing the square roots of the eigenvalues from a dbRDA analysis (a capital gamma)
$ \lambda _ i $ = the *i*th eigenvalue from a PCO analysis
$ \Lambda $ = diagonal matrix of eigenvalues from a PCO analysis (a capital lambda)
$ \nu$ = number of parameters in a particular model during model selection
$ \rho $= Spearman rank correlation (rho)
$ \sum $ = sum over the relevant index
# A4 Index to data sets used in examples
Below is an index to the data sets used in examples, listed in order of appearance in the text. With each dataset are given the name and location of the data file, the original reference, a description of its use as an example in the manual and the page number where this can be found (italicised and in parentheses).
1. Ekofisk oil-field macrofauna (
ekma.pri in Examples v6\Ekofisk), {{@324#bkmrk-gray1990a}} - demonstrate one-way PERMANOVA ([*1.8*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/18-one-way-example-ekofisk-oil-field-macrofauna)), model selection procedures, diagnostics and building models in DISTLM ([*4.10*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/410-ekofisk-macrofauna)) and visualising models using dbRDA ([*4.11*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/411-visualising-models-dbrda)).
2. Victorian avifauna (
vic.pri in Examples add-on\VictAvi), {{@324#bkmrk-macnally2005a}} – demonstrate Monte Carlo *P* values ([*1.12*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/112-monte-carlo-p-values-victorian-avifauna)). Also used at the level of individual surveys (
vicsurv.pri) to demonstrate a repeated measures design ([*1.32*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/132-repeated-measures-victorian-avifauna-revisited)) and also PCO ([*3.4*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/34-example-victorian-avifauna)), negative eigenvalues ([*3.5*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/35-negative-eigenvalues)), scree plots ([*3.5*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/35-negative-eigenvalues)) and vector overlays ([*3.6*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/36-vector-overlays)).
3. Subtidal epibiota (
sub.pri in Examples add-on\SubEpi), {{@324#bkmrk-glasby1999a}} – demonstrate a two-way crossed design ([*1.14*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/114-two-way-crossed-design-subtidal-epibiota)) and contrasts ([*1.19*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/119-contrasts)) in PERMANOVA.
4. Tasmanian meiofauna (
tas.pri in Examples add-on\TasMei), {{@324#bkmrk-warwick1990a}} – demonstrate fixed *versus* random factors ([*1.20*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/120-fixed-vs-random-factors-tasmanian-meiofauna)), components of variation ([*1.21*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/121-components-of-variation)), expected mean squares ([*1.22*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/122-expected-mean-squares-ems)), constructing F from EMS ([*1.23*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/123-constructing-f-from-ems)), exchangeable units ([*1.24*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/124-exchangeable-units)), inference space and power ([*1.25*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/125-inference-space-and-power)), and testing the design ([*1.26*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/126-testing-the-design)).
5. Holdfast invertebrates (
hold.pri,
holdenv.pri and
Mollusca.agg in Examples add-on\HoldNZ), {{@324#bkmrk-anderson2005a}} – demonstrate a nested design ([*1.27*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/127-nested-design-holdfast-invertebrates)), estimating components of variation ([*1.28*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/128-estimating-components-of-variation)), and pooling or excluding terms ([*1.29*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/129-pooling-or-excluding-terms)). Also used later to demonstrate analyses with covariates in PERMANOVA ([*1.35*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/135-designs-with-covariates-holdfast-invertebrates-revisited)) and marginal and conditional tests with DISTLM ([*4.6*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/46-holdfast-invertebrates)).
6. Plankton net study (
plank.pri in Examples add-on\Plankton), {{@324#bkmrk-winsor1940a}} – demonstrate designs that lack replication ([*1.30*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/130-designs-that-lack-replication-plankton-net-study)) and increased power as a result of blocking ([*1.30*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/130-designs-that-lack-replication-plankton-net-study)).
7. Woodstock plants (
wsk.pri in Examples add-on\Woodstock), {{@324#bkmrk-prober2007a}} – demonstrate a split-plot design ([*1.31*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/131-split-plot-designs-woodstock-plants)).
8. Birds from Borneo (
born.pri in Examples add-on\BorneoBirds), {{@324#bkmrk-cleary2005a}} – demonstrate an unbalanced design ([*1.34*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/134-types-of-sums-of-squares-birds-from-borneo)).
9. New Zealand fish (
fishNZ.pri in Examples add-on\FishNZ), {{@324#bkmrk-anderson2004b}} – demonstrate analyses involving linear combinations of mean squares ([*1.36*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/136-linear-combinations-of-mean-squares-nz-fish-assemblages)).
10. Mediterranean molluscs (
medmoll.pri in Examples add-on\MedMoll), {{@324#bkmrk-terlizzi2005a}} – demonstrate an asymmetrical design ([*1.37*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/137-asymmetrical-designs-mediterranean-molluscs)).
11. Bumpus’ sparrows (
spar.pri in Examples add-on\BumpSpar), {{@324#bkmrk-bumpus1898a}} – demonstrate test of dispersion in Euclidean space ([*2.3*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/23-multivariate-levenes-test-bumpus-sparrows)).
12. Tikus Island corals (
tick.pri in Examples v6\Corals), {{@324#bkmrk-warwick1990b}} – demonstrate test of dispersion for ecological data ([*2.7*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/27-ecological-example-tikus-island-corals)) and how choice of dissimilarity measure matters ([*2.8*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/28-choice-of-measure)). Also used later to demonstrate how CAP tells you nothing about relative within-group dispersions ([*5.9*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/59-caveats-on-using-cap-tikus-island-corals)).
13. Norwegian macrofauna (
norbio.pri and
norenv.pri in Examples add-on\NorMac), {{@324#bkmrk-ellingsen2002a}} – demonstrate use of the test of dispersion to investigate beta diversity ([*2.9*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/29-dispersion-as-beta-diversity-norwegian-macrofauna)).
14. Okura macrofauna (
okura.pri, in Examples add-on\Okura), {{@324#bkmrk-anderson2004a}} – demonstrate tests of dispersion in nested designs ([*2.11*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/211-dispersion-in-nested-designs-okura-macrofauna)). Also used to demonstrate PCO of distances among centroids ([*3.8*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/38-distances-among-centroids-okura-macrofauna)) and PCO *versus* MDS when samples are split into groups ([*3.9*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/39-pco-versus-mds)).
15. Cryptic fish assemblages (
cryptic.pri in Examples add-on\Cryptic), {{@324#bkmrk-willis2003a}} – demonstrate PERMDISP for a two-factor crossed design, in conjunction with PERMANOVA ([*2.12*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/212-dispersion-in-crossed-designs-cryptic-fish)).
16. Clyde macrofauna and environmental data (
clma.pri and
clev.pri, in Examples v6\Clydemac), {{@324#bkmrk-pearson1984a}} – demonstrate PCO *versus* PCA for environmental data ([*3.7*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/37-pco-versus-pca-clyde-environmental-data)) and simple linear regression using DISTLM ([*4.4*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/44-simple-linear-regression-clyde-macrofauna)).
17. Thau lagoon bacteria (
thbac.pri and
thevsp.pri in Examples add-on\Thau), {{@324#bkmrk-amanieu1989a}} – demonstrate analysing variables in sets using DISTLM ([*4.14*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/414-analysing-variables-in-sets-thau-lagoon-bacteria)).
18. Oribatid mites (
ormites.pri and
orenvgeo.pri in Examples add-on\OrbMit), {{@324#bkmrk-borcard1992a}} – demonstrate analysing categorical predictor variables using DISTLM ([*4.15*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/415-categorical-predictor-variables-oribatid-mites)).
19. Flea-beetles (
flea.pri in Examples add-on\FleaBeet), {{@324#bkmrk-lubischew1962a}} – demonstrate the rationale for CAP by comparing unconstrained vs constrained ordination ([*5.2*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/52-rationale-flea-beetles)).
20. Poor Knights Islands fish (
pkfish.pri in Examples add-on\PKFish), {{@324#bkmrk-willis2000a}} – demonstrate discriminant analysis based on Bray-Curtis using CAP ([*5.4*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/54-discriminant-analysis-poor-knights-islands-fish)).
21. Iris data (
iris.pri in Examples add-on\Irises), {{@324#bkmrk-anderson1935a}} – demonstrate classical discriminant analysis and MANOVA test statistics using CAP ([*5.7*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/57-test-by-permutation-andersons-irises)). Also used later to show how the positions of new samples are added into a discriminant-type analysis, with prediction of group membership ([*5.10*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/510-adding-new-samples)).
22. Fal estuary biota (
Fa.xls in Examples v6\Fal;
falbio.pri and
falenv.pri in Examples add-on\FalEst), {{@324#bkmrk-somerfield1994a}} – demonstrate canonical correlation analysis with CAP based on the Bray-Curtis measure relating biota to a single environmental gradient ([*5.11*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/511-canonical-correlation-single-gradient-fal-estuary-biota)).
23. Hunting spiders (
hspi.pri and
hspienv.pri in Examples add-on\Spiders), {{@324#bkmrk-vanderaart1975a}} – demonstrate a canonical correlation-type analysis using CAP on the basis of chi-squared distances ([*5.16*](https://learninghub.primer-e.com/books/permanova-for-primer-guide-to-software-and-statistical-methods/page/516-hunting-spiders)).