# Clustering and Classification methods for Biologists

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### Background

This is a 'constrained ordination' technique that is an extension of the original correspondence analysis technique (CA). CCA differs from CA in that rather than looking for implicit relationships between our ordination of, for example, a species matrix and some environmental variables, we look for some explicit relationships. It is, therefore, a direct gradient analysis method.

The reciprocal averaging algorithm is used in the calculation but at each cycle the sample scores are regressed on the environmental variables. This constrains the scores to be linear combinations of the environmental variables.

The example analysis uses the classical data set from the Jongman et al. (1995) book. There are 30 species and 10 environmental variables (some continuous, some ordinal). The question is 'are there any gradients in the species data that can be explained by the environmental variables'?

A brief (very!) explanation of the results is given (See Jongman et al. (1995) for an in depth description). There is also a detailed example on the ordination web site.

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### Example analysis

```	CANONICAL CORRESPONDENCE ANALYSIS
Data file - E:\Stats\MVSP\Dune.mvs
DUNE MEADOW SPECIES DATA (M. BATTERINK AND G. WIJFFELS, 1983)
Analysing 30 variables x 20 cases
Environmental data file - E:\Stats\MVSP\Dune.mve
ENVIRONMENTAL DATA IN FULL FORMAT - DUNE MEADOW DATA
Analysing 10 variables x 20 cases
Tolerance of eigenanalysis set at 1E-9
Scores scaled by species ```
Variable Weighted mean Weighted SD Inflation Factor
A1 4.685 1.861 1.781
Moisture 2.801 1.731 1.850
Manure 1.902 1.363 8.303
Hayfield 0.339 0.473 2.849
Haypasture 0.415 0.493 0.000 ***
Pasture 0.247 0.431 1.694
SF 0.298 0.457 3.193
BF 0.171 0.376 2.070
HF 0.311 0.463 0.000 ***
NM 0.220 0.415 3.661
```Multicollinearity detected. Variables marked with "***" will be ignored
in analysis

Axis 1 Axis 2  Axis 3  Axis 4
Eigenvalues             0.461  0.298  0.160   0.134
Percentage             21.804 14.092  7.567   6.320
Cum. Percentage        21.804 35.896 43.462  49.783
Cum.Constr.Percentage  37.815 62.254 75.377  86.338
Spec-env. correlations  0.958  0.902  0.855   0.889>```

The species - env. correlations tell us how much of the variation in species composition is 'explained' by the environmental variables. The large figure of 0.958 tells us that we can account for most of the variaiton in species composition by taking account of the environmental varaibles. Later we will discover the contribution that each environmental variable makes to this 'explanation'.

#### CCA variable scores

These are the scores that will be plotted later.

Species Axis 1 Axis 2 Axis 3 Axis 4
Achmil -0.840 0.382 0.028 -0.334
Agrsto 0.770 -0.500 -0.114 -0.080
Airpra 0.740 1.787 -1.077 0.532
Alogen 0.354 -0.970 -0.347 0.139
Antodo -0.386 0.778 -0.041 0.226
Belper -0.712 0.169 -0.258 -0.718
Brohor -0.832 0.018 -0.161 -0.858
Chealb 0.931 -1.647 -0.693 0.027
Cirarv -0.398 -0.845 -0.988 -0.667
Elepal 1.497 -0.090 0.575 -0.383
Elyrep -0.638 -0.382 -0.372 -0.004
Empnig 1.206 1.549 -1.452 0.558
Junart 0.961 -0.044 0.047 0.329
Junbuf 0.207 -0.804 -0.124 1.106
Leoaut 0.000 0.431 0.032 -0.067
Lolper -0.624 -0.042 -0.065 -0.281
Plalan -0.802 0.570 0.588 0.185
Poapra -0.459 -0.040 -0.129 -0.156
Poatri -0.264 -0.532 -0.130 -0.034
Potpal 2.029 0.399 2.139 -0.860
Ranfla 1.374 0.071 0.117 -0.158
Rumace -0.824 -0.207 0.790 0.962
Sagpro 0.299 -0.432 -0.447 0.288
Salrep 0.785 1.759 -1.093 0.520
Tripra -1.112 -0.059 1.015 0.933
Trirep -0.023 0.110 0.301 -0.125
Viclat -0.537 1.015 0.319 -0.681
Brarut 0.135 0.253 0.138 0.225
Calcus 1.657 0.451 0.386 -0.254

#### CCA case scores

 Axis 1 Axis 2 Axis 3 Axis 4 1 -1.219 -0.497 -0.935 -1.252 2 -0.864 -0.250 -0.536 -1.703 3 -0.315 -1.010 -0.900 -0.638 4 -0.237 -0.931 -1.271 -1.047 5 -1.146 0.240 0.914 0.624 6 -1.031 0.368 1.799 1.495 7 -1.034 0.151 0.963 0.649 8 0.697 -0.717 -0.108 -0.276 9 -0.053 -1.005 -0.489 1.241 10 -0.963 0.600 0.236 -1.458 11 -0.507 0.932 0.146 -0.515 12 0.354 -1.442 -0.524 1.825 13 0.472 -1.619 -0.907 0.655 14 2.021 0.266 2.543 -1.804 15 1.958 0.049 2.169 -0.860 16 1.932 -0.691 0.802 -0.528 17 -0.388 2.770 -1.065 0.905 28 -0.311 1.495 -0.147 -0.074 29 0.665 2.873 -2.664 1.724 30 2.001 1.003 -0.264 0.328

#### Site scores, constrained by env. data

These are the scores that will be plotted later.

Axis 1 Axis 2 Axis 3 Axis 4
1 -0.886 -0.433 -1.277 -0.233
2 -1.043 0.105 -0.228 -1.649
3 -0.391 -0.859 -0.948 -0.697
4 -0.398 -0.845 -0.988 -0.667
5 -1.192 -0.074 1.290 0.069
6 -1.193 -0.224 0.984 1.562
7 -0.829 0.371 0.815 0.223
8 0.845 -0.680 0.287 -0.236
9 -0.157 -0.250 -0.178 2.101
10 -0.770 0.447 -0.686 -1.215
11 -0.532 0.671 0.982 -0.913
12 0.545 -1.314 -0.112 1.362
13 0.931 -1.647 -0.693 0.027
14 2.199 0.792 2.109 -0.791
15 1.858 0.006 2.169 -0.930
16 1.398 -1.131 0.007 -0.411
17 0.040 2.145 -0.515 0.492
28 -0.315 2.272 -0.003 0.317
29 1.206 1.549 -1.452 0.558
30 1.192 1.577 -1.532 0.619

#### Canonical coefficients

These are the 'regression coefficients' that are used to calculate the scores. We can use these, and the intraset correlations to interpret our axes. Thus, the first axis is primarily related to soil moisture, whilst the second axis is related to manuring.

Spec. Axis 1 Spec. Axis 2 Spec. Axis 3 Spec. Axis 4
A1 0.124 -0.265 0.746 -0.566
Moisture 0.684 -0.368 -0.471 -0.014
Manure -0.031 -0.126 -0.531 -1.725
Hayfield -0.062 0.206 -0.235 -0.419
Haypastu 0.000 0.000 0.000 0.000
Pasture 0.210 0.204 0.354 -0.228
SF 0.207 -0.109 -0.403 0.129
BF 0.077 0.081 -0.336 -1.300
HF 0.000 0.000 0.000 0.000
NM 0.392 0.796 -0.577 -1.161

We are given information about interset and intraset correlations. This can be a little confusing.

Intraset correlations are the correlations between the environmental variables and the axis scores (similar to the structure matrix in a discriminant analysis)
Interset correlations are the correlations between the site scores (derived from the species scores) and the environmental variables. You can derive the interset correaltion from the intraset correlations if you multiply the latter by R, the species-environment correlation. For example, the intraset correlation between A1 and axis 1 is 0.563. The species-environment correlation for axis 1 is 0.958. 0.563 x 0.958 is 0.593, the interset correlation between A1 and axis 1.

Interset correlations between env. variables and site scores

Envi. Axis 1 Envi. Axis 2 Envi. Axis 3 Envi. Axis 4
A1 0.539 -0.156 0.504 -0.097
Moisture 0.883 -0.153 -0.120 0.151
Manure -0.296 -0.690 -0.169 -0.160
Hayfield -0.072 0.545 -0.216 0.251
Haypastu -0.165 -0.499 -0.113 -0.077
Pasture 0.268 -0.028 0.366 -0.187
SF 0.142 -0.627 -0.360 -0.077
BF -0.349 0.158 -0.025 -0.519
HF -0.346 -0.105 0.376 0.464
NM 0.546 0.666 0.001 0.038

Intraset correlations between env. variables and constrained site scores

Envi. Axis 1 Envi. Axis 2 Envi. Axis 3 Envi. Axis 4
A1 0.563 -0.173 0.589 -0.109
Moisture 0.922 -0.170 -0.140 0.170
Manure -0.309 -0.765 -0.197 -0.180
Hayfield -0.076 0.605 -0.252 0.282
Haypastu -0.172 -0.554 -0.132 -0.086
Pasture 0.279 -0.031 0.428 -0.211
SF 0.148 -0.696 -0.421 -0.086
BF -0.364 0.175 -0.030 -0.584
HF -0.361 -0.116 0.439 0.522
NM 0.570 0.738 0.001 0.043

#### Biplot scores for env. variables

These are the intraset correlations, they are used to place arrows on the plots (see later). These are equivalent to the loading plots used in PCA.

Axis 1 Axis 2 Axis 3 Axis 4
A1 0.563 -0.173 0.589 -0.109
Moisture 0.922 -0.170 -0.140 0.170
Manure -0.309 -0.765 -0.197 -0.180
Hayfield -0.076 0.605 -0.252 0.282
Haypastu -0.172 -0.554 -0.132 -0.086
Pasture 0.279 -0.031 0.428 -0.211
SF 0.148 -0.696 -0.421 -0.086
BF -0.364 0.175 -0.030 -0.584
HF -0.361 -0.116 0.439 0.522
NM 0.570 0.738 0.001 0.043

#### Centroids of env. variables

Axis 1 Axis 2 Axis 3 Axis 4
A1 0.224 -0.069 0.234 -0.043
Moisture 0.570 -0.105 -0.087 0.105
Manure -0.222 -0.548 -0.141 -0.129
Hayfield -0.106 0.845 -0.353 0.394
Haypastu -0.204 -0.658 -0.157 -0.103
Pasture 0.488 -0.055 0.748 -0.369
SF 0.228 -1.068 -0.646 -0.133
BF -0.803 0.386 -0.066 -1.288
HF -0.538 -0.173 0.654 0.778
NM 1.073 1.388 0.002 0.080

The lengths and positions of the arrows provide information about the relationship between the original environmental variables and the derived axes. Arrows that are parallel to an axis (e.g. moisture and axis 1) indicate a correlation, the length of the arrow tells us about the strength of that correlation. Thus, pasture is related to axis 1 but not as strongly as moisture. Neither of these is related to axis 2.

On the above plot only the site scores are shown, as blue dots. (We could also plot the species scores). Consider site 17, it has a high score of axis 2. What does this mean? Axis 2 is associated with manuring, however because the correlation between manure and axis 2 is negative (also shown the direction of the manure arrow), large positive scores on axis 2 should have low values for manure, whilst large negative values on axis 2 should have lots of manure. Thus we would expect site 17 to have little manuring, whilst site 13 would have lots. Similarly, sites 5 and 16 are at opposite ends of the moistrure gradient.

The second output is from the free plrcacca program. Although the scores differ from those produced by MVSP they are almost perfectly correlated and hence they 'tell the same story'.

```Linear ordination and canonical analysis Vladimir Makarenkov
- Pierre Legendre, Universite de Montreal
Input file: dune.dat
20 objects
30 response variables (matrix Y)
10 explanatory variables (matrix X)
Maximum number of canonical eigenvalues = 30
Maximum number of non-canonical eigenvalues = 30
Total inertia (total CA variance) = 2.11526
Mean coefficient of multiple determination R^2 = 0.56612
(CCA) Percentage of the total variance of Q (xi-square of Y) accounted for = 57.98499
Mean squared difference = 0.00148121559
*** Canonical correspondence analysis ***
Canonical eigenvalues
0.4646 0.2983 0.1586 0.1372 0.0667 0.0410 0.0325 0.0268 0.0005
% of total variance of CA
21.9658 14.1037 7.4999 6.4882 3.1537 1.9407 1.5398 1.2680 0.0248
Cumulative % of total variance of CA
21.96585 36.06958 43.56954 50.05776 53.21155 55.15226 56.69207 57.96014 57.98495
Cumulative % of canonical variance
37.8819 62.2051 75.1394 86.3289 91.7679 95.1148 97.7703 99.9572 100.000
Sum of all canonical eigenvalues 1.22653
Scaling = 1: sites at centroids of species
Species scores (matrice V)```

The first two axes have correlations of -0.999 and -0.995 with the CCA variable scores from MVSP

 1.21565 -0.78363 -0.19893 -0.79069 0.43849 -1.28333 -0.12902 0.13564 1.54783 -1.14829 0.87988 0.27379 -0.28977 0.22047 0.62641 0.83905 0.27188 -0.40198 -1.05655 -3.75029 2.81111 0.97933 0.92634 1.40561 1.29766 -0.81923 3.45419 -0.4972 1.65646 1.07352 0.2468 -0.81682 0.02558 0.44513 0.213 -0.01585 0.53383 -1.58228 0.18821 0.67246 1.32327 -0.46331 0.59752 2.06375 1.11378 1.03117 -0.33883 0.14021 -1.95396 1.39977 -0.95132 2.15569 0.80608 -1.85766 1.15929 -0.05188 -0.09296 -2.21447 0.90476 -2.3091 -0.92637 0.73299 -1.94803 -1.31802 2.83783 2.26299 -0.26115 0.62653 -2.58721 2.45388 -1.3534 3.49615 0.4961 1.32809 1.73922 -2.3253 1.73357 4.26325 3.20684 -0.85762 -5.13821 -2.21725 0.2586 -1.49957 -0.78658 0.36583 0.74747 -0.83948 1.16241 0.45139 0.97044 0.698 0.91123 -0.32295 1.58675 0.41997 1.1992 0.5562 1.47328 -1.79363 -2.80624 3.7553 0.6967 1.77031 -0.95151 -2.69889 -1.99534 -2.91277 -0.79251 -2.95311 1.75187 0.26687 -1.61935 -0.02451 -0.41158 -0.74493 1.61222 -1.37821 0.08261 0.09134 0.89111 1.62026 -1.26754 -3.20518 0.66986 0.44244 -0.27525 1.43853 0.95938 2.99808 -1.69221 -1.93412 0.72045 0.39065 1.48854 0.01179 -0.78947 -0.17693 -0.11584 -0.26215 -0.42628 -0.05183 -0.73154 -0.22464 0.94528 0.13211 0.01178 -0.89589 -0.81375 1.26357 -0.88564 -0.89847 0.60622 1.16305 -0.98804 -1.44401 0.81191 -0.38997 1.01399 0.33839 1.81208 -0.12225 0.7069 0.06445 0.20818 -0.5425 -0.19691 0.86578 -0.61789 -0.15328 0.56336 0.38322 0.94328 0.39658 -0.14167 0.5944 -0.336 -0.2809 0.06489 -0.02941 -3.03005 -0.37537 -5.81484 -1.21222 2.53338 -1.18835 4.94791 -5.41526 3.35132 -2.02387 -0.08055 -0.26429 -0.38995 1.05243 -0.10776 -1.13099 -0.72685 0.35691 1.16021 0.5156 -1.39755 2.95236 0.97938 0.27522 -0.04154 0.43887 -0.79921 -0.41799 0.69535 1.25801 0.6025 -1.1993 -0.51082 0.14818 -0.59436 -0.73879 -1.12004 -3.15158 2.685 0.78511 1.08951 0.70348 0.37957 -0.9016 -2.60232 1.54497 0.29594 -1.97287 2.89017 1.12479 1.95718 -1.33612 -2.53525 -1.84652 0.003 -0.15742 -0.83743 -0.12601 -0.69555 -1.47527 -0.17901 -0.74859 -0.43475 0.85737 -1.7698 -1.3366 -1.5987 -5.85915 -2.53949 0.89542 2.7212 0.04874 -0.18508 -0.39085 -0.28432 0.65657 -0.53802 0.05001 1.05423 -0.35396 -0.58319 -2.54556 -0.70573 -1.11287 -0.5876 -1.76386 1.99832 -0.14557 2.74732 -1.12204

#### Site scores (matrice F)

 0.85047 0.28208 0.28635 -0.6004 0.0828 0.66848 -0.25362 -0.24512 0.77106 0.58513 0.12659 0.09142 -0.63668 0.19912 -0.38781 -0.04664 -0.08676 -0.02568 0.22589 0.5332 0.33572 -0.31849 -0.02454 0.21633 0.17846 -0.07551 0.10629 0.16155 0.47912 0.41926 -0.4824 0.11736 0.21684 0.32462 -0.08335 -0.39549 0.76303 -0.11787 -0.32591 0.30391 0.4794 -0.0183 0.11865 0.32324 -0.07011 0.68332 -0.15464 -0.61344 0.66777 0.07936 0.27146 -0.13999 -0.17026 -0.19457 0.69349 -0.06145 -0.33838 0.31219 -0.01913 0.07664 -0.20437 0.04683 0.01841 -0.4633 0.38727 0.06273 -0.11721 -0.01176 0.059 -0.45263 -0.0377 0.07438 0.04889 0.54069 0.30566 0.39603 0.09223 -0.23024 -0.06804 0.05287 0.30079 0.64557 -0.33381 -0.21947 -0.47219 -0.06725 -0.41751 -0.08464 0.19443 0.00338 0.3611 -0.49437 -0.14389 -0.1811 -0.95497 0.08885 -0.09285 -0.18818 0.10045 -0.23423 0.75704 0.38116 0.64492 -0.50415 -0.36975 0.36461 -0.05745 -0.12966 -0.31402 0.8421 0.47945 0.18484 -0.14198 -0.34734 0.14682 -0.0934 0.14097 -1.40461 -0.07101 -1.12047 -0.45199 -0.09318 0.04968 0.24467 -0.06301 0.00284 -1.33198 0.0369 -0.89632 -0.14584 0.49441 -0.09911 0.04832 -0.30446 0.26852 -1.31993 0.39838 -0.29464 -0.14728 0.05798 0.36278 -0.23397 0.59292 -0.09039 0.26167 -1.65292 0.42982 0.2967 0.21886 0.02557 0.2573 0.49371 1.17021 0.22276 -0.77837 -0.03521 -0.03802 -0.33738 0.039 0.34978 -0.09151 -0.53443 -0.44119 -1.66293 1.08696 0.43681 -0.04489 -0.14329 0.09369 -0.43078 0.09881 -1.3604 -0.50243 0.10562 0.06345 0.34632 0.30583 -0.35268 0.21407 -0.52151

#### Site scores (matrice Z)/ linear comb. of explanatory var.

The first two axes have correlations of -0.997 and -0.987 with the site scores (constrained by env. data) from MVSP

 0.78772 0.35179 0.57119 -0.31768 -0.03759 0.40864 0.16853 -0.10279 0.03124 0.70857 -0.085 -0.01239 -0.57622 -0.01781 -0.23383 -0.11018 -0.25914 0.02236 0.24642 0.42746 0.27907 -0.34797 0.1319 0.18341 0.10899 -0.01854 0.02134 0.23428 0.39579 0.27689 -0.3208 0.11742 0.17515 0.09859 -0.02167 0.02011 0.80152 0.09674 -0.46026 0.12651 0.43838 -0.06349 0.11487 0.32058 0.0228 0.76596 0.16261 -0.24008 0.60904 -0.01088 0.02805 -0.03339 -0.25505 0.02166 0.53089 -0.10787 -0.34358 0.12834 -0.06555 0.35552 -0.25361 0.01702 0.02131 -0.50186 0.30564 -0.08768 -0.05616 0.22459 0.10151 -0.47324 0.01405 0.0223 0.11792 0.10934 0.19432 0.75816 0.04485 -0.23795 -0.11001 0.1248 0.0236 0.48362 -0.26589 0.155 -0.45656 -0.02368 -0.39851 -0.02952 0.18532 0.02219 0.40741 -0.33068 -0.45904 -0.26849 -0.75695 -0.10959 -0.08395 0.01987 0.02361 -0.33027 0.70334 0.2053 0.53765 -0.40588 -0.11786 0.23848 -0.0687 0.02392 -0.60862 0.84619 0.35999 -0.03751 0.04357 -0.10607 0.07407 -0.03497 0.02464 -1.58806 -0.34502 -0.92953 -0.15475 -0.13299 0.12638 0.05337 0.00463 0.02282 -1.22013 0.12021 -0.91349 -0.18132 0.47454 -0.22368 0.25723 -0.29247 0.02631 -0.98762 0.61183 0.02419 -0.17952 -0.31217 0.13894 -0.00269 0.3062 0.02238 0.02657 -1.54171 0.22226 0.19092 -0.02088 0.20298 0.23166 0.02668 0.02963 0.31012 -1.18637 -0.0812 0.10922 -0.01881 0.20123 0.29063 0.07222 0.02192 -0.82961 -0.83761 0.59673 0.09395 0.11987 -0.03893 -0.09377 -0.05219 0.02128 -0.8249 -0.85501 0.62993 0.11148 0.10316 -0.03355 -0.10381 -0.06229 0.02125

#### Biplot scores of explanatory variables; multiple linear correlations

between the explanatory variables X and the site scores Z

 -0.26628 0.14186 -0.16881 0.06005 0.04934 -0.01928 0.06065 -0.0464 0.00492 -0.43487 0.08712 0.12015 0.0404 0.0182 -0.03644 -0.03866 0.02102 0.00119 0.22172 0.30155 0.09256 -0.10161 0.04274 0.09264 -0.04212 0.01393 -0.00203 0.03567 -0.23375 0.07206 0.06409 0.05387 -0.05018 0.0119 0.05406 -0.00023 0.08216 0.20558 0.04001 -0.03088 0.01275 0.00263 0.04887 -0.07611 0.00168 -0.13306 0.02168 -0.12484 -0.03507 -0.0737 0.05208 -0.0689 0.02764 -0.00167 -0.06679 0.25036 0.11758 -0.04846 -0.02974 0.04281 0.05211 0.00594 0.00133 0.16844 -0.06811 -0.02309 -0.13931 -0.06987 -0.08129 -0.02108 -0.0065 -0.00055 0.16598 0.05325 -0.08807 0.15127 0.05681 0.01491 -0.06316 0.01679 -0.00232 -0.26457 -0.27381 -0.01041 0.01101 0.03281 0.00993 0.03217 -0.01941 0.00163

#### Biplot scores of centroids of binary explanatory variables

 0.07627 -0.47548 0.14743 0.12859 0.11124 -0.10185 0.01831 0.11119 0.02276 0.14577 0.35487 0.07012 -0.05504 0.0238 0.00468 0.07855 -0.13021 0.02308 -0.33436 0.05467 -0.31627 -0.09078 -0.18553 0.1325 -0.18088 0.07157 0.0224 -0.14538 0.55869 0.26357 -0.10989 -0.06464 0.09575 0.11045 0.0146 0.02309 0.54353 -0.21868 -0.07303 -0.44807 -0.22213 -0.26037 -0.07336 -0.01948 0.02264 0.36312 0.11488 -0.18969 0.32584 0.12476 0.03241 -0.14252 0.03769 0.02233 -0.71983 -0.7493 -0.02749 0.02846 0.0915 0.02728 0.08208 -0.05175 0.02323

Percentage of variance associated with matrix Y (not permuted)
Percentage of variance (linear regression) = 57.66018
Permutation tests begin

P(Lin)= 0.00990

## Examples from the literature.

Rottenborn, SC. 1999. Predicting the impacts of urbanization on riparian bird communities. Biological Conservation, 88: 289-299.
French, BW, Elliott, NC. 1999. Temporal and spatial distribution of ground beetle (Coleoptera:Carabidae) assemblages in grasslands and adjacent wheat fields. Pedobiologia, 43: 73-84.
French, BW, Elliott, NC, Berberet, RC. 1998. Reverting conservation reserve program lands to wheat and livestock production: Effects on ground beetle (Coleoptera: Carabidae) assemblages. Environmental Entomology, 27: 1323-1335.
Lara, EN, Gonzalez, EA. 1998. The relationship between reef fish community structure and environmental variables in the southern Mexican Caribbean. Journal of Fish Biology, 53: 209-221.
Syms, C. 1998. Disturbance and the structure of coral reef fish communities on the reef slope. Journal of Experimental Marine Biology and Ecology, 230: 151-167.
OliveiraFilho, AT, Curi, N, Vilela, EA, Carvalho, DA. 1998. Effects of canopy gaps, topography, and soils on the distribution of woody species in a central Brazilian deciduous dry forest. Biotropica, 30: 362-375.
Lammerts, EJ, Grootjans, AP. 1998. Key environmental variables determining the occurrence and life span of basiphilous dune slack vegetation. Acta Botanica Neerlandica, 47: 369-392.
Read, HJ, Martin, MH, Rayner, JMV. 1998. Invertebrates in woodlands polluted by heavy metals - An evaluation using canonical correspondence analysis. Water Air and Soil Pollution, 106: 17-42.
Clenaghan, C, Giller, PS, OHalloran, J, Hernan, R. 1998. Stream macroinvertebrate communities in a conifer-afforested catchment in Ireland: relationships to physico-chemical and biotic factors. Freshwater Biology, 1998,40: 175-193.
Izhaki, I, Adar, M. 1997. The influence of land use and physico-chemical factors on the effects of post-fire management on bird community succession. International Journal of Wildland Fire, 7: 335-342.

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### Resources

Jongman, R.H.G., ter Braak, C.J.F. and van Tongeren, O.F.R. (eds) 1995. Data analysis in community and landscape ecology. Cambridge University Press. (ter Braak developed the CCA algorithm)

Pierre Legendre's free plrcacca software can be used to carry out a standard CCA (and redundancy analysis) analysis. In addition it is able to carry out an extension to the normal CCA algorithm in that the linear regression model can be replaced by a polynomial regression.

The ordination web site has a detailed list of available software.

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