Workshop 14: R-mode MVA

Murray Logan

06 Aug 2016

> # might want to put this in to make the vector arrows and text look nicer..
> geom_text(data=vectors, aes(y=RDA2,x=RDA1,label=Species,
+               hjust=0.5*(1-sign(RDA2)),vjust=0.5*(1-sign(RDA1))), color="red", size=4)
Error in eval(expr, envir, enclos): could not find function "geom_text"

R-mode

Sp1 Sp2 Sp3 Sp4
Site1 2 0 0 5
Site2 13 7 10 5
Site3 9 5 55 93
Site4 10 6 76 81
Site5 0 2 6 0

R-mode

Sp1 Sp2 Sp3 Sp4
Site1 2 0 0 5
Site2 13 7 10 5
Site3 9 5 55 93
Site4 10 6 76 81
Site5 0 2 6 0

R-mode

Sp1 Sp2 Sp3 Sp4
Site1 2 0 0 5
Site2 13 7 10 5
Site3 9 5 55 93
Site4 10 6 76 81
Site5 0 2 6 0

Axis rotation - eigenanalysis

Sp1 Sp2 Sp3
1 18.0 12.0 3.0
2 23.0 10.0 4.0
3 21.0 10.0 6.0
4 18.0 13.0 4.0
5 22.0 6.0 9.0
6 21.0 9.0 7.0
7 22.0 9.0 7.0
8 20.0 11.0 1.0
9 21.0 9.0 5.0
10 17.0 10.0 5.0

Axes rotation - eigenanalysis

Axes rotation - eigenanalysis

Axes rotation - eigenanalysis

Axes rotation - eigenanalysis

Axes rotation - eigenanalysis

Axes rotation - eigenanalysis

Principle components analysis


Principle components analysis


Sites Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10
Site1 5 0 0 65 5 0 0 0 0 0
Site2 0 0 0 25 39 0 6 23 0 0
Site3 0 0 0 6 42 0 6 31 0 0
Site4 0 0 0 0 0 0 0 40 0 14
Site5 0 0 6 0 0 0 0 34 18 12
Site6 0 29 12 0 0 0 0 0 22 0
Site7 0 0 21 0 0 5 0 0 20 0
Site8 0 0 0 0 13 0 6 37 0 0
Site9 0 0 0 60 47 0 4 0 0 0
Site10 0 0 0 72 34 0 0 0 0 0

Principle components analysis

> library(vegan)
> data.rda <- rda(data[,-1], scale=TRUE)
> summary(data.rda, scaling=2)

Call:
rda(X = data[, -1], scale = TRUE) 

Partitioning of correlations:
              Inertia Proportion
Total              10          1
Unconstrained      10          1

Eigenvalues, and their contribution to the correlations 

Importance of components:
                         PC1    PC2    PC3    PC4     PC5
Eigenvalue            3.8220 2.4205 1.6753 1.1701 0.66872
Proportion Explained  0.3822 0.2420 0.1675 0.1170 0.06687
Cumulative Proportion 0.3822 0.6242 0.7918 0.9088 0.97567
                          PC6     PC7    PC8      PC9
Eigenvalue            0.14643 0.06784 0.0280 0.001045
Proportion Explained  0.01464 0.00678 0.0028 0.000100
Cumulative Proportion 0.99031 0.99710 0.9999 1.000000

Scaling 2 for species and site scores
* Species are scaled proportional to eigenvalues
* Sites are unscaled: weighted dispersion equal on all dimensions
* General scaling constant of scores:  3.08007 


Species scores

         PC1       PC2      PC3      PC4      PC5       PC6
Sp1  -0.1300 -0.494334 -0.63551  0.08911 -0.50512  0.112154
Sp2   0.5000 -0.157276  0.11701 -0.79900 -0.06343 -0.104375
Sp3   0.9007 -0.170843  0.27779  0.16895 -0.01456  0.038032
Sp4  -0.5235 -0.714165 -0.27820  0.04967  0.27670  0.027303
Sp5  -0.7564 -0.196662  0.47970 -0.01385  0.26323  0.128107
Sp6   0.6212 -0.213901  0.31704  0.63259 -0.04035 -0.101082
Sp7  -0.6311  0.231834  0.61283 -0.01782 -0.32040  0.085889
Sp8  -0.2207  0.925419 -0.04869  0.06432 -0.17279  0.006936
Sp9   0.9110 -0.009745  0.12221 -0.14486  0.05298  0.274048
Sp10  0.1964  0.702341 -0.54197  0.08330  0.30368  0.063979


Site scores (weighted sums of species scores)

            PC1     PC2     PC3       PC4     PC5     PC6
Site1  -0.38988 -1.4830 -1.9065  0.267341 -1.5154  0.3365
Site2  -0.88111  0.1838  0.8579 -0.024947 -0.2562  0.4380
Site3  -0.85013  0.5434  0.9979 -0.028670 -0.5856  0.4768
Site4   0.08351  1.5556 -1.2827  0.242057  0.5852 -1.5801
Site5   0.75662  1.2398 -0.8607  0.079914  0.6381  2.1217
Site6   1.50003 -0.4718  0.3510 -2.396989 -0.1903 -0.3131
Site7   1.86374 -0.6417  0.9511  1.897781 -0.1210 -0.3032
Site8  -0.53994  0.8632  0.5845  0.000059 -1.3634 -0.8814
Site9  -0.93105 -0.7920  0.5521 -0.042154  1.1052  0.5402
Site10 -0.61180 -0.9973 -0.2446  0.005608  1.7033 -0.8354

Principle components analysis

Component loadings

> scores(data.rda, choices=1:4,display="species",
+        scaling=0)
             PC1          PC2         PC3         PC4
Sp1  -0.06824908 -0.326218616 -0.50409421  0.08457936
Sp2   0.26258604 -0.103789073  0.09281608 -0.75834235
Sp3   0.47302827 -0.112742127  0.22034809  0.16035284
Sp4  -0.27494466 -0.471288231 -0.22067476  0.04713954
Sp5  -0.39723499 -0.129780438  0.38050677 -0.01314949
Sp6   0.32625468 -0.141156180  0.25148155  0.60040642
Sp7  -0.33140801  0.152990372  0.48610356 -0.01691299
Sp8  -0.11589293  0.610697873 -0.03862036  0.06104665
Sp9   0.47844118 -0.006430814  0.09693508 -0.13748918
Sp10  0.10312211  0.463485199 -0.42989602  0.07906481
attr(,"const")
[1] 3.08007

Principle components analysis

Component loadings

> scores(data.rda, choices=1:4,display="sites",
+        scaling=0)
               PC1         PC2         PC3           PC4
Site1  -0.12658001 -0.48148365 -0.61898528  8.679697e-02
Site2  -0.28606925  0.05966779  0.27853408 -8.099421e-03
Site3  -0.27600945  0.17642036  0.32398109 -9.308103e-03
Site4   0.02711372  0.50505835 -0.41646556  7.858829e-02
Site5   0.24564899  0.40253243 -0.27942593  2.594538e-02
Site6   0.48701231 -0.15318789  0.11397034 -7.782255e-01
Site7   0.60509708 -0.20834002  0.30879819  6.161486e-01
Site8  -0.17530145  0.28025505  0.18975585  1.915695e-05
Site9  -0.30228092 -0.25712656  0.17925008 -1.368605e-02
Site10 -0.19863101 -0.32379586 -0.07941286  1.820656e-03
attr(,"const")
[1] 3.08007

Principle components analysis

Species scores

> summary(data.rda, scaling=2)$species
            PC1          PC2         PC3         PC4
Sp1  -0.1299584 -0.494334499 -0.63550605  0.08911359
Sp2   0.5000107 -0.157276491  0.11701221 -0.79899645
Sp3   0.9007303 -0.170843477  0.27779043  0.16894923
Sp4  -0.5235437 -0.714165347 -0.27820226  0.04966665
Sp5  -0.7564064 -0.196662436  0.47970072 -0.01385442
Sp6   0.6212472 -0.213900635  0.31704004  0.63259371
Sp7  -0.6310600  0.231833546  0.61282545 -0.01781968
Sp8  -0.2206808  0.925419372 -0.04868826  0.06431931
Sp9   0.9110374 -0.009744917  0.12220500 -0.14485986
Sp10  0.1963629  0.702341045 -0.54196521  0.08330341
             PC5          PC6
Sp1  -0.50512223  0.112154283
Sp2  -0.06343012 -0.104375378
Sp3  -0.01455567  0.038032290
Sp4   0.27669707  0.027303319
Sp5   0.26323433  0.128106871
Sp6  -0.04034569 -0.101081509
Sp7  -0.32040479  0.085888650
Sp8  -0.17279101  0.006935947
Sp9   0.05297717  0.274047718
Sp10  0.30367672  0.063978952

Principle components analysis

Species scores

> summary(data.rda, scaling=1)$species
            PC1         PC2        PC3         PC4
Sp1  -0.2102120 -1.00477627 -1.5526456  0.26051036
Sp2   0.8087835 -0.31967764  0.2858800 -2.33574773
Sp3   1.4569603 -0.34725367  0.6786876  0.49389803
Sp4  -0.8468489 -1.45160088 -0.6796938  0.14519308
Sp5  -1.2235117 -0.39973287  1.1719876 -0.04050134
Sp6   1.0048873 -0.43477095  0.7745809  1.84929399
Sp7  -1.0207599  0.47122110  1.4972331 -0.05209320
Sp8  -0.3569584  1.88099237 -0.1189534  0.18802797
Sp9   1.4736325 -0.01980736  0.2985669 -0.42347634
Sp10  0.3176233  1.42756699 -1.3241099  0.24352519
             PC5         PC6
Sp1  -1.95331854  0.92683727
Sp2  -0.24528563 -0.86255279
Sp3  -0.05628709  0.31429690
Sp4   1.06999353  0.22563323
Sp5   1.01793280  1.05866864
Sp6  -0.15601764 -0.83533243
Sp7  -1.23901222  0.70977941
Sp8  -0.66818654  0.05731831
Sp9   0.20486387  2.26471634
Sp10  1.17432441  0.52871879

Principle components analysis

Site scores

> summary(data.rda, scaling=2)$sites
               PC1        PC2        PC3           PC4
Site1  -0.38987532 -1.4830035 -1.9065182  2.673408e-01
Site2  -0.88111339  0.1837810  0.8579046 -2.494679e-02
Site3  -0.85012850  0.5433871  0.9978845 -2.866961e-02
Site4   0.08351215  1.5556152 -1.2827432  2.420575e-01
Site5   0.75661614  1.2398282 -0.8606515  7.991361e-02
Site6   1.50003214 -0.4718295  0.3510366 -2.396989e+00
Site7   1.86374152 -0.6417019  0.9511201  1.897781e+00
Site8  -0.53994078  0.8632053  0.5844614  5.900474e-05
Site9  -0.93104649 -0.7919679  0.5521028 -4.215398e-02
Site10 -0.61179747 -0.9973140 -0.2445972  5.607748e-03
              PC5        PC6
Site1  -1.5153667  0.3364629
Site2  -0.2561519  0.4380054
Site3  -0.5855919  0.4768201
Site4   0.5851709 -1.5800521
Site5   0.6380907  2.1216597
Site6  -0.1902904 -0.3131261
Site7  -0.1210371 -0.3032445
Site8  -1.3633582 -0.8813587
Site9   1.1052341  0.5401866
Site10  1.7033003 -0.8353532

Principle components analysis

Site scores

> summary(data.rda, scaling=1)$sites
               PC1         PC2        PC3           PC4
Site1  -0.24103094 -0.72961495 -0.7803480  9.145008e-02
Site2  -0.54472693  0.09041742  0.3511449 -8.533624e-03
Site3  -0.52557128  0.26733811  0.4084394 -9.807103e-03
Site4   0.05162936  0.76533880 -0.5250336  8.280135e-02
Site5   0.46775954  0.60997643 -0.3522692  2.733629e-02
Site6   0.92735841 -0.23213286  0.1436812 -8.199456e-01
Site7   1.15221289 -0.31570749  0.3892985  6.491799e-01
Site8  -0.33380527  0.42468374  0.2392231  2.018393e-05
Site9  -0.57559686 -0.38963603  0.2259786 -1.441974e-02
Site10 -0.37822892 -0.49066318 -0.1001149  1.918260e-03
               PC5         PC6
Site1  -0.39186922  0.04071454
Site2  -0.06624011  0.05300195
Site3  -0.15143228  0.05769882
Site4   0.15132341 -0.19119820
Site5   0.16500833  0.25673679
Site6  -0.04920851 -0.03789062
Site7  -0.03129982 -0.03669487
Site8  -0.35256027 -0.10665104
Site9   0.28581018  0.06536664
Site10  0.44046828 -0.10108402

Axes retention

        PC1         PC2         PC3         PC4         PC5 
3.822030054 2.420488947 1.675308207 1.170140005 0.668723871 
        PC6         PC7         PC8         PC9 
0.146428210 0.067836468 0.027999675 0.001044563

Axes retention

        PC1         PC2         PC3         PC4         PC5 
3.822030054 2.420488947 1.675308207 1.170140005 0.668723871 
        PC6         PC7         PC8         PC9 
0.146428210 0.067836468 0.027999675 0.001044563 
   PC1    PC2    PC3    PC4    PC5    PC6    PC7    PC8 
 38.22  62.43  79.18  90.88  97.57  99.03  99.71  99.99 
   PC9 
100.00

Axes retention

        PC1         PC2         PC3         PC4         PC5 
3.822030054 2.420488947 1.675308207 1.170140005 0.668723871 
        PC6         PC7         PC8         PC9 
0.146428210 0.067836468 0.027999675 0.001044563 
   PC1    PC2    PC3    PC4    PC5    PC6    PC7    PC8 
 38.22  62.43  79.18  90.88  97.57  99.03  99.71  99.99 
   PC9 
100.00

Scree plot

Ordination plot

> plot(data.rda)

Biplot

Environmental correlates


Site pH Slope Pressure Altitude Substrate
Site1 6 4 101325 2 Quartz
Site2 7 9 101352 510 Shale
Site3 7 9 101356 546 Shale
Site4 7 7 101372 758 Shale
Site5 7 6 101384 813 Shale
Site6 8 8 101395 856 Quartz
Site7 8 0 101396 854 Quartz
Site8 7 12 101370 734 Shale
Site9 8 8 101347 360 Quartz
Site10 6 2 101345 356 Quartz

Environmental correlates

PC1

Environmental correlates

PC2

Environmental correlates

PC1

> library(car)
> vif(lm(data.rda$CA$u[,1]~Slope+Altitude+Pressure+
+   Substrate+pH, data=enviro))
    Slope  Altitude  Pressure Substrate        pH 
 2.187796 52.754368 45.804821  5.118418  1.976424

Environmental correlates

PC1

> library(car)
> vif(lm(data.rda$CA$u[,1]~Slope+Altitude+Pressure+
+   Substrate+pH, data=enviro))
    Slope  Altitude  Pressure Substrate        pH 
 2.187796 52.754368 45.804821  5.118418  1.976424 
> vif(lm(data.rda$CA$u[,1]~Slope+Altitude+Substrate+
+   pH, data=enviro))
    Slope  Altitude Substrate        pH 
 2.116732  1.830014  2.636302  1.968157

Environmental correlates

Multivariate regression

Three responses

> data.lm<-lm(data.rda$CA$u[,1:3]~Slope+Altitude+Substrate+
+   pH, data=enviro)

Environmental correlates

> summary(data.lm)[[1]]$coef
                   Estimate   Std. Error    t value
(Intercept)     0.180787455 0.8121647657  0.2225995
Slope          -0.021324431 0.0259195942 -0.8227147
Altitude        0.001127743 0.0003074246  3.6683553
SubstrateShale -0.326063277 0.1951991407 -1.6704135
pH             -0.075386061 0.1322828004 -0.5698856
                 Pr(>|t|)
(Intercept)    0.83265462
Slope          0.44811475
Altitude       0.01446816
SubstrateShale 0.15570393
pH             0.59340544

Environmental correlates

> summary(data.lm)[[1]]$coef
                   Estimate   Std. Error    t value
(Intercept)     0.180787455 0.8121647657  0.2225995
Slope          -0.021324431 0.0259195942 -0.8227147
Altitude        0.001127743 0.0003074246  3.6683553
SubstrateShale -0.326063277 0.1951991407 -1.6704135
pH             -0.075386061 0.1322828004 -0.5698856
                 Pr(>|t|)
(Intercept)    0.83265462
Slope          0.44811475
Altitude       0.01446816
SubstrateShale 0.15570393
pH             0.59340544
> summary(data.lm)[[2]]$coef
                    Estimate  Std. Error    t value
(Intercept)    -0.8361050965 0.513155076 -1.6293420
Slope          -0.0074535542 0.016376937 -0.4551250
Altitude        0.0003381004 0.000194242  1.7406140
SubstrateShale  0.5480944974 0.123333878  4.4439898
pH              0.0589810882 0.083581058  0.7056753
                  Pr(>|t|)
(Intercept)    0.164169419
Slope          0.668099646
Altitude       0.142232378
SubstrateShale 0.006739835
pH             0.511901060

Environmental correlates

> summary(data.lm)[[1]]$coef
                   Estimate   Std. Error    t value
(Intercept)     0.180787455 0.8121647657  0.2225995
Slope          -0.021324431 0.0259195942 -0.8227147
Altitude        0.001127743 0.0003074246  3.6683553
SubstrateShale -0.326063277 0.1951991407 -1.6704135
pH             -0.075386061 0.1322828004 -0.5698856
                 Pr(>|t|)
(Intercept)    0.83265462
Slope          0.44811475
Altitude       0.01446816
SubstrateShale 0.15570393
pH             0.59340544
> summary(data.lm)[[2]]$coef
                    Estimate  Std. Error    t value
(Intercept)    -0.8361050965 0.513155076 -1.6293420
Slope          -0.0074535542 0.016376937 -0.4551250
Altitude        0.0003381004 0.000194242  1.7406140
SubstrateShale  0.5480944974 0.123333878  4.4439898
pH              0.0589810882 0.083581058  0.7056753
                  Pr(>|t|)
(Intercept)    0.164169419
Slope          0.668099646
Altitude       0.142232378
SubstrateShale 0.006739835
pH             0.511901060
> summary(data.lm)[[3]]$coef
                    Estimate   Std. Error    t value
(Intercept)    -1.1148132894 1.6441465097 -0.6780498
Slope           0.0225742272 0.0524716316  0.4302177
Altitude        0.0003341175 0.0006223505  0.5368638
SubstrateShale -0.0908322738 0.3951611784 -0.2298613
pH              0.1162957219 0.2677933269  0.4342742
                Pr(>|t|)
(Intercept)    0.5278434
Slope          0.6849494
Altitude       0.6143806
SubstrateShale 0.8273072
pH             0.6821907

Horseshoe effect

Horseshoe effect

Horseshoe effect

Principle components analysis


Sites Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10
Site1 5 0 0 65 5 0 0 0 0 0
Site2 0 0 0 25 39 0 6 23 0 0
Site3 0 0 0 6 42 0 6 31 0 0
Site4 0 0 0 0 0 0 0 40 0 14
Site5 0 0 6 0 0 0 0 34 18 12
Site6 0 29 12 0 0 0 0 0 22 0
Site7 0 0 21 0 0 5 0 0 20 0
Site8 0 0 0 0 13 0 6 37 0 0
Site9 0 0 0 60 47 0 4 0 0 0
Site10 0 0 0 72 34 0 0 0 0 0

Principle components analysis

Sites Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10
Site1 5 0 0 65 5 0 0 0 0 0
Site2 0 0 0 25 39 0 6 23 0 0
Site3 0 0 0 6 42 0 6 31 0 0
Site4 0 0 0 0 0 0 0 40 0 14
Site5 0 0 6 0 0 0 0 34 18 12
Site6 0 29 12 0 0 0 0 0 22 0
Site7 0 0 21 0 0 5 0 0 20 0
Site8 0 0 0 0 13 0 6 37 0 0
Site9 0 0 0 60 47 0 4 0 0 0
Site10 0 0 0 72 34 0 0 0 0 0
Site pH Slope Pressure Altitude Substrate
Site1 6 4 101325 2 Quartz
Site2 7 9 101352 510 Shale
Site3 7 9 101356 546 Shale
Site4 7 7 101372 758 Shale
Site5 7 6 101384 813 Shale
Site6 8 8 101395 856 Quartz
Site7 8 0 101396 854 Quartz
Site8 7 12 101370 734 Shale
Site9 8 8 101347 360 Quartz
Site10 6 2 101345 356 Quartz

Redundancy analysis

Redundancy analysis

Output summary


Call:
rda(formula = data.sp ~ Slope + Altitude + Substrate + pH, data = enviro,      scale = TRUE) 

Partitioning of correlations:
              Inertia Proportion
Total          10.000     1.0000
Constrained     6.803     0.6803
Unconstrained   3.197     0.3197

Eigenvalues, and their contribution to the correlations 

Importance of components:
                        RDA1   RDA2    RDA3    RDA4    PC1
Eigenvalue            3.2736 2.3246 0.94753 0.25721 1.6686
Proportion Explained  0.3274 0.2325 0.09475 0.02572 0.1669
Cumulative Proportion 0.3274 0.5598 0.65457 0.68029 0.8471
                          PC2     PC3     PC4     PC5
Eigenvalue            0.98370 0.42545 0.08546 0.03393
Proportion Explained  0.09837 0.04255 0.00855 0.00339
Cumulative Proportion 0.94552 0.98806 0.99661 1.00000

Accumulated constrained eigenvalues
Importance of components:
                        RDA1   RDA2   RDA3    RDA4
Eigenvalue            3.2736 2.3246 0.9475 0.25721
Proportion Explained  0.4812 0.3417 0.1393 0.03781
Cumulative Proportion 0.4812 0.8229 0.9622 1.00000

Scaling 2 for species and site scores
* Species are scaled proportional to eigenvalues
* Sites are unscaled: weighted dispersion equal on all dimensions
* General scaling constant of scores:  3.08007 


Species scores

        RDA1     RDA2     RDA3      RDA4      PC1      PC2
Sp1  -0.4625  0.53068 -0.06866  0.253791 -0.45077 -0.42289
Sp2   0.4496  0.17767  0.70674  0.047286 -0.20609  0.09076
Sp3   0.8605  0.14459 -0.07160 -0.073481  0.01511 -0.39443
Sp4  -0.5038  0.73807 -0.17492 -0.053209  0.09486  0.28160
Sp5  -0.5412  0.06265 -0.02421 -0.300536  0.66564  0.28803
Sp6   0.6378  0.16243 -0.37542 -0.090719  0.25610 -0.50570
Sp7  -0.5134 -0.42926  0.28574 -0.206257  0.58746 -0.17367
Sp8  -0.1998 -0.91143 -0.04662  0.156707 -0.11938  0.04756
Sp9   0.8211  0.01396  0.04385 -0.058407 -0.23106 -0.21288
Sp10  0.1332 -0.53007 -0.36293  0.008524 -0.63767  0.30795


Site scores (weighted sums of species scores)

                RDA1    RDA2     RDA3    RDA4     PC1
Site Site1  -0.79327  1.6804 -0.57398  3.8613 -1.3523
Site Site2  -0.83161 -0.4019  0.46703 -2.1620  1.0017
Site Site3  -0.78597 -0.7893  0.56036 -1.9250  1.0466
Site Site4   0.03055 -1.3306 -1.05657  2.8638 -1.2801
Site Site5   0.75712 -1.0386 -0.86818  1.9420 -1.2800
Site Site6   1.58269  0.5814  2.55098  0.9117 -0.6183
Site Site7   2.06536  0.6400 -1.36534 -1.2030  0.7683
Site Site8  -0.52322 -1.0342  0.62009  0.1174  0.5119
Site Site9  -0.88977  0.6559  0.08146 -3.1438  0.1980
Site Site10 -0.61188  1.0369 -0.41586 -1.2624  1.0043
                PC2
Site Site1  -1.2687
Site Site2  -0.3103
Site Site3  -0.2950
Site Site4   0.9220
Site Site5   0.2638
Site Site6   0.2723
Site Site7  -1.5171
Site Site8  -0.5805
Site Site9   0.5848
Site Site10  1.9287


Site constraints (linear combinations of constraining variables)

               RDA1    RDA2     RDA3    RDA4     PC1
Site Site1  -1.3876  1.5920 -0.20599  0.7614 -1.3523
Site Site2  -0.8984 -0.7156  0.06891  0.3099  1.0017
Site Site3  -0.7330 -0.7750 -0.18965  0.1650  1.0466
Site Site4   0.1311 -1.0148 -0.39673  0.1773 -1.2801
Site Site5   0.4262 -1.1215 -1.11563 -0.1698 -1.2800
Site Site6   1.3488  0.5330  2.12023  0.1419 -0.6183
Site Site7   1.9135  0.4873 -1.12627 -0.2722  0.7683
Site Site8  -0.2790 -0.9677  1.35583  0.3537  0.5119
Site Site9  -0.6631  0.7367  0.11148 -2.6608  0.1980
Site Site10  0.1414  1.2455 -0.62218  1.1935  1.0043
                PC2
Site Site1  -1.2687
Site Site2  -0.3103
Site Site3  -0.2950
Site Site4   0.9220
Site Site5   0.2638
Site Site6   0.2723
Site Site7  -1.5171
Site Site8  -0.5805
Site Site9   0.5848
Site Site10  1.9287


Biplot scores for constraining variables

                  RDA1    RDA2     RDA3     RDA4 PC1 PC2
Slope          -0.4012 -0.5892  0.66815 -0.21310   0   0
Altitude        0.7686 -0.6192  0.15152 -0.05368   0   0
SubstrateShale -0.2779 -0.9434 -0.05693  0.17170   0   0
pH              0.3768 -0.1196  0.18204 -0.90031   0   0


Centroids for factor constraints

                   RDA1    RDA2     RDA3    RDA4 PC1 PC2
SubstrateQuartz  0.2706  0.9189  0.05545 -0.1672   0   0
SubstrateShale  -0.2706 -0.9189 -0.05545  0.1672   0   0

Redundancy analysis

> summary(data.rda, scaling=2)$biplot
                     RDA1       RDA2        RDA3
Slope          -0.4012090 -0.5892359  0.66814874
Altitude        0.7686367 -0.6191599  0.15151633
SubstrateShale -0.2778501 -0.9434386 -0.05693409
pH              0.3768405 -0.1195525  0.18203818
                      RDA4 PC1 PC2
Slope          -0.21309525   0   0
Altitude       -0.05367921   0   0
SubstrateShale  0.17170182   0   0
pH             -0.90031137   0   0

Redundancy analysis

biplot

Redundancy analysis

Permutation ANOVA

Permutation test for rda under reduced model
Permutation: free
Number of permutations: 999

Model: rda(formula = data.sp ~ Slope + Altitude + Substrate + pH, data = enviro, scale = TRUE)
         Df Variance      F Pr(>F)   
Model     4   6.8029 2.6598  0.007 **
Residual  5   3.1971                 
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Permutation ANOVA (Type III SS)

Permutation test for rda under reduced model
Marginal effects of terms
Permutation: free
Number of permutations: 999

Model: rda(formula = data.sp ~ Slope + Altitude + Substrate + pH, data = enviro, scale = TRUE)
          Df Variance      F Pr(>F)   
Slope      1   1.0065 1.5741  0.159   
Altitude   1   2.2199 3.4718  0.002 **
Substrate  1   1.5685 2.4531  0.033 * 
pH         1   0.4273 0.6683  0.694   
Residual   5   3.1971                 
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Redundancy analysis

Stepwise model selection

          Df    AIC      F Pr(>F)   
<none>       20.569                 
Slope      1 21.306 1.5741  0.170   
Altitude   1 23.842 3.4718  0.005 **
Substrate  1 22.561 2.4531  0.025 * 
pH         1 19.823 0.6683  0.645   
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correspondence analysis

PCA

Correspondence analysis

Correspondence analysis

Correspondence analysis

Correspondence analysis

> data <- decostand(data[,-1],method="total",MARGIN=2)
> data
       Sp1 Sp2       Sp3        Sp4        Sp5 Sp6
Site1    1   0 0.0000000 0.28508772 0.02777778   0
Site2    0   0 0.0000000 0.10964912 0.21666667   0
Site3    0   0 0.0000000 0.02631579 0.23333333   0
Site4    0   0 0.0000000 0.00000000 0.00000000   0
Site5    0   0 0.1538462 0.00000000 0.00000000   0
Site6    0   1 0.3076923 0.00000000 0.00000000   0
Site7    0   0 0.5384615 0.00000000 0.00000000   1
Site8    0   0 0.0000000 0.00000000 0.07222222   0
Site9    0   0 0.0000000 0.26315789 0.26111111   0
Site10   0   0 0.0000000 0.31578947 0.18888889   0
             Sp7       Sp8       Sp9      Sp10
Site1  0.0000000 0.0000000 0.0000000 0.0000000
Site2  0.2727273 0.1393939 0.0000000 0.0000000
Site3  0.2727273 0.1878788 0.0000000 0.0000000
Site4  0.0000000 0.2424242 0.0000000 0.5384615
Site5  0.0000000 0.2060606 0.3000000 0.4615385
Site6  0.0000000 0.0000000 0.3666667 0.0000000
Site7  0.0000000 0.0000000 0.3333333 0.0000000
Site8  0.2727273 0.2242424 0.0000000 0.0000000
Site9  0.1818182 0.0000000 0.0000000 0.0000000
Site10 0.0000000 0.0000000 0.0000000 0.0000000

Correspondence analysis

Tile plot

Correspondence analysis

Iterative re-weighting

Minimize \(\chi^2\) residuals

Correspondence analysis

CA

> data.ca <- cca(data)
> summary(data.ca)

Call:
cca(X = data) 

Partitioning of mean squared contingency coefficient:
              Inertia Proportion
Total           3.256          1
Unconstrained   3.256          1

Eigenvalues, and their contribution to the mean squared contingency coefficient 

Importance of components:
                         CA1    CA2    CA3    CA4     CA5
Eigenvalue            0.9463 0.7823 0.6214 0.5784 0.26286
Proportion Explained  0.2906 0.2402 0.1908 0.1776 0.08073
Cumulative Proportion 0.2906 0.5309 0.7217 0.8994 0.98009
                          CA6     CA7      CA8
Eigenvalue            0.04315 0.01551 0.006164
Proportion Explained  0.01325 0.00476 0.001890
Cumulative Proportion 0.99334 0.99811 1.000000

Scaling 2 for species and site scores
* Species are scaled proportional to eigenvalues
* Sites are unscaled: weighted dispersion equal on all dimensions


Species scores

         CA1     CA2      CA3      CA4      CA5       CA6
Sp1  -1.4446  1.8842 -0.86530  0.03061 -0.47720 -0.006432
Sp2   1.1313  0.5380  0.52836  1.74914 -0.05360  0.247953
Sp3   1.0302  0.3407  0.16160 -0.32148  0.00978 -0.135905
Sp4  -1.1395  0.2409  0.47543 -0.02254  1.06125  0.022650
Sp5  -0.9094 -0.6375  0.85632 -0.03372  0.35896 -0.014035
Sp6   1.1250  0.5162  0.37542 -1.61102 -0.05869  0.265755
Sp7  -0.7780 -0.9688  0.78276 -0.02684 -0.79494 -0.033065
Sp8  -0.2565 -1.0628 -0.43367  0.02821 -0.55703  0.053469
Sp9   0.9388  0.1688 -0.07722  0.11955  0.07364 -0.513328
Sp10  0.3026 -1.0196 -1.80371  0.08808  0.43783  0.112936


Site scores (weighted averages of species scores)

           CA1     CA2     CA3      CA4      CA5       CA6
Site1  -1.4446  1.8842 -0.8653  0.03061 -0.47720 -0.006432
Site2  -0.8156 -0.9072  0.8514 -0.03082 -0.51678 -0.066579
Site3  -0.7373 -1.0761  0.7693 -0.02515 -1.10800 -0.053106
Site4   0.1364 -1.3205 -2.2180  0.12014  0.49068  2.189249
Site5   0.4965 -0.6686 -1.3203  0.05068  0.37618 -2.309429
Site6   1.1313  0.5380  0.5284  1.74914 -0.05360  0.247953
Site7   1.1250  0.5162  0.3754 -1.61102 -0.05869  0.265755
Site8  -0.6227 -1.2320  0.5034 -0.01041 -2.11065  0.079747
Site9  -1.0159 -0.5055  1.1190 -0.04803  1.23098 -0.121948
Site10 -1.1131 -0.1123  0.9944 -0.04620  3.03739  0.206699

Correspondence analysis

CA

> data.ca <- cca(data)
> summary(data.ca)

Call:
cca(X = data) 

Partitioning of mean squared contingency coefficient:
              Inertia Proportion
Total           3.256          1
Unconstrained   3.256          1

Eigenvalues, and their contribution to the mean squared contingency coefficient 

Importance of components:
                         CA1    CA2    CA3    CA4     CA5
Eigenvalue            0.9463 0.7823 0.6214 0.5784 0.26286
Proportion Explained  0.2906 0.2402 0.1908 0.1776 0.08073
Cumulative Proportion 0.2906 0.5309 0.7217 0.8994 0.98009
                          CA6     CA7      CA8
Eigenvalue            0.04315 0.01551 0.006164
Proportion Explained  0.01325 0.00476 0.001890
Cumulative Proportion 0.99334 0.99811 1.000000

Scaling 2 for species and site scores
* Species are scaled proportional to eigenvalues
* Sites are unscaled: weighted dispersion equal on all dimensions


Species scores

         CA1     CA2      CA3      CA4      CA5       CA6
Sp1  -1.4446  1.8842 -0.86530  0.03061 -0.47720 -0.006432
Sp2   1.1313  0.5380  0.52836  1.74914 -0.05360  0.247953
Sp3   1.0302  0.3407  0.16160 -0.32148  0.00978 -0.135905
Sp4  -1.1395  0.2409  0.47543 -0.02254  1.06125  0.022650
Sp5  -0.9094 -0.6375  0.85632 -0.03372  0.35896 -0.014035
Sp6   1.1250  0.5162  0.37542 -1.61102 -0.05869  0.265755
Sp7  -0.7780 -0.9688  0.78276 -0.02684 -0.79494 -0.033065
Sp8  -0.2565 -1.0628 -0.43367  0.02821 -0.55703  0.053469
Sp9   0.9388  0.1688 -0.07722  0.11955  0.07364 -0.513328
Sp10  0.3026 -1.0196 -1.80371  0.08808  0.43783  0.112936


Site scores (weighted averages of species scores)

           CA1     CA2     CA3      CA4      CA5       CA6
Site1  -1.4446  1.8842 -0.8653  0.03061 -0.47720 -0.006432
Site2  -0.8156 -0.9072  0.8514 -0.03082 -0.51678 -0.066579
Site3  -0.7373 -1.0761  0.7693 -0.02515 -1.10800 -0.053106
Site4   0.1364 -1.3205 -2.2180  0.12014  0.49068  2.189249
Site5   0.4965 -0.6686 -1.3203  0.05068  0.37618 -2.309429
Site6   1.1313  0.5380  0.5284  1.74914 -0.05360  0.247953
Site7   1.1250  0.5162  0.3754 -1.61102 -0.05869  0.265755
Site8  -0.6227 -1.2320  0.5034 -0.01041 -2.11065  0.079747
Site9  -1.0159 -0.5055  1.1190 -0.04803  1.23098 -0.121948
Site10 -1.1131 -0.1123  0.9944 -0.04620  3.03739  0.206699
Inertia - \(\chi^2\) value

Correspondence analysis

CA

> data.ca <- cca(data)
> summary(data.ca)

Call:
cca(X = data) 

Partitioning of mean squared contingency coefficient:
              Inertia Proportion
Total           3.256          1
Unconstrained   3.256          1

Eigenvalues, and their contribution to the mean squared contingency coefficient 

Importance of components:
                         CA1    CA2    CA3    CA4     CA5
Eigenvalue            0.9463 0.7823 0.6214 0.5784 0.26286
Proportion Explained  0.2906 0.2402 0.1908 0.1776 0.08073
Cumulative Proportion 0.2906 0.5309 0.7217 0.8994 0.98009
                          CA6     CA7      CA8
Eigenvalue            0.04315 0.01551 0.006164
Proportion Explained  0.01325 0.00476 0.001890
Cumulative Proportion 0.99334 0.99811 1.000000

Scaling 2 for species and site scores
* Species are scaled proportional to eigenvalues
* Sites are unscaled: weighted dispersion equal on all dimensions


Species scores

         CA1     CA2      CA3      CA4      CA5       CA6
Sp1  -1.4446  1.8842 -0.86530  0.03061 -0.47720 -0.006432
Sp2   1.1313  0.5380  0.52836  1.74914 -0.05360  0.247953
Sp3   1.0302  0.3407  0.16160 -0.32148  0.00978 -0.135905
Sp4  -1.1395  0.2409  0.47543 -0.02254  1.06125  0.022650
Sp5  -0.9094 -0.6375  0.85632 -0.03372  0.35896 -0.014035
Sp6   1.1250  0.5162  0.37542 -1.61102 -0.05869  0.265755
Sp7  -0.7780 -0.9688  0.78276 -0.02684 -0.79494 -0.033065
Sp8  -0.2565 -1.0628 -0.43367  0.02821 -0.55703  0.053469
Sp9   0.9388  0.1688 -0.07722  0.11955  0.07364 -0.513328
Sp10  0.3026 -1.0196 -1.80371  0.08808  0.43783  0.112936


Site scores (weighted averages of species scores)

           CA1     CA2     CA3      CA4      CA5       CA6
Site1  -1.4446  1.8842 -0.8653  0.03061 -0.47720 -0.006432
Site2  -0.8156 -0.9072  0.8514 -0.03082 -0.51678 -0.066579
Site3  -0.7373 -1.0761  0.7693 -0.02515 -1.10800 -0.053106
Site4   0.1364 -1.3205 -2.2180  0.12014  0.49068  2.189249
Site5   0.4965 -0.6686 -1.3203  0.05068  0.37618 -2.309429
Site6   1.1313  0.5380  0.5284  1.74914 -0.05360  0.247953
Site7   1.1250  0.5162  0.3754 -1.61102 -0.05869  0.265755
Site8  -0.6227 -1.2320  0.5034 -0.01041 -2.11065  0.079747
Site9  -1.0159 -0.5055  1.1190 -0.04803  1.23098 -0.121948
Site10 -1.1131 -0.1123  0.9944 -0.04620  3.03739  0.206699
Eigenvalues - component of \(\chi^2\)

Axes retention

  1. Eigenvalues greater than average
> data.ca$CA$eig
        CA1         CA2         CA3         CA4         CA5 
0.946305134 0.782293787 0.621442179 0.578436060 0.262857400 
        CA6         CA7         CA8 
0.043153975 0.015510825 0.006164366 
> mean(data.ca$CA$eig)
[1] 0.4070205
> data.ca$CA$eig>mean(data.ca$CA$eig)
  CA1   CA2   CA3   CA4   CA5   CA6   CA7   CA8 
 TRUE  TRUE  TRUE  TRUE FALSE FALSE FALSE FALSE

Axes retention

  1. Eigenvalues greater than 0.6
> data.ca$CA$eig
        CA1         CA2         CA3         CA4         CA5 
0.946305134 0.782293787 0.621442179 0.578436060 0.262857400 
        CA6         CA7         CA8 
0.043153975 0.015510825 0.006164366 
> data.ca$CA$eig>0.6
  CA1   CA2   CA3   CA4   CA5   CA6   CA7   CA8 
 TRUE  TRUE  TRUE FALSE FALSE FALSE FALSE FALSE

Axes retention

  1. Elbow in screeplot
> screeplot(data.ca)

Ordination plots

Ordination plots

Ordination plots

The arch effect

Environmental correlates


Site pH Slope Pressure Altitude Substrate
Site1 6.1 4.2 101325 2 Quartz
Site2 6.7 9.2 101352 510 Shale
Site3 6.8 8.6 101356 546 Shale
Site4 7.0 7.4 101372 758 Shale
Site5 7.2 5.8 101384 813 Shale
Site6 7.5 8.4 101395 856 Quartz
Site7 7.5 0.5 101396 854 Quartz
Site8 7.0 11.8 101370 734 Shale
Site9 8.4 8.2 101347 360 Quartz
Site10 6.2 1.5 101345 356 Quartz

Environmental correlates

> envfit(data.ca,env=enviro[,-1])

***VECTORS

              CA1      CA2     r2 Pr(>r)    
pH        0.91731 -0.39818 0.4044  0.244    
Slope    -0.41261 -0.91091 0.3135  0.355    
Pressure  0.98437 -0.17609 0.9833  0.001 ***
Altitude  0.90763 -0.41978 0.9828  0.001 ***
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Permutation: free
Number of permutations: 999

***FACTORS:

Centroids:
                    CA1     CA2
SubstrateQuartz  0.1358  0.6470
SubstrateShale  -0.2098 -0.9992

Goodness of fit:
              r2 Pr(>r)  
Substrate 0.3375  0.033 *
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Permutation: free
Number of permutations: 999

Environmental correlates

> plot(data.ca)
> plot(envfit(data.ca,env=enviro[,-1]))

Environmental correlates

> data.lm <- lm(data.ca$CA$u[,1:3] ~ enviro$Altitude + enviro$Slope+enviro$pH+enviro$Substrate)
> summary(data.lm)
Response CA1 :

Call:
lm(formula = CA1 ~ enviro$Altitude + enviro$Slope + enviro$pH + 
    enviro$Substrate)

Residuals:
    Site1     Site2     Site3     Site4     Site5     Site6 
 0.422144  0.061252 -0.002323  0.112485  0.232025  0.287221 
    Site7     Site8     Site9    Site10 
-0.002549 -0.403439 -0.161979 -0.544838 

Coefficients:
                        Estimate Std. Error t value
(Intercept)           -1.5720999  1.7269003  -0.910
enviro$Altitude        0.0033945  0.0006537   5.193
enviro$Slope          -0.0367321  0.0551127  -0.666
enviro$pH             -0.0241178  0.2812720  -0.086
enviro$SubstrateShale -0.5363786  0.4150506  -1.292
                      Pr(>|t|)   
(Intercept)            0.40438   
enviro$Altitude        0.00349 **
enviro$Slope           0.53461   
enviro$pH              0.93500   
enviro$SubstrateShale  0.25275   
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4042 on 5 degrees of freedom
Multiple R-squared:  0.8972,    Adjusted R-squared:  0.815 
F-statistic: 10.91 on 4 and 5 DF,  p-value: 0.01098


Response CA2 :

Call:
lm(formula = CA2 ~ enviro$Altitude + enviro$Slope + enviro$pH + 
    enviro$Substrate)

Residuals:
    Site1     Site2     Site3     Site4     Site5     Site6 
 0.870600  0.004159 -0.109777 -0.243666  0.519301  0.252775 
    Site7     Site8     Site9    Site10 
 0.257422 -0.170017 -0.318529 -1.062268 

Coefficients:
                        Estimate Std. Error t value
(Intercept)            4.214e+00  3.014e+00   1.398
enviro$Altitude       -5.495e-06  1.141e-03  -0.005
enviro$Slope           3.346e-03  9.619e-02   0.035
enviro$pH             -5.270e-01  4.909e-01  -1.073
enviro$SubstrateShale -1.623e+00  7.244e-01  -2.240
                      Pr(>|t|)  
(Intercept)             0.2209  
enviro$Altitude         0.9963  
enviro$Slope            0.9736  
enviro$pH               0.3321  
enviro$SubstrateShale   0.0752 .
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7054 on 5 degrees of freedom
Multiple R-squared:  0.7305,    Adjusted R-squared:  0.5149 
F-statistic: 3.388 on 4 and 5 DF,  p-value: 0.1066


Response CA3 :

Call:
lm(formula = CA3 ~ enviro$Altitude + enviro$Slope + enviro$pH + 
    enviro$Substrate)

Residuals:
  Site1   Site2   Site3   Site4   Site5   Site6   Site7 
-1.2780  1.0341  1.0361 -1.7686 -0.6495 -0.3962  0.5326 
  Site8   Site9  Site10 
 0.3479  0.1555  0.9860 

Coefficients:
                        Estimate Std. Error t value
(Intercept)           -0.2674575  5.6952905  -0.047
enviro$Altitude       -0.0001025  0.0021558  -0.048
enviro$Slope           0.1369508  0.1817607   0.753
enviro$pH              0.0172449  0.9276307   0.019
enviro$SubstrateShale -1.2384816  1.3688304  -0.905
                      Pr(>|t|)
(Intercept)              0.964
enviro$Altitude          0.964
enviro$Slope             0.485
enviro$pH                0.986
enviro$SubstrateShale    0.407

Residual standard error: 1.333 on 5 degrees of freedom
Multiple R-squared:  0.2334,    Adjusted R-squared:  -0.3799 
F-statistic: 0.3805 on 4 and 5 DF,  p-value: 0.8148

Worked Examples

Worked Examples

> veg <- read.csv('../data/veg.csv', strip.white=TRUE)
> head(veg) 
  SITE HABITAT SP1 SP2 SP3 SP4 SP5 SP6 SP7 SP8
1    1       A   4   0   0  36  28  24  99  68
2    2       B  92  84   0   8   0   0  84   4
3    3       A   9   0   0  52   4  40  96  68
4    4       A  52   0   0  52  12  28  96  24
5    5       C  99   0  36  88  52   8  72   0
6    6       A  12   0   0  20  40  40  88  68
> data <- read.csv('../data/data.csv', strip.white=TRUE)
> head(data) 
  Sites Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10
1 Site1   5   0   0  65   5   0   0   0   0    0
2 Site2   0   0   0  25  39   0   6  23   0    0
3 Site3   0   0   0   6  42   0   6  31   0    0
4 Site4   0   0   0   0   0   0   0  40   0   14
5 Site5   0   0   6   0   0   0   0  34  18   12
6 Site6   0  29  12   0   0   0   0   0  22    0
> enviro <- read.csv('../data/enviro.csv', strip.white=TRUE)
> head(enviro)
   Site  pH Slope Pressure Altitude Substrate
1 Site1 6.1   4.2   101325        2    Quartz
2 Site2 6.7   9.2   101352      510     Shale
3 Site3 6.8   8.6   101356      546     Shale
4 Site4 7.0   7.4   101372      758     Shale
5 Site5 7.2   5.8   101384      813     Shale
6 Site6 7.5   8.4   101395      856    Quartz