06 May, 2022

Overview

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
  • Measures of association
    • variance covariance matrix
var(Y)
##        Sp1   Sp2     Sp3     Sp4
## Sp1  30.70 15.00   96.35  117.70
## Sp2  15.00  8.50   56.25   62.50
## Sp3  96.35 56.25 1153.80 1477.85
## Sp4 117.70 62.50 1477.85 2122.20

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
  • Measures of association
    • correlation matrix
cor(Y)
##           Sp1       Sp2       Sp3       Sp4
## Sp1 1.0000000 0.9285656 0.5119379 0.4611202
## Sp2 0.9285656 1.0000000 0.5679993 0.4653475
## Sp3 0.5119379 0.5679993 1.0000000 0.9444347
## Sp4 0.4611202 0.4653475 0.9444347 1.0000000

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 0.6243 0.7918 0.9088 0.97567
##                           PC6      PC7    PC8       PC9
## Eigenvalue            0.14643 0.067836 0.0280 0.0010446
## Proportion Explained  0.01464 0.006784 0.0028 0.0001045
## Cumulative Proportion 0.99031 0.997096 0.9999 1.0000000
## 
## 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

  • Eigenvalues greater than 1
##         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

  • Eigenvalues greater than 1
##         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
  • Cummunaltive percentages greater than 80%
##    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

  • Eigenvalues greater than 1
##         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
  • Cummunaltive percentages greater than 80%
##    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

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

  • Combines (multiple) linear regression with PCA
  • Regress each of the species against linear predictor
  • Perform PCA on fitted values
  • Axes (PC’s) explain the communities rather than gradients

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 0.085455 0.033927
## Proportion Explained  0.09837 0.04255 0.008546 0.003393
## Cumulative Proportion 0.94552 0.98806 0.996607 1.000000
## 
## 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  0.003 **
## 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  0.177   
## Altitude   1   2.2199 3.4718  0.006 **
## Substrate  1   1.5685 2.4531  0.043 * 
## pH         1   0.4273 0.6683  0.697   
## 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   0.17   
## Altitude   1 23.842 3.4718   0.01 **
## Substrate  1 22.561 2.4531   0.04 * 
## pH         1 19.823 0.6683   0.69   
## ---
## Signif. codes:  
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correspondence analysis

PCA

  • Sites projected onto rotated species space
  • Assumes multivariate linearity and species abundances linearly related to enviro gradients

Correspondence analysis

  • Major environmental gradients should cause associations between sites and species abundances


Correspondence analysis

  • Major environmental gradients should cause associations between sites and species abundances
  • axes represent axes of greatest association between sites and species

Correspondence analysis

  • projects sites and species into new ordination space
    • representing enviro gradients
  • data must be expressed as frequencies

Correspondence analysis

  • data must be expressed as frequencies
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 scaled Chi-square:
##               Inertia Proportion
## Total           3.256          1
## Unconstrained   3.256          1
## 
## Eigenvalues, and their contribution to the scaled Chi-square 
## 
## Importance of components:
##                          CA1    CA2    CA3    CA4     CA5
## Eigenvalue            0.9463 0.7823 0.6214 0.5784 0.26286
## Proportion Explained  0.2906 0.2403 0.1909 0.1776 0.08073
## Cumulative Proportion 0.2906 0.5309 0.7217 0.8994 0.98009
##                           CA6      CA7      CA8
## Eigenvalue            0.04315 0.015511 0.006164
## Proportion Explained  0.01325 0.004764 0.001893
## Cumulative Proportion 0.99334 0.998107 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 scaled Chi-square:
##               Inertia Proportion
## Total           3.256          1
## Unconstrained   3.256          1
## 
## Eigenvalues, and their contribution to the scaled Chi-square 
## 
## Importance of components:
##                          CA1    CA2    CA3    CA4     CA5
## Eigenvalue            0.9463 0.7823 0.6214 0.5784 0.26286
## Proportion Explained  0.2906 0.2403 0.1909 0.1776 0.08073
## Cumulative Proportion 0.2906 0.5309 0.7217 0.8994 0.98009
##                           CA6      CA7      CA8
## Eigenvalue            0.04315 0.015511 0.006164
## Proportion Explained  0.01325 0.004764 0.001893
## Cumulative Proportion 0.99334 0.998107 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 scaled Chi-square:
##               Inertia Proportion
## Total           3.256          1
## Unconstrained   3.256          1
## 
## Eigenvalues, and their contribution to the scaled Chi-square 
## 
## Importance of components:
##                          CA1    CA2    CA3    CA4     CA5
## Eigenvalue            0.9463 0.7823 0.6214 0.5784 0.26286
## Proportion Explained  0.2906 0.2403 0.1909 0.1776 0.08073
## Cumulative Proportion 0.2906 0.5309 0.7217 0.8994 0.98009
##                           CA6      CA7      CA8
## Eigenvalue            0.04315 0.015511 0.006164
## Proportion Explained  0.01325 0.004764 0.001893
## Cumulative Proportion 0.99334 0.998107 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

  • scaling=1, sites scaled
    • close sites have similar patterns of species abundances
    • sites close to species have high abundance of that species

Ordination plots

  • scaling=2, species scaled
    • close species have similar patterns of abundances across sites
    • species close to sites have highest abundance at that site

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  0.253    
## Slope    -0.41261 -0.91091 0.3135  0.396    
## 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  0.041 *
## ---
## 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