Multivariate Statistics

This post provides a comprehensive overview of multivariate statistical methods frequently used in ecology and marine science. Each method is accompanied by the corresponding R packages for implementation.

1. Principal Component Analysis (PCA)

Purpose: Reduces data dimensionality and identifies patterns.

R Packages: - vegan: rda() - stats: prcomp(), princomp() - FactoMineR, factoextra: For enhanced visualization ### Example: Principal Component Analysis (PCA) Principal Component Analysis (PCA) is a multivariate statistical technique that uses an orthogonal transformation to convert a set of correlated variables into a set of orthogonal, uncorrelated axes called principal components. The primary motivation behind PCA is to reduce a large number of variables into a smaller number of derived variables that can be visualized in 2 or 3-dimensional space.

Applications of PCA include: - Reducing strongly correlated data groups - Evaluating aquatic habitat suitability - Analyzing fish abundance, seasonal and spatial variation - Studying ecosystem organization changes - Investigating intra- and interspecific variability, niche differentiation, and partitioning

# Load necessary library
library(ggplot2)

# Create a sample dataset
data <- data.frame(
  Variable1 = c(2.5, 0.5, 2.2, 1.9, 3.1, 2.3, 2, 1, 1.5, 1.1),
  Variable2 = c(2.4, 0.7, 2.9, 2.2, 3.0, 2.7, 1.6, 1.1, 1.6, 0.9)
)

# Perform PCA
pca_result <- prcomp(data, scale. = TRUE)

# Print PCA summary
summary(pca_result)

Importance of components:
                         PC1     PC2
Standard deviation     1.388 0.27216
Proportion of Variance 0.963 0.03704
Cumulative Proportion  0.963 1.00000

# Plot PCA
biplot(pca_result, main = "PCA Biplot")

2. Non-metric Multidimensional Scaling (NMDS)

Purpose: Ordination based on ranks (non-parametric).

R Package: - vegan: metaMDS()

Applications of Non-metric Multidimensional Scaling (NMDS)

NMDS is widely used in ecological and environmental studies to visualize the similarity or dissimilarity of data points in a low-dimensional space. Applications include: - Analyzing community composition and structure - Studying species-environment relationships - Visualizing temporal changes in ecological data - Comparing biodiversity across sites

library(vegan)

Warning: package 'vegan' was built under R version 4.3.3

Loading required package: permute

Warning: package 'permute' was built under R version 4.3.3

Loading required package: lattice

data(dune)
nmds <- metaMDS(dune)

Run 0 stress 0.1192678 
Run 1 stress 0.1192679 
... Procrustes: rmse 0.0001218127  max resid 0.0003736083 
... Similar to previous best
Run 2 stress 0.1192678 
... New best solution
... Procrustes: rmse 7.269566e-06  max resid 2.137083e-05 
... Similar to previous best
Run 3 stress 0.1192678 
... Procrustes: rmse 4.629212e-05  max resid 0.0001436214 
... Similar to previous best
Run 4 stress 0.2035424 
Run 5 stress 0.1192679 
... Procrustes: rmse 9.770457e-05  max resid 0.0002999795 
... Similar to previous best
Run 6 stress 0.1192679 
... Procrustes: rmse 0.0001655233  max resid 0.0005082011 
... Similar to previous best
Run 7 stress 0.1192679 
... Procrustes: rmse 0.000153753  max resid 0.0004721524 
... Similar to previous best
Run 8 stress 0.1192678 
... Procrustes: rmse 4.858184e-05  max resid 0.0001484307 
... Similar to previous best
Run 9 stress 0.1192678 
... Procrustes: rmse 6.195451e-05  max resid 0.0001903387 
... Similar to previous best
Run 10 stress 0.1809581 
Run 11 stress 0.1808911 
Run 12 stress 0.1192678 
... Procrustes: rmse 1.237881e-05  max resid 3.616359e-05 
... Similar to previous best
Run 13 stress 0.1192678 
... Procrustes: rmse 6.937557e-06  max resid 1.78123e-05 
... Similar to previous best
Run 14 stress 0.1183186 
... New best solution
... Procrustes: rmse 0.02027193  max resid 0.06497369 
Run 15 stress 0.1808911 
Run 16 stress 0.1192679 
Run 17 stress 0.1183186 
... New best solution
... Procrustes: rmse 1.364961e-05  max resid 4.380806e-05 
... Similar to previous best
Run 18 stress 0.1192679 
Run 19 stress 0.1183186 
... Procrustes: rmse 5.076165e-06  max resid 1.40165e-05 
... Similar to previous best
Run 20 stress 0.1886532 
*** Best solution repeated 2 times

plot(nmds)

3. Canonical Correspondence Analysis (CCA)

Purpose: Ordination constrained by environmental variables.

Applications of Canonical Correspondence Analysis (CCA)

CCA is commonly applied in ecological research to explore relationships between species and environmental variables. Applications include: - Identifying key environmental drivers of species distribution - Analyzing community-environment interactions - Studying habitat preferences of species

data(dune.env)
cca_model <- cca(dune ~ ., data = dune.env)


Some constraints or conditions were aliased because they were redundant. This
can happen if terms are linearly dependent (collinear): 'Manure^4'

plot(cca_model)

4. Redundancy Analysis (RDA)

Purpose: Linear constrained ordination.

Applications of Redundancy Analysis (RDA)

RDA is used to summarize and visualize the variation in response variables explained by explanatory variables. Applications include: - Investigating the influence of environmental factors on species abundance - Analyzing multivariate ecological datasets - Studying the effects of land use on biodiversity

rda_model <- rda(dune ~ ., data = dune.env)


Some constraints or conditions were aliased because they were redundant. This
can happen if terms are linearly dependent (collinear): 'Manure^4'

plot(rda_model)

5. Cluster Analysis

Purpose: Groups similar observations.

R Packages: - stats: hclust(), kmeans() - cluster: agnes(), diana()

Applications of Cluster Analysis

Cluster analysis is a versatile tool for grouping similar observations. Applications include: - Classifying vegetation types - Grouping sampling sites based on species composition - Identifying patterns in genetic or molecular data

dist_matrix <- dist(scale(dune))
hc <- hclust(dist_matrix)
plot(hc)

6. Discriminant Analysis (LDA/QDA)

Purpose: Classify observations into predefined groups.

R Package: - MASS

Applications of Discriminant Analysis (LDA/QDA)

Discriminant analysis is used for classification and prediction. Applications include: - Classifying species based on morphological traits - Predicting habitat types from environmental variables - Identifying key variables that differentiate groups

library(MASS)
# Example assumes 'Species' is a factor in iris
lda_model <- lda(Species ~ ., data = iris)
plot(lda_model)

7. PERMANOVA

Purpose: Permutation-based multivariate ANOVA.

R Package: - vegan: adonis(), adonis2()

Applications of PERMANOVA

PERMANOVA is a robust method for testing differences in multivariate data. Applications include: - Comparing community composition across treatments - Testing the effects of environmental gradients on species assemblages - Analyzing experimental data with complex designs

adonis_model <- adonis(dune ~ Management, data = dune.env)

Warning: 'adonis' is deprecated.
Use 'adonis2' instead.
See help("Deprecated") and help("vegan-deprecated").

adonis_model

$aov.tab
Permutation: free
Number of permutations: 999

Terms added sequentially (first to last)

           Df SumsOfSqs MeanSqs F.Model      R2 Pr(>F)   
Management  3    1.4686 0.48953  2.7672 0.34161  0.004 **
Residuals  16    2.8304 0.17690         0.65839          
Total      19    4.2990                 1.00000          
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$call
adonis(formula = dune ~ Management, data = dune.env)

$coefficients
              Achimill   Agrostol   Airaprae   Alopgeni  Anthodor   Bellpere
(Intercept)  1.0083333  2.0583333  0.2083333  1.6500000 1.1166667  0.7666667
Management1  1.3250000 -2.0583333 -0.2083333 -0.9833333 0.2166667  0.9000000
Management2  0.1916667 -0.6583333 -0.2083333 -0.0500000 0.6833333 -0.3666667
Management3 -0.6750000  0.1083333  0.6250000 -1.6500000 0.2166667 -0.4333333
              Bromhord    Chenalbu    Cirsarve   Comapalu  Eleopalu
(Intercept)  0.9916667  0.04166667  0.08333333  0.1666667  1.075000
Management1  1.6750000 -0.04166667 -0.08333333 -0.1666667 -1.075000
Management2 -0.1916667 -0.04166667 -0.08333333 -0.1666667 -0.275000
Management3 -0.9916667 -0.04166667 -0.08333333  0.5000000  1.091667
                 Elymrepe    Empenigr   Hyporadi   Juncarti   Juncbufo
(Intercept)  1.333333e+00  0.08333333  0.4583333  0.8166667  0.5916667
Management1 -5.315636e-16 -0.08333333  0.2083333 -0.8166667 -0.5916667
Management2  6.666667e-01 -0.08333333 -0.4583333  0.7833333  0.6083333
Management3 -1.333333e+00  0.25000000  0.7083333  0.3500000 -0.5916667
              Lolipere   Planlanc    Poaprat    Poatriv   Ranuflam   Rumeacet
(Intercept)  3.3333333  1.4583333  2.6416667  3.2833333  0.6000000  0.8833333
Management1  2.6666667  0.5416667  1.3583333  0.3833333 -0.6000000 -0.8833333
Management2  0.6666667  1.5416667  0.7583333  1.5166667 -0.2000000  2.3166667
Management3 -3.0000000 -0.6250000 -1.9750000 -3.2833333  0.7333333 -0.8833333
              Sagiproc   Salirepe   Scorautu Trifprat   Trifrepe   Vicilath
(Intercept)  0.9500000  0.4583333  2.9083333     0.45  2.6583333  0.2916667
Management1 -0.2833333 -0.4583333  1.4250000    -0.45  2.0083333  0.7083333
Management2 -0.1500000 -0.4583333 -0.1083333     1.35  0.1416667 -0.2916667
Management3 -0.4500000  1.3750000  0.2583333    -0.45 -0.8250000 -0.1250000
              Bracruta   Callcusp
(Intercept)  2.4083333  0.4166667
Management1 -0.4083333 -0.4166667
Management2  0.3916667 -0.4166667
Management3  0.4250000  0.7500000

$coef.sites
                      1           2           3           4           5
(Intercept)  0.67574520  0.53161494  0.50736459  0.53767659  0.54776275
Management1 -0.14226615 -0.25339555 -0.05173176 -0.06500654 -0.08533989
Management2 -0.05922431 -0.05667973 -0.07145964 -0.05140339 -0.21375164
Management3  0.26701574  0.27967840  0.25476356  0.23760571  0.20010780
                     6          7           8           9          10
(Intercept)  0.5439066  0.5146877  0.50369055  0.53529312  0.50417436
Management1 -0.1173092 -0.1278286  0.03182005  0.02163523 -0.26835736
Management2 -0.2009296 -0.2208891 -0.08938326 -0.15314104 -0.08681523
Management3  0.1817377  0.2478188  0.12351081  0.21045063  0.21427188
                     11          12          13          14          15
(Intercept)  0.54326809  0.61282294  0.62050552  0.71853509  0.68579075
Management1 -0.22531013  0.08846914  0.07639056  0.07163204  0.14808648
Management2 -0.01457139 -0.06318735 -0.04763974  0.05771058  0.03074555
Management3  0.11608592  0.11575703  0.13515185 -0.14997406 -0.17737191
                     16          17           18          19          20
(Intercept)  0.71914202  0.76273816  0.597040253  0.71241400  0.71952832
Management1  0.16918907 -0.04694525 -0.129723226 -0.02358819  0.16267874
Management2  0.02385609 -0.01041765 -0.014764304  0.02718627  0.04425268
Management3 -0.04953049 -0.09068176 -0.002636699 -0.13054729 -0.20361567

$f.perms
             [,1]
   [1,] 1.4770127
   [2,] 1.6741268
   [3,] 0.5841741
   [4,] 1.0793058
   [5,] 1.3939571
   [6,] 0.5312787
   [7,] 1.2214243
   [8,] 0.6273901
   [9,] 0.8102732
  [10,] 1.4203213
  [11,] 0.7830450
  [12,] 0.4712738
  [13,] 0.7274186
  [14,] 0.9329011
  [15,] 0.9334849
  [16,] 0.4632469
  [17,] 0.4831136
  [18,] 1.3909926
  [19,] 0.6054234
  [20,] 1.1386544
  [21,] 0.8940494
  [22,] 1.2806116
  [23,] 1.9641093
  [24,] 0.9741948
  [25,] 0.6531695
  [26,] 1.3248005
  [27,] 1.4124523
  [28,] 1.3773216
  [29,] 1.0519740
  [30,] 1.1525879
  [31,] 0.8695853
  [32,] 1.2351863
  [33,] 0.8724930
  [34,] 0.6903864
  [35,] 1.3501681
  [36,] 1.1824076
  [37,] 0.7419446
  [38,] 1.4360489
  [39,] 1.0690261
  [40,] 0.9301971
  [41,] 0.7623953
  [42,] 1.0511879
  [43,] 1.1721067
  [44,] 0.6674446
  [45,] 1.1962906
  [46,] 0.7492973
  [47,] 0.8001220
  [48,] 1.2280829
  [49,] 0.2880842
  [50,] 0.4022638
  [51,] 0.6737903
  [52,] 1.1783578
  [53,] 0.6936991
  [54,] 1.1631255
  [55,] 0.6775807
  [56,] 0.2893694
  [57,] 0.6653908
  [58,] 1.4210496
  [59,] 1.5012472
  [60,] 0.6638606
  [61,] 1.3021709
  [62,] 0.5785438
  [63,] 1.1420707
  [64,] 0.7380980
  [65,] 0.4966893
  [66,] 0.5924792
  [67,] 0.9808446
  [68,] 0.4963282
  [69,] 0.7565734
  [70,] 0.9940520
  [71,] 0.7483531
  [72,] 1.0850450
  [73,] 0.9988767
  [74,] 0.5043698
  [75,] 0.8972336
  [76,] 0.7039330
  [77,] 0.9657978
  [78,] 1.0284344
  [79,] 0.7344814
  [80,] 1.0063138
  [81,] 1.6712571
  [82,] 1.0358732
  [83,] 1.1810227
  [84,] 2.4980652
  [85,] 0.9193928
  [86,] 1.0921393
  [87,] 1.3260082
  [88,] 1.2823243
  [89,] 2.1462794
  [90,] 0.8752580
  [91,] 0.7103119
  [92,] 1.3050155
  [93,] 0.7557751
  [94,] 0.5923924
  [95,] 0.8419388
  [96,] 2.0546380
  [97,] 1.4433198
  [98,] 0.9948003
  [99,] 1.1990555
 [100,] 0.7197558
 [101,] 0.6681426
 [102,] 1.1277194
 [103,] 0.9402664
 [104,] 0.8360112
 [105,] 1.2427407
 [106,] 0.7076889
 [107,] 0.7154894
 [108,] 1.5948556
 [109,] 1.3095611
 [110,] 0.9337607
 [111,] 0.8970954
 [112,] 0.9544802
 [113,] 1.9199597
 [114,] 0.8123536
 [115,] 0.7468794
 [116,] 0.9622113
 [117,] 0.2663180
 [118,] 0.8779799
 [119,] 0.5472752
 [120,] 1.9629848
 [121,] 1.4655385
 [122,] 0.4355588
 [123,] 1.1348573
 [124,] 1.2765454
 [125,] 1.0064938
 [126,] 0.9185663
 [127,] 1.1743829
 [128,] 0.8495841
 [129,] 1.4028903
 [130,] 1.8667872
 [131,] 1.0533410
 [132,] 0.4666848
 [133,] 0.9977309
 [134,] 0.6676392
 [135,] 0.7978343
 [136,] 0.8246253
 [137,] 1.5295156
 [138,] 1.3533550
 [139,] 0.7923841
 [140,] 2.0718269
 [141,] 1.5044331
 [142,] 0.6403341
 [143,] 0.8390345
 [144,] 0.3248323
 [145,] 0.4747476
 [146,] 2.4191610
 [147,] 0.9908124
 [148,] 0.6973857
 [149,] 1.1067674
 [150,] 1.7477629
 [151,] 0.8553347
 [152,] 0.8154237
 [153,] 1.0061334
 [154,] 1.2086589
 [155,] 1.3321821
 [156,] 0.9336491
 [157,] 0.3682444
 [158,] 0.6032235
 [159,] 1.0119507
 [160,] 0.4058727
 [161,] 0.6800951
 [162,] 1.0304500
 [163,] 0.7032275
 [164,] 0.8425849
 [165,] 0.7981298
 [166,] 0.8403245
 [167,] 0.5825641
 [168,] 0.8772570
 [169,] 0.6738811
 [170,] 1.5591647
 [171,] 0.4762378
 [172,] 0.9215314
 [173,] 0.8388783
 [174,] 0.6199798
 [175,] 0.6565736
 [176,] 2.1060625
 [177,] 1.3489550
 [178,] 1.0062732
 [179,] 1.1919056
 [180,] 0.4609944
 [181,] 0.7502184
 [182,] 1.4279961
 [183,] 1.4781943
 [184,] 1.0519368
 [185,] 1.6619095
 [186,] 0.4739645
 [187,] 0.6738266
 [188,] 1.0003629
 [189,] 1.0771126
 [190,] 1.7081498
 [191,] 0.9302083
 [192,] 1.2599451
 [193,] 0.6606229
 [194,] 1.3758623
 [195,] 0.6238083
 [196,] 0.4068884
 [197,] 0.7085488
 [198,] 0.5056951
 [199,] 0.9601244
 [200,] 1.1108904
 [201,] 1.1790648
 [202,] 0.9693691
 [203,] 0.8400174
 [204,] 1.8135628
 [205,] 0.8523361
 [206,] 0.9223765
 [207,] 0.8702576
 [208,] 1.0758547
 [209,] 1.0101402
 [210,] 1.1017830
 [211,] 0.7452655
 [212,] 0.9013168
 [213,] 1.3029965
 [214,] 0.8205975
 [215,] 0.9645074
 [216,] 0.9345365
 [217,] 1.1337161
 [218,] 0.7369172
 [219,] 1.2837005
 [220,] 0.7697555
 [221,] 0.7783000
 [222,] 0.8023562
 [223,] 1.3639488
 [224,] 1.2673809
 [225,] 0.8001389
 [226,] 1.2582407
 [227,] 1.5006700
 [228,] 0.7652815
 [229,] 0.6640557
 [230,] 0.9377710
 [231,] 0.6535452
 [232,] 1.0500841
 [233,] 1.0768823
 [234,] 2.6332037
 [235,] 0.5858923
 [236,] 0.5960403
 [237,] 0.8137275
 [238,] 0.7397481
 [239,] 1.4897358
 [240,] 1.4026573
 [241,] 1.4366856
 [242,] 0.8597631
 [243,] 0.8066541
 [244,] 0.8044416
 [245,] 1.2362615
 [246,] 1.0821938
 [247,] 1.2111738
 [248,] 0.5439102
 [249,] 0.9361084
 [250,] 1.5508510
 [251,] 0.7175440
 [252,] 1.8257393
 [253,] 1.6070306
 [254,] 0.7039476
 [255,] 0.9591631
 [256,] 0.9814181
 [257,] 1.0474431
 [258,] 1.4948642
 [259,] 1.1846257
 [260,] 0.7772480
 [261,] 0.8201491
 [262,] 0.7828198
 [263,] 1.0798898
 [264,] 0.8728824
 [265,] 0.5473524
 [266,] 0.7452955
 [267,] 2.3839572
 [268,] 1.0123776
 [269,] 0.9346628
 [270,] 1.4210620
 [271,] 0.7026874
 [272,] 0.9224575
 [273,] 1.1474375
 [274,] 1.1292296
 [275,] 0.6998635
 [276,] 0.7763794
 [277,] 2.6533640
 [278,] 0.8259252
 [279,] 1.2755326
 [280,] 1.5899695
 [281,] 0.5814523
 [282,] 1.2370601
 [283,] 1.4848072
 [284,] 1.0604471
 [285,] 0.9492693
 [286,] 2.0029945
 [287,] 1.3647031
 [288,] 0.3773867
 [289,] 1.0488958
 [290,] 1.0414270
 [291,] 0.4160364
 [292,] 0.7416273
 [293,] 0.6785256
 [294,] 1.2160278
 [295,] 0.8776552
 [296,] 1.2948961
 [297,] 0.8161454
 [298,] 0.7670753
 [299,] 0.7581265
 [300,] 0.9536657
 [301,] 0.6757823
 [302,] 0.7529551
 [303,] 0.6625711
 [304,] 0.8492308
 [305,] 1.0948772
 [306,] 1.0211374
 [307,] 1.1225175
 [308,] 0.7004408
 [309,] 0.6943875
 [310,] 0.5761640
 [311,] 0.3944804
 [312,] 0.6041955
 [313,] 0.5616060
 [314,] 0.4347134
 [315,] 1.7510698
 [316,] 0.8216853
 [317,] 1.1183855
 [318,] 1.3944784
 [319,] 1.3092314
 [320,] 1.2653304
 [321,] 0.5144483
 [322,] 0.3861568
 [323,] 0.6571438
 [324,] 0.4778806
 [325,] 1.8443280
 [326,] 0.7905290
 [327,] 1.3643009
 [328,] 0.4526702
 [329,] 0.9294289
 [330,] 0.4619874
 [331,] 0.7534640
 [332,] 1.4867032
 [333,] 0.4736031
 [334,] 0.9352602
 [335,] 1.4226953
 [336,] 1.7456812
 [337,] 0.9558558
 [338,] 1.1741469
 [339,] 1.0515656
 [340,] 1.8289292
 [341,] 1.4863810
 [342,] 0.4136271
 [343,] 1.3661146
 [344,] 1.1698071
 [345,] 0.8921639
 [346,] 0.9789351
 [347,] 0.9428690
 [348,] 1.0009387
 [349,] 1.1333190
 [350,] 1.1280314
 [351,] 0.7879399
 [352,] 0.7143198
 [353,] 0.6974031
 [354,] 0.6677926
 [355,] 0.9368312
 [356,] 0.7752936
 [357,] 0.6613949
 [358,] 0.7096997
 [359,] 0.5278780
 [360,] 1.5979675
 [361,] 1.0102512
 [362,] 0.9657582
 [363,] 0.6788591
 [364,] 1.2513459
 [365,] 0.8923123
 [366,] 0.5935933
 [367,] 1.4005758
 [368,] 0.9663236
 [369,] 0.7198910
 [370,] 1.4308977
 [371,] 0.7303822
 [372,] 0.9000348
 [373,] 0.8130237
 [374,] 1.1551960
 [375,] 0.6321509
 [376,] 1.2640836
 [377,] 0.6367746
 [378,] 0.9342290
 [379,] 0.7619839
 [380,] 0.6642945
 [381,] 0.9393655
 [382,] 0.4842257
 [383,] 1.0335787
 [384,] 0.2640405
 [385,] 1.9530058
 [386,] 0.7716831
 [387,] 1.8781195
 [388,] 0.7438069
 [389,] 2.2227582
 [390,] 1.1514433
 [391,] 0.9942989
 [392,] 1.0074796
 [393,] 0.6961640
 [394,] 0.7870044
 [395,] 1.3591964
 [396,] 0.8059156
 [397,] 0.8355524
 [398,] 0.6160495
 [399,] 1.0430180
 [400,] 0.8523824
 [401,] 1.5934819
 [402,] 0.6773612
 [403,] 0.8013232
 [404,] 1.1993369
 [405,] 1.0512708
 [406,] 1.8519746
 [407,] 0.8757457
 [408,] 1.0569673
 [409,] 1.0775598
 [410,] 1.1199711
 [411,] 1.1922530
 [412,] 0.7866440
 [413,] 0.9744095
 [414,] 1.3679371
 [415,] 1.6040984
 [416,] 0.5837127
 [417,] 1.3503217
 [418,] 1.6066318
 [419,] 0.6603888
 [420,] 0.6193697
 [421,] 0.8862458
 [422,] 1.1215359
 [423,] 1.8969206
 [424,] 0.9385519
 [425,] 1.2601149
 [426,] 0.8316728
 [427,] 1.1814472
 [428,] 2.8395517
 [429,] 0.7996109
 [430,] 0.9794073
 [431,] 0.5975595
 [432,] 0.7908001
 [433,] 1.6127786
 [434,] 0.8268354
 [435,] 0.9649756
 [436,] 1.7133208
 [437,] 1.2635960
 [438,] 1.2866071
 [439,] 1.0576726
 [440,] 0.5306406
 [441,] 0.8742260
 [442,] 1.1792678
 [443,] 0.9612873
 [444,] 1.0988587
 [445,] 1.5794592
 [446,] 1.0438684
 [447,] 0.6670132
 [448,] 1.9153712
 [449,] 1.9637715
 [450,] 0.7159528
 [451,] 0.8119893
 [452,] 1.1489600
 [453,] 0.9023244
 [454,] 0.9857213
 [455,] 0.6793446
 [456,] 1.0284229
 [457,] 0.9268201
 [458,] 1.1315809
 [459,] 0.8589509
 [460,] 0.8169525
 [461,] 1.4750788
 [462,] 0.7496458
 [463,] 0.5848952
 [464,] 0.7189014
 [465,] 0.6204754
 [466,] 0.7702430
 [467,] 3.0760004
 [468,] 1.7608209
 [469,] 0.5577154
 [470,] 1.4365379
 [471,] 1.4839914
 [472,] 0.9486829
 [473,] 0.7956295
 [474,] 1.0041222
 [475,] 1.0792992
 [476,] 0.8408217
 [477,] 1.6118840
 [478,] 1.2312092
 [479,] 1.3609367
 [480,] 1.2067170
 [481,] 1.6461551
 [482,] 1.1098814
 [483,] 0.5815324
 [484,] 1.2958019
 [485,] 0.5281246
 [486,] 0.8585710
 [487,] 0.9438420
 [488,] 1.0361399
 [489,] 0.8841896
 [490,] 1.6976918
 [491,] 1.1097510
 [492,] 0.6784978
 [493,] 1.0010389
 [494,] 1.7725987
 [495,] 0.5220205
 [496,] 1.0089502
 [497,] 1.4860288
 [498,] 1.8923380
 [499,] 1.2821390
 [500,] 0.8784554
 [501,] 0.8777686
 [502,] 1.6479702
 [503,] 0.3985830
 [504,] 0.3962295
 [505,] 0.9262719
 [506,] 0.9455292
 [507,] 0.9251755
 [508,] 2.0208831
 [509,] 1.4155221
 [510,] 0.7336200
 [511,] 0.7850498
 [512,] 1.5910079
 [513,] 1.0185668
 [514,] 1.0460104
 [515,] 1.1437914
 [516,] 0.9050149
 [517,] 0.8035864
 [518,] 1.0520923
 [519,] 1.6509403
 [520,] 0.9485036
 [521,] 0.7115430
 [522,] 1.3963897
 [523,] 0.7604124
 [524,] 0.7997896
 [525,] 0.8465368
 [526,] 0.9852758
 [527,] 0.6827786
 [528,] 1.1769061
 [529,] 0.6879249
 [530,] 0.7899654
 [531,] 0.5698013
 [532,] 0.8842289
 [533,] 2.6513424
 [534,] 1.0939446
 [535,] 0.7898021
 [536,] 0.8147453
 [537,] 1.2140595
 [538,] 0.6591184
 [539,] 0.8393913
 [540,] 1.0233121
 [541,] 1.2381719
 [542,] 0.5110001
 [543,] 0.7646791
 [544,] 0.7475627
 [545,] 1.1081737
 [546,] 1.3478255
 [547,] 1.1151663
 [548,] 0.8014322
 [549,] 1.0767288
 [550,] 1.2122135
 [551,] 0.9519276
 [552,] 1.6617904
 [553,] 0.7743693
 [554,] 1.3326022
 [555,] 0.4702571
 [556,] 0.7436965
 [557,] 1.6217353
 [558,] 1.4318648
 [559,] 1.0274740
 [560,] 0.6377227
 [561,] 1.3193315
 [562,] 0.5786926
 [563,] 0.7231989
 [564,] 0.2851996
 [565,] 0.3650425
 [566,] 1.0522937
 [567,] 0.5703126
 [568,] 0.8050937
 [569,] 0.4503331
 [570,] 0.7122370
 [571,] 0.8819020
 [572,] 0.7714385
 [573,] 0.7233526
 [574,] 1.6450777
 [575,] 1.4985357
 [576,] 1.4503597
 [577,] 0.7815944
 [578,] 1.1468518
 [579,] 1.8714636
 [580,] 0.7827182
 [581,] 1.2273104
 [582,] 0.4884806
 [583,] 0.8828140
 [584,] 0.6788610
 [585,] 1.0419636
 [586,] 1.0158278
 [587,] 0.5389564
 [588,] 0.7930568
 [589,] 1.3048304
 [590,] 1.0945713
 [591,] 0.6550372
 [592,] 1.6274987
 [593,] 1.2229634
 [594,] 0.8408857
 [595,] 0.9556498
 [596,] 0.6906298
 [597,] 2.0029042
 [598,] 0.7516475
 [599,] 0.9414026
 [600,] 1.0010650
 [601,] 0.9780873
 [602,] 1.2716579
 [603,] 1.1449026
 [604,] 1.0553143
 [605,] 1.1130427
 [606,] 0.7841182
 [607,] 1.1578110
 [608,] 1.6123949
 [609,] 0.9265977
 [610,] 1.0198548
 [611,] 0.6647458
 [612,] 1.3470644
 [613,] 0.9191686
 [614,] 1.0483869
 [615,] 0.9601701
 [616,] 0.7284904
 [617,] 0.8887999
 [618,] 0.9655551
 [619,] 1.0021245
 [620,] 1.6736188
 [621,] 1.1036328
 [622,] 0.6383364
 [623,] 0.7497011
 [624,] 0.3991816
 [625,] 1.1098361
 [626,] 1.5052506
 [627,] 1.4220487
 [628,] 0.7768215
 [629,] 0.9654913
 [630,] 1.6320503
 [631,] 0.8470110
 [632,] 0.6143199
 [633,] 0.4797388
 [634,] 1.2245252
 [635,] 0.4933556
 [636,] 0.5159171
 [637,] 1.5322786
 [638,] 0.9297709
 [639,] 1.2096926
 [640,] 0.6647641
 [641,] 0.7603668
 [642,] 1.0126839
 [643,] 0.6917319
 [644,] 0.9987544
 [645,] 0.8710559
 [646,] 0.7159691
 [647,] 1.7395091
 [648,] 1.5052827
 [649,] 1.1549154
 [650,] 0.8475777
 [651,] 0.5540226
 [652,] 0.6446049
 [653,] 0.8764935
 [654,] 2.2613992
 [655,] 1.0756627
 [656,] 0.7827463
 [657,] 1.7351834
 [658,] 1.3879341
 [659,] 1.0248647
 [660,] 1.2710529
 [661,] 1.9995982
 [662,] 0.9530648
 [663,] 1.0264613
 [664,] 1.3211122
 [665,] 1.4020787
 [666,] 1.4861293
 [667,] 1.0650711
 [668,] 0.3969514
 [669,] 1.1036307
 [670,] 0.9497661
 [671,] 1.1720965
 [672,] 1.2555416
 [673,] 0.5254982
 [674,] 0.6059300
 [675,] 0.7542074
 [676,] 1.1376381
 [677,] 0.7507373
 [678,] 0.5777125
 [679,] 1.0049130
 [680,] 0.8775843
 [681,] 1.5811939
 [682,] 1.3096946
 [683,] 1.0128141
 [684,] 0.9821165
 [685,] 0.8278430
 [686,] 0.4447845
 [687,] 1.1821780
 [688,] 0.7649361
 [689,] 0.9999616
 [690,] 0.9191673
 [691,] 0.6168196
 [692,] 2.0148695
 [693,] 0.9906245
 [694,] 0.6074741
 [695,] 0.5113902
 [696,] 0.9757688
 [697,] 1.1030728
 [698,] 0.6872698
 [699,] 1.2680956
 [700,] 0.8835791
 [701,] 1.5787880
 [702,] 0.6347134
 [703,] 0.7344958
 [704,] 0.5509849
 [705,] 1.1617276
 [706,] 1.0678092
 [707,] 1.4692061
 [708,] 1.0881524
 [709,] 0.6695910
 [710,] 0.7753518
 [711,] 1.4115232
 [712,] 0.3822089
 [713,] 0.9080424
 [714,] 0.8734658
 [715,] 1.5312500
 [716,] 0.8855532
 [717,] 0.8922694
 [718,] 0.8540800
 [719,] 0.5648934
 [720,] 0.8925795
 [721,] 1.6835659
 [722,] 1.2699480
 [723,] 0.9480144
 [724,] 0.4873501
 [725,] 1.4660703
 [726,] 1.3403206
 [727,] 0.4708188
 [728,] 0.6231866
 [729,] 2.6275301
 [730,] 0.5811493
 [731,] 0.4350171
 [732,] 1.0436309
 [733,] 1.2708625
 [734,] 0.9385576
 [735,] 0.9926020
 [736,] 0.5727729
 [737,] 0.2981709
 [738,] 0.6905867
 [739,] 0.9933417
 [740,] 1.4648702
 [741,] 1.0903207
 [742,] 1.0901867
 [743,] 1.1373228
 [744,] 1.1236182
 [745,] 0.7860639
 [746,] 0.9151859
 [747,] 1.1365619
 [748,] 1.0043211
 [749,] 1.1384638
 [750,] 0.9089713
 [751,] 1.2716192
 [752,] 0.9856708
 [753,] 1.9829178
 [754,] 0.8601711
 [755,] 0.3547898
 [756,] 1.2491089
 [757,] 0.6064096
 [758,] 1.3701582
 [759,] 0.7666979
 [760,] 1.1465875
 [761,] 1.6485164
 [762,] 0.5601745
 [763,] 0.7640381
 [764,] 0.7462475
 [765,] 0.9058168
 [766,] 1.4594132
 [767,] 0.7058350
 [768,] 0.4929972
 [769,] 0.9717025
 [770,] 0.5865832
 [771,] 0.5135598
 [772,] 0.7528624
 [773,] 0.7647569
 [774,] 1.5494439
 [775,] 1.5321672
 [776,] 1.2723336
 [777,] 1.3750043
 [778,] 0.6450450
 [779,] 1.4593406
 [780,] 0.4568231
 [781,] 1.0768763
 [782,] 0.7946096
 [783,] 1.0275471
 [784,] 1.2840721
 [785,] 0.8653640
 [786,] 1.4459419
 [787,] 1.1485698
 [788,] 1.2635760
 [789,] 1.7116361
 [790,] 0.4255087
 [791,] 1.4096811
 [792,] 0.5967885
 [793,] 0.9246016
 [794,] 0.7766885
 [795,] 0.4195362
 [796,] 0.6385803
 [797,] 0.7249236
 [798,] 0.7517525
 [799,] 1.0521368
 [800,] 1.0759493
 [801,] 1.5092213
 [802,] 0.8963566
 [803,] 0.7120206
 [804,] 0.6711344
 [805,] 1.4962775
 [806,] 0.6045987
 [807,] 0.6317447
 [808,] 1.2567686
 [809,] 1.5991375
 [810,] 1.3640191
 [811,] 0.9880754
 [812,] 0.9671767
 [813,] 1.4740601
 [814,] 1.3092224
 [815,] 0.3311038
 [816,] 0.7378053
 [817,] 0.8897675
 [818,] 0.9491255
 [819,] 0.7235911
 [820,] 0.8625198
 [821,] 1.0424529
 [822,] 0.9827649
 [823,] 1.1378564
 [824,] 0.7709857
 [825,] 0.8860889
 [826,] 1.4839919
 [827,] 0.8049073
 [828,] 1.1658418
 [829,] 0.6535602
 [830,] 1.6945666
 [831,] 0.7122305
 [832,] 0.8354899
 [833,] 0.7410459
 [834,] 1.0333930
 [835,] 0.8956504
 [836,] 3.3265944
 [837,] 0.4607808
 [838,] 2.0016576
 [839,] 0.9664852
 [840,] 0.8591136
 [841,] 0.4436233
 [842,] 1.2676282
 [843,] 1.8280519
 [844,] 1.5233518
 [845,] 1.5538160
 [846,] 1.1909282
 [847,] 0.6403338
 [848,] 1.1079572
 [849,] 0.9946946
 [850,] 1.6265167
 [851,] 2.2160460
 [852,] 0.7047208
 [853,] 0.9520784
 [854,] 0.6697161
 [855,] 1.3412183
 [856,] 1.1564769
 [857,] 1.4735548
 [858,] 1.1963289
 [859,] 0.8908181
 [860,] 1.1058037
 [861,] 1.0176669
 [862,] 0.6967462
 [863,] 1.1055883
 [864,] 1.2232689
 [865,] 1.5669295
 [866,] 0.9927264
 [867,] 0.7922969
 [868,] 0.8924227
 [869,] 2.2273688
 [870,] 1.9035866
 [871,] 1.0281940
 [872,] 0.6114104
 [873,] 1.1185919
 [874,] 1.2752002
 [875,] 1.3628293
 [876,] 1.6860143
 [877,] 0.9036242
 [878,] 1.5502328
 [879,] 2.1304006
 [880,] 0.7798978
 [881,] 0.8582456
 [882,] 0.7763562
 [883,] 1.1338050
 [884,] 0.9490513
 [885,] 1.8062837
 [886,] 0.6245626
 [887,] 0.9137725
 [888,] 0.3311868
 [889,] 1.2194877
 [890,] 1.0160828
 [891,] 1.0822496
 [892,] 0.8741492
 [893,] 1.0376285
 [894,] 0.5971172
 [895,] 0.9930797
 [896,] 1.1978342
 [897,] 0.9820470
 [898,] 0.8871070
 [899,] 2.0435166
 [900,] 0.8820640
 [901,] 0.6704312
 [902,] 0.5520564
 [903,] 0.7753335
 [904,] 0.8594861
 [905,] 0.8798928
 [906,] 0.5390696
 [907,] 0.5915295
 [908,] 1.2222008
 [909,] 1.0178717
 [910,] 0.9276021
 [911,] 2.5061406
 [912,] 0.9791406
 [913,] 0.9181410
 [914,] 1.8180531
 [915,] 0.9761132
 [916,] 0.7540974
 [917,] 1.0614900
 [918,] 0.6780670
 [919,] 0.8102315
 [920,] 1.1964871
 [921,] 0.5333357
 [922,] 0.7619583
 [923,] 0.7831024
 [924,] 1.1944872
 [925,] 0.8343710
 [926,] 1.2736611
 [927,] 1.0380704
 [928,] 0.6784847
 [929,] 0.9797288
 [930,] 0.8117795
 [931,] 0.5183487
 [932,] 0.7292961
 [933,] 0.8450584
 [934,] 0.6513820
 [935,] 0.7342797
 [936,] 0.8651456
 [937,] 0.7114826
 [938,] 0.7565788
 [939,] 1.1157545
 [940,] 0.9362141
 [941,] 1.4258037
 [942,] 1.6709660
 [943,] 0.8322643
 [944,] 1.0377285
 [945,] 1.2729434
 [946,] 0.9317816
 [947,] 0.7424998
 [948,] 0.5997798
 [949,] 1.7143862
 [950,] 0.8865491
 [951,] 1.1110944
 [952,] 0.4839150
 [953,] 1.1751674
 [954,] 1.4343744
 [955,] 0.8499582
 [956,] 0.7519829
 [957,] 1.1345566
 [958,] 1.1807313
 [959,] 0.8559385
 [960,] 0.7042612
 [961,] 1.0249730
 [962,] 0.5327833
 [963,] 1.9667756
 [964,] 1.4416414
 [965,] 1.3192209
 [966,] 1.2340020
 [967,] 0.7627219
 [968,] 0.8285015
 [969,] 0.6307524
 [970,] 0.5265525
 [971,] 1.2354141
 [972,] 2.2840114
 [973,] 0.8072222
 [974,] 0.7202883
 [975,] 0.5839934
 [976,] 0.9968098
 [977,] 1.0683461
 [978,] 0.4833957
 [979,] 1.2610925
 [980,] 1.6876075
 [981,] 0.6734502
 [982,] 0.9832715
 [983,] 0.9257128
 [984,] 0.6049874
 [985,] 0.9722589
 [986,] 0.7110103
 [987,] 1.2555990
 [988,] 0.5739538
 [989,] 0.5208723
 [990,] 0.7150321
 [991,] 0.5798138
 [992,] 1.5866842
 [993,] 1.1645795
 [994,] 0.7675501
 [995,] 1.0302099
 [996,] 0.6968812
 [997,] 1.3375302
 [998,] 0.6388878
 [999,] 0.9733554

$model.matrix
   (Intercept) Management1 Management2 Management3
1            1          -1          -1          -1
2            1           1           0           0
3            1          -1          -1          -1
4            1          -1          -1          -1
5            1           0           1           0
6            1           0           1           0
7            1           0           1           0
8            1           0           1           0
9            1           0           1           0
10           1           1           0           0
11           1           1           0           0
12           1          -1          -1          -1
13           1          -1          -1          -1
14           1           0           0           1
15           1           0           0           1
16           1          -1          -1          -1
17           1           0           0           1
18           1           0           0           1
19           1           0           0           1
20           1           0           0           1

$terms
dune ~ Management
attr(,"variables")
list(dune, Management)
attr(,"factors")
           Management
dune                0
Management          1
attr(,"term.labels")
[1] "Management"
attr(,"order")
[1] 1
attr(,"intercept")
[1] 1
attr(,"response")
[1] 1
attr(,".Environment")
<environment: R_GlobalEnv>

attr(,"class")
[1] "adonis"

8. Multivariate GLMs

Purpose: Model multivariate responses.

R Package: - mvabund

Applications of Multivariate GLMs

Multivariate GLMs are used to model multiple response variables simultaneously. Applications include: - Analyzing species abundance data - Modeling the effects of environmental variables on community structure - Studying interactions between multiple ecological factors

9. Distance-based RDA (db-RDA)

Purpose: Constrained ordination using distance matrices.

R Package: - vegan: capscale()

Applications of Distance-based RDA (db-RDA)

db-RDA is a flexible method for constrained ordination. Applications include: - Analyzing genetic or molecular data - Studying the effects of environmental gradients on community composition - Visualizing relationships between response and explanatory variables

db_rda <- capscale(dune ~ ., data = dune.env, distance = "bray")


Some constraints or conditions were aliased because they were redundant. This
can happen if terms are linearly dependent (collinear): 'Manure^4'

plot(db_rda)

10. Indicator Species Analysis

Purpose: Identifies species associated with site groups.

R Packages: - labdsv: indval() - vegan: multipatt()

Applications of Indicator Species Analysis

Indicator species analysis identifies species that are characteristic of specific groups or conditions. Applications include: - Identifying species associated with particular habitats - Studying the effects of environmental changes on species distribution - Monitoring biodiversity and conservation status

This post aims to provide a clear and concise guide to multivariate statistical methods, making it easier for researchers and students to apply these techniques in their work.