Correlations and Associations
| Type | Name | Link |
|---|---|---|
| Slides | Correlation lecture slides | 💾 BCB743_06_correlations.pdf |
| Data | The Doubs River data | 💾 Doubs.RData |
You were introduced to correlations in BCB744, and you will now revisit this concept in the context of environmental data.
Set-Up the Analysis Environment
The Doubs River Data
The background to the data is described by David Zelený on his excellent website and in the book Numerical Ecology with R by Borcard et al. (2011). These data are a beautiful example of how gradients structure biodiversity. It will be in your own interest to fully understand how the various environmental factors used as explanatory variables vary along a riverine gradient from the source to the terminus of the river.
Correlations between environmental variables
Correlation refers to the statistical (non-causal) relationship between two continuous variables. It measures the extent to which changes in one variable correspond to changes in another variable. Correlations are quantified into values ranging from -1 and +1, with -1 indicating a perfect negative correlation, +1 indicating a perfect positive correlation, and 0 indicating no correlation. A positive correlation implies that as one variable increases, the other variable also increases. Conversely, a negative correlation implies that as one variable increases, the other decreases. Correlation can be calculated using several methods, the most common one being the Pearson correlation coefficient. Non-parametric correlations can be applied to ordinal or non-normal data.
dfs ele slo dis pH har pho nit amm oxy bod
1 0.3 934 48.0 0.84 7.9 45 0.01 0.20 0.00 12.2 2.7
2 2.2 932 3.0 1.00 8.0 40 0.02 0.20 0.10 10.3 1.9
3 10.2 914 3.7 1.80 8.3 52 0.05 0.22 0.05 10.5 3.5
4 18.5 854 3.2 2.53 8.0 72 0.10 0.21 0.00 11.0 1.3
5 21.5 849 2.3 2.64 8.1 84 0.38 0.52 0.20 8.0 6.2We use correlations to establish how the environmental variables relate to one another across the sample sites. We do not need to standardise as one would do for the calculation of Euclidian distances, but in some instances data transformations might be necessary:
dfs ele slo dis pH har pho nit amm oxy bod
dfs 1.00 -0.94 -0.38 0.95 0.01 0.70 0.48 0.75 0.41 -0.51 0.39
ele -0.94 1.00 0.44 -0.87 -0.04 -0.74 -0.44 -0.76 -0.38 0.36 -0.34
slo -0.38 0.44 1.00 -0.34 -0.22 -0.53 -0.19 -0.31 -0.17 0.31 -0.18
dis 0.95 -0.87 -0.34 1.00 0.02 0.70 0.39 0.61 0.29 -0.36 0.25
pH 0.01 -0.04 -0.22 0.02 1.00 0.09 -0.08 -0.05 -0.12 0.18 -0.15
har 0.70 -0.74 -0.53 0.70 0.09 1.00 0.36 0.51 0.29 -0.38 0.34
pho 0.48 -0.44 -0.19 0.39 -0.08 0.36 1.00 0.80 0.97 -0.72 0.89
nit 0.75 -0.76 -0.31 0.61 -0.05 0.51 0.80 1.00 0.80 -0.63 0.64
amm 0.41 -0.38 -0.17 0.29 -0.12 0.29 0.97 0.80 1.00 -0.72 0.89
oxy -0.51 0.36 0.31 -0.36 0.18 -0.38 -0.72 -0.63 -0.72 1.00 -0.84
bod 0.39 -0.34 -0.18 0.25 -0.15 0.34 0.89 0.64 0.89 -0.84 1.00Or if we want to see the associated p-values to establish a statistical significance:
dfs ele slo dis pH har pho nit amm oxy bod
dfs 1.00 -0.94 -0.38 0.95 0.01 0.70 0.48 0.75 0.41 -0.51 0.39
ele -0.94 1.00 0.44 -0.87 -0.04 -0.74 -0.44 -0.76 -0.38 0.36 -0.34
slo -0.38 0.44 1.00 -0.34 -0.22 -0.53 -0.19 -0.31 -0.17 0.31 -0.18
dis 0.95 -0.87 -0.34 1.00 0.02 0.70 0.39 0.61 0.29 -0.36 0.25
pH 0.01 -0.04 -0.22 0.02 1.00 0.09 -0.08 -0.05 -0.12 0.18 -0.15
har 0.70 -0.74 -0.53 0.70 0.09 1.00 0.36 0.51 0.29 -0.38 0.34
pho 0.48 -0.44 -0.19 0.39 -0.08 0.36 1.00 0.80 0.97 -0.72 0.89
nit 0.75 -0.76 -0.31 0.61 -0.05 0.51 0.80 1.00 0.80 -0.63 0.64
amm 0.41 -0.38 -0.17 0.29 -0.12 0.29 0.97 0.80 1.00 -0.72 0.89
oxy -0.51 0.36 0.31 -0.36 0.18 -0.38 -0.72 -0.63 -0.72 1.00 -0.84
bod 0.39 -0.34 -0.18 0.25 -0.15 0.34 0.89 0.64 0.89 -0.84 1.00
n= 30
P
dfs ele slo dis pH har pho nit amm oxy
dfs 0.0000 0.0365 0.0000 0.9771 0.0000 0.0076 0.0000 0.0251 0.0040
ele 0.0000 0.0146 0.0000 0.8447 0.0000 0.0144 0.0000 0.0376 0.0493
slo 0.0365 0.0146 0.0625 0.2362 0.0028 0.3067 0.0997 0.3593 0.1006
dis 0.0000 0.0000 0.0625 0.9147 0.0000 0.0355 0.0004 0.1136 0.0522
pH 0.9771 0.8447 0.2362 0.9147 0.6405 0.6619 0.7976 0.5134 0.3494
har 0.0000 0.0000 0.0028 0.0000 0.6405 0.0481 0.0039 0.1191 0.0370
pho 0.0076 0.0144 0.3067 0.0355 0.6619 0.0481 0.0000 0.0000 0.0000
nit 0.0000 0.0000 0.0997 0.0004 0.7976 0.0039 0.0000 0.0000 0.0002
amm 0.0251 0.0376 0.3593 0.1136 0.5134 0.1191 0.0000 0.0000 0.0000
oxy 0.0040 0.0493 0.1006 0.0522 0.3494 0.0370 0.0000 0.0002 0.0000
bod 0.0309 0.0677 0.3546 0.1770 0.4232 0.0619 0.0000 0.0001 0.0000 0.0000
bod
dfs 0.0309
ele 0.0677
slo 0.3546
dis 0.1770
pH 0.4232
har 0.0619
pho 0.0000
nit 0.0001
amm 0.0000
oxy 0.0000
bod We can also do a visual exploration (see Question 1, below).
Association between species
Species associations refer to the relationships or interactions between different species within an ecosystem or community. The term can be used to describe the outcome of a wide range of relationships, including competition, predation, symbiosis (mutualism, commensalism, parasitism), or simply the tendency for different species to occur in the same habitats or microhabitats.
When two or more species are frequently found in the same area or under the same conditions, they are positively associated. This could be due to similar environmental preferences, mutualistic relationships, or one species depending on the presence of another. For example, bees and flowering plants have a mutualistic relationship where the bees gather nectar for food, and in the process, they pollinate the flowers. In this sense, bees would be positively associated with some flowering plants.
Conversely, if two species are rarely found in the same area or under the same conditions, they are negatively associated. This can be due to competition for resources, predation, or differing environmental preferences.
Analyses of species associations can help us understand the complex dynamics of ecological communities, including how species interact with each other and their environment, the roles they play in their ecosystems, and the effects of environmental changes on species distributions and community composition. A first glance insight into the existence of some of these types of interactions can be found by examining tables of association among species.
The Doubs River fish species dataset is an example of abundance data and it will serve well to examine the properties of an association matrix:
Cogo Satr Phph Babl Thth Teso Chna Pato Lele Sqce Baba Albi Gogo Eslu Pefl
1 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 5 4 3 0 0 0 0 0 0 0 0 0 0 0
3 0 5 5 5 0 0 0 0 0 0 0 0 0 1 0
4 0 4 5 5 0 0 0 0 0 1 0 0 1 2 2
5 0 2 3 2 0 0 0 0 5 2 0 0 2 4 4
6 0 3 4 5 0 0 0 0 1 2 0 0 1 1 1
Rham Legi Scer Cyca Titi Abbr Icme Gyce Ruru Blbj Alal Anan
1 0 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 1 0 0 0 0 0 0 0
5 0 0 2 0 3 0 0 0 5 0 0 0
6 0 0 0 0 2 0 0 0 1 0 0 0In order to calculate an association matrix for the fish species we first need to transpose the data:
Now we can calculate the association matrix:
spe_assoc1 <- vegdist(spe_t, method = "jaccard")
# display only a portion of the data...
as.matrix((spe_assoc1))[1:10, 1:10] Cogo Satr Phph Babl Thth Teso Chna
Cogo 0.0000000 0.7368421 0.7794118 0.7945205 0.3333333 0.4545455 0.9354839
Satr 0.7368421 0.0000000 0.3108108 0.4705882 0.7368421 0.7333333 0.9583333
Phph 0.7794118 0.3108108 0.0000000 0.2804878 0.7794118 0.7571429 0.9113924
Babl 0.7945205 0.4705882 0.2804878 0.0000000 0.8108108 0.7397260 0.8481013
Thth 0.3333333 0.7368421 0.7794118 0.8108108 0.0000000 0.5833333 0.9000000
Teso 0.4545455 0.7333333 0.7571429 0.7397260 0.5833333 0.0000000 0.8787879
Chna 0.9354839 0.9583333 0.9113924 0.8481013 0.9000000 0.8787879 0.0000000
Pato 0.8918919 0.9078947 0.7948718 0.7307692 0.9210526 0.7500000 0.4827586
Lele 0.8627451 0.8235294 0.7386364 0.6666667 0.9056604 0.7346939 0.6136364
Sqce 0.8360656 0.7978723 0.7346939 0.6562500 0.8730159 0.8281250 0.7017544
Pato Lele Sqce
Cogo 0.8918919 0.8627451 0.8360656
Satr 0.9078947 0.8235294 0.7978723
Phph 0.7948718 0.7386364 0.7346939
Babl 0.7307692 0.6666667 0.6562500
Thth 0.9210526 0.9056604 0.8730159
Teso 0.7500000 0.7346939 0.8281250
Chna 0.4827586 0.6136364 0.7017544
Pato 0.0000000 0.5000000 0.6774194
Lele 0.5000000 0.0000000 0.4531250
Sqce 0.6774194 0.4531250 0.0000000 Cogo Satr Phph Babl Thth Teso Chna
Cogo 0.0000000 0.5294118 0.6000000 0.6666667 0.2222222 0.4000000 0.8888889
Satr 0.5294118 0.0000000 0.2380952 0.3600000 0.5294118 0.6111111 0.8846154
Phph 0.6000000 0.2380952 0.0000000 0.1666667 0.6000000 0.6000000 0.7692308
Babl 0.6666667 0.3600000 0.1666667 0.0000000 0.6666667 0.6666667 0.6153846
Thth 0.2222222 0.5294118 0.6000000 0.6666667 0.0000000 0.4000000 0.8235294
Teso 0.4000000 0.6111111 0.6000000 0.6666667 0.4000000 0.0000000 0.7500000
Chna 0.8888889 0.8846154 0.7692308 0.6153846 0.8235294 0.7500000 0.0000000
Pato 0.8125000 0.8333333 0.7083333 0.6000000 0.8125000 0.6428571 0.2307692
Lele 0.8181818 0.6538462 0.5384615 0.3846154 0.8181818 0.7000000 0.4210526
Sqce 0.7307692 0.5517241 0.3928571 0.2500000 0.7307692 0.7307692 0.5200000
Pato Lele Sqce
Cogo 0.8125000 0.8181818 0.7307692
Satr 0.8333333 0.6538462 0.5517241
Phph 0.7083333 0.5384615 0.3928571
Babl 0.6000000 0.3846154 0.2500000
Thth 0.8125000 0.8181818 0.7307692
Teso 0.6428571 0.7000000 0.7307692
Chna 0.2307692 0.4210526 0.5200000
Pato 0.0000000 0.3888889 0.5600000
Lele 0.3888889 0.0000000 0.2800000
Sqce 0.5600000 0.2800000 0.0000000References
Reuse
Citation
@online{j._smit2021,
author = {J. Smit, Albertus},
title = {Correlations and {Associations}},
date = {2021-01-01},
url = {http://tangledbank.netlify.app/BCB743/correlations.html},
langid = {en}
}