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en:rda_cca_examples [2019/01/23 08:09]
David Zelený created
en:rda_cca_examples [2019/03/07 21:16] (current)
David Zelený
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-====== Constrained ​ordination ​====== +Section: [[en:ordination]] 
-===== Redundancy Analysis (RDA) & transformation-based version (tb-RDA)Canonical Correspondence Analysis ​(CCA) =====+===== RDAtb-RDA, ​CCA & db-RDA ​(constrained ordination) ===== 
 [[{|width: 7em; background-color:​ white; color: navy}rda_cca|Theory]] [[{|width: 7em; background-color:​ white; color: navy}rda_cca|Theory]]
 [[{|width: 7em; background-color:​ white; color: navy}rda_cca_R|R functions]] [[{|width: 7em; background-color:​ white; color: navy}rda_cca_R|R functions]]
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 [[{|width: 7em; background-color:​ white; color: navy}rda_cca_exercise|Exercise {{::​lock-icon.png?​nolink|}}]] [[{|width: 7em; background-color:​ white; color: navy}rda_cca_exercise|Exercise {{::​lock-icon.png?​nolink|}}]]
  
-==== Example 1: Vltava river valley dataset ==== +==== Example 1: Vltava river valley dataset ​(tb-RDA) ​==== 
-In this example, we will apply constrained ordination on [[en:​data:​vltava|Vltava river valley dataset]]. We will ask how much variance in species composition can be explained by two variables, soil pH and soil depth. Both are important factors for plant growth, and moreover, in the study area, they are somewhat correlated (shallower soils have lower pH since the prevailing geological substrate is acid).+In this example, we will apply constrained ordination ​(tb-RDA) ​on [[en:​data:​vltava|Vltava river valley dataset]]. We will ask how much variance in species composition can be explained by two variables, soil pH and soil depth. Both are important factors for plant growth, and moreover, in the study area, they are somewhat correlated (shallower soils have lower pH since the prevailing geological substrate is acid).
  
 First, upload the Vltava river valley data: First, upload the Vltava river valley data:
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 </​code>​ </​code>​
   ​   ​
-Upload library ''​vegan''​ and calculate tb-RDA based on Hellinger pre-transformed species composition data. Note that since the original data represent estimates of percentage cover, it is better to log transform these values first before Hellinger transformation is done:+Upload library ''​vegan''​ and calculate tb-RDA based on Hellinger pre-transformed species composition data. Note that since the original data represent estimates of percentage cover, it is better to log transform these values first before Hellinger transformation is done (using function ''​log1p'',​ which calculates //log (x+1)// to avoid //log (0)//):
 <code rsplus> <code rsplus>
 library (vegan) library (vegan)
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 tbRDA tbRDA
 </​code>​ </​code>​
 +
 +The result printed by ''​rda''​ function is the following:
 <​code>​ <​code>​
 Call: rda(formula = spe.hell ~ pH + SOILDPT, data = env) Call: rda(formula = spe.hell ~ pH + SOILDPT, data = env)
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 </​code>​ </​code>​
  
-The two variables explain 8.9% of the variance (the row ''​Constrained''​ and column ''​Proportion''​ in the table above, can be calculated also as the sum of eigenvalues for the constrained axes divided by total variance (inertia): (0.04023+0.02227) /0.70476=0.08868. The first constrained axis (RDA1) explains 0.04023/​0.70476=5.7% of the variance, while the second (RDA2) explains 0.02227/​0.70476=3.2%. Note that the first unconstrained axis (PC1) represents 0.07321/​0.70476=10.4% of total variance, which is more than both explanatory variables together; the first two unconstrained explain (0.07321+0.04857)/​0.70476=17.3%. This means that the dataset may be structured by some strong environmental variable(s) different from pH and soil depth (we will check this below).+and, the same with comments: 
 + 
 +{{:​obrazky:​rda-vltava-dataset-ph_soildpt.jpg?​direct|}} 
 + 
 +The two variables explain 8.9% of the variance (the row ''​Constrained''​ and column ''​Proportion''​ in the table above, can be calculated also as the sum of eigenvalues for the constrained axes divided by total variance (inertia): (0.04023+0.02227) /0.70476=0.08869. The first constrained axis (RDA1) explains 0.04023/​0.70476=5.7% of the variance, while the second (RDA2) explains 0.02227/​0.70476=3.2%. Note that the first unconstrained axis (PC1) represents 0.07321/​0.70476=10.4% of total variance, which is more than both explanatory variables together; the first two unconstrained explain (0.07321+0.04857)/​0.70476=17.3%. This means that the dataset may be structured by some strong environmental variable(s) different from pH and soil depth (we will check this below). 
 + 
 +The relationship between the variation represented by individual (constrained and unconstrained) ordination axes can be displayed using the barplot on eigenvalues:​ 
 +<code rsplus>​ 
 +constrained_eig <- tbRDA$CCA$eig/​tbRDA$CA$tot.chi*100 
 +unconstrained_eig <- tbRDA$CA$eig/​tbRDA$CA$tot.chi*100 
 +barplot (c(constrained_eig,​ unconstrained_eig),​ col = c(rep ('​red',​ length (constrained_eig)),​ rep ('​black',​ length (unconstrained_eig))),​ las = 2, ylab = '% variation'​) 
 +</​code>​ 
 + 
 +(note that all information about the eigenvalues and total inertia is in the object calculated by ''​vegan'''​s ordination function (''​rda''​ in this case, stored in the list ''​tbRDA''​),​ you just need to search a bit inside to find it - consider using the function ''​str''​ to check the structure of tbRDA first). 
 + 
 +{{:​obrazky:​tb_rda_vltava_barplot_eig.png?​direct|}} 
  
 Let's see the ordination diagram: Let's see the ordination diagram:
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 Note one more thing: when applying the function ''​envfit''​ on mean Ellenberg indicator values, I did not test for the significance (I set the argument ''​permutation = 0''​). This has a meaning: both mean Ellenberg indicator values and sample scores on ordination axes are calculated from the same matrix of species composition,​ and to directly test their relationship would be wrong (they are not independent,​ and we get high probability to get significant result even if the species Ellenberg indicator values are randomly generated). Check the section [[en:​traits]] for detail explanation on how to solve this FIXME. Note one more thing: when applying the function ''​envfit''​ on mean Ellenberg indicator values, I did not test for the significance (I set the argument ''​permutation = 0''​). This has a meaning: both mean Ellenberg indicator values and sample scores on ordination axes are calculated from the same matrix of species composition,​ and to directly test their relationship would be wrong (they are not independent,​ and we get high probability to get significant result even if the species Ellenberg indicator values are randomly generated). Check the section [[en:​traits]] for detail explanation on how to solve this FIXME.
 +
 +==== Example 2: How different are cookies from pastries and pizzas (CCA)? ====
 +
 +This example is using [[en:​data:​cookies-pastry-pizza]] dataset, which I found on reddit.com, posted by author everest4ever. As the post on [[https://​www.reddit.com/​r/​dataisbeautiful/​comments/​7ke5a6/​the_christmas_cookie_competition_at_my_office/​drdkcgz/​|reddit.com]] goes, the author attended the Christmas party at his office with "​Christmass cookie competition",​ which sparked a "huge debate about what are eligible entries for the cookie competition (e.g. are mini-pizzas cookies?​)"​. The author decided to approach the discussion rigorously and did the following: "I scraped 1931 recipes from the Food Network that contain the keywords cookies (my group of interest), pastry, or pizza (two control groups). Next, I extracted the ingredient list and pooled similar ingredients together (e.g. //salt//, //​seasalt//,​ //Kosher salt//), coming up with a total of 133 unique ingredients. I ended up with a 1931x133 matrix, where each row is one recipe, and each column is whether this recipe contains a certain ingredient (0 or 1)". The author did PCA analysis on the data accompanied by some clustering and predictions,​ just to prove that "NO IAN AND JOSEPH YOUR FUCKING EGG TARTS AREN'T COOKIES, NO MATTER HOW GOOD THEY WERE!!"​. I think we can also use this dataset for a simple constrained ordination exercise.
 +First import the data:
 +<code rsplus>
 +recipes.ingr <- read.delim ('​https://​raw.githubusercontent.com/​zdealveindy/​anadat-r/​master/​data/​cookie_dataset_everest4ever_composition.txt',​ row.names = 1)
 +recipes.type <- read.delim ('​https://​raw.githubusercontent.com/​zdealveindy/​anadat-r/​master/​data/​cookie_dataset_everest4ever_type.txt',​ row.names = 1)
 + 
 +</​code>​
 +
 +Data represent a matrix of presence/​absence of different ingredients in individual recipes (each row of ''​recipes.ingr''​ matrix is a recipe, each column one ingredients). To know which recipe is classified how, we need a variable ''​type_of_food''​ in the data frame ''​recipes.type''​.
 +
 +To get familiar with data, let's first calculate DCA:
 +<code rsplus>
 +library (vegan)
 +DCA <- decorana (recipes.ingr)
 +DCA
 +</​code>​
 +<​code>​
 +Call:
 +decorana(veg = recipes.ingr) ​
 +
 +Detrended correspondence analysis with 26 segments.
 +Rescaling of axes with 4 iterations.
 +
 +                  DCA1   ​DCA2 ​  ​DCA3 ​  DCA4
 +Eigenvalues ​    ​0.5633 0.2598 0.2224 0.2185
 +Decorana values 0.5918 0.2622 0.2352 0.2138
 +Axis lengths ​   6.2276 4.1302 5.6396 3.5449
 +</​code>​
 +
 +The output shows that the length of the first axis is 6.2 S.D. units, so unimodal ordination methods is advisable. The ordination diagram which displays recipes of cookies, pastries and pizzas by different symbols and colours, is more informative:​
 +
 +<code rsplus>
 +type_num <- as.numeric (recipes.type$type_of_food)
 +
 +ordiplot (DCA, type = '​n'​)
 +points (DCA, display = '​sites',​ col = type_num, pch = type_num)
 +legend ('​topright',​ col = 1:3, pch = 1:3, legend = levels (recipes.type$type_of_food))
 +</​code>​
 +{{:​obrazky:​cookies-pastries-pizza-dca.png?​direct|}}
 +(the variable ''​type_num''​ contains numerical values 1, 2 and 3 in place of Cookies, Pastries and Pizzas from the original ''​type_of_food''​ variable in ''​recipes.type''​ data frame, so as we can use these values as colors and symbols in ordination diagram).
 +
 +It seems that pizzas are somewhat different from the rest (although part of pastries is close), while pastries and cookies form a cloud with big overlap. Let's try to ask the following question: can the classification of a recipe into cookies/​pastries/​pizzas (done largely subjectively by authors of that recipes based on their opinion how each category item should look like) explain the difference in "​ingredients composition"​ of individual recipes? This is task for constrained ordination. Since the first DCA axis is long, we use CCA for it, with recipes dependent variable and assignment into the type as explanatory. Note that explanatory variable is categorical with three levels (''​pastry'',​ ''​cookie'',​ ''​pizza''​):​
 +
 +<code rsplus>
 +type <- recipes.type$type_of_food
 +CCA <- cca (recipes.ingr ~ type)
 +CCA
 +</​code>​
 +<​code>​
 +Call: cca(formula = recipes.ingr ~ type)
 +
 +               ​Inertia Proportion Rank
 +Total         ​14.28961 ​   1.00000 ​    
 +Constrained ​   0.60311 ​   0.04221 ​   2
 +Unconstrained 13.68650 ​   0.95779 ​ 132
 +Inertia is scaled Chi-square ​
 +
 +Eigenvalues for constrained axes:
 +  CCA1   ​CCA2 ​
 +0.4649 0.1382 ​
 +
 +Eigenvalues for unconstrained axes:
 +    CA1     ​CA2 ​    ​CA3 ​    ​CA4 ​    ​CA5 ​    ​CA6 ​    ​CA7 ​    ​CA8 ​
 +0.30447 0.28409 0.26249 0.25278 0.23942 0.21521 0.21069 0.20674 ​
 +(Showed only 8 of all 132 unconstrained eigenvalues)
 +</​code>​
 +(note that I saved the column ''​type_of_food''​ from the data frame ''​recipes.type''​ into a variable ''​type'',​ not really because I want to simplify the ''​cca''​ (it would need to be ''​CCA <- cca (recipes.ingr ~ type_of_food,​ data = recipes.type)'',​ but that is still fine), but because this will make it simple to display individual factor levels onto ordination diagram (the levels are displayed as the Name_of_variableName_of_category,​ which would be too long with the original names).
 +
 +We got two constrained axes (explanatory variable is qualitative with three factor levels -> number of contrained axes = number of levels - 1), the first exlaining more than 3 time more than the second (eig<​sub>​CCA1</​sub>​ = 0.4649, eig<​sub>​CCA2</​sub>​ = 0.1382, which means that the first axis represents 0.4649/​14.28961 = 3.3 % of variance (eigenvalue/​total inertia), while the second 0.1382/​14.28961 = 1.0%. Ordination diagram shows that the first axis is mostly separating pizzas (right) and cookies+pastries (left), while the second axis is mostly separating cookies (up) from pastries (bottom):
 +<code rsplus>
 +ordiplot (CCA, display = c('​si',​ '​cn'​),​ type = '​n'​)
 +points (CCA, display = '​si',​ col = type_num, pch = type_num)
 +text (CCA, display = '​cn',​ col = '​navy',​ cex = 1.5)
 +legend ('​topright',​ col = 1:3, pch = 1:3, legend = levels (recipes.type$type_of_food))
 +</​code>​
 +{{:​undefined:​cookies-pastries-pizza-cca.png?​direct|}}
 +
 +Few more things. First, we should ask whether the CCA ordination is significant and whether it is worth to interpret it:
 +<code rsplus>
 +anova (CCA)
 +</​code>​
 +<​code>​
 +Permutation test for cca under reduced model
 +Permutation:​ free
 +Number of permutations:​ 999
 +
 +Model: cca(formula = recipes.ingr ~ type)
 +           Df ChiSquare ​     F Pr(>​F) ​   ​
 +Model       ​2 ​   0.6031 42.479 ​ 0.001 ***
 +Residual 1928   ​13.6865 ​                 ​
 +---
 +Signif. codes: ​ 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
 +</​code>​
 +
 +Indeed, it is. And how about individual axes, are they both significant?​ We will use the argument ''​by = "​axis"''​ in the function ''​anova''​ - see [[en:​monte_carlo_r|this explanation]] what it means:
 +<code rsplus>
 +anova (CCA, by = '​axis'​)
 +</​code>​
 +<​code>​
 +Permutation test for cca under reduced model
 +Forward tests for axes
 +Permutation:​ free
 +Number of permutations:​ 999
 +
 +Model: cca(formula = recipes.ingr ~ type)
 +           Df ChiSquare ​     F Pr(>​F) ​   ​
 +CCA1        1    0.4649 65.493 ​ 0.001 ***
 +CCA2        1    0.1382 19.465 ​ 0.001 ***
 +Residual 1928   ​13.6865 ​                 ​
 +---
 +Signif. codes: ​ 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
 +</​code>​
 +
 +Both axes are significant,​ which means that even the distinction between cookies and pastries along the second axis is important. So I would say, without further testing, that not only pizza is (quite obviously) different from cookies, but also much more ambiguous category pastries are different regarding ingredients they use. Btw, let's see these ingredients (display species in CCA ordination diagram):
 +<code rsplus>
 +ordiplot (CCA, display = c('​sp',​ '​cn'​),​ type = '​n'​)
 +orditorp (CCA, display = '​sp',​ priority = colSums (recipes.ingr))
 +</​code>​
 +{{:​undefined:​cookies-pastries-pizza-cca-w-species.png?​direct|}}
 +
 +Note that I did not display all 133 ingredients ("​species"​);​ otherwise the diagram gets too cluttered. I used the low-level graphical function ''​orditorp''​ which is adding only some labels and draws others as symbols. It has argument ''​priority''​ (the species with the highest priority will be more likely plotted as text, with lower as symbols, if there is not enough space); the priority here is the overall frequency of ingredence in the dataset (''​colSums''​ applied on the ''​recipes.ingr''​ data frame). The diagram shows clear triangle, with each corner representing one type of food. There is a gradient of ingredients connecting pizzas with pastries and pastries with cookies, but almost no ingredients connecting pizzas and cookies (except ''​pine nuts'',​ which can perhaps make it in both pizzas and cookies - but I have no idea). Some ingredients are shared among all three (obviously //water// and seems that also //honey//), some are only for that type of food (//basil// for pizza, //puff pastry// for pastries and //​cookies//​(??​) and //ice cream// for cookies. Please, see the diagram and guess which item in your opinion should be where (//​carrots//​ in pastry? not sure...).
en/rda_cca_examples.1548202151.txt.gz · Last modified: 2019/01/23 08:09 by David Zelený