# ecoCopula package for model based ordination and graphical modelling in ecology

## Overview

The ecoCopula package allows you to visualise multivariate discrete data with graphical models and ordination. The package was designed primarily for multivariate abundance data in ecology, however it can be applied to any multivariate discrete data.

The two main functions are:

1. Copula ordination (`cord`) to visualise how samples (sites) and variables (taxa) are located along several latent variables (unobserved environmental gradient).
2. Copula graphical models (`cgr`) to plot a graph which distinguishes between direct and indirect associations between variables (e.g taxa).

The ecoCopula package used model-based methods (rather than distance-based, e.g. nMDS). Both `cord` and `cgr` will work on either a `stackedsdm` object (stacked species distribution model, from ecoCopula) or a `manyglm` object (from mvabund).

## Under the hood

### Copulas for discrete data

Copulas are a way to construct a multivariate distribution, and can be used as an alternative to hierarchical models (as in gllvm, HMSC, boral) and generalised estimating equations (GEE; as in mvabund). Like hierarchical models, but unlike GEEs, copulas directly model covariance between variables (taxa). The main advantage of copulas relative to hierarchical models is speed, with copula ordination usually being at least 10 times faster than ordination with hierarchical models, and for large sample sizes faster non-metric multidimensional scaling (nMDS). For more details about copulas and how models are fit in ecoCopula see Popovic et. al. (2018,2019).

### Copula ordination

Model based ordination methods, including copula ordination using the `cord` function as well as `gllvm`, `boral` and `HMSC`, are implemented with latent variables. These can be interpreted as unobserved environmental covariates. Biplots are created by plotting the scores of sites and loadings of taxa. By directly modelling the data, model-based methods account for both the natural variation mean-variance relationships in multivariate abundance data. We can use standard statistical tools to check the assumptions and perform model selection, quantify the uncertainty in the estimated correlations between taxa, and predict to existing and/or new sites, all of which are generally more challenging when using dissimilarity-based ordination methods (like nMDS).

### Copula graphical models

Graphical models look at direct and indirect associations between variables (taxa). They try to answer the question: Is the abundance of this pair of taxa related, after accounting for the effect of the abundances of all the other taxa in the data? A simple and crude way to do this might be regressing each of the two taxa separately (as response) against all the other taxa in the data (as predictors), and then looking for any residual correlation between the pair. Graphical models do something like this, but using modern statistical techniques to improve efficiency and model the community jointly.

Install `ecoCopula` in the usual way.

    install.packages("ecoCopula")

Then attach the library.

    library(ecoCopula)

## Hunting spiders (counts and presence/absence)

The hunting `spider` dataset (van der Aart & Smeenk-Enserink, 1975) has counts of 12 hunting spiders at 28 sites, collected using pit traps, as well as 6 environmental covariates. Species codes are first four letters of genus then first four letters of species. We will analyse it both as counts and presence-absence.

    # spider data is stored in ecoCopula
    data(spider)
    X <- as.data.frame(spider$x) # environmental covariates

Counts

    abund <- spider$abund # abundance of spiders
    abund[1:5,1:6]
    #>      Alopacce Alopcune Alopfabr Arctlute Arctperi Auloalbi
    #> [1,]       25       10        0        0        0        4
    #> [2,]        0        2        0        0        0       30
    #> [3,]       15       20        2        2        0        9
    #> [4,]        2        6        0        1        0       24
    #> [5,]        1       20        0        2        0        9

Presence-absence

    pa=(abund>0)*1 # presence-absence of spiders
    pa[1:4,1:6]
    #>      Alopacce Alopcune Alopfabr Arctlute Arctperi Auloalbi
    #> [1,]        1        1        0        0        0        1
    #> [2,]        0        1        0        0        0        1
    #> [3,]        1        1        1        1        0        1
    #> [4,]        1        1        0        1        0        1

### Ordination

To plot an ordination biplot for presence-absence data `pa`, we first fit a marginal model using `stackedsdm` or `manyglm` with `family="binomial"`, then use the copula ordination function `cord` on the resulting object, and `plot` the output.

    # fit marginal model
    spider_pa <- stackedsdm(pa,~1, data = X, family="binomial",ncores = 2) #eqiv. manyglm()
    # fit copula ordination
    spid_lv=cord(spider_pa)
    # biplot
    plot(spid_lv,biplot = TRUE)

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To colour the sites by a predictor, we can create a colour variable.

    sand<-ifelse(X$bare.sand==0,"red","blue")
    plot(spid_lv,site.col = sand)

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To check that we fit a sensible model we can plot residuals `plot(spider_pa)`. For fancier graphics with `ggplot` see `?plot.cgr`.

### Graphical model

To plot a graph of the spider counts `abund`, we again first fit a marginal model using `stackedsdm` or `manyglm` with `family="negative.binomial"` (for counts).For graphical models it makes the most sense to first control for environmental variables, so that associations between taxa control for those. Then we use the copula graph function `cgr` on the resulting object, and `plot` the output.

    # fit marginal model
    spider_nb <- stackedsdm(abund,~., data = X, family="negative.binomial", ncores = 2) #eqiv. manyglm()
    # fit copula ordination
    spid_gr=cgr(spider_nb, seed=3)
    # biplot
    plot(spid_gr, pad=1)

plot of chunk unnamed-chunk-10

Graphs are interpreted as maps of direct and indirect associations between taxa.

- taxa pairs that have no direct edge between them (e.g Pardmont and Trocterr) have no direct association, so any co-occurrence patterns they have are due to associations they both have with other taxa in the data (mediator species, e.g. Alopcune).

- taxa with direct edges between them have positive associations (blue; e.g. Alopacce and Alopfabr), or negative associations (pink; none found) even after controlling for mediation effects of all the other taxa in the data.

One interpretation is that taxa with direct edges are interacting, though this can only be a hypothesis, as there may be unobserved taxa or environmental variables which could be causing the apparent interaction. The main objective of this analysis is to visualise relationships in order to generate hypotheses to be experimentally tested.

To check that we fit a sensible model we can plot residuals `plot(spider_nb)`. For fancier plots with `ggplot` and `igraph` see `?plot.cgr`.

## Bryce Canyon vegetation ordinal cover

These data contain estimates of cover class for all non-tree vascular plant species in 160 375m^2 circular sample plots and associated environmental variables in Bryce Canyon National Park, Utah, U.S.A. Species codes are first three letters of genus then first three letters of specific epithet. We can download the data from the `labdsv` package, it is in two parts `brycesite` gives the site data and `bryceveg` the cover data.

    library(labdsv)
    # site data
    data(brycesite)
    brycesite$plotcode=substr(brycesite$plotcode,3,5)
    #species data
    data(bryceveg)
    bryceveg<- bryceveg[,-which(colSums(bryceveg>0) <= 20)]  #most abundant species

To model ordinal data in `stackedsdm` we need the categories to be integers starting at 1.

    #recode data to integer categories
    old <- c(0,0.2,0.5,1,2,3,4,5,6) #existing categories
    bryceord=bryceveg
    for(i in 1:length(old)){
      bryceord[bryceveg==old[i]]=i
    }
    #marginal model
    bryce_marg <- stackedsdm(bryceord, formula_X = ~ 1, data = brycesite, family="ordinal",ncores = 2)

### Ordination

    bryce_ordi=cord(bryce_marg)

We plot the ordinations with `ggplot`, colouring sites by elevation gradient. If we had traits we could color species by traits.

    library(RColorBrewer)
    library(ggplot2)
    # data frames for plotting
    site_res <- data.frame(bryce_ordi$scores,brycesite)
    sp_res <- data.frame(bryce_ordi$loadings,
                         species = colnames(bryceord))
    alpha= 2.5 # scaling parameter
    ggplot()+
      geom_point(aes(x=Factor1,y=Factor2,color = elev ),site_res)+ #sites
      geom_text(aes(x = Factor1*alpha, y = Factor2*alpha,label = species),data=sp_res)+ #species
      scale_color_gradientn(colours = brewer.pal(n = 10, name = "PuOr"))+
      theme_classic()

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Low elevation and high elevation sites have quite different species, with low elevations associated with e.g. cermon, and high elevation with e.g. pacmyr.

### Graphical model of veg species

We will control for slope.

    cont_preds=sapply(brycesite, class)%in%c("int","num")
    brycesite[,cont_preds]=scale(brycesite[,cont_preds])
    bryce_marg_all <- stackedsdm(bryceord, formula_X = ~ slope, data = brycesite,
                                 family="ordinal",ncores = 2)

    bryce_graph<-cgr(bryce_marg_all, seed = 1) #seed for demonstration

For more detailed plots `cgr` has an igraph output, which you can manipulate and plot with the `tidygraph` and `ggraph` packages as well as the `igraph` package.

    igraph_out<-bryce_graph$best_graph$igraph_out

    library(tidyr)
    library(tidygraph)
    library(ggraph)

    igraph_out %>% ggraph('fr') + # see ?layout_tbl_graph_igraph
        geom_edge_fan0(aes( colour = partcor, width=partcor)) +
        scale_edge_width(range = c(0.5, 3))+
        scale_edge_color_gradient2(low="#b2182b",mid="white",high="#2166ac")+
        geom_node_text(aes(label=name), repel = TRUE)+
        geom_node_point(size=2)+
        theme_void() +
        theme(legend.position = 'none')

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There is a strong positive association between e.g. ceamar and arcpat after controlling for all the other species.

## Other ecoCopula functionality and notes

- `stackedsdm` can take a vector of families, this means each column can have a different distribution. This is most often useful when modelling traits rather than abundance (some of which are binary, and some continuous), but could also be useful if you have abundance data collected differently for different taxa (some presence/absence, some counts, some ordinal)

- Residual ordination: by including predictors in the `stackedsdm` or `manyglm` model, you are controlling for these. Any ordination you will control for the effect of these variables. This might make it easier to see the effect of other variables.

- It is not uncommon for `cgr` to output a graph with no edges. The final model you plot is the best model chosen by BIC. With a small amount of data this is often a model with no associations. The models are indexed by the shrinkage parameter `lambda`, and given graphical modelling is exploratory, you can further explore different models by changing lambda manually. You can find the shrinkage parameter for the best model with `cgr_fit$all_graphs$lambda.opt` where cgr_fit is your fitted `cgr` object. Smaller values of lambda give more dense graphs with more edges. You can then fit a model with a smaller lambda with the `lambda` argument in `cgr`.

- Both `cord` and `cgr` have a `n.samp` argument. If you find different runs of these functions give different results, try increasing `n.samp`.

### References

Aart, P. J. M. van der & N. Smeenk-Enserink 1975. Correlations between distributions of hunting spiders (Lycosidae, Ctenidae) and environmental characteristics in a dune area. Neth. J. Zool, 25, 1-45.

Popovic, G. C., Hui, F. K., & Warton, D. I. (2018). A general algorithm for covariance modeling of discrete data. Journal of Multivariate Analysis, 165, 86-100.

Popovic, G. C., Warton, D. I., Thomson, F. J., Hui, F. K., & Moles, A. T. (2019). Untangling direct species associations from indirect mediator species effects with graphical models. Methods in Ecology and Evolution, 10(9), 1571-1583.

Roberts D.W. (1992): Plant Community Distribution and Dynamics in Bryce Canyon National Park: Final Report for Project PX 1200-7-0966